2012 nicolaus copernicus university press. all rights reserved. http://www.dem.umk.pl/dem dynamic econometric models vol. 12 ( 2012) 105−110 submitted october 25, 2012 issn accepted december 20, 2012 1234-3862 joanna górka* the formula of unconditional kurtosis of sign-switching garch(p,q,1) processes a b s t r a c t. in the paper we argue that a general formula for the unconditional kurtosis of signswitching garch(p,q,k) processes proposed by thavaneswaran and appadoo (2006) does not give correct results. to show that we revised the original theorem given by thavaneswaran and appadoo (2006) for the special case of the garch(p,q,k) process, i.e. garch(p,q,1). we show that the formula for the unconditional kurtosis basing on the original theorem and the revised version is different. k e y w o r d s: kurtosis, sign-switching garch models. j e l classification: c22. introduction in the article „properties of a new family of volatility sing models” thavaneswaran and appadoo (2006) proposed a general formula for the unconditional kurtosis of the sign-switching garch(p,q,k) process (fornari, mele, 1997). unfortunately, the proposed general formula of kurtosis does not give correct results. the formula for the unconditional kurtosis of the process derived from the theorem 2.1 a) in thavaneswaran and appadoo (2006) is not the same as the formula obtained without using this theorem (see equation 9 in fornari and mele (1997) or equation 27 in górka (2008)). 1. introductory remarks the general sign-switching garch(p,q,k) model is described by equations (fornari, mele, 1997): * correspondence to: department of econometrics and statistics, nicolaus copernicus university, gagarina 13a, toruń, poland e-mail: joanna.gorka@umk.pl. joanna górka dynamic econometric models 12 (2012) 105–110 106 t t ty σ ε= , (1) 2 2 2 1 1 1 q p k t i t i j t j l t l i j l y sσ ω α β σ− − − = = = = + + + φ ,∑ ∑ ∑ (2) where ~ . . .(0,1)t i i dε , 0 0 0i jω α β> , ≥ , ≥ , l ωφ ≤ ,∑ 1 for 0 0 for 0 1 for 0 t t t t y s y y >  = = . − < if 2 2t t tu y σ= − is the martingale difference with variance 2var( )t uu σ= , the model (1)–(2) can be interpreted as arma(m,q) with the sign function for the 2 ty and can be written as: 2 2 1 1 1 ( ) pm k t i i t i j t j l t l t i j l y y u s uω α β β− − − = = = = + + − + φ + ,∑ ∑ ∑ (3) or 2 1 ( ) ( ) k t t l t l l b y b u sφ ω β − = = + + φ ,∑ (4) where 1 1 ( ) 1 ( ) 1 m m i i i i i i i b b bφ α β φ = = = − + = −∑ ∑ , 1 ( ) 1 p j j j b bβ β = = −∑ , maxm {p q}= , , 0iα = for i q> and 0iβ = for j p> . the stationarity assumptions for 2ty specified by (4) are the following (thavaneswaran, appadoo, 2006): (z.1) all roots of the polynomial ( ) 0bφ = lie outside the unit circle. (z.2) 2 0 i i ψ ∞ = < ∞∑ , where the iψ are coefficients of the polynomial 1 ( ) 1 ii i b bψ ψ ∞ = = +∑ satisfying the equation ( ) ( ) ( )b b bψ φ β= . assumptions (z.1)–(z.2) ensure that the variance of tu is finite and that the 2 ty process is weakly stationary. assume that 1k = . then the equation (4) has the form: 2 1 1( ) ( )t t tb y b u sφ ω β −= + + φ . (5) if the assumptions (z.1)–(z.2) are satisfied, then the above equation can be converted to the form: 2 1 1( ) ( ) ( )t t ty b b u b sπ ω ψ π −= + + φ , (6) the formula of unconditional kurtosis of sign-switching garch(p,q,1) processes dynamic econometric models 12 (2012) 105–110 107 where 1 ( ) 1 ii i b bψ ψ ∞ = = +∑ satisfies the equation ( ) ( ) ( )b b bψ φ β= , and 1 ( ) 1 ii i b bπ π ∞ = = +∑ satisfies the equation ( ) ( ) 1b bπ φ = . 2. author’s results the theorem presented below is the revised version of the part a) of the theorem 2.1 presented in thavaneswaran and appadoo (2006) but for the special case of the garch(p,q,k) process, i.e garch(p,q,1). theorem. suppose the ty is a sign-switching garch(p,q,1) process specified by (1)–(2) and satisfying the assumptions (z.1)–(z.2), with a finite fourth moment and a symmetric distribution of tε . then the unconditional kurtosis of the process ty is given by: 2 2 2 2 41 0 2 2 4 4 2 0 1 t i ti t t t i i e e k e e e σ π ε σ ε ε ψ ∞               = ∞                   = + φ = ⋅ .  − −  ∑ ∑ (7) proof. a kurtosis of the process ty described by equations (1)–(2) can be written as: 4 4 4 4 4 2 2 2 2 2 2 2 t t t t t t t t t e y e e k e e y e e ε σ σ ε ε σ σ                                                        = = = . (8) we note that by definition of the tu ( 2 2 t t tu y σ= − ) it follows that: ( ) ( ) ( ) 22 2 4 4 4 4 4 4 4 4 4 4 0 var 1 t t u t t t t t t t t t t t t e u u e u e u e y e e e e e e e e σ σ ε σ σ σ ε σ σ ε                                                                             = , = = − = − = − = −  = − .   let us indicate that the variance of the process 2ty , satisfying the assumptions of the theorem and described by the equation (6), is given by: joanna górka dynamic econometric models 12 (2012) 105–110 108 2 2 2 2 2 1 0 0 4 4 2 2 2 1 0 0 var 1 t u i i i i t t i i i i y e e σ ψ π σ ε ψ π ∞ ∞       = = ∞ ∞             = = = + φ  = − +φ .   ∑ ∑ ∑ ∑ (9) on the other hand, this variance can be calculated from the equation (1). we get then 2 2 4 2 24 4 2 2 24 4 2 var t t t t t t t t t t y e y e y e e e e e ε σ ε σ σ ε σ                                                            = − = − = − . (10) comparing the results of (9) and (10) we receive: 2 4 4 2 2 2 4 4 2 1 0 0 1t t i i t t t i i e e e e eσ ε ψ π σ ε σ ∞ ∞                                  = =  − + φ = − . ∑ ∑ hence, 2 4 4 4 2 2 2 2 1 0 0 2 2 2 2 4 1 4 4 2 0 2 2 2 20 1 1 t t t i i t i i i t t i t t i i t t e e e e ee e e e e σ ε ε ψ π σ π σσ ε ε ψ σ σ ∞ ∞                                 = =  ∞        ∞            =               =            − − = φ + ,  φ +  − − =  ∑ ∑ ∑ ∑      , 2 2 2 2 4 1 0 2 2 2 2 4 4 2 0 1 1 t i t i t t t t i i ee e e e e σ πσ σ σ ε ε ψ ∞              = ∞                               = + φ = ⋅ .  − −  ∑ ∑ (11) substituting (11) to (8) we obtain: 2 2 2 2 41 0 2 2 4 4 2 0 1 t i ti t t t i i e e k e e e σ π ε σ ε ε ψ ∞               = ∞                   = + φ = ⋅ .  − −  ∑ ∑  if 1 0φ = , then the formula (7) of the unconditional kurtosis process is reduced to the formula of the unconditional kurtosis processes generated by appropriate garch models (see the theorem 2 1. in thavaneswaran et al., (2005)). the formula of unconditional kurtosis of sign-switching garch(p,q,1) processes dynamic econometric models 12 (2012) 105–110 109 example. the example concerns the sign-switching garch(1,1,1) model with normal distribution, i.e. 2 2 2 1 1 1 1 1 1. t t t t t t t y y s σ ε σ ω α β σ− − − = , = + + + φ (12) if 2 2t t tu y σ= − is the martingale difference with variance 2var( )t uu σ= , the model (12) is following 2 2 1 1 1 1 1 1 1( )t t t t ty y u u sω α β β− − −= + + + − + φ . (13) then the polynomials (see the equation (5)) have the form: 1 1( ) 1 ( )b bφ α β= − + , 1( ) 1b bβ β= − . the individual weights ψ are following: 1 1ψ α= , 2 1 1 1( )ψ α α β= + , …, 1 1 1 1( ) i iψ α α β −= + , …. the weights π are: 1 1 1π α β= + , 2 2 1 1( )π α β= + , …, 1 1( ) i iπ α β= + , …. if condition (z.2) is satisfied, then 21 1( ) 1α β+ < and then: 2 1 2 1 1 2 2 2 2 2 4 1 1 1 1 1 1 10 1 ( ) 1 ( ) ( ) 1ii … α α β ψ α α α β α α β ∞ = − + = + + + + + + = +∑ , 2 1 1 2 2 4 6 1 1 1 1 1 1 10 1 ( ) 1 ( ) ( ) ( )ii … α βπ α β α β α β ∞ = − + = + + + + + + + =∑ . assuming that (0 1)t nε , and substituting into (7) we obtain: ( ) ( ) ( ) ( ) 2 1 2 1 1 1 1 2 1 2 1 1 1 1 2 1 1 ( ) 2 1 1 ( ) 22 2 2 1 1 1 1 1 2 2 2 1 1 1 3 3 2 1 1 ( ) 1 3 1 ( ) 2 k ω α β α β αω α β α β ω α β α β ω α β α φ − − − + − − − +       + = ⋅ − + − + + φ − − = ⋅ − + − ( )22 2 21 1 1 1 1 2 2 2 1 1 1 1 3 1 ( ) 1 1 2 3 ω α β α β ω α β β α                  − + + φ − − = . − − − (14) this result is the same like the formula of the unconditional kurtosis obtained by fornari and mele (1997) and by górka (2008) but it is different from the result obtained by thavaneswaran and appadoo (2006). nonetheless, in each case, if 1 0φ = then the formula (14) reduces to a formula for the unconditional kurtosis of the garch (1,1) process. references fornari, f., mele, a. (1997), signand volatility-switching arch models: theory and applications to international stock markets, journal of applied econometrics, 12, 49–65. górka, j. (2008), description the kurtosis of distributions by selected models with sing function, dynamic econometric models, 8, 39–49. joanna górka dynamic econometric models 12 (2012) 105–110 110 thavaneswaran, a., appadoo, s. s. (2006), properties of a new family of volatility sing models, computers & mathematics with applications, 52, 809–818. thavaneswaran, a., appadoo, s. s., samanta, m. (2005b), random coefficient garch models, mathematical & computer modelling, 41, 723–733. wzór na bezwarunkową kurtozę procesu generowanego przez model sign-switching garch(p,q,1) z a r y s t r e ś c i. w artykule zauważono, że na podstawie wzoru na bezwarunkową kurtozę procesu garch(p,q,k) zaproponowanego przez thavaneswarana i appadoo (2006) nie otrzymujemy poprawnych wyników. dlatego też w niniejszej pracy przedstawiono poprawioną formułę twierdzenia thavaneswarana i appadoo (2006) dla szczególnego przypadku procesu garch(p,q,k), tzn. garch(p,q,1). wykazano, że formuła na bezwarunkową kurtozę procesu generowanego przez model sign-switching garch(1,1,1) bazująca na oryginalnym twierdzeniu i poprawionej wersji jest inna. s ł o w a k l u c z o w e: kurtoza, model sign-switching garch. introduction 1. introductory remarks 2. author’s results references © 2016 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2016.003 vol. 16 (2016) 37−47 submitted november 30, 2016 issn (online) 2450-7067 accepted december 17, 2016 issn (print) 1234-3862 józef stawicki * using the first passage times in markov chain model to support financial decisions on the stock exchange a b s t r a c t. the purpose of this article is to present the possibilities of using such a tool as markov chain to analyse the dynamics of returns observed at the warsaw stock exchange. process analysis is the basis for decision-making with regard to the accepted horizon. expected times for achieving specified states, understood as intervals of rates of return, in particular those describing negative rates of return, are extremely important. in this context, there is a possibility of determining easily the value at risk with the accepted probability. k e y w o r d s: markov chain, first passage times, normal white noise, var. j e l classification: c58; f47. introduction markov chain can be found a useful tool when describing the phenomenon of changes in the financial market (stock listings, currency exchange rates, trading volume, etc.). this description may serve to identify the mere phenomenon, its character, or to perform comparative analysis of markets. constructing markov chain model begins with a precise determination of states. in the case of analysis of the return rates process, the classes may be intervals in which the rate of return can be contained. determination of these intervals is defined by the need of research or the consequences of decisions made based on the process that has been identified and described with * correspondence to: józef stawicki, nicolaus copernicus university, faculty of economic sciences and management, 13a gagarina street, 87-100 toruń, poland, e-mail: stawicki@umk.pl. józef stawicki dynamic econometric models 16 (2016) 37–47 38 a relevant model. another very important stage in the construction of markov chain model is the choice of an estimation method. observation of long time series allows applying a method based on microdata, i.e., direct observation of a change of a class. predictions derived based on the estimated markov chain model are of particular significance. predictions can serve individual or institutional investors, as well as institutions evaluating and supervising a given market. comparing the behaviours of real stock market processes with classical ones (for instance, with gaussian white noise) and comparing ergodic distributions is interesting both from a theoretical point of view and practice of decision making processes. this article aims to show how the markov chain model through the construction of process states, understood as a range which may contain a particular indicator, can be helpful when describing the phenomenon treated as random and identifying the risk of an investment decision. the simplicity of the process of finite markov chain and its natural interpretation creates opportunities for popularising the proposed analyses. 1. markov chain and average passage times between states 1.1. homogenous markov chain the markov process with a discrete time parameter and discrete phase space is called markov chain (ching, ng, 2006; stawicki, 2004). it is determined by a sequence of stochastic matrices of the following form:   rrij tp   )((t)p , (1) i.e., matrices with positive elements and satisfying additional requirements in the following form:   j ijit tp 1)( . (2) when designating with td the unconditional decomposition vector of random variable t y , that is:  rtttt ddd ,,, 21 d , where }pr{ iyd tit  , (3) we determine the probability with which the process at time t reaches the phase state i. the elements of the vector td satisfy the conditions below: using the first passage times in markov chain model... dynamic econometric models 16 (2016) 37–47 39 0 itit d (4) and 1  i itt d . (5) the dependence between unconditional decompositions of random variables ty and 1ty is shown by the equation resulting from the theorem of total probability: )(1 ttt pdd   , (6) thus    t k t k 1 0 )(pdd . (7) matrices   rrij tp   )((t)p reflect the mechanism of changes in the distribution of the analysed random variable ty over time. the markov chain },{ ntyt  with the phase space }...,,2,1{ rs  is referred to as a homogeneous markov chain, if conditional probabilities )(tpij of transition from phase state i to state j in a time unit, which means in the period from )1( t to t , are not dependent on the choice of time t , i.e., ijijt ptp  )( . (8) in the case of a homogeneous markov chain dependence (5) and (6) take the following form: pdd  1tt , (9) thus t t pdd  0 . (10) for homogeneous markov chains described by means of transition matrix  ijpp , the expected first passage and return times can be designated. the matrix of these times is determined by the formula (decewicz, 2011): , (11) where is a fundamental matrix and is an ergodic matrix. the elements of matrix can be interpreted simply as the expected number of steps józef stawicki dynamic econometric models 16 (2016) 37–47 40 needed to achieve state after abandoning state . if then we obtain the expected time of the first return. 1.2. parameter estimation of the transition matrix mased on microdata due to the nature of the studied phenomenon, microdata is understood as the observations of the object in subsequent units and registration of the state of the object in a specific unit. observation of the state change in the period from 1t to t allows applying a maximum likelihood estimator in the following form:       t t i t t ij ij tn tn p 2 2 )1( )( ˆ , where:      sitation adverse in the0 statein wasat time andstatein was1at timeobject thewhen1 )( jtit tnij     sit uat ion adverse in t he0 st at ein wasat t ime object was when t he1 )( it tni this estimator has the desirable properties of fitting, asymptotic unbiasedness and has an asymptotic normal distribution with the expected value: ijij ppe )ˆ( , and the variance:      t t i ijij ij tn pp p 2 )1( )1( )ˆvar( . 2. the return rate process – comparison with gaussian white noise research on the rate of return as a stochastic process has been the subject of a vast number of publications. the fundamental question about the nature of this process mostly concerned the issue of the properties of gaussian white noise. in the case of the generated process of the rate of return, it is using the first passage times in markov chain model... dynamic econometric models 16 (2016) 37–47 41 easy to obtain the property of normality for both the unconditional distribution and conditional distributions. the transition matrix for a markov chain being the model in this process takes the form:                      0228.01359.03413.03413.01359.00228.0 0228.01359.03413.03413.01359.00228.0 0228.01359.03413.03413.01359.00228.0 0228.01359.03413.03413.01359.00228.0 0228.01359.03413.03413.01359.00228.0 0228.01359.03413.03413.01359.00228.0 p , where the states of this chain are defined by intervals σ. the rate of return may belong to one of the following states: ).,2[ ),2,[ ),,0[ ),0,[ ),,2[ ),2,( 6 5 4 3 2 1             s s s s s s this matrix is composed of ergodic vectors which are characterized by gaussian distribution. adoption of the intervals in accordance with the threesigma rule facilitates the comparison of the selected processes. this method of determining the states of the observed stock market processes will be maintained in further parts of the work. to provide an empirical example, analysis was performed of returns of the wig index as well as of the rate of return for the budimex company listed on the warsaw stock exchange for the period from 02 february 2003 to 11 august 2016. it provided a series of daily observations amounting to 3475. despite the many advantages of logarithmic returns, simple returns were analysed due to their clear interpretation. for the wig index, the expected value and standard deviation in the examined period were as follows: e(x) = 0.000424, std(x) = 0.0123. the matrix of transition probabilities takes the form: józef stawicki dynamic econometric models 16 (2016) 37–47 42                      . 0526.02105.03553.03289.00395.00132.0 0266.00976.04053.03876.00533.00296.0 0214.00885.03842.04034.00833.00192.0 0133.00951.04162.03515.01013.00226.0 0350.01051.03408.03248.01306.00637.0 0565.01321.02736.02642.00849.01887.0 p . when comparing this matrix with the transition matrix for gaussian white noise, hypothesis assuming the equality of matrices must be rejected (decewicz, 2011, p. 51). test 67.287 2 p relative to 67.5305,0,30 2  . the comparison of the ergodic distribution for such a design of the chain with the process of gaussian white noise is presented in figure 1. figure 1. the comparison of white noise with the ergodic distribution of the rate of return obtained from the wig index for the rate of return obtained from budimex, the expected value and standard deviation in the examined period were as follows: e(x) = 0.000787, std(x) = 0.0224. the matrix of transition probabilities takes the following form: 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 -3σ -2σ -σ σ 2σ 3σ gaussian wn wig using the first passage times in markov chain model... dynamic econometric models 16 (2016) 37–47 43                      0841.01495.01869.03832.01589.00374.0 0459.01311.02787.04295.00918.00230.0 0459.00897.03672.04389.00640.00128.0 0174.00622.03507.04485.01011.00201.0 0505.01325.02808.03817,01199.00347.0 1266.01139.02911.02152.01013.01519.0 p . when comparing this matrix with the transition matrix for gaussian white noise, hypothesis assuming the equality of matrices must be rejected (decewicz, 2011, p. 51). test 87.375 2 p relative to 67.5305,0,30 2  . comparing the different rows of the matrix, i.e., comparing the conditional distributions, it must be noted that in the case each of the conditional distribution must be rejected. for 05.0 and the fifth order the table value 07.1105,0,5 2  . the comparison of the ergodic distribution for such a design of the chain with the process of gaussian white noise is presented in figure 2. figure 2. the comparison of gaussian white noise with the ergodic distribution of the rate of return obtained from the budimex company index 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 -3σ -2σ -σ σ 2σ 3σ gaussian wn budimex józef stawicki dynamic econometric models 16 (2016) 37–47 44 3. first passage times analysis of the expected times of transition from state i to state and j is extremely important from the point of view of an individual or institutional investor. the investor is able to determine the horizon for the portfolio being constructed or decide about the position that should be taken. calculations were made using the winqsb package. the matrix for the wig index takes the following form:                      7.451.96.27.22.127.39 9.463.105.26.29.111.39 1.474.106.26.26.114.39 5.473.105.27.24.112.39 3.461.107.28.20.116.37 1.458.99.20.36.116.32 m . it is worth noting that the expected passage times from any state to the state which is characterized by a value greater than twice the standard deviation are very large and in excess of one month. it should be noted, however, that negative returns are faster achieved than non-negative ones. somewhat different is the situation of the single company tested. the analysed rate of return for budimex was characterized by the following first passage times matrix:                      4.320.116.35.24.107.49 9.334.112.34.23.115.50 7.349.119.23.26.111.51 9.342.120.33.22.117.50 7.333.112.35.29.109.49 5.304.113.39.21.119.43 m . here, the expected passage times to the class of rates lower than twice the standard deviation are larger than the expected times of the first passage times to the state with a rate of return higher than twice the standard deviation. using the first passage times in markov chain model... dynamic econometric models 16 (2016) 37–47 45 4. determination of var using markov chain quantification of risk usually occurs by determining the measure known as var. there are many ways of determining the value at risk depending on the model which is used by analysts or investors. literature on this subject is really abundant. this article proposes to use the markov chain methodology to determine var. when referring to the classic definition of value at risk (osińska, 2006; doman, doman, 2009), one should pay attention to the fact that the market value is linked to the present rate of return and the horizon, i.e., the accepted time interval. these two elements are exposed in a markov chain. one takes the form of state as a range of the rate of return, the other is linked to the time aggregation (chains based on daily, weekly, and monthly data) and the expected time of reaching a specified state for the first time. the idea is to construct states appropriately including the state of risk ),,(1 vars  as well as the state which contains the rate of return at the present time t ),[ yxs b  . state 2s can be determined in the form )0,[2 vars  as long as the current value of the rate satisfies the condition 0x . this structure state of state 2s is not necessary and may be determined as ),[2 zvars  providing zx  . value at risk is determined empirically by changing the range of state 1s and thus at the same time the interval of state 2s until the time when the transition probability 1bp in matrix p reaches the assumed level of risk. analysis of the transition matrix of the budimex company allows to determine var in accordance with the above manner. it was assumed that the last observed rate of return was contained in the interval )01.0;0.0[4  ssb . the rate of return as of 8 th of november, 2016 was r = 0.004149. other classes were determined in as follows: józef stawicki dynamic econometric models 16 (2016) 37–47 46 ).;03.0[ ),03.0;01.0[ ),01.0;0[ ),0;01.0[ ),01.0;[ ),;( 6 5 4 3 2 1       s s s s vars vars the current state, i.e., state bs , is state 4s . the transition matrix allowing 05.01 bp is the following matrix: , 1564.01673.01964.01418.02545.00836.0 0835.01967.03010.01699.02072.00.0417 0639.01957.03069.02091.01751.0 0456.01626.03191.02219.01900,00608.0 0687.02311.02456.01944.01915.00687.0 1729.01729.01822.01122.02290.01308.0                      0.0494 p for which state )03.0;(1 s . that means that var = –0.03 . the matrix determined in that way has also interesting expected times of achieving individual states. this matrix takes the following form:                      6.123.51.48.57.41.17 7.132.56.36.50.5 0.142.56.33.52.5 2.144.55.33.51.55.17 9.130.58.34.51.54.17 3,123.51.49.58.42.16 17.9 17.8 m . the expected time of the first attainment of the state of risk from the present state is relatively high and amounts to 17.8 days. conclusions the proposals do not exhaust all the possibilities offered by the tool in the form of markov chain. a comparison of the processes with gaussian white noise showed a considerable discrepancy. this only confirms the results already described by the subject literature. using the first passage times in markov chain model... dynamic econometric models 16 (2016) 37–47 47 especially important is the proposal of determining var with a very simple interpretation in the markov chain model. the result in the form of expected time of achieving the appropriate states is significant. this applies to both processes where states were constructed by standard deviation and to the construction of states in search of value at risk. the above results are an incentive for undertaking further research. in particular, this concerns the influence of aggregation of states in markov chain on the results presented in the work. references osińska, m. (2006), ekonometria finansowa, polskie wydawnictwo ekonomiczne, warszawa. doman, m., doman, r. (2009), modelowanie zmienności i ryzyka,wolter kluwer polska, kraków. decewicz, a. (2011), probabilistyczne modele badań operacyjnych, oficyna wydawnicza sgh, warszawa. podgórska, m., śliwka, p., topolewski, m., wrzosek, m. (2002), łańcuchy markowa w teorii i w zastosowaniach, oficyna wydawnicza sgh, warszawa. ching, w., ng, m.,k. (2006) markov chains models, algorithms and applications, springer science+business media. stawicki, j. (2004), wykorzystanie łańcuchów markowa w analizie rynku kapitałowego, wydawnictwo umk, toruń. wykorzystanie oczekiwanych czasów pierwszego przejścia i powrotu w modelu łańcuchów markowa do wspomagania decyzji finansowych na giełdzie z a r y s t r e ś c i. artykuł prezentuje możliwość wykorzystania narzędzia jakim są łańcuchy markowa do analizy dynamiki stóp zwrotu obserwowanych na gpw. analiza procesu jest podstawą podejmowania decyzji w zadanym horyzoncie. oczekiwane czasy osiągnięcia zadanych stanów, w szczególności opisujących ujemne stopy zwrotu, jest niezwykle ważne. w tym kontekście pojawia się możliwość łatwego wyznaczania wartości narażonej na ryzyko z zadanym prawdopodobieństwem. s ł o w a k l u c z o w e: łańcuchy markowa, oczekiwane czasy pierwszego przejścia i powrotu, gaussowski biały szum, var. microsoft word 14_muller_pietrzak dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 iwona müller-frączek, michał bernard pietrzak nicolaus copernicus university in toruń space-time modelling of the unemployment rate in polish poviats a b s t r a c t. the purpose of the article is to model the unemployment rate in poland in its spatial and time dimensions. the spatial lag models with the neighbourhood matrix based on the common border of poviats were used in the study. the analysis of the changes over time of the parameters in the space models gave a foundation to apply the space-time model with the parameters linear dependent on time. such an approach enabled us to work out models with good statistical properties and with a possibility of providing a correct economic interpretation of the parameters. k e y w o r d s: spatial econometrics, spatial model, space-time model. introduction the authors show interest in modelling economic phenomena that are characterized by spatial dependencies. the spatial approach requires the use of specific methods and models taken from spatial econometrics. the term of spatial econometrics was introduced by j. h. p. paelinck in 1974. since that year the problems of spatial econometrics have been discussed in abundant works, such as, for instance, clif and ord (1981), anselin (1988), zeliaś (1991), anselin, florax and rey (2004), lesage and pace (2004), haining (2005), arbia (2006), szulc (2007), bivand, pebesma and gómez-rubio (2008), lesage and pace (2009), suchecki (2010). in the present paper the authors will conduct a spatial analysis of the unemployment in poland with a view to researching its properties and existing dependencies within the selected determinants. a correct description of this phenomenon, regarded as one of the major social and economic problems, constitutes a significant task of spatial econometrics. the article is an attempt to present two research approaches used in modelling of economic processes, i.e. the spatial and space-time modelling the processes. the models falling into the former category were applied to provide iwona müller-frączek, michał bernard pietrzak 204 a spatial description of the character of the unemployment rate, though only in its static aspect. the models consider the spatial dependencies showing the influence of the unemployment observed in one poviat on the unemployment level in other poviats. in addition, the global spatial tendency of the phenomenon was characterized by means of the spatial trend. the dependencies between the unemployment rate and the spatial processes defined as ‘determinants’ were adequate interpreted. the purely spatial analysis of the unemployment rate carried out in the years 2004–2009 constituted a starting point for conducting a dynamic spatial analysis. it was observed that the models in specific years possessed some common features such as the same spatial trend and a similar parameter of the spatial autocorrelation. these settlements formed a basis for the space-time approach also referred to as ‘dynamic’. the next step was to estimate the space-time model for the unemployment rate in polish poviats in the years 2004–2009, with time-varying parameters of the space-time trend and the structural parameters. the analysis of the parameters showed linear dependence with regard to time. consequently, the final version of the hypothetical space-time model with parameters that change linearly relative to time was selected. this choice allowed the number of the parameters of the model to be significantly limited. the space-time approach is infrequently used in spatial econometrics, though it allows the enriching of spatial analyses with a time dimension. the methodology relating to space-time modelling and few of applications can be found e.g. in the works of anselin (1988), szulc (2007), lesage and pace (2009). 1. methodology the models used for the purposes of the paper are to reflect the spatial and space-time dependencies relative to the unemployment rate. the models that were applied included static ones as well as two types of dynamic spatial models. throughout the present paper n denotes the number of the ordered units of space. the letters x and y denote the processes observed in space only (the static version) or in space and time (the dynamic version) and they possess either a single or double index dependent on their use. one of the methods of taking into consideration the spatial relations observed in economic processes is to introduce into the formal description the so-called the spatial weights matrix (denoted as )( ijww , nji ,...,1,  ). this matrix determines the intensity of connections between separated areas of space. usually the form of the matrix can reflect only the geographical structure of the researched area; however, it may consider some other characteristics (properties) of the space, such as economic ones (see anselin, 1988). for the dynamic models our assumption is that the spatial matrix is constant over time. space-time modelling of the unemployment rate in polish poviats 205 with the specified w matrix, we can determine a lag operator for the spatial process y, which determines the influence on the observation of the process in the given location 1,...,i n of the observations in other locations, that is:    nj jjii ywy ,...,1 ,)(w – in the spatial version,    nj jtjiit ywy ,...,1 ,)(w , where niitt yy ,...,1)(  – in the space-time version. moreover, it is assumed that for a given area it is possible to determine the global force of the relations occurring in the process between various locations. this is the so-called spatial autocorrelation of the process, incorporated into the model by adding the spatial autoregression component with the parameter ρ. when defining models the so-called spatial or space-time white noise ε, i.e. collections of the uncorrelated random variables with the following properties: ),,0(~ 2 ni ,,...,1 ni  (1) ),,0(~ 2 nit ni ,...,1 and ,,...,1 tt  (2) are used. in the static part of the research the spatial lag model was applied and it had the following form (see anselin,1988): ,)( iiii xyy   w (3) where: ni ,...,1 stands for the number of the location, )( ijww , nji ,...,1,  is the weight matrix, i is the spatial white noise. for the dynamic part of the research two types of models were applied. the first model is autoregressive with time-varying structural parameters and with constant spatial autocorrelation and it is defined as follows: ,)( itittitit xyy   w (4) where: ni ,...,1 is the number of the location, tt ,...,1 is the time index, )( ijww , nji ,...,1,  is the weight matrix, it is the white space-time noise. due to its slight research value the above model constitutes rather a starting point for further research and not a final form of the model that would be ready for adaptation. iwona müller-frączek, michał bernard pietrzak 206 the other model under consideration is a peculiar case of the model (4), in which changes of coefficients for explanatory variables are linear relative to time, that is: ,)()( 10 itititit xtyy   w (5) where: ni ,...,1 is the number of the location, tt ,...,1 is the time index, )( ijww , nji ,...,1,  is the weight matrix, it is the white space-time noise. in the empirical part of the research all the spatial and space-time models were extended by the introduction of the second explanatory variable and by the spatial time-varying trend.1 2. empirical results the empirical research was conducted based on the statistical data obtained from the official central statistical office website and they concerned the registered unemployment rate in polish poviats ( 379n ) at the end of the years 2004–2009 ( 6t ). the research conducted earlier by the authors (see müllerfrączek and pietrzak (2011a, 2011b)) showed that the unemployment rate is characterized by strong spatial interdependencies, which justified the use of the models presented in this section for the purpose of the description of the phenomenon. also, it was observed that the unemployment rate is subject to a global spatial trend; therefore, the models used in the empirical analyses were extended by the addition of the spatial trend. for the purpose of this research the normalized weights matrix (w) was used and this matrix reflects the geographical neighbourhood of poviats in the sense of the common border line. moreover, centres of gravity were determined for the poviats whose geographic coordinates were taken for the estimation of spatial trends. the analysis of statistical data showed that among potential unemployment determinants only two were statistically significant. table 1 presents economic processes and their designations used in the research. table 1. economic spatial processes used in modelling the unemployment rate process designation unemployment rate y investment outlays in thousands of pln per capita x1 one hundred of entities of the national economy per 10000 inhabitants x2 1 the calculations were performed in r-cran. space-time modelling of the unemployment rate in polish poviats 207 2.1. spatial models in the static approach the spatial model of the unemployment rate was estimated separately for each considered year. in all the cases of the estimation of the static models contained in this section, extended by the spatial trend, only the coefficients of a first degree trend turned out to be statistically significant. therefore, the static models used for successive years took the following hypothetical form expressed by the following equation: ,21   xxcybxayy w (6) where: x and y are geographic coordinates for the centres of gravity of poviats, )( ijww , nji ,...,1,  is the weight matrix,  stands for the white noise. table 2 shows the estimated parameters of the models determined by the equation (6) for individual years, where the designations used were consistent with the equation. table 2. the results of the estimation of the models determined by the equation (6) parameter static models 2004 2005 2006 2007 2008 2009 a estimates 16,51 15,69 13,90 11,42 9,86 12,30 p-value ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 b estimates 0,61 0,58 0,48 0,42 0,41 0,52 p-value ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 c estimates -0,84 -0,73 -0,62 -0,43 -0,30 -0,45 p-value ≈0,00 ≈0,00 ≈0,00 ≈0,00 0,02 ≈0,00 α estimates -1,20 -1,27 -1,07 -0,68 -0,49 -0,42 p-value ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 β estimates -0,73 -0,67 -0,59 -0,55 -0,53 -0,66 p-value ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 ≈0,00 ρ estimates 0,66 0,65 0,65 0,64 0,64 0,60 p-value ≈0,00 ≈0,00 ≈0,00 ≈0,000 ≈0,00 ≈0,00 the statistical quality of the models was examined by calculating the respective characteristics for the residuals. these were the following: the value of the determination coefficient, the value of the global moran’s statistics, and significance of the parameters. the values of the above-mentioned characteristics are shown in table 3. all of them prove a proper fit each model to the statistical data. the estimates of the parameters of the static models changes over time. however, prior to the year 2008 the changes had been subject to some regularity with the exclusion of the 2009 model. its being different can be spotted especially in the parameters of the trend that reflects the main spatial tendency of the examined phenomenon. the years 2004–2008 in poland were the period of economic prosperity. the unemployment rate was decreasing every iwona müller-frączek, michał bernard pietrzak 208 subsequent year, investment and salaries were increasing. at that time poland was perceived as a quickly developing country and was attracting foreign investors. the 2008 crisis slowed down all world economies significantly. obviously, this affected the polish economy. because of that two models for the years 2004–2008 and 2004–2009 will be considered in the dynamic approach. table 3. the selected measures of quality of the models determined by the equation (6) characteristics static models 2004 2005 2006 2007 2008 2009 r2 0,640 0,630 0,600 0,570 0,550 0,536 i 0,032 0,028 0,036 0,013 -0,003 0,003 (i-e(i))/s(i) -0,003 -0,003 -0,003 -0,003 -0,003 -0,003 p-value 0,157 0,185 0,128 0,320 0,509 0,439 when analyzing the models presented in table 2, it can be stated that up to the year 2008 the intercept was decreasing. that means that the unemployment rate was, globally, becoming lower and lower in the period 2004–2008. moreover, the slope of the surface of the trend, determined by the b and c values, becomes less steep, so the disproportions in the unemployment rate between individual poviats decrease. these positive effects, as seen from an economic point of view, started to disappear in the year 2009. although the signs of the coefficients do not change with the spatial trend, their absolute values increase in relation to the previous year. the regression parameters α, β for all the examined years had negative values. this reflects the beneficial impact of the growth of investments made and the number of economic subjects on the decrease of the unemployment rate in poviats. however, it can be seen that in course of time the influence of the 1x process is constantly decreasing and the slightly decreasing impact of the 2x process changed its character in the year 2009. in all the six models there can be noticed strong spatial autodependence, which gives the average value of the autoregression parameter ρ at the level of 0.63. moreover, this dependency appeared to be quite stable over time, which resulted in the constant spatial autocorrelation being accepted in the dynamic models. a slight decrease in the autoregression parameter over time indicates the weakening of spatial dependencies of the unemployment rate along with the improvement in the level of social and economic development. this should be read as a positive phenomenon since the existing spatial dependencies create a negative mechanism countering changes in the spatial system of the unemployment rate. 2.2. space-time models due to the change in the developmental trend of the major phenomena in poland both space-time models used for the purposes of the present paper were space-time modelling of the unemployment rate in polish poviats 209 estimated twice – with and without the inclusion of the observations from the year 2009 ( 6t or 5t ). first the spatial lag model with time-varying structural parameters, the trend parameters and the stable over time spatial autocorrelation was considered. what turned out to be statistically significant during the estimation was the parameters for the trend model of the first degree; hence the final hypothetical form of the model was as follows: ,21 tttttttttt xxycxbayy   w (7) where: ,,...,1 tt  )( ijww , nji ,...,1,  is the weight matrix, x and y are the geographic coordinates of the gravity centres of the poviats, t stands for the space-time noise. tables 4 and 5 include the results of the estimation of the parameters of the models designated by the equation (7) with and without the inclusion of the observations from the year 2009. the analysis of the results presented allows a statement to be formulated that the autoregression parameter and the parameters for the determinants model of the unemployment rate are statistically valid. also, in the case of the spatial trend all the parameters proved to be statistically significant. it needs to be emphasized that in order to identify a major spatial trend it is enough to have a significant parameter at least for one of the spatial coordinates. table 4. the results of the estimation of the models determined by the equation (7) in the years 2004–2009 parameter dynamic model 2004 2005 2006 2007 2008 2009 at estimates 16.87 13.71 12.45 11.60 11.30 11.69 p-value ≈0.00 0.590 0.170 0.010 0.001 0.015 bt estimates 0.68 0.63 0.52 0.44 0.43 0.50 p-value ≈0.00 ≈0.00 ≈0.00 ≈0.00 0.039 0.003 ct estimates -0.86 -0.74 -0.63 -0.44 -0.31 -0.45 p-value ≈0.00 ≈0.00 ≈0.00 0.003 0.003 ≈0.00 αt estimates -1.20 -1.27 -1.07 -0.68 -0.48 -0.41 p-value ≈0.00 ≈0.00 ≈0.00 ≈0.00 ≈0.00 ≈0.00 βt estimates -0.73 -0.67 -0.59 -0.56 -0.53 -0.65 p-value ≈0.00 ≈0.00 ≈0.00 ≈0.00 ≈0.00 ≈0.00 ρ estimates 0.65 p-value ≈0.00 the statistical quality of both dynamic models was evaluated applying the same measures that were used for static models. the aggregate results are iwona müller-frączek, michał bernard pietrzak 210 shown in table 7 and they show that the models were well fitted to the empirical data and that the autocorrelation in the residuals did not occur. table 5. the results of the estimation of the models determined by the equation (7) in the years 2004–2008 parameter dynamic model 2004 2005 2006 2007 2008 at estimates 16.68 16.14 15.74 15.33 15.28 p-value ≈0.00 0.600 0.180 0.011 0.001 bt estimates 0.67 0.62 0.51 0.43 0.42 p-value ≈0.00 ≈0.00 ≈0.00 0.005 0.041 ct estimates -0.88 -0.74 -0.63 -0.43 -0.31 p-value ≈0.00 ≈0.00 0.001 0.003 0.004 αt estimates -1.20 -1.27 -1.07 -0.67 -0.48 p-value ≈0.00 ≈0.00 ≈0.00 ≈0.00 ≈0.00 βt estimates -0.73 -0.66 -0.59 -0.55 -0.52 p-value ≈0.00 ≈0.00 ≈0.00 ≈0.00 ≈0.00 ρ estimates 0.68 p-value ≈0.00 the comparison of the results presented in table 4 and table 5 with the results contained in table 2 shows that the values of the parameters of the two dynamic models are similar to the corresponding parameters of the static models. consequently, their economic interpretation will be similar. the analysis of the changes in the time of the parameters of the dynamic models indicates that for the model in which the last observations made (i.e., from the year 2009) were omitted, the changes are linear relative to time. the linear property concerns both the trend and the structural parameters of the model. the consequence of the assumption of the linear changes of the parameters over time was the acceptance of the theoretical model reflecting this property: 0 1 0 1 0 1 0 1 1 0 1 2 ( ) ( ) ( ) ( ) , t t t t t y y a a t b bt x c c t y t x t x                     w (8) where: ,,...,1 tt  )( ijww , nji ,...,1,  is the weight matrix, x and y are the geographic coordinates of the gravity centres of the poviats, t stands for the space-time noise. despite the fact that the analysis of the results presented in table 4 and table 5 showed that in 2009 the linear character of the changes of the parameters relative to time was altered, the estimation of the models designated by the equation (8) was made both with and without the inclusion of the observations space-time modelling of the unemployment rate in polish poviats 211 from that year. the estimation results are shown in table 6 and the residuals of both models are contained in table 7. for both models the autoregression parameter and the parameter for the per capita investment are statistically significant. in the case of the number of economic subjects the linear variation over time is insignificant for the model estimated in the years 2004-2009. however, for the years 2004–2008 it is just on the edge of significance. attention must be paid to the fact that the analysis covered five or six years. increasing the number of periods would certainly enhance the significance of the parameters in the estimated models. the estimate of the autoregression parameter indicates strong spatial dependencies of the unemployment rate and the estimates obtained for the parameters of the determinants show negative dependencies between the unemployment rate, investment, and the number of economic subjects per capita. table 6. the results of the estimation of the models determined by the equation (8) parameter model for the years 2004-2009 model for the years 2004-2008 estimates p-value estimates p-value a0 17.44 ≈0.00 18.81 ≈0.00 a1 -1.28 ≈0.00 -1.81 ≈0.00 b0 -0.91 ≈0.00 -1.01 ≈0.00 b1 0.10 0.005 0.14 0.004 c0 0.65 ≈0.00 0.74 ≈0.00 c1 -0.04 0.210 -0.07 0.140 α0 -1.57 ≈0.00 -1.58 ≈0.00 α1 0.20 ≈0.00 0.21 ≈0.00 β0 -0.70 ≈0.00 -0.76 ≈0.00 β1 0.02 0.300 0.05 0.110 ρ 0.66 ≈0.00 0.65 ≈0.00 table 7. the selected measures of quality of the models determined by the equation (7) and (8) property models described by the equation (7) models described by the equation (8) 2004-2009 2004-2008 2004-2009 2004-2008 r2 0.69 0.70 0.69 0.69 i 0.019 0.023 0.012 0.022 (i-e(i))/s(i) 1.436 1.546 0.915 1.483 p-value 0.075 0.061 0.180 0.069 when analyzing the results presented in table 6 and table 7 it can be inferred that the observations from the year 2009 did not affect considerably the obtained models either in the statistical or the interpretational aspect. the acceptance of the observation for the years following 2009, where the economic slow down would be continued, would undoubtedly affect the evaluation of the parameters and the properties of the model. both models are characterized by a high degree of fitting to the empirical data and the differences in the estimates of the parameters remain slight. if the change of the character of the unemployment rate observed in the year 2009 were to be of a lasting nature, then the iwona müller-frączek, michał bernard pietrzak 212 model created on the basis of all the observations would certainly not be correct from a forecasting point of view, either. conclusions the article discussed a spatial analysis of the registered unemployment rate in polish poviats at the end years 2004–2009. within the first approach, a static one, spatial models of the unemployment rate in subsequent years were estimated. next, the analysis was enhanced by a time dimension and within the dynamic approach estimations were made of appropriate space-time models. a spatial linear trend as well as two determinants of the unemployment rate, investment made and the number of economic subjects per capita were assumed for all the models. the analysis conducted allows the identification of a linear spatial trend and negative dependencies occurring between the unemployment rate, investment made and the number of economic subjects per capita. for the trend parameters and assumed processes of determinants a linear character of the time-varying changes was determined. all of the estimated models were characterized by good statistical properties and economic interpretability of the parameters. the question of the dissimilarity of the observations from the year 2009 remains unsettled. if the economic cycle in poland and the related unemployment, level of investment and general condition of enterprises changed permanently, then the model with a linear dependency of time-varying parameters could result in an erroneous simplification of changes of the parameters. references anselin, l. (1988), spatial econometrics: method and models, kluwer academic publishers, netherlands. anselin, l., florax, r. j. g. m., rey, s. j. (2004), advances in spatial econometrics. methodology, tools and applications, springer-verlag, berlin. arbia, g. (2006), spatial econometrics, springer-verlag gmbh. bivand, r. s., pebesma, e. j., gómez-rubio, v. (2008), applied spatial data analyses with r, springer, new york. clif, a., ord, j. (1981), spatial processes, models and applications, pion, london. müller-frączek, i., pietrzak, m. b. (2011a), analiza stopy bezrobocia w polsce z wykorzystaniem przestrzennego modelu mess (model mess of the unemployment rate in poland), folia oeconomica 253, wydawnictwo uniwersytetu łódzkiego, łódź, 215–223. müller-frączek, i., pietrzak, m. b. (2011b), przestrzenna analiza stopy bezrobocia w polsce w latach 2004–2008 (spatial analysis of the unemployment rate in poland in the years 2004–2008), economic development and management of regions, part ii, hradec kralove, 205–209. haining, r. p. (2005) spatial data analysis. theory and practice, cambridge university press, cambridge. lesage, j. p., pace, r. k. (2004), advances in econometrics: spatial and spatiotemporal econometrics, elsevier, amsterdam. space-time modelling of the unemployment rate in polish poviats 213 lesage, j. p, pace, r. k. (2009), introduction to spatial econometrics, crc press. suchecki, b. (2010), ekonometria przestrzenna (spatial econometrics), wydawnictwo c.h.beck, warsaw. szulc, e. (2007), ekonometryczna analiza wielowymiarowych procesów gospodarczych (econometric analysis of multidimensional economic processes), wydawnictwo umk, toruń. zeliaś, a. (1991), ekonometria przestrzenna (spatial econometrics), pwe, warsaw. przestrzenno-czasowe modelowanie stopy bezrobocia w polsce z a r y s t r e ś c i. celem artykułu jest modelowanie stopy bezrobocia w ujęciu przestrzennym i przestrzenno-czasowym. dane wykorzystane w badaniach empirycznych dotyczyły powiatów polski w latach 2004-2009. wykorzystano modele o charakterze regresyjno-autoregresyjnym z macierzą sąsiedztwa opartą na zasadzie wspólnej granicy powiatów. analiza zmian w czasie parametrów modeli przestrzennych dała podstawy do zastosowania modeli czasowoprzestrzennych z parametrami liniowo zależnymi od czasu. podejście takie pozwoliło na otrzymanie modeli o dobrych własnościach statystycznych oraz charakteryzujących się poprawną ekonomiczną interpretowalnością parametrów. s ł o w a k l u c z o w e: ekonometria przestrzenna, model przestrzenny, model przestrzennoczasowy. microsoft word 09_osińska_m.docx dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 magdalena osińska nicolaus copernicus university in toruń on the interpretation of causality in granger’s sense a b s t r a c t. the concept of causality formulated in 1969 by c.w.j. granger is mostly popular in the econometric literature. the central assumption of the concept is the fact that the cause precedes the effect and can help in forecasting the effect. years of application of granger causality idea have resulted in many misunderstandings related with the interpretation of the empirical findings. the paper focuses on systematization of the definitions based on granger concept and their proper interpretation. k e y w o r d s: granger causality, systematic causality, informational causality, nonlinear causality. introduction the concept of causality formulated in 1969 by clive granger, based on earlier paper by wiener (1956), is mostly popular in the econometric literature. the central assumption of the concept is the fact that the cause precedes the effect and can help in forecasting it. it is further assumed that the cause includes unique information of the effect, which is not available in any other way. the potential causes are chosen from all information connected with the effect. the general definition of granger’s causality is formulated in the framework of conditional probability distribution. let )|( xyf denote conditional distribution of y given x, and t represents all information in the universe in time t. it is said that xt does not cause yt if for all k > 0 the following relation is true: ),\|()|( ttkttkt xyfyf   where: tt x\ denotes all information in the universe except for those included in tx . otherwise tx cause ty (granger, newbold, 1986). the above definition cannot be called operational for the reason of using the phrase ‘all information magdalena osińska 130 in the universe’ that cannot be identified in practice. since 1969 several operational definitions have been formulated. they were the subject of statistical verification. the word ‘causality’ present in all these definitions is the source of misunderstanding. in the paper the attention is focused on the operational definitions of causality in granger’s sense and their interpretation in the two contexts: philosophical as well as empirical. 1. systematic granger causality modelling conditional mean of the endogenous variable via the econometric model means that we are looking for a systematic, repeatable relation which can be used, among others, in forecasting. we call granger causality defined for linear representation of time series ‘the systematic granger causality’ because it refers to such cases. operational definition of granger causality, corresponding to systematic causality concept, is formulated for wide sense stationary time series and it is measured in terms of forecasting errors. it is assumed that autoregressive (or vector autoregressive) representation of the time series that constitutes ‘all the information in the universe’ can be used in forecasting. the direction of causality is the subject of testing. granger causality tests were discussed widely in osinska (2008) and will not be the subject of further analysis. the attention is concentrated at the definitions and their understanding. we say that tx granger-cause ty if: ),\|()|( ''2'2 ttttt xyy   (1) where )|( '2 tty  is the prediction error in the case where all information from the past are used and )\|( ''2 ttt xy  is the prediction error corresponding to the situation when xt was excluded from the information set. key function in the above definition is played by the information set t , because the results of testing for granger causality strongly depend on it. granger did not define explicitly t , leaving it to the researchers. if the variables included into the information set come directly from the economic theory then the definition, despite of its limitations, is closer to the meaning of the word ‘causality’. if however, the information set is based on the data, we cannot expect finding out any unknown relations which will occur a new economic law in future. as cartwright (2007) wrote ‘no cause in no cause out’. so we should not expect anything more than it comes out from the data. furthermore, the formula (1) turns our attention to the understanding of causal relations in terms of forecasting. this is not satisfactory from the philosophical point of view but very practical. forecasting ability is definitely one of on the interpretation of causality in granger’s sense 131 the desired characteristics of causal relation, and when we are able to identify the cause and its effect we expect that the cause (or rather set of causes) will result in the occurrence of a given effect. but causality in granger’s sense cannot be identified as the relation determining that the cause is able to induce the effect. it is obvious that in the case when economic theory states that there is the causal relation between two variables, say, the magnitude of money supply and the inflation rate, testing for causality in granger’s sense will confirm that statement. in many cases however the theory says nothing and the researchers are looking for any relation confirmed by the data. a good example of such an investigation is testing for granger causality between variables characterizing financial markets, for example indices of two or more stock exchanges. the question arises then what we should expect using granger causality tests. first of all let us turn the attention to the case when granger causality can be thought as an idea of finding causal relations using structural econometric models. this was explained by leroy (2004). assume simple structural econometric model of the form: ,1121211112121 ttttt ybybyay   (2) ,2122211211212 ttttt ybybyay   (3) where the explanatory variables (on the rhs of each equation) are not correlated with the residuals. the model is not identified and lagged values of variables are earlier than those observed in time t. the condition 012 a defines conditional causality of the form 121121 ,|  tttt yyyy (leroy, 2004). this condition cannot be tested directly because observable implications for it do not exist. on the other hand, granger causality is defined for reduced form of the model: 1 11 1 1 12 2 1 1t t t ty c y c y u ,    (4) ,2122211212 tttt uycycy   (5) where:    211222121212 1/ aababc  . we say that ty2 does not granger cause ty1 , if 012 c . this condition is neither necessary nor sufficient for conditional causality 121121 ,|  tttt yyyy . that is why granger causality does not allow stating what is prior: ty1 or ty2 . only if 012 b and 022 b , 012 c is equal to 012 a and then granger causality is equivalent to the condition 121121 ,|  tttt yyyy . such a case can occur very rarely in practice. respecting the explanation given above, one may ask whether the structural econometric model is able to state causal relation between variables. the answer we often put is ‘no’. to say ‘yes’ we need some additional information of interventions (hoover, 2001). magdalena osińska 132 granger causality concept was the subject of criticism in econometric literature (basmann, 1988; zellner, 1988 and leroy, 2004). on the other hand the explosion of causality tests based on granger’s idea shows great support for this concept (see for example: caporale, pittis, spagnolo, 2002; geweke 1984; hsiao, 1979; pierce, haugh, 1977)). the main objection is that in granger’s definition no stress is put to the role of the economic theory. granger did not also take into account the definition of causality formulated by feigl. feigl understood causality as forecasting according to a law or a set of laws that joins prediction with the theory (basmann, 1988; zellner, 1979). hoover (2001), in his classification of different concepts of causality in econometrics, includes causality in granger’s sense to the process approach, defined as the inference based on the data. this puts granger in line with the continuators of hume’s philosophy who rejected any metaphysics and indicated observation as the main source of scientific finding. important limits of granger causality were indicated by zellner (1979). he argues, among others, that minimum prediction error cannot be used as a criterion of causal relation because the prediction error can be reduced using many techniques, not necessarily by including causes into the model. having defined the limits of granger causality concept let’s take a glance at its advantages: 1. the definition is operational and allows for testing such important aspects of causal relations as: time sequence of variables, asymmetry of relationships and forecasting ability in the sample and out of the sample. 2. the definition resulted in many statistical tests, developed for different classes of data. 3. the definition was extended for causality in conditional variance as well as causality in risk and nonlinear causality. this allows finding out many relationships of different nature present in the observable economic reality. 4. the information set included into the model is chosen by the researcher. this is a chance for combing of data analysis with a certain theory and developing an individual approach to a specific problem. 5. it is easy to be used in practice. granger’s definition and its modifications, as well as the tests based on that concept should be used with care and interpreted only in the domain they are specified for. we cannot interpret any relation confirmed by granger causality test as ‘causal’ in the broad philosophical sense. analyzing the data without any theoretical background one cannot avoid spurious causality or symptomatic causality resulting from the presence of the third variable not included into the information set. if, however a certain theory is tested using granger causality tests it can be stated that it is accepted or rejected by the data in the sense of on the interpretation of causality in granger’s sense 133 asymmetry and forecasting ability. the question whether it ensures repeatability in future cannot be answer directly because it depends on the economic policy in a given country, which – in normal conditions changes from one economic regime to another. it should be stated that attempts at formulating operational definition of causality are not very popular because it is very difficult to do (cartwright, 2007). for that reason granger’s definition, although far from excellence, is so much important. there are some further implications that can be applied jointly with forecasting ability like the analysis of interventions (hoover, 2001), thick causality (cartwright, 2007) or modular concept (pearl, 2000) but none of these concepts can be thought of as universal. at first granger causality concept was formulated for stationary time series possessing autoregressive representations. further, it was developed for cointegrated time series (granger, 1981; toda, yamamoto, 1995) and, what is very important, granger causality considered in the sample and out of the sample that allows determining whether granger cause in the sample is still able to help predicting the effect out of the sample (ashey, granger, schmalensee, 1980; chao, corradi, swanson, 2001). all the mentioned concepts refer to the direct causality in granger’s sense such as xt  yt+1. in the recent years dufour and renault (1998) turned their attention to the indirect causality concept, based on the definition given by hsiao (1979). it is not the subject of further analysis here because hsiao definition of causality can be decomposed into granger understanding of causality in the terms of forecasting. 2. informational granger causality in the previous part the attention was concentrated on causality in mean based on linear systems. however granger causality can be also considered in terms of causality in conditional variance, and the related concept of causality in risk. we will call it ‘the informational granger causality’ because it is related mainly with financial markets where relations between different stocks, portfolios or derivatives are usually the subject of influence of the information affecting both: the cause and the effect. cheung and ng (1996), basing on granger’s definition have proposed the following formulation for causality in variance. let tx and ty denote stochastic processes which are covariance stationary and  0; ,   jyx jtjtt is a set of all information from the past, available at time t and tt x\ is the corresponding information set excluding tx . magdalena osińska 134 it can be said that tx granger does not cause ty in variance, if:       ttyttttyt yexye  |\| 22  , (6) where: ty  is a conditional mean of ty , assuming the information set tt x\ . further development, made by hong (2001), corresponding directly with the above definition concerns causality in risk. granger causality in risk is defined as follows. let  tt yy 21 , is a bivariate not necessarily stationary stochastic time series. let   1 tlltit iaa l = 1,2 be the value at risk (var) at level  1;0 for ylt predicted using the information set    )2()1(1 ,   tltltl yyi available at time 1t  . lta satisfies     1,| tlltlt iayp . in the case of granger non-causality the null hypothesis takes the form:     1111,1110 ||:   tttttt iaypiayph almost surely (7) where  1 1 1 2 1t ( t ) ( t )i i ,i ,    with the alternative     1111,1111 ||:   tttttt iaypiayph . (8) comparing the above definition with the original one we may state that it concentrates only on the violations of value at risk computed for a given portfolio represented by y1t. so we interpret it as if information about the second portfolio represented by y2t could help change the probability of breaking the var. the definition captures the general characteristics of granger causality concept. it can be extended for other risk measures, belonging to the class of coherent risk measures (artzner, delbaen, eber, heath, 1998). as it was mentioned above, the concept of granger causality was often criticized because it depreciates the philosophical nature of causal relation. on the other hand it is widely known and popular in econometric literature. in fact granger’s definition is related with predictability of one variable using previous values of another one. such an approach takes into consideration only one of many characteristics of causal relation, however in practice it is often the unique possibility of measuring interdependencies between variables. it is particularly important when causality in conditional variance is considered. the number of factors that cause the volatility of financial returns is enormous. furthermore, they change in time and occur only in some periods such as they cannot be observed systematically. their nature is also very much complicated, starting from fundamental causes coming from company itself, through causes located in the macroeconomic surroundings, ending at those of social and psychological nature. however the results they cause are very important, observable and spread all over the world. very similar situation takes place in the case of granger causality in risk, where specified risk measures are applied. the causes, which on the interpretation of causality in granger’s sense 135 evoke the failure of the risk measures are rarely of systematic nature. so if such a raise in risk occurs at one market it is very likely to be moved to another one. it is due to the risk-transferring procedure realized by many market participants including banks. avoiding the risk by closing positions and moving financial capital from one market to another are the main characteristics of contemporary markets. it changes the liquidity preference in the markets that cannot be avoided without the intervention. such a situation is called the contagion phenomenon (allen, gale, 2000). 3. nonlinear granger causality nonlinear granger causality is analyzed separately since it may consider other aspects of granger causality that the ones discussed above. baek and brock (1992) formulated the general definition of granger causality for nonlinear case. it is expressed in terms of the correlation integral that is a measure of local spatial correlation of time series which belong to a specified space. formally, for a multivariate random vector w , the associated correlation integral ( )wc  is the probability of finding two independent realizations of the vector at a distance smaller (or equal) than , i.e.          1 2 1 2 1 2 2 1w ww w wc p i s s f s f s ds ds ,        (9) where: 1 2w , w are independent realizations of w , the integrals are taken over the sample of w , is the supremum norm and i() is the indicator function, which is equal to one if its argument is true, and is zero otherwise. denote the m-length lead vector (m-history) of ty by 1 1( , ,..., ) m t t t t my y y y   and the p-length and q-length lag vectors of tx and ty , respectively by 1 1( , ,..., ) p t p t p t p tx x x x     and 1 1( , ,..., ) q t q t q t q ty y y y     . baek and brock proposed the following definition of granger nonlinear causality: tx does not nonlinearly granger cause ty if     m m p p q q t s t p s p t q s q m m q q t s t q s q p y y | x x , y y p y y | y y .                        (10) where is the supremum-norm distance. the definition says that, given  , p lags of tx does not incrementally help predict next period’s value of ty , given q lags of ty . it seems to be clear why the event ‘ p ptx  is close to p psx  ’ may help incrementally predict ty close to magdalena osińska 136 sy , in case when ),( q qt p ptt yxfy  for some deterministic and continuous function f. while ),( q qt p ptt yxgy  , where g is a stochastic function, the definition is motivated by a hope that at least part of the deterministic relation is present, especially when the conditional variance of ty , given q qt p pt yx  , is smaller than the unconditional variance of ty . it is worth noting that the definition is based on the assumptions concerning the choice of  , the lags number p and q as well as the forecasting horizon m . note that from the definition of conditional probability      . , |          q qs q qt q qs q qt m s m tq qs q qt m s m t yyp yyyyp yyyyp (11) moreover, since the supremum norm implies that    ,,    qmqsqmqtq qsq qtmsmt yypyyyyp (12) the identity      m q m q t q s qm m q q t s t q s q q q t q s q p y y p y y , y y , p y y                      (13) is satisfied. by analogy       m m p p q q t s t p s p t q s q m q m q p p t q s q t p s p q q p p t q s q t p s p p y y | x x , y y p y y , x x , p y y , x x                                      (14) thus the null hypothesis of granger nonlinear noncausality given by (10) may be expressed as follows:            ,4 ,3 ,,2 ,,1 qc qmc pqc pqmc    , (15) where:    ,,,,1    p psp ptqmqsqmqt xxyyppqmc    ,,,,2    p psp ptq qsq qt xxyyppqc       qmqsqmqt yypqmc ,3 and on the interpretation of causality in granger’s sense 137    .,4    q qsq qt yypqc in practice, nonlinear causality is tested above the linear one, so first of all the linear relation should be excluded, for example by estimating a bivariate stationary var model. the tests are usually carried out on the residuals of the linear model. the same indication can be applied for causality in variance. if the source of nonlinearity is known and can be modeled, for example by garch models, it should also be filtered out to show whether other possible source of nonlinear granger causality can be found. conclusions in the paper we discussed the concept of granger causality and its main developments present in the econometric literature. it should be emphasized that the concept refers to predictability as the one of the characteristics of causal relation. that is why it is too narrow to satisfy all the important attributes of philosophical nature of causal relations. on the other hand it is very practical and popular in economic as well as econometric applications. the user of granger causality concept and the tests based on this background should bear in mind their limitations and not to expect finding out any unlikely relations between economic variables. from granger definition it comes out only such characteristics of causal relation like: sequence in time, asymmetry of the cause and the effect and forecasting ability. however it can be analyzed in very many aspects: in the conditional mean and in the conditional variance, in linear and in nonlinear framework, in the short and in the long run as well as in the sample and out of the sample,. this variety of applications gives a possibility of wide empirical analyses which should be put in a theoretical framework if we want to consider them as causal in a broader philosophical sense. in the paper the types of granger causality were classified. we discriminated: systematic granger causality, informational granger causality as well as nonlinear granger causality. the first one refers to linear causality in conditional mean, the second one corresponds to two related concepts of causality: i.e. causality in variance as well as causality in risk and the last one defines the relation which can happen over the two mentioned earlier. although granger causality has many disadvantages from the philosophical point of view its usefulness can be compared with the vector autoregression model defined by c. sims in 1980 who won the nobel prize in 2011. the var model was thought of as atheoretical remedy for disadvantages of traditional, structural econometric models. in fact it became structural, among others, because of testing for granger causality. that example shows that even a very narrow tool can be used in solving much more complicated and sophisticated theoretical problems. in the same sense proper use of granger causality can be magdalena osińska 138 a method of verification of the economic theory and possibly the source of some new empirical facts. references allen, f., gale, d. (2000), financial contagion, the journal of political economy, 108(1), 1–33. artzner, p., delbaen, f., eber, j.m., heath, d. (1998), coherent measures of risk, mathematical finance, 9, 203–228. ashley, r., granger, c. w. j., schmalensee, r. (1980), advertising and aggregate consumption: an analysis of causality, econometrica, 48, 1149–1167. baek, e.g., brock, w.a. (1992), a general test for nonlinear granger causality: bivariate model. technical report. iowa state university and university of wisconsin, madison basmann, r. l. (1988), causality tests and observationally equivalent representations of econometric models, journal of econometrics, 39, 69–104. cartwright, n. (2007), hunting causes and using them (approaches in philosophy and economics), cambridge university press. caporale, g. m., pittis, n., spagnolo, n. (2002), testing for causality-in-variance: an application the east asian markets, international journal of finance and economics, 7(3), 235–245. chao, j. c., corradi, v., swanson, n. r. (2001), an out-of-sample test for granger causality, macroeconomic dynamics, 5, 598–620. cheung, y. v., ng, l. k. (1996), a causality-in-variance test and its application to financial market prices, journal of econometrics, 72(1-2), 33–48. dufour, j-m., renault, e. (1998), short run and long run causality in time series: theory, econometrica, 66, 1099–1112. geweke, j. (1984), inference and causality in economic time series models, handbook of econometrics, vol. ii, 1101–1144. granger, c.w.j. (1969), investigating causal relations by econometric models and crossspectral methods. econometrica, 37, 424–438. granger, c.w.j. (1988), some recent developments in a concept of causality, journal of econometrics, 39, 199–211 granger, c. w. j., newbold, p. (1986), forecasting economic time series, 2nd edition, academic press, orlando, florida. hiemstra, c., jones, j.d. (1994), testing for linear and nonlinear granger causality in the stock price volume relation, journal of finance, 49, 1639–1664. hong, y. (2001), a test for volatility spillover with applications to exchange rates, journal of econometrics, 103(1-2), 183–224. hong, y., liu, y., wang, s. (2009), granger causality in risk and detection of extreme risk spillover between financial markets, journal of econometrics, 150(2), 271–287. hoover, k. d. (2001), causality in macroeconomics, cambridge university press. hsiao, c. (1979), causality tests in econometrics, journal of economic dynamics and control, 1, 321–346. leroy, s. (2004), causality in economics, ms, university of california, santa barbara. osinska, m. (2008), ekonometryczna analiza zależności przyczynowych (econometric analysis of causal relationships), nicolaus copernicus university in torun. pearl, j. (2000), causality, cambridge university press. pierce, d. a, haugh, l.d. (1977), causality in temporal systems, journal of econometrics, 5(3), 265–293. toda, h.y., yamamoto, t. (1995), statistical inferences in vector autoregressions with possibly integrated processes, journal of econometrics, 66, 225–250. wiener, n. (1956), the theory of prediction, [in:] e. f. backenback (ed.), modern mathematics for engineers, series i. on the interpretation of causality in granger’s sense 139 zellner, a. (1979), causality and econometrics, [in:] k. brunner, i a. m. meltzer (eds.), three aspects of policy and policymaking, north-holland, amsterdam. o interpretacji przyczynowości w sensie grangera z a r y s t r e ś c i. koncepcja przyczynowości sformułowana w 1969 roku przez c.w.j. grangera jest najbardziej popularna w literaturze ekonometrycznej. centralnym założeniem tej koncepcji jest fakt, że przyczyna poprzedza skutek i jest pomocna w prognozowaniu skutku w przyszłości. lata stosowania koncepcji przyczynowości w sensie grangera zaowocowały wieloma nieporozumieniami związanymi z interpretacją wyników empirycznych. artykuł dotyczy systematyzacji definicji przyczynowości w sensie grangera i ich właściwej interpretacji. s ł o w a k l u c z o w e: przyczynowość w sensie grangera, przyczynowość systematyczna, przyczynowość informacyjna, przyczynowość w zależnościach nieliniowych. microsoft word 07_będowska-sojka_b.docx dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 barbara będowska-sójka poznan university of economics the impact of macro news on volatility of stock exchanges† a b s t r a c t. the vast of literature concerning the reaction to macroeconomic announcements focus on american releases and their impact on returns and volatility. we are interested if the news from the german and the polish economy are significant for the stock exchanges in these two countries. using high-frequency 5-minute returns from 2009-2010 we show that the periodical patterns of the german and the polish main indices is very similar and their reaction to the macroeconomic announcements too. in both cases the domestic and neighbor-country announcements are much less important comparing to american releases. k e y w o r d s: high frequency data, macroeconomic announcement, flexible fourier form, intraday periodicity, volatility modeling. introduction a growing literature has documented the significance of macroeconomic news announcements in price formation process. the literature on the effect of macro news on returns and volatility is huge and includes surveys concerning foreign exchange market (bauwens, omrane, giot, 2003), bond market (dominguez, 2003) and equity market (hanousek, kocenda, kutan, 2008). it is worth to stress that surveys focused on forex are most popular and exhaustive (see for instance works of andersen and bollerslev, 1998, faust et al., 2007). the literature considering high frequency returns of the european stock markets in the presence of us macroeconomic announcements is very much limited. harju and hussein (2011) examine four major european equity markets in the aspect of us announcements. they find that us fundamentals have an impact on europeans investor’s behavior. both equity returns and volatility are sensitive to american macro releases. moreover, the indices (cac40, dax, dmi and ftse100) show similar strong intraday seasonality pattern and react in the similar direction to the macroeconomic information. † this work was financed from the polish science budget resources in the years 2010-2012 as the research project n n111 346039. barbara będowska-sójka 100 routinely the announcements considered in the literature are from us, mainly due to the importance of american economy and the timing of us macroeconomic releases. these releases are characterized by specific features that make them useful: they are periodically publicized with timing of announcements being strictly predetermined to the date and the hour. additionally the releases are preceded by the expectations which are obtained as a consensus between different financial analytics (li, engle, 1998). the crucial is the fact that the announcements are released at the time when european stock exchanges are open providing the area for the research of market reaction to the news. contrary to this, majority of european macroeconomic announcements is released before the opening or after the closing of the session, only some of them being announced within the session time. the impact of news releases may be observed on returns and in volatility with the latter more popular. what in fact is influencing the prices is not a news itself, but the surprise content of the news – the more surprising it is, the more volatility increases. in the world of continuously flowing information the only way to measure the reaction to announcements is to focus on intraday data. hanousek, kocenda and kutan (2008) estimate the impact of eu-wide macroeconomic news from different countries on composite stock returns of three markets, the czech, the polish and the hungarian. they conclude that the emerging markets react similarly to foreign news and this reaction is in line with the reaction of more advanced western european markets. however, in their paper all observations from the opening and the closing of the sessions are removed from the sample. our analysis contributes to the existing works in several ways. we study the reaction to several macroeconomic announcements, domestic and neighbor-country as well as the american and compare which of them have a stronger influence on intraday volatility. we focus on the polish and the german stock markets and macroeconomic releases from these two markets. from the previous works we know that on both markets the reaction to american macro releases is quite strong (będowska-sójka, 2010), but the size or the strength of the reaction to polish or german releases is unknown. from the seminal paper of wood et al. (1985) it is known that high-frequency series are characterized by the strong periodical pattern in volatility and that the reasonable intraday dynamic analysis requires the estimation of intraday periodic component. following works of andersen and bollerslev (1998) we use flexible fourier form framework to model intraday series and find out what is the reaction to the announcements on both european markets. two approaches are adopted: in the first the announcement effects are estimated within the flexible fourier form regression, whereas in the second the regression is used only for the purpose of filtering from periodicity and the filtered series are introduced to figarch models. considering the reaction to macro releases from different countries in the short 5-minute interval our main finding is that volatility of indices react stronger to american announcements than to similar in category macro news from germany or from poland. for the domestic releases the reaction is weak in germany and not significant in poland. the intraday periodical pattern is quite similar on both markets and the reaction to american news is similar in size and direction. the rest of the paper is as follows: in section 1 we describe the data, in section 2 periodical pattern is considered. section 3 is devoted to the intraday effect of macroeconomic announcements. in section 4 we present the methodology of volatility modeling the impact of macro news on volatility of stock exchanges 101 and in section 5 the results of ar-figarch models are presented. section 6 concludes. 1. the data we use the data of the german and the polish indices in conjunction with the data on expectations and realizations of scheduled macroeconomic announcements from the american, the german and the polish markets. 1.1. the return series the sample consists of 5-minute intraday percentage logarithmic returns of the german dax and the polish wig20 main stock exchange indices within the period 5.01.2009-30.12.2010. we consider the percentage logarithmic returns, observed with frequency 1/δ (with 1/δ being an integer bigger than 0). as quite common when dealing with intraday data the overnight return is excluded from the analysis. the length of the trading day is normalized to unity and therefore the time that elapses between two consecutive returns is equal (boudt et al., 2010). the dax index is quoted from 9am to 5.30pm, and wig20 from 9am to 4.10pm. after excluding overnight return we obtain 101 observations per day on the german market and 85 on the polish. we use the data available at database www.stooq.pl. the estimation and charts are made in oxmetrics 6.0, in particular g@rch 6 software and ox codes (laurent, peters, 2010). the sample mean of the five minute returns in both series is not distinguishable different from zero with standard deviation higher for wig20 series. the distribution of both series is asymmetric and the kurtosis is very high. hence, the distribution is not gaussian. the significant autocorrelation in the series is observable only for low lags. this changes diametrically when we move to absolute returns which are characterized by very strong autocorrelation function – we will focus on this issue when describing periodical pattern in the intraday series. table 1. descriptive statistics of dax and wig20 returns dax wig20 mean -0.0002 -0.0005 standard deviation 0.1235 0.1569 minimum -1.8979 -1.9750 maximum 1.1926 1.9750 skewness -0.2765 -0.2319 excess kurtosis 7.0612 11.1270 observations 51510 42755 1.2. the macroeconomic announcements from the broad spectrum of macroeconomic announcements we choose only few that are very often used in the papers and are regularly released together with forecasts on the three markets: the american, the german and the polish. we consider only announcement surprises, that means that the release is treated as a news only when it is different from the value of previously announced expectation. the macroeconomic data are from websites: ww.macronext.pl and www.deltastock.com. barbara będowska-sójka 102 some announcements are released regularly before the opening of the markets. these could not be included in the study (and are marked in the table 2 with grey color). therefore we get only 5 types of announcements from germany, 8 from poland and 10 from united states. table 2. the macroeconomic announcement and the timing on the three markets germany poland united states gross domestic product gdp 08:00 10:00 14:30 consumer price index cpi 08:00 14:00 14:30 producer price index ppi 08:00 14:00 14:30 unemployment rate un 09:55 10:00 14:30 industrial production ip 12:00 14:00 15:15 retail sales rs 08:00 10:00 14:30 economic sentiment indicator esi* 11:00 10:00 16:00 durable goods order dgo** 12:00 na 14:30 trade balance tb 08:00 14:00 14:30 purchasing manager index pmi 09:30 09:00 15:45 note: * for germany we take zew (zentrum fűr europaische wirtschaftsforschung economic sentiment) that measures institutional investor sentiment. in poland it is consumers’ confidence indicator published by gus on 10:00 or 14:00. in case of us it is conference board consumer confidence; ** in germany the factory orders are taken into account. there is no such announcement that could stand for proxy in poland. for the united states it is durable goods order ex transportation. as the daylight saving time is changing in different time in europe and america, we control for that when modeling the reaction to american announcements. 2. the intraday pattern in volatility the intraday periodical pattern is very well described in number of papers (see e.g. dacorogna et al., 2001, rossi, fantanzzini, 2008) and usually described as u-shaped or inverted j curve of averages of absolute returns. what is characteristic for the shapes of averages of absolute returns for european stock markets is a sharp increase in volatility at the time of american macroeconomic announcements at 14:30 and 16:00 (będowskasójka, 2010, harju, hussein, 2011). figure 1. average absolute returns for dax and wig20 the impact of macro news on volatility of stock exchanges 103 both the u-shape of autocorrelation of absolute returns (figure 1) and the inverted j shape in averages of absolute returns (figure 2) are visible for dax and wig20. in case of averages of absolute returns they reach higher values at the opening of the markets and then decrease in the lunch time. finally they go up at the closing. this pattern is very similar in both series – the only difference is that in case of wig20 at the end of the session volatility goes up higher. the repeating pattern of autocorrelation function is observed every 101 lags for dax and 85 lags for wig20 (figure 2). this structure of the acf and shape of intraday volatility demand the proper treatment of periodicity. additionally numerous studies have found the day-of-the-week effects, that should be accounted for (bauwens et al., 2000). figure 2. sample autocorrelation function (acf) of the dax and wig20 series of absolute returns in the paper the periodicity removal is achieved with gallants’ (1981) flexible fourier form regression adopted by andersen and bollerslev (1998). we consider the 5-minute returns, where n refers to the number of intraday returns per day (n = 1,….n), and t is the number of trading days in the sample (t = 1, …, t): , , , .( ) , t t n t n t n t n s z r e r n    (1) where σt,n is daily volatility factor, st,n is periodicity factor and zt,n is i.i.d. mean zero unit variance innovation term. the daily volatility component is measured as realized volatility, rv, which means it is a sum of squares of intraday returns, whereas periodicity component is estimated with flexible fourier form regression:   2 2 2, , , , ,2 log ( ) log log log log ,t n t n t n t t n t nx r e r n s z      (2) barbara będowska-sójka 104 2 2 , , , , , ,log (log ) ,t n t n t n t n t n t nx f z e z f u     (3) 2 , 0 1 2 11 2 2 2 cos sin , p t n p p p n n p p f n n n n n                    (4) where pp  ,,,, 210 are estimated parameters and 1 ( 1) / 2,n n  2 ( 1)( 2) / 6n n n   are normalized constants (andersen, bollerslev, 1998). after some experimentation, we found that the order of expansion p=8 is sufficient to capture the basic shape of the series. the estimator of periodic component on day t and interval n: . )2/ˆexp( )2/ˆexp( ˆ 1 1 , , ,      t t n n nt nt nt f ft s (5) finally we obtain periodically filtered series by dividing original series by the estimated seasonal pattern: .~ , , , nt nt nt s r r  (6) we show both the average absolute returns with periodical pattern of volatility in figure 3. after periodicity filtering the intraday pattern for both series is removed, while the effects of macroeconomic announcements remain in the series. figure 3. average absolute returns and absolute filtered returns for dax series the impact of macro news on volatility of stock exchanges 105 figure 4. average absolute returns and absolute filtered returns for wig20 series the descriptive statistics of series before and after filtering are presented in table 3. the filtering of the data with fff regression does not substantially change the descriptive statistics of the series. table 3. descriptive statistics of dax and wig20 returns before and after periodicity removal dax dax after fff wig20 wig20 after fff mean -0.0002 -0.0001 -0.0005 -0.0005 standard deviation 0.1235 0.1224 0.1569 0.1506 minimum -1.8979 -1.9688 -1.9750 -1.4531 maximum 1.1926 1.6331 1.9750 1.5642 skewness -0.2765 -0.2176 -0.2319 -0.2319 excess kurtosis 7.0612 7.7792 11.1270 11.1270 observations 51510 42755 after filtering from periodicity we expect that the strong u-shaped autocorrelation observed previously in absolute returns is removed from the data. in fact, after filtering the periodical pattern is not observed in the series of absolute returns, but in both cases they are still characterized by long memory (figure 5). this slow decay in autocorrelation function is typical for long memory process. therefore we will model volatility in section 5 with an appropriate garch model that allows for such a long memory. 3. the intraday effects of macroeconomic announcements on volatility – fff regression our approach is aimed to study the influence of macroeconomic announcements of the same type from three markets on the volatility of two indices, the german and the polish. the regression specification is than: barbara będowska-sójka 106 , 2 sin 2 cos 12 2 2 1 10, ii p p ppnt x n p n n p n n n n f               (7) where λi is the estimated coefficient and xi is the time-stamped to the nearest 5-minute return announcements dummy variable. in some works it is suggested to filter series using only the control days – which means days without events under study (conrad and lamla 2007, dominguez 2003). we rather agree with boudt et al. (2010) that “conditioning on the days without any news, would lead to a too small sample”. what is actually important is the difference between releases and the expected values – it is not the event itself, but the surprise that is causing the price change. figure 5. sample autocorrelation function of the series of absolute filtered returns in our approach the macroeconomic announcements take the value 1 if they were different from previously released forecasts and 0 otherwise. additionally the dummies representing day-of-the-week effect are also included in the regression. the estimates of fff regression are shown in table 4 (dax) and 5 (wig20). we consider the reaction in first 5 minutes after macro news releasing. for both series, dax and wig20, american announcements do increase volatility of returns in such a short period. in case of dax the domestic announcements increase volatility (ip, es, dgo), whereas in poland domestic releases play no role in very short time interval. the neighbor-country announcements have no impact on volatility within first 5-minutes. the reaction to announcements on 14:35 which are clearly visible in average absolute returns (figure 3) are now confirmed by the estimated parameters. the most powerful announcement in a short run is the american unemployment rate. the estimated coefficients for day-of-the-week effect are omitted, however all of them are statistically significant. the considered announcements together with day-ofthe impact of macro news on volatility of stock exchanges 107 the-week dummies explain only 4% and 5% of intraday volatility in case of dax and wig20 respectively. table 4. parameters estimated in fff regressions for dax united states germany poland gdp 3.1392 (0.5588) 1.2857 (0.8425) cpi 1.7249 (0.5462) 0.7035 (0.5504) ppi 1.4704 (0.5020) 0.1714 (1.6936) un 4.104 (0.5446) -0.3629 (0.5938) 0.3127 (1.6935) ip 1.3241 (0.4947) 1.4878 (0.4957) -0.2941 (0.9892) rs 2.5896 (0.4932) 0.1797 (0.6346) es 2.3163 (0.4952) 1.7297 (0.4858) 0.8400 (0.6461) dgo 2.1283 (0.4946) 0.9873 (0.4853) tb 1.0291 (0.4952) 0.3455 (0.5006) pmi -0.1474 (0.4949) 0.4309 (0.4857) observations 51510 r2 0.044905 0.0418 0.0415 adj. r2 0.044312 0.0413 0.0409 note: the estimated parameters together with standard errors (in italics) are in the upper part. the bolded parameters are statistically significant at α=0.05. 5. modeling with figarch models for the purpose of modeling with figarch models we filter the series again with fff regression but this time including only the day-of-the-week dummies. the series filtered from periodicity with only day-of-the week dummies are introduced into figarch models with dummy variables in the conditional variance equations. for these models the sample is restricted to 2009 only. the conditional mean equations are modeled with the ar(2) process: 1 1 2 2 .t t t tr c r c r a    (8) due to the long memory in series of absolute filtered returns, the innovations are modeled with figarch (p, d, q) process with specification given by bbm’s method (1999): 2 2 2(1 ) ( ) ( )( ),d t t tl l a l a       (9) where lag polonymials ,1)( 1    q i i i ll  ,1)( 1    p i i i llb  and 10  d being the fractional differencing parameter. bollerslev and mikkelsen (1996) define the sufficient condition of non-negative conditional variance in (9) as 31 f with jdjf j /)1(  . an extensive discussion of the properties of figarch model can be found in conrad and haag (2006) where the necessary and sufficient conditions have been described in details. barbara będowska-sójka 108 table 5. the parameters estimated in fff regressions for wig20 united states germany poland gdp 2.7633 (0.5358) 0.4863 (0.8511) cpi 0.5254 (0.5520) 0.3784 (0.5558) ppi 1.4518 (0.5076) -0.2171 (1.7103) un 4.0206 (0.5504) 0.8649 (0.6000) 1.2194 (1.7103) ip 1.792 (0.4999) 0.2251 (0.5011) -0.6301 (0.9989) rs 2.2259 (0.5093) 0.7275 (0.6411) es 1.6942 (0.5014) 0.2867 (0.4912) -0.1671 (0.6526) dgo 2.2502 (0.4999) tb 1.4016 (0.5004) -0.4574 (0.4951) pmi 1.239 (0.5005) -0.1594 (0.5013) observations 42925 42925 42925 r2 0.0572 0.0535 0.0535 adj. r2 0.0564 0.0529 0.0529 note: the estimated parameters together with standard errors (in italics) are in the upper part. the bolded parameters are statistically significant at α=0.05. we introduce dummy variables into conditional variance equation: .))()1()(()( 2 3 1 , t d i iti alllxl     (10) these dummy variables are defined in the way that they take the value of 1 at the time of macroeconomic surprise release on the particular market (american, german or polish) and 0 otherwise. table 6. the estimates of ar(2)-figarch(1, d, 1). dax wig20 c1 0.0052 (0.0064) 0.0011 (0.0112) c2 -0.0201 (0.0082) -0.0312 (0.0121) ω 0.0014 (0.0005) 0.0034 (0.0010) φ 0.0997 (0.1178) 0.0791 (0.0999) β 0.2936 (0.1375) 0.2599 (0.1144) d 0.2681 (0.0211) 0.2425 (0.0172) ω1 (germany) 0.0142 (0.0051) 0.0108 (0.0074) ω2 (poland) 0.0032 (0.0039) 0.0084 (0.0080) ω3 (united states) 0.0936 (0.0293) 0.1540 (0.0424) note: the estimated parameters together with standard errors (in italics). the bolded parameters are statistically significant at α=0.05. 6. results of ar-figarch estimations we estimate ar(2)-figarch(1,d,1) model including aggregated dummy variables standing for announcements. as can be seen in table 6 the autoregressive parameters c2 in conditional mean equation are statistically significant. in conditional variance equation ω, β and d are statistically significant and the values are reasonable according to suggestions in the literature (bollerslev and mikkelsen, 1996), but φ estimates are not the impact of macro news on volatility of stock exchanges 109 statistically significant. it suggest that squares of previous shocks do not impact volatility. the estimated persistence parameters β are highly significant and around the value of 0.25. when we consider the aggregate dummy variables standing for announcements, it is visible that in germany domestic announcements do increase volatility, but the parameter standing at the variable for american news is higher and that confirms our earlier findings. in case of polish market only american announcements increase volatility and this impact is stronger than in germany. conclusions the periodical pattern in high frequency index returns on two european stock exchange markets, the german and the polish, is very strong and might be successfully removed with flexible fourier form (fff). our results of fff regression indicate that us announcements have definitely stronger impact on volatility of both european indices, dax and wig20 than domestic and neighbor country macroeconomic news. it might be partly due to the fact that the number of german news released within the session is very limited. in poland the only significant macro releases are those from america. in both countries, germany and poland, neighbor country news releases generally have no effect on volatility. the reaction to american announcements is immediate and recognized in first five minutes after announcements. if there is any crossreaction to the announcements from the neighbor countries, it is not observable in such a short time. references andersen, t., bollerslev, t. (1998), dm-dollar volatility: intraday activity patterns, macroeconomic announcements and longer run dependencies, the journal of finance, 53, 219–265. andersen, t. g., bollerslev, t., diebold, f. x., vega, c. (2003), micro effects of macro announcements: real-time price discovery in foreign exchange, american economic review, 93, 38–62. bauwens, l., ben omrane, w., giot, p. (2005), news announcements, market activity and volatility in the euro/dollar foreign exchange market, journal of international money and finance, 24, 1108–1125. będowska, b. (2010), intraday cac40, dax and wig20 returns when the american macro news is announced, bank i kredyt 41, 2, 7–20. bollerslev, t., mikkelsen, h. (1996), modeling and pricing long memory in stock market volatility, journal of econometrics, 73, 151–184. boudt, k., croux, ch., laurent, s. (2011), robust estimation of intraweek periodicity in volatility and jump detection, journal of empirical finance 18, 353–367. baillie, r., bollerslev, t., mikkelsen, h. o. (1996), fractionally integrated generalized autoregressive conditional heteroskedasticity, journal of econometrics, 74, 3–30. conrad c., haag b. r., (2006), inequality constraints in the fractionally integrated garch model, journal of financial econometrics, 4, 413–449. conrad, c., lamla, m. (2008), the high-frequency response of the eur-us dollar exchange rate to ecb communication, http://www.fmpm.ch/docs/11th/papers_2008_web/e2c.pdf. (september 2011) dacorogna, m., gençay, r., müller, u., olsen, r., pictet, o. (2001), an introduction to highfrequency finance, academic press, london. dominguez, c. (2003), when do central banks interventions influence intra-daily and longerterm exchange rate movements, university of michigan and nber working paper. barbara będowska-sójka 110 faust, j., rogers, j. h., wang, s. b., wright, j. h. (2007), the high-frequency response of exchange rates and interest rates to macroeconomic announcements, journal of monetary economics, 54, 1051–1068. gallant, r. (1981), on the bias in flexible functional forms and an essentially unbiased form: the flexible fourier form, journal of econometrics, 211–245. hanousek, j., kocenda, e. and kutan, a. (2008), the reaction of asset prices to macroeconomic announcements in new eu markets: evidence from intraday data, working paper series issn 1211-3298, 349, charles university, centre for economic research and graduate education. harju, k., hussain, s. (2011), intraday seasonalities and macroeconomic news announcements, european financial management, volume 17, issue 2, 367–390. lahaye, j., laurent, s., neely, ch. (2007), jumps, cojumps and macroannouncements, federal reserve bank of saint louis, working paper 2007–032a. laurent, s., peters, j.-p. (2004), estimating and forecasting arch models using g@rch 4.2, timberlake consultants ltd, www.timberlake.co.uk,. li, l., engle, r. (1998), macroeconomic announcements and volatility of treasury futures, university of california at san diego, economics working paper series 98–27. rossi, e., fantanzzini, d. (2008), long memory and periodicity in intraday volatilities of stock index futures, working paper, http://economia.unipv.it/docs/dipeco/quad/ps/q210.pdf (24.10.2011). wood, r., mcinish, t., ord, j.k. (1985), an investigation of transactions data for nyse stocks, the journal of finance 40, 723–739. wpływ ogłoszeń makroekonomicznych na zmienność rynków akcji z a r y s t r e ś c i. celem artykułu jest zbadanie wpływu ogłoszeń makroekonomicznych z trzech krajów, stanów zjednoczonych, niemiec i polski na zmienność śróddziennych indeksów dax i wig20. dla obu indeksów opisano wzorzec zmienności i zastosowano elastyczną postać fouriera w modelowaniu szeregów. w stosunkowo krótkim przedziale czasowym pięciu minut w obu indeksach zaobserwowano silną reakcję na ogłoszenia ze stanów, a w niemieckim indeksie dax słabszą reakcję na ogłoszenia niemieckie. dla polskiego wig20 nie wychwycono w tak krótkim interwale czasowym reakcji na polskie ogłoszenia. dodatkowo oba indeksy nie reagują na ogłoszenia z rynku kraju sąsiadującego. s ł o w a k l u c z o w e: dane śróddzienne, ogłoszenia makroekonomiczne, elastyczna forma fouriera, cykliczność śróddzienna, modelowanie zmienności. microsoft word 06_fiszeder_p.docx dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 piotr fiszeder nicolaus copernicus university in toruń minimum variance portfolio selection for large number of stocks – application of time-varying covariance matrices a b s t r a c t. an evaluation of the efficiency of different methods of the minimum variance portfolio selection was performed for seventy stocks from the warsaw stock exchange. eight specifications of multivariate garch models and six other methods were used. the application of all considered garch-class models was more efficient in stocks allocation than the implementation of the other analyzed methods. the simple specifications of multivariate garch models, whose parameters were estimated in two stages, like the dcc and ccc models were the best performing models. k e y w o r d s: multivariate garch models, time-varying covariance matrix, portfolio selection. introduction the selection of estimators of the population mean, variance and covariance of financial returns and closely connected with it the selection of forecasting methods of the population mean, variance and covariance of returns plays a vital role in the construction of efficient portfolios. there is a consensus among both financial market practitioners and scientists, that financial returns are difficult to forecast and the portfolio construction process according to the criteria proposed by markowitz (1952, 1959) is very sensitive to the selection of an estimator of the expected value of returns (see michaud, 1989; best, grauer, 1991; chopra, ziemba, 1993). small differences in estimates of the expected returns often lead to a meaningful portfolio reconstruction (see e.g. jobson, korkie, 1980). it is believed that the estimation of population variances and covariances of returns is easier than the estimation of population means (merton 1980; nelson, 1992), however the selection of estimators of population variances and covariances of returns has also a significant impact on an asset allocation (see e.g. litterman, winkelmann, 1998; chan, karceski, lakonishok, 1999). traditionally used estimators of the population mean, variance and copiotr fiszeder 88 variance, namely the arithmetic mean and empirical variance and covariance calculated on the basis of available historical data do not give the best results in efficient portfolios construction (see e.g. jorion, 1991; sheedy, trevor, wood, 1999; johannes, polson, stroud, 2002; flavin, wickens, 2006). the classical approach to the selection of efficient portfolios is static, i.e. a chosen period or moment is considered, but not their sequence. such an approach ignores therefore the variability of conditional variances and covariances of returns. a dynamic approach to the selection of efficient portfolios for homogeneous assets, i.e. stocks, based on the forecasts of variances and covariances of returns constructed from multivariate garch models is presented in this paper. there is an extensive literature on time-varying conditional variances of returns but also on time-varying conditional covariances and correlation coefficients between financial series. the multivariate garch process allows to describe both time-varying conditional variances and covariances of returns. if variances and covariances of returns are not constant, then the forecasts based on multivariate garch models should give additional benefits in the selection of efficient portfolios. the literature on the application of garch models in construction of portfolios is poor, and the results of such analyses, due to the complexity of the problem, are still fragmentary. in most of the studies the univariate garch model is applied or a very limited number of assets are used. among the few investigations in which multivariate garch models were applied for large portfolios the following papers can be mentioned: engle, sheppard (2001), engle, colacito (2006), osiewalski, pajor (2010). the primary purpose of this study is to evaluate the effectiveness of different methods of the minimum variance portfolio selection, mainly with the application of various multivariate specifications of garch models. the adopted approaches to the portfolio construction differ only by forecasting methods of the covariance matrix of returns. the paper is an extension of the author's earlier work presented in fiszeder (2004, 2007). the plan for the rest of the paper is as follows. section 1 outlines the way in which efficient portfolios are selected. in section 2 competing methods of the covariance matrix estimation are presented. section 3 contains the effectiveness analysis of the minimum variance portfolio selection for seventy stocks quoted on the warsaw stock exchange (wse) and section 4 presents the conclusions. 1. dynamic process of portfolio selection with application of garch models it is difficult to evaluate the influence of the choice of the covariance matrix estimator against the selection of the population mean estimator in construction of efficient portfolios. one of the ways to eliminate the influence of the choice of the mean estimator is the minimum variance portfolio selection. shares of minimum variance portfolio selections for large number of stocks – application… 89 individual assets in the minimum variance portfolio depend solely on the covariance matrix, that is why such procedure is used in the paper. for a given t, based on all the available data from the period ],1[ t parameters of a garch model are estimated1. the forecast of the covariance matrix based on the estimated model is formulated at time t , where the  is the forecast horizon. the constructed forecast is used for the selection of efficient portfolio. let )...,,,(' 21 pttnpttpttptt www   w , where pttiw  is the share of asset i in the portfolio at time t , ptt h is the forecast of the conditional covariance matrix of returns at time t . the variance of portfolio returns is then equal to pttpttptt   whw' . in order to select the minimum variance portfolio (the global minimum) the following quadratic programming problem has to be solved: min,'  pttpttptt  whw (1) subject to the constraint: ,1'  lw ptt  (2) where l is the 1n vector of ones. when short selling is not allowed the boundary conditions have to be additionally imposed: 0 pttiw  for ....,,2,1 ni  (3) the whole procedure is repeated for successive periods (with the arrival of subsequent data). if short selling is allowed, then shares of assets in the minimum variance portfolio are defined by the following formula: , 1 1 lhw     ptt ptt ptt c   (4) where lhl 1'   pttpttc  . the variance of the minimum variance portfolio is then equal to pttptt cv    /1 . other efficient portfolios can be selected by finding the minimum variance portfolio subject to a minimum expected return. 1 if conditional expected values of returns are different from zero, then parameters of conditional mean equations should also be estimated. piotr fiszeder 90 2. specifications of multivariate garch models eight parameterizations of multivariate garch models were applied in the analysis: scalar bekk, integrated, ccc, orthogonal, dcc, integrated dcc, deco-dcc and additionally scalar bekk with student-t innovations. the results obtained for the garch models were compared with the outcomes for six other methods: equal shares in all stocks, the unconditional covariance matrix of returns, the rolling covariance matrix2, the 25-day rolling covariance matrix, the exponentially weighted moving average estimate of the covariance matrix (hereafter the ewma covariance matrix)3, the ewma covariance matrix with a smoothing parameter set to 0.94 (hereafter the riskmetrics4). only some basic information about the considered multivariate models is given below. more details can be found for example in bauwens, laurent, rombouts (2006) or silvennoinen, teräsvirta (2009). let assume that tε can be either returns with the mean zero or residuals from filtered time series: ,),(~ 1 ttt n h0ε  (5) where 1t is the set of all information available at time t-1 and th is the nn  symmetric conditional covariance matrix. the estimation of parameters of the general form of a multivariate garch model, vech model5 (kraft, engle, 1983), is very difficult even for a small number of assets. for this reason simpler parameterizations of multivariate garch models were used in the study. considered were only those specifications which ensure the positive definiteness of the covariance matrix. baba, engle, kraft, kroner (1990) introduced the following form the so-called bekk(p,q) model6 (see engle, kroner, 1995): ,'' 1 1 ''      p j jjtjiitit q i it ehedεεdcch (6) where c, id and je are nn  parameter matrices and c is an upper triangular matrix. 2 the value of a rolling window was chosen to minimize the variance of a portfolio in a presample. 3 the value of a smoothing parameter was chosen to minimize the variance of a portfolio in a pre-sample. 4 the riskmetrics methodology was developed in the investment bank j. p. morgan for measuring market risk with var. the value of a smoothing parameter in the ewma model was often set by financial market practitioners to 0.94 for daily data. 5 the name of the model comes from the application of the vech operator for the conditional covariance matrix. 6 the name of the model is formed by the first letters of the authors surnames. minimum variance portfolio selections for large number of stocks – application… 91 the bekk model is also too complex for large portfolios and the imposition of restrictions on parameters in estimation is necessary. the so-called scalar bekk model can be obtained by replacing the matrices id and je by the scalars 2/1id and 2/1 je . additionally the variance targeting approach (see engle, mezrich, 1996) was used. for example for 1 pq instead of the product 'cc the following formula can be substituted: ,)1(' 11 scc ed  (7) where s is the sample covariance matrix given as    t t tt t 1 '1 εεs . two specifications of the conditional distribution of tε in (5) were considered for the scalar bekk model, namely multivariate normal and student-t. a further simplification of the model can be obtained by the assumptions 0'cc and      q i p j ji ed 1 1 1 . this formulation is often called the integrated multivariate model7. for 1 pq the model has only one parameter. in the constant conditional correlations (ccc) model of bollerslev (1990), which is outside the bekk class, the time-varying conditional covariances are parameterized to be proportional to the product of the corresponding conditional standard deviations: , ttt dγdh  (8) where td is a nn  diagonal matrix ),...,,(diag 2/12/1 2 2/1 1 ntttt hhhd with ith defined as any univariate garch model, and γ is a nn  matrix of the time–invariant conditional correlations. alexander and chibumba (1996) introduced the orthogonal garch model defined as: ,2/1 tmtt fλuεv   (9) ,2/12/1 vvvh tt  (10) where ),...,,(diag 21 nvvvv , with iv the population variance of it , mλ is a matrix of dimension mn  given by ),...,,diag( 2/12/12 1/2 1 mmm lllpλ  , 0...21  mlll being the m largest eigenvalues of the population correlation matrix of tu , and mp the mn  matrix of associated (mutually orthogonal) eigenvectors, )'...( 21 mtttt ffff is a random process such that 0)(1  tte f , the 7 the name of the model comes from an analogous parameterization of the univariate igarch model. piotr fiszeder 92 conditional covariance matrix of tf is equal to ),...,,(diag 222 t 21 mttt fff q , 2 itf  is defined as a univariate garch model (for mi ,...,2,1 ) and the conditional covariance matrix of tu is equal to mtmt 'λqλv  . the dynamic conditional correlation (dcc) model of engle (2002) can be defined as: ,tttt drdh  (11) ,1*1*  tttt qqqr (12) ,)'()1( 111 1          p j jtj q i ititi q i p j jit qzzsq  (13) where ),...,,(diag 2/12/12 2/1 1 ntttt hhhd , kth can be defined as a univariate garch model (for nk ,...,2,1 ), tz is a vector of standardized values of kt , i.e. ktktkt hz / , tr is the time-varying conditional correlation matrix of tz , s is the sample covariance matrix of tz , * tq is a diagonal matrix composed of the square root of the diagonal elements of tq and the parameters have to satisfy the condition 10 1 1      q i p j ji  . besides the standard dcc model two other modifications were also considered. when the sum of the parameters is equal to one 1 1 1      q i p j ji  the dcc model is called the integrated dcc model. engle and kelly (2008) introduced the deco-dcc (dynamic equicorrelation) model. in this parameterization of the dcc model the equality of all pairwise conditional correlations at each time is assumed. 3. evaluation of portfolio performance for seventy stocks quoted on the wse the presented approach for the minimum variance portfolio selection was evaluated for polish stocks. the investigated period was november 17, 20008 to june 30, 2009 (2158 daily returns). all stocks quoted in the specified period on the warsaw stock exchange were considered in the analysis. the companies for which the percent of non-trading days was higher than 5% were omitted in order to avoid the problem of non-synchronous trading. in total seventy stocks were analyzed, however all the models used in the paper can be applied for 8 since the introduction of the new trading system warset. minimum variance portfolio selections for large number of stocks – application… 93 a much larger number of companies. an evaluation of portfolio performance was based on data from january 2004 to june 2009 (1380 observations). the following five steps were performed: (1) the estimation of parameters of all considered models9 (at the beginning for data from november 17, 2000 to december 30, 2003)10, (2) the construction of one-day ahead forecasts of the conditional covariance matrix, (3) the minimum variance portfolio selection (4) the ex post calculation of the portfolio variance as a square of the realized portfolio return, (5) the extension of the sample with one observation. all the steps were repeated 1380 times. every time the estimation of parameters was performed for increasing sample size. for each model and method the mean of the portfolio variances was calculated. the results are presented in table 1 (a square root of the mean is given). table 1. estimates of the standard deviations of returns for the minimum variance portfolios portfolio designation standard deviation (× 10-2) rank integrated dcc 0.9257 1 dcc 0.9316 2 ccc 0.9326 3 deco-dcc 0.9843 4 scalar bekk student-t 1.0030 5 scalar bekk 1.0065 6 orthogonal 70 factors 1.0854 7 integrated 1.1155 8 unconditional covariance matrix 1.1155 8 rolling covariance matrix 1.2634 10 equal shares in all stocks 1.3538 11 riskmetrics 1.6549 12 ewma covariance matrix 2.9846 13 25-day rolling covariance matrix 29.7629 14 the best performing model in the selection of the minimum variance portfolio was the integrated dcc model. on the other hand the worst performing method was the 25-day rolling covariance matrix. an application of all the garch-class models was more effective in allocation of stocks than the implementation of the other analyzed methods. the first four positions in the ranking (see table 1) were occupied by the models, whose parameters were estimated in two steps and in the first stage parameters of a univariate garch model were estimated. the integrated model had the worst rank among all the applied garch models, however in this case the model was reduced to the unconditional covar 9 codes written by the author in the gauss programming language were applied. 10 logarithmic returns were used in the study. because daily data were used, the logarithmic return on a portfolio is very close to the weighted average of the logarithmic returns on the individual assets. piotr fiszeder 94 iance matrix of returns11. the orthogonal garch model, which assumes that returns depend on common independent factors extracted by principal component analysis ranked the one before last among the garch models. this result follows, first of all, from the loss of important information. the methods used by financial market practitioners, i.e. the rolling covariance matrix and the exponentially weighted moving average estimate of the covariance matrix took a distant positions in the ranking, even behind the unconditional covariance matrix of returns. the very poor result of the rolling covariance matrix with a low value of the rolling window equal to 25 deserves to be highlighted. the value of the smoothing parameter, set often for daily data in the riskmetrics methodology to 0.94 is not the optimal value for the polish stock market (it is definitely too low). for a large number of assets like in this study, probably none of the restrictions, assumed by the considered garch models, is met. however, one can attempt to indicate the restrictions, which have a stronger negative influence on the performance of portfolio, in the case when they are not met. for example, the assumption that models describing conditional variances and covariances have the same parameters for all series12 (like in the scalar bekk and integrated models) is probably too strong. it also seems that the presupposition of the constancy of conditional correlation coefficients is less restrictive. the statistical significance of the observed differences between the performance of the different models was not verified, however some of them are relevant from the economic point of view. for example, the difference between the first in the ranking, namely the integrated dcc model and the unconditional covariance matrix of returns means a decrease of about 3 percentage points of the standard deviation per year. additionally, values of the schwarz information criterion (sic) were calculated, when it was possible to evaluate a joint likelihood function (see table 2). the ranking of the models according to the sic is similar to the one for the portfolio performance evaluation (table 1). the exceptions are the scalar bekk model with student-t innovations and the integrated model which take better position in the sic ranking. the better performance of garch models with student-t innovations in the rankings based on the information criteria is common in other studies (see fiszeder, 2009). the rankings of models constructed on the basis of the information criteria provide useful clues for the selection of models for the minimum variance portfolio construction. it has to be remembered, however, that some characteristics are crucial in the evaluation of the general fit of the model in a sample (like for instance a type of conditional density), but their influence on the performance of portfolio is not so important. 11 estimates of the parameter on the lagged conditional covariance matrix were equal to one. 12 it means, that the volatility dynamics of all series is very similar. minimum variance portfolio selections for large number of stocks – application… 95 table 2. ranking of the models based on the schwarz information criterion portfolio designation sic rank integrated dcc -679060.02 1 dcc -666933.43 2 scalar bekk student-t -661928.17 3 deco-dcc -658854.85 4 integrated -656773.56 5 scalar bekk -645459.24 6 riskmetrics 2052174.1 7 it is interesting to compare the results of this analysis with the similar study for twenty stocks (fiszeder, 2007). the garch models, whose parameters were estimated in one stage (like the scalar bekk and integrated models) took further positions in the ranking. it seems that the assumption that models describing conditional variances have the same parameters for all series becomes more and more restrictive for a larger number of assets. furthermore, the differences between evaluations of portfolios, obtained with the use of the applied methods, were greater for the seventy stocks. the reason for such results is probably the higher differentiation of the seventy stocks, among which are both huge but also very small companies. in the study for the twenty stocks only the biggest companies were considered. this comparison for different numbers of stocks clearly shows that the received results depend on the properties of financial time series. conclusions the dynamic approach to the selection of efficient portfolios for a large number of homogeneous assets, i.e. stocks, based on the forecasts of variances and covariances of returns constructed from multivariate garch models has been presented in this paper. an evaluation of the efficiency for different methods of the minimum variance portfolio selection was performed for the seventy stocks from the warsaw stock exchange. the eight specifications of multivariate garch models and the six other methods were used. capturing time-varying variances and covariances of stock returns does not always increase the efficiency of the asset allocation process. the application of all the considered garch-class models was more efficacious in the allocation of stocks than the implementation of the other analyzed methods, including the methods employed often by financial market practitioners. the simple specifications of multivariate garch models, whose parameters were estimated in two stages, like the dcc and ccc models, were the best performing models. piotr fiszeder 96 references alexander, c., chibumba, a. (1996), multivariate orthogonal factor garch, university of sussex discussion papers in mathematics. baba, y., engle, r. f., kraft, d. f., kroner, k. f. (1990), multivariate simultaneous generalized arch, department of economics, university of california at san diego, working paper. bauwens, l., laurent, s., rombouts, j., v., k. (2006), multivariate garch models: a survey, journal of applied econometrics, 21, 1, 79–110. best, m. j., grauer, r. r. (1991), on the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results, review of financial studies, 4, 315–342. bollerslev, t. (1990), modelling the coherence in short-run nominal exchange rates: a multivariate generalized arch approach, review of economics and statistics, 72, 498–505. chan, l. k. c., karceski, j., lakonishok, j. (1999), on portfolio optimization: forecasting covariances and choosing the risk model, review of financial studies, 12, 937–974. chopra, v. k., ziemba, w. t. (1993), the effect of errors in means, variances and covariances on optimal portfolio choice, journal of portfolio management, 19, 6–11. engle, r. f. (2002), dynamic conditional correlation – a simple class of multivariate garch models, journal of business and economic statistics, 20, 339–350. engle, r. f., colacito, r. (2006), testing and valuing dynamic correlations for asset allocation, journal of business and economic statistics, 24, 2, 238–253. engle, r. f., kelly, b. (2008), dynamic equicorrelation, working paper, stern school of business, new york. engle, r. f., kroner, k. f. (1995), multivariate simultaneous generalized arch, econometric theory, 11, 122–150. engle, r. f., mezrich, j. (1996), garch for groups, risk, 9, no. 8, 36–40. engle, r. f., sheppard, k. (2001), theoretical and empirical properties of dynamic conditional correlation multivariate garch, nber working paper no. 8554. fiszeder, p. (2004), dynamiczna alokacja aktywów – model markowitza (dynamic assets allocation – markowitz model), rynki finansowe – prognozy a decyzje (financial markets – forecasts and decisions), acta universitatis lodziensis, folia oeconomica, 177, uniwersytet łódzki, łódź. fiszeder, p. (2007), konstrukcja portfeli efektywnych z zastosowaniem wielorównaniowych modeli garch (efficient portfolios construction with application of multivariate garch models), folia oeconomica cracoviensia, 48, 47–68. fiszeder, p. (2009), modele klasy garch w empirycznych badaniach finansowych, (the class of garch models in empirical finance), wydawnictwo umk, toruń. flavin, t. j., wickens, m. r. (2006), optimal international asset allocation with time-varying risk, scottish journal of political economy, 53, 5, 543–564. jobson, j. d., korkie, b. (1980), estimation for markowitz efficient portfolios, journal of american statistical association, 75, 544–554. johannes, m., polson, n., stroud, j. (2002), sequential optimal portfolio performance: market and volatility timing, columbia university, working paper. jorion, p. (1991), bayesian and capm estimators of the means: implications for portfolio selection, journal of banking and finance, 15, 717–727. kraft, d. f., engle, r. f. (1983), autoregressive conditional heteroskedasticity in multiple time series, department of economics, ucsd, working paper. litterman, r., winkelmann, k. (1998), estimating covariance matrices, risk management series, goldman sachs. markowitz, h. m. (1952), portfolio selection, journal of finance, 7, 77–91. markowitz, h. m. (1959), portfolio selection: efficient diversification of investments, yale university press, new haven, ct. merton, r. c. (1980), on estimating the expected return on the market: an exploratory investigation, journal of financial economics, 8, 323–361. minimum variance portfolio selections for large number of stocks – application… 97 michaud, r. o. (1989), the markowitz optimization enigma: is ‘optimized’ optimal?, financial analysts journal, 45, 31–42. nelson, d. b. (1992), filtering and forecasting with misspecified arch models i. getting the right variance with the wrong model, journal of econometrics, 52, 61–90. osiewalski, j., pajor, a. (2010), bayesian value-at-risk and expected shortfall for a portfolio (multiand univariate approaches), paper presented at the v symposium fens 2010, warsaw. sheedy, e., trevor, r., wood, j. (1999), asset-allocation decisions when risk is changing, journal of financial research, 22, 3, 301–315. silvennoinen, a., teräsvirta, t. (2009), multivariate garch models, in andersen t. g., davis r. a., kreiss j. p., mikosch t. (ed.), handbook of financial time series, 201–229, springer, new york. konstrukcja portfeli o minimalnej wariancji dla dużej liczby spółek – zastosowanie zmiennych w czasie macierzy kowariancji z a r y s t r e ś c i. w pracy dokonano oceny efektywności różnych metod tworzenia portfeli o minimalnej wariancji, w tym przede wszystkim z wykorzystaniem różnych specyfikacji wielorównaniowych modeli garch. badanie zostało przeprowadzone dla 70 spółek notowanych na gpw w warszawie. zastosowano osiem parametryzacji modelu garch: skalarny bekk, zintegrowany, ccc, ortogonalny dla 70 czynników, dcc, zintegrowany dcc, deco-dcc, skalarny bekk z warunkowym rozkładem t studenta oraz sześć innych metod: równe udziały dla wszystkich aktywów, bezwarunkowa macierz kowariancji stóp zwrotu, ruchoma macierz kowariancji, ruchoma macierz kowariancji ze stałą wygładzania równą 25, metoda wyrównywania wykładniczego dla macierzy kowariancji oraz metoda wyrównywania wykładniczego dla macierzy kowariancji z parametrem wygasania równym 0,94. s ł o w a k l u c z o w e: wielorównaniowe modele garch, zmieniająca się w czasie macierz kowariancji, konstrukcja portfela. dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 mariola piłatowska nicolaus copernicus university in toruń information and prediction criteria in selecting the forecasting model a b s t r a c t. the purpose of the paper it to compare the performance of both information and prediction criteria in selecting the forecasting model on empirical data for poland when the data generating model is unknown. the attention will especially focus on the evolution of information criteria (aic, bic) and accumulated prediction error (ape) for increasing sample sizes and rolling windows of different size, and also the impact of initial sample and rolling window sizes on the selection of forecasting model. the best forecasting model will be chosen from the set including three models: autoregressive model, ar (with or without a deterministic trend), arima model and random walk (rw) model. k e y w o r d s: information and prediction criteria, accumulated prediction error, model selection. introduction the model selection literature has recently emphasized the necessity of considering the choice of model depending on the purpose of econometric modeling. in modeling approach the two aims are mentioned the most frequently, namely searching 'true' model and selecting the best forecasting model (optimizing prediction). that first aim of modeling is hard to realize because the economic reality is seen as a complex and dynamically evolving structure whose mechanism is hidden and almost impossible to uncover. therefore the model is an approximation (or simplification) of reality which represents the relevance of a particular phenomenon. it is advocated to assume that each model is not true by definition (taub, 1993; deleew, 1998) or that "all models are wrong, but some are useful" (box, 1976). having in mind that none of models cannot reflect all of reality, the debate concerning true models should be completed because it seems to be unproductive. hence, the second aim of modeling, i.e. selecting the best forecasting model acquires relevance from the practical point of view. mariola piłatowska 22 in predictive approach the goal of selecting the true model is abandoned and the attention focuses on seeking a model with as small predictive errors as possible. it should be emphasized that in forecasting situation the misspecified model are allowed because such a model may yield excellent forecasts. on the other hand good prediction is treated as a test of any subsidiary aim, i.e. if the purpose of an analysis is to estimate parameters, then the best estimated model should give the best prediction; if the purpose of an analysis is hypothesis testing, then any rejected model should give worse forecasts than any accepted model (clarke, 2001; de luna, skouras, 2003; kunst, 2003). to select the forecasting model the different model selection methods can be used, for instance information and prediction criteria. however, question may arise whether the performance of both criteria is the same with regard to the choice of model. it has been shown on simulated data1 (kunst, 2003) that information criteria should be rather used if one is interested in finding 'true' model. if the purpose of analysis is to choose a forecasting model, the prediction criteria are preferred because they select the model yielding the smallest prediction error, although sometimes it may be an incorrect choice (not true model). however, in economic reality the true model is unknown, therefore it is worth checking the performance of information and predictiion criteria in practical context (empirical data). the purpose of the paper it to compare the performance of both information and prediction criteria in selecting the forecasting model on empirical data when the data generating model is unknown. the attention will especially focus on the evolution of information criteria (aic, bic) and accumulated prediction error (ape) for increasing sample sizes and rolling windows of different size, and also the impact of sample and rolling window sizes on the selection of forecasting model. the best forecasting model will be chosen from the set including three models: autoregressive model, ar (with or without a deterministic trend), arima model and random walk (rw) model on the basis of empirical data for poland. the choice of model will be carried out using information (aic, bic) and prediction (ape and mse, mape, u) criteria. the decision of selecting a model by information and prediction criteria is checked in out-of-sample forecasting by comparing accuracy measures for given forecast models. 1 the true data were generated from arma(1, 1) models with 100+100+10 observations (first 100 observations were discarded; 1000 replication were conducted). the set of models included: arma(1, 1), ar(1) and ma(1) models. to select a model the aic information criterion and the mean squared error (mse) based on prediction error from 10 one-step-ahead forecasts were applied (see kunst, 2003). information and prediction criteria in selecting the forecasting model 23 1. information and prediction criteria generally, information criterion takes the form: ,)ˆ(ln2 qlic   where )ˆ(l is the likelihood function, and q is a penalty term that is a function of the number of parameters k and the number of observations n; this penalty guards against overfitting, i.e. using too many parameters. for akaike's information criterion (aic) the penalty is equal to ,2kq  for schwartz (bayes) information criterion − ),ln(nkq  for hannan-quinn information criterion − ).ln(ln2 nkq  for small samples the aic criterion is biased and may suggest a model with a high number of parameters compared with the number of observations ).40/( kn thus a bias-corrected version, aicc, is increasingly used. the latter is given by adding the quantity )1/()1(2  knkk to the ordinary aic. the bic (like the aicc criterion) penalizes the addition of extra parameters more severely than the aic, and should be preferred to the ordinary aic in timeseries analysis especially when the number of parameters is high compared with the number of observations. applying information criteria in model selection the model with the minimum of a given information criterion is chosen. to traditional prediction criteria, used both in the accuracy evaluation and selection of forecasting model, belong: mean absolute error , || 1 t e mae t t t  mean square error ,1 2 t e mse t t t  root mean square error ,msermse  mean absolute percentage error %,100 |/| 1 t ye mape t t t t  theil's inequality coefficient , model)benchmark''( )modelnew''( rmse rmse u  where te denotes prediction error, ,ˆttt yye  ty − realization of y in period t, tŷ − forecast of y for period t. applying usual accuracy measures the model with the smallest value of given measure is selected what corresponds to the smallest prediction error. mariola piłatowska 24 the theil's inequality coefficient u indicates whether a given model is worse (u > 1) or better (u < 1) than the random walk model ( tt yy 1ˆ ) considered as a benchmark model. it is worth highlighting that the choice of accuracy measure can affect the ranking of forecasting methods, and also models (armstrong, 2001; armstrong, fildes, 1995). for instance, the mse depends on the scale in which the variable is measured. this means that the mse is appropriate only for assessing the results for a single time series, and should be avoided to assess accuracy across (many) different series. in that case the scale-independent measures are required2, e.g. mape or theil's inequality coefficient (u). besides, for the reason that the loss function may be asymmetric (e.g. underforecasting is worse than overforecasting), it may be important to forecast the direction of movement or to predict large movements (chatfield, 2000). however, it is not possible to select a measure of forecast accuracy that is scale-independent and yet satisfies the demands of the appropriate loss function. in conclusion, there is no measure suitable for all types of data and all contexts. there are many empirical evidence that a method which is 'best' under one criterion need not be 'best' under alternative criteria (e.g. swanson and white, 1997). the choice of forecasting model may also be carried out by the accumulated prediction error, ape, (rissanen, 1986). according to the ape the most useful model is the model with the smallest out-of-sample one-step-ahead prediction error. the ape method proceeds by calculating sequential one-step-ahead forecasts based on gradually increasing sample. for model mj the ape is calculated as follows (wagenmaker, grünwald, steyvers, 2006): 1. determine the smallest number s of observations that makes the model identifiable. set ,1 si so that .1 si  2. based on the first 1i observations, calculate a prediction ip̂ for the next observation .i 3. calculate the prediction error for observation i, e.g. squared difference between the predicted value ip̂ and the observed value .ix 4. increase i by 1 and repeat steps 2 and 3 until .ni  5. sum all of the one-step-ahead prediction errors as calculated in step 3. the result is the ape. for model jm the accumulated prediction error is given by: )],ˆ(,[)( 11  ii n si ij xpxdmape 2 mentzer and kahn (1995) found in a survey of 207 forecasting executives in us that mape was the most commonly used measure (52%) while only 10% used mse. information and prediction criteria in selecting the forecasting model 25 where d indicates the specific loss function that quantifies the discrepancy between observed and predicted values. in the case of point predictions one typically uses the squared error 2)ˆ( ii px  , but another choice would be to compute the absolute value loss ii px ˆ , or more generally, an α-loss function,  ii px ˆ , ]2,1[ (rissanen, 2003). 2. selection of forecasting model – empirical examples to compare the performance of information and prediction criteria in selecting the best forecasting model the monthly data on consumer price index cpi (corresponding period of previous year =100) and industry production ip (in billion pln zl) in poland were used3 (in the period 2002:01−2010:12, 108 observations, data are seasonally adjusted). the reason for such selection of time series is the desire to check the performance of selection criteria with regard to time series of different properties, i.e. in the above case cpi is expected to be rather an integrated process, and ip − rather a stationary process around deterministic trend4. the set of candidate models for cpi includes: autoregressive model, ar(12), arima(12,1,0) model and random walk model (rw) as a benchmark model; for industry production ip this set was as follows: linear trend model with autoregression of twelfth order (further denoted as ar for the convenience of presentation), arima(12,1,0) and random walk model as a benchmark model5. two versions of estimation procedures are considered: version i: the models are iteratively estimated beginning with the initial sample size n(s) which is being increased by one until n = 108 (until 2010:12); three sizes of initial sample n(s) are taken: 40, 60, 80; version ii: the models are iteratively estimated for rolling window size of 40, 60 and 80 observations. the question is to what extent the size of initial sample and rolling window have the impact on the choice of forecast model when the information criteria 3 data have been taken from the statistical bulletin of the central statistical office in poland. 4 many empirical studies conclude that the consumer price index (cpi) as a financial series is rather integrated and then the arima model is more appropriate than the ar model (with or without deterministic trend). whereas, the industry production (ip) is treated rather as a stationary process around deterministic trend and then the ar model (with or without deterministic trend) is often taken as a more correct model. however, it may occur in forecasting that an inappropriate model will yield better forecasts. to take into account this possibility, both models are used in empirical study. 5 the order of autoregression was fixed at 12 as the potentially highest order reflecting monthly frequency of data. mariola piłatowska 26 (aic, bic) and prediction criteria (ape_se, ape_ae) are used as selection criteria. notations ape_se and ape_ae denote the accumulated prediction error (ape) that uses the squared error and absolute error respectively as a loss function. to make a choice of best forecasting model the information criteria (aic and bic) and prediction criteria (ape_se and ape_ae) were calculated at each iteration. the results are presented in figures 1−8 as difference in a given criterion for pairs of models, i.e. aic(mi)−aic(mj), bic(mi)−bic(mj), ape_se(mi)−ape_se(mj), ape_ae(mi)−ape_ae(mj), and also in tables presenting the choices of forecasting model for all pairs of models. these differences in selection criteria are interpreted as follows: the positive differences favor the second model over the first one in a pair of models (this means either a lower value of information criterion or smaller prediction error for the second model), and the negative differences indicate that the first model outperforms the second one. the sign of differences in criteria may change in time which indicates that one model has become outdated. however, from the forecasting point of view the most important is the sign of differences in criteria at the end of studied period, therefore the choice of the best forecasting model has been made basing on the performance of differences in criteria at the end of sample (at least three observations with the same sign of difference in a given criterion). figure 1 (row 1) demonstrates that independently of initial sample size the performance of differences in aic criterion for pairs of models is similar, i.e. the aic criterion favors the ar model over the arima and rw models, and the arima model is better in sense of aic criterion than the rw model. however, the results for different rolling window sizes are different (fig. 1, row 2). while the dominance of the ar model over arima model is maintained, the aic criterion prefers the rw model over ar and arima model (differences in aic for pairs of models, aic(arima)−aic(rw) and aic(ar)−aic(rw), are positive) what is opposite to results obtained for increasing by one sample size (fig. 1, row 1). the different results for window size of 80 observations in comparison with those obtained for window size of 40 and 60 observations could suggest the influence of window size on a choice of model by the aic criterion, but in that case it is rather the problem of too large window size with regard to the number of observations. the results of model selection for consumer price index (cpi) by the bic criterion seem to be more stable and insensitive as well to the initial sample size (fig. 2, row 1) as rolling window size (fig. 2, row 2) in comparison with the selection by the aic criterion. the bic criterion favors the rw model over the ar and arima models. as previously the ar is preferred over the arima model. summing up, the choices by information criterion differ, i.e. the aic criterion prefers the ar model for cpi in the case of increasing sample size (version information and prediction criteria in selecting the forecasting model 27 i), and model rw − in the case of rolling window size, version ii (except the window size of 80 observations), while the bic criterion prefers the rw model in both version. figures 3 and 4 demonstrate the differences in prediction criteria ape_se and ape_ae respectively. the ape_se criterion favors the rw model over the ar and arima model (fig. 3, row 1 and 2) except the initial sample size and rolling window size of 80 observations when the arima model is preferred over the rw and ar models. the lack of support for the rw model in sample of 80 observations shows rather the influence of initial sample size and rolling window size on selecting the model. hence, the size of initial sample or rolling window should not be to large with regard to total number of observations. observing the differences in ape_ae the influence of the size of initial sample and rolling window is much more distinct (fig. 4, row 1 and 2). for initial sample size and window size of 40 observations the ape_ae criterion prefers the rw model over the arima and ar models. however, for sample (or window) of 60 observations this criterion favors the arima model over the ar and rw models, and for sample (or window) of 80 observations – the ar model over the arima and rw models. table 1. results of model selection for cpi in poland using information (aic, bic) and prediction criteria (ape_se, ape_ae) pairs of models selection criteria version i version ii n=40 n=60 n=80 n=40 n=60 n=80 arima vs. ar aic ar ar ar ar ar ar bic ar ar ar ar ar ar ape_se arima arima arima ar arima arima ape_ae ar arima ar ar arima ar arima vs. rw aic arima arima arima rw rw arima bic rw rw rw rw rw rw ape_se rw rw arima rw rw arima ape_ae rw arima arima rw arima arima ar vs. rw aic ar ar ar rw rw ar bic rw rw rw rw rw rw ape_se rw rw ar rw rw ar ape_ae rw ar ar rw ar ar mariola piłatowska 28 table 2. accuracy measures for one-step-ahead forecasts of cpi from different models in the period 2011:01−20011:06 − version i accuracy measures models arima ar rw mse 0.34 0.35 0.26 rmse 0.58 0.59 0.51 u 1.31 1.36 1.00 mape (%) 0.48 0.46 0.41 table 3. accuracy measures for one-step-ahead forecasts of cpi from different models in the period 2011:01−2011:06 − version ii accuracy measures n=40 n=60 n=80 arima ar rw arima ar rw arima ar rw mse 0.314 0.292 0.265 0.305 0.365 0.276 0.297 0.342 0.279 rmse 0.560 0.540 0.515 0.552 0.604 0.526 0.545 0.585 0.528 u 1.183 1.099 1.000 1.104 1.322 1.000 1.063 1.224 1.000 mape 0.443% 0.434% 0.423% 0.453% 0.470% 0.421% 0.444% 0.450% 0.432% generally, the rw model should be chosen as the best model for cpi because it is preferred by all criteria in the case of rolling windows (except window of 80 observations for ape_se and ape_ae and window of 60 observations for ape_ae which seem rather too large with regard to number of observations) and also in the case of increasing sample size except the aic criterion which favored the ar model and prediction criteria for initial sample size of 80 observations (for both criteria ape) and of 60 observations (for ape_ae) − see figure 1−4 and table 1. if the rw model is really the best one, its predictive performance should be also confirmed in out-of-sample evaluation. out-of-sample forecast evaluation (i.e. in the period 2011:01−2011:06) has been realized by the measures of accuracy (mse, rmse, u, mape) − see table 2 and 3. the results in table 2 and 3 indicate that one-step-ahead forecasts of cpi made from the rw model have the smallest prediction errors independently of the versions (iteratively increasing sample size by one − version i, or rolling window of given size − version ii). the out-of-sample performance of rw model is in rough agreement with choices received by ape criteria but also bic criterion, thus providing evidence of correct model selection by these criteria. the results of selecting the best model for industry production (ip) in poland are presented in figures 5−8. figure 5 shows that the aic criterion favors the ar model over the arima and rw models independently of initial sample size and rolling windows. hence, in that case the selection of a model by the aic criterion seems to be insensitive to the size of initial sample and rolling information and prediction criteria in selecting the forecasting model 29 windows. when the bic criterion has been used the results were similar but only in version i (initial sample size increased by one) − see figure 6, row 1 and for window size of 80 observations (version ii, see figure 6, row 2). for window sizes of 40 and 60 observations the bic criterion prefers the rw model over the arima and ar model (figure 6, row 2). the choice of model of industry production (ip) in poland by prediction criteria is different than by information criteria. namely, the ape_se criterion favors the arima model over the ar and rw model except the initial sample of 80 observations (version i) and window size of 80 observations (version ii) − figure 7. this dominance of model arima is maintained when the ape_ae criterion is used to select the best model (except the initial sample size of 60 observations, version i) − figure 8. summing up, according to information criteria (aic, bic) the ar model should be chosen as the best model for industry production, ip, (except the choice of bic criterion for rolling window of 40 and 60 observations when the rw model is preferred) − see figure 5−6 and table 4. whereas the choice of prediction criteria (ape_se, ape_ae) is the arima model (except initial sample of 60 observations) − see figure 7-8 and table 4. table 4. results of model selection for ip in poland using information (aic, bic) and prediction criteria (ape_se, ape_ae) pairs of models selection criteria version i version ii n=40 n=60 n=80 n=40 n=60 n=80 arima vs. ar aic ar ar ar ar ar ar bic ar ar ar ar ar ar ape_se arima arima ar arima arima ar ape_ae arima ar arima arima arima ar arima vs. rw aic arima arima arima arima arima arima bic arima arima arima rw rw arima ape_se arima arima arima arima arima arima ape_ae arima arima arima arima arima arima ar vs. rw aic ar ar ar ar ar ar bic ar ar ar rw rw ar ape_se ar ar ar rw ar ar ape_ae ar ar ar ar ar ar out-of-sample evaluation of ip forecasts (i.e. in the period 2011:01−2011:06) has been realized by the measures of accuracy (mse, rmse, u, mape) − see table 5 and 6. the results in table 5 and 6 show that both the arima model and ar model have similar predictive value because the accuracy measures of one-stepmariola piłatowska 30 ahead forecasts of ip do not differ much, so these model may complete with each other. in case of version i (iteratively increasing sample size by one) the rmse and u measure prefer the arima model as a model with the smallest prediction error but the mape indicates the ar model (table 5). the opposite result is obtained in version ii for rolling window of 40 observations (table 6). table 5. accuracy measures for one-step-ahead forecasts of ip from different models in the period 2011:01−20011:06 − version i accuracy measures models arima ar rw rmse 2673.30 2679.50 3390.70 u 0.622 0.624 1.00 mape (%) 2.12 2.03 3.52 table 6. accuracy measures for one-step-ahead forecasts of ip from different models in the period 2011:01−2011:06 − version ii accuracy measures n=40 n=60 n=80 arima ar rw arima ar rw arima ar rw rmse 2916.6 2908.4 3411.8 2716.42 2863.65 3386.73 2787.6 2697.1 3395.6 u 0.731 0.727 1.000 0.643 0.715 1.000 0.674 0.631 1.00 mape (%) 2.390 2.460 3.510 2.12% 2.210 3.500 2.220 1.990 3.520 for the window of 60 and 80 observations the arima model and ar model give the smallest prediction errors. it is worth noting that the arima and ar models substantially outperform the rw model. on the whole, the predictive performance of arima and ar models is in agreement with choices obtained by ape criteria and aic criterion. conclusions the results of choosing the forecasting model on empirical data for poland showed that the size of initial sample (version i) and rolling windows (version ii) do have an impact on the choice of forecasting model (within a given version), especially it concerns the ape. the size of initial sample and rolling windows should be relatively small with regard to sample size because too large initial sample (or window) enables to follow the evolution of ape for sufficient number of periods. there are no relevant differences between the selection of forecasting model for the initial sample and rolling windows (within the comparative number of observations). therefore it seems sufficient to calculate the ape only for some small initial sample size increasing iteratively by one observation. both prediction and information criteria are useful in selecting the forecasting model, however the choice of models by prediction criteria is supposed to be much more in agreement with their best out-of-sample performance. information and prediction criteria in selecting the forecasting model 31 references armstrong, j. s. (2001), principles of forecasting, springer, new york. armstrong, j. s., fildes, r. (1995), on the selection of error measures for comparisons among forecasting methods, journal of forecasting, 14, 67−71. box, g. e. p. (1976), science and statistics, journal of the american association, 71, 791−799. burnham, k. p., anderson, d. r. (2002), model selection and multimodel inference, springer. chatfield, c. (2000), time-series forecasting, chapman & hall/crc, boca raton-london-new york-washington. clarke, b. (2001), combining model selection procedures for online prediction, sankhya: the indian journal of statistics, 63, series a, 229−249. de leeuw, j. (1988), model selection in multinomial experiments, in: t. k. dijkstra (ed.), on model uncertainty and its statistical implication, lecture notes in economics and mathematical systems, springer-verlag, new york. dawid, a. p. (1984), statistical theory: the prequential approach, journal of royal statistical society series b, 147, 278−292. de luna, x., skouras, k. (2003), choosing a model selection strategy, scandinavian journal of statistics, 30, 113−128. kunst, r. m. (2003), testing for relative predictive accuracy: a critical viewpoint, economics series, 130, 1−45, department of economics and finance, vienna. mentzer, j. t., kahn, k. b. (1995), forecasting technique familiarity, satisfaction, usage, and application, journal of forecasting, 14, 465−476. rissanen, j. (1986), order estimation by accumulated prediction errors, journal of applied probability, 23a, 55-61. skouras, k., dawid, a. p. (1998), on efficient point prediction systems, journal of royal statistical society b, 60, 765−780. swanson, n. r., white, h. (1997), forecasting economic time series using flexible versus fixed specification and linear versus nonlinear econometric models, journal of forecasting, 13, 439−461. taub, f. b. (1993), book review: estimating ecological risks, ecology, 74, 1290−1291. wagenmaker, e-j., grünwald, p., steyvers, m. (2006), accumulative prediction error and the selection of time series models, journal of mathematical psychology, 50, 149−166. kryteria informacyjne i predykcyjne w wyborze modelu prognostycznego z a r y s t r e ś c i. celem artykułu jest porównanie zachowania się kryteriów informacyjnych i predykcyjnych w wyborze modelu prognostycznego na podstawie danych empirycznych dla polski, przy założeniu nieznajomości modelu generującego dane. uwaga będzie poświęcona śledzeniu zmian kryteriów informacyjnych (aic, bic) oraz skumulowanego błędu prognoz (ape) dla próby powiększanej iteracyjnie o jedną obserwację i ruchowych okien (o różnej wielkości), a także ocenie wpływu wielkości próby (startowej) i ruchomego okna na wybór modelu prognostycznego. wybór najlepszego modelu prognostycznego jest dokonywany spośród następującego zestawu modeli: model autoregresyjny (ar, z trendem i bez trendu deterministycznego), model arima, model błądzenia przypadkowego (rw). s ł o w a k l u c z o w e: kryteria informacyjne, kryteria predykcyjne, skumulowany błąd predykcji, wybór modelu. figure 1. differences in aic information criterion (version i row 1, version ii row 2) for pairs of models (arima vs. ar, arima vs. rw, ar vs. rw) depending on starting sample size and size of rolling window for cpi in poland figure 2. differences in bic information criterion (version i row 1, version ii row 2) for pairs of models (arima vs. ar, arima vs. rw, ar vs. rw) depending on starting sample size and size of rolling window for cpi in poland figure 3. differences in prediction criterion ape_se (version i row 1, version ii row 2) for pairs of models (arima vs. ar, arima vs. rw, ar vs. rw) depending on starting sample size and size of rolling window for cpi in poland figure 4. differences in prediction criterion ape_ae (version i row 1, version ii row 2) for pairs of models (arima vs. ar, arima vs. rw, ar vs. rw) depending on starting sample size and size of rolling window for cpi in poland figure 5. differences in aic criterion (version i row 1, version ii row 2) for pairs of models (arima vs. ar, arima vs. rw, ar vs. rw) depending on starting sample size and size of rolling window for ip in poland figure 6. differences in bic criterion (version i row 1, version ii row 2) for pairs of models (arima vs. ar, arima vs. rw, ar vs. rw) depending on starting sample size and size of rolling window for ip in poland figure 7. differences in prediction criterion ape_se (version i row 1, version ii row 2) for pairs of models (arima vs. ar, arima vs. rw, ar vs. rw) depending on starting sample size and size of rolling window for ip in poland figure 8. differences in prediction criterion ape_ae (version i row 1, version ii row 2) for pairs of models (arima vs. ar, arima vs. rw, ar vs. rw) depending on starting sample size and size of rolling window for ip in poland 02_piłatowska_m piłatowska dem 2011 wykresy microsoft word 12_kliber_p.docx dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 paweł kliber poznan university of economics jumps activity and singularity spectra for instruments in the polish financial market† a b s t r a c t. in the paper we try to measure the activity of jumps in returns of some instruments from the polish financial market. we use blumenthal-getoor index β for lévy processes as a measure of jumps’ activity. this allows us to distinguish between processes with rare and sharp jumps and the processes with infinitely-active jump component. we use three different methods. first we use activity signature plots to estimate the activity patterns of jumps. then we estimate the blumenthal-getoor index with aït-sahalia and jacod threshold estimator.then we use methods based on singularity spectra of lévy processes. finally, we compare the results. k e y w o r d s: blumenthal-getoor index, singularity spectrum, lévy exponential models. introduction the classical models of assets’ returns are based on the assumption of the normality of returns. this is for example the case of classical portfolio theory, developed by markowitz (1952) and sharpe (1963), and the option pricing formula, derived by black and scholes (1973) and by merton (1973). however it is well-known that the normality assumption does not hold. many well-established stylized facts about assets’ returns contradict this. the observed returns reveal characteristics such as heavy tails, high kurtosis or volatility clustering, which is not consistent with gaussian distribution1. there are several methods to deal with non-normality of returns and to make models better fitted to observed data. the most popular approach is the modeling the conditional volatility with some kind of arch/garch model. † the research was undertaken in the project sponsored by the polish ministry of science and higher education, n n111 436 534. the author would like to thank two anonymous referees for valuable comments. 1 the survey of stylized facts concerning assets’ returns can be found in (cont, 2001). paweł kliber 172 when using non-gaussian distributions of error terms, such models seem to be well-fitted to the data. in this work we take another approach, which lately becomes more and more popular, namely we use lévy processes in the modeling. although this line of modeling is as old as the seminal paper of maldenbrot (1963), in which models with stable distributions were proposed, it became more popular at the beginning of xxi century. lévy processes can be represented as a sum of continuous diffusion (wiener process) and discontinuous jumps. thus, one of the main questions is if there are jumps in the returns of financial instruments and if there are, how much intensive are the jumps. formally the intensity of jumps of lévy process is described by blumenthal-getoor index. in the paper we try to estimate the value of this index for the suitably-chosen sample of instrument from polish financial market. we use three different methods of estimation and then we compare the results. the paper is organized as follows. in the section 1 we present basic facts about lévy processes and lévy exponential models of prices. in this section we also introduce the concept of blumenthal-getoor index of jumps. in section 2 we try to analyze the type of processes using activity signature plots. in section 3 we estimate blumenthal-getoor indexes using threshold estimator. in section 4 we analyze the type of processes using its singularity spectrum. section 5 concludes. 1. lévy processes and lévy exponential models the lévy process l is the stochastic process with continuous time that starts at zero (i.e. 0 0l  ) and fulfils the following conditions: 1. for any 1 2 3 40 t t t t    the random variables 12t tl l and 4 3t tl l are independent and the distribution of t h tl l  depends only on h and not on t , 2. the process is stochastically continuous, i.e. for all 0t  and all 0    0 lim 0 h t h tp l l      3. the trajectories of the process are cadlag (i.e. they are right-continuous with left limits). we should stress that the second condition does not imply that the process is continuous. in fact the lévy processes typically have discontinuities (of “jumps”) and some of them are discontinuous at each point t . the condition 2 means only that the process l does not have the jump of arbitrary size  at any pre-specified moment t . jumps activity and singularity spectra for instruments in the polish financial market 173 the lévy processes can be seen as the extension of wiener process. in fact if we change the condition 2 and require the process l to be continuous, then the only processes that fulfill the definition are wiener processes with drift. there is also another connection between wiener process and lévy processes. the increments of a wiener process are normally-distributed, while the increments of lévy process belong to the family of infinitely-divisible distributions. it is the biggest family of distributions that can serve as limits for the sums of independent variables (see feller, 1971). thus, if we believe that the returns of the assets in financial markets result from many independent shock, then the lévy processes are natural choice as an engine in the model. there are two fundamental theorems that reveal the structure of the lévy processes. the first one is the lévy-itô decomposition, which states that any lévy process l can be uniquely represented as the sum of wiener process with drift, poisson process and purely discontinuous martingale: l s t t t tl xw xt    , (1) where  and 0  are constants, w is a standardized wiener process, lx is a compound poisson process of large jumps and sx is purely discontinuous martingale (i.e. it is discontinuous at each point t ). the process sx represents small jumps – in every finite interval there are infinitely many of them, but they are very small, so that the process sx does not explode. the second theorem characterizes the characteristic function of the lévy processes. according to lévy-khintchine representation the characteristic function of the lévy process is: [ ] exp[ ( )]tiule e t u , (2) where the characteristic exponent  equals:   2 2 11 1( ) ( )2 iux x r e iuxu i u u xd         . (3) the first two terms in the sum (3) are the same as in the characteristic function for the gaussian distribution. these terms describes continuous part of the process (diffusion). the measure  (called lévy measure) describes the jumps of the process l . the value of ( )r is the intensity of jumps. if finite, it is the average number of jumps in the unit of time – the process is then said to be “finitely active”. if ( )v r   , then the process is infinitely active – in any interval the number of jumps is infinite. the values (( , ))v a b give the relative intensity of jumps with sizes between a and b (i.e. jumps such that ( , )t t tll al b   ). thus the lévy measure contains information of both the paweł kliber 174 intensity of jumps and distribution of jumps’ sizes. the triple ( , , )   , called “characteristic triple”, gives unique characterization of the process. the measure  , which contains all information of jumps and its structure, can vary, depending on the type of the process or on the probability distributions of increments of the process. there exists however one synthetic index that divides measures  into certain categories and gives characteristics of jump behavior. blumenthal-getoor index is defined as 1 1 inf 0 : ( ) b db x x               . (4) the index  takes values in the interval [0, 2) . higher values of  mean that jumps are more intensive and smaller and the process l resemble continuous process. if 0  the process is finitely active. all other values mean that the activity of the process is infinite. in the case 0  the discontinuous part of the process l is the compound poisson process, i.e. 0sx  in the decompo sition (1). the models of assets prices driven by lévy process usually take the form of exponential lévy models: 0 tl ts s e , (5) where ts denotes the asset price at time t . the models can be also formulated as stochastic differential equation (as in classis black-scholes model): t t tds s dv , (6) where v is a lévy process, whose characteristics can be derived from l 2. the logarithms of the prices are described by lévy process l and thus the logarithmic returns are increments of lévy process. according to (1) the logarithmic price is given by: ln l st t t t tts xws x     . (7) in the literature one considers also the generalization of (7), assuming that the volatility of continuous part is not constant. such a model can be specified as follows: d t t t ts wt l   , (8) where t is a process or a function representing volatility and dl is discontinuous part of the process l and represents jumps. 2 kallsen (2000) has shown that specifications (5) and (6) are equivalent and gave the formulae how to express v in terms of l and vice versa. jumps activity and singularity spectra for instruments in the polish financial market 175 let us select some time-scale, i.e. choose some frequency in which we sample the process. suppose that we sample every h units of time (in practice it can be a period from several seconds to one day). the increments of the process at the specified frequency are logarithmic returns of the assets. we denote them by ( )ix h : ( 1) ( 1)( )i i h ih i h ihh s sx l l    . (9) if there is no ambiguity about frequency we omit brackets and denote returns simply by ix . in the next three sections we will use data about returns to calculate index  , using three different methods. 2. estimating jumps activity with activity signatures the first method is based on power variation of stochastic process, defined as: 1 ( , ) ( ) p i i n v p h x h    . (10) in the case 2p  , the (2, )v h is well-known realized volatility, which tends to the quadratic variation of the process, as the sampling frequency tends to infinity. the behavior of ( , )v p h in other cases depends on the type of the process. as barndorff-nielsen and shephard (2002) have shown, if the process l contains no jumps (i.e. 0dl  in (8)), then: 1 / 2 0 0 plim ( , ) t pp s h pph shv d      , (11) where p is appropriate constant. if the process l contains no diffusion part ( 0t  ), then as 0h  the sum (10) diverge for p  . on the other hand for p  the sum is convergent: 0 0 (plim , ) t p s h s p h lv     . (12) if the process contains both diffusion and jump parts, then the following equations holds: paweł kliber 176 1 2 / 2 0 0 2 0 00 0 0 plim fo( , ) , (0, 2)r pli , (2, ) , ,m for 2 pli (m , for 2, .) t pp p s t t s s s t p s h h s h p h h h v ds p l p h l v ds p v p                               (13) based on these limit behavior of the power variation, todorov and tauchen (2009) proposed a qualitative test of the type of the process. they have defined activity signature function ˆ ( ; , )p k h as: lnˆ ( ; , ) ln ln ( , ) ln ( , ) p k p k h k v p kh v p h     . (14) the graph of ˆ ( ; , )p k h with respect to p are called “activity signature plot”. if the process is continuous, then for all 0p  : 0 ˆplim ( ; , ) 2 h p k h   . (15) for processes of pure jumps we have: 0 , for (0, ),ˆplim ( ; , ) , for .h p p k h p p         (16) in the case, when the process contains both jumps and diffusion: 0 2, for (0, 2),ˆplim ( ; , ) , for .h p p k h p p       (17) we apply method of activity signatures to several instruments from polish financial markets. the sample contains of four stocks: two liquid ones (pkn orlen and kghm) and two less liquid (agora and bre bank), three stock market indexes (wig, wig20 and mwig40), one future contract (fwig) and two currencies (euro and us dollar). the sample was chosen as to contain possibly wide range of different instruments. we have used intraday data for the period from the beginning of 2009 to the march of 2011. in the computation we used 10-minutes returns. we have tried some other frequencies and decided that this frequency is high enough to justify the usage of the limits (15)-(17). on the other hand it is not so high, that the market microstructure noise affects the results3. in the computations of ˆ ( ; , )p k h we took 2k  , as in the original work 3 we have used methods of zhang, mykland, aït-sahalia (2005) to control the microstructure noise. for sampling period of 10 minutes the difference between “two time scales” estimator of jumps activity and singularity spectra for instruments in the polish financial market 177 of todorov and tauchen (2009). thus we have worked with two time scales: 10 minutes and 20 minutes returns. we do not present results for each instrument, but only show three typical cases. figure 1 shows activity signature plot for agora. such a graph is typical for less-liquid stocks. it represents the case (16) of pure jump process with 0  , which means that the prices are driven by compound poisson processes. figure 1. the activity signature plot for agora (vertical axis – exponents p , horizontal axis – the values of ˆ ( ; 2, )p h ) figure 2. the activity signature plot for pkn orlen (vertical axis – exponents p , horizontal axis – the values of ˆ ( ; 2, )p h ) quadratic variance and realized variance was small, so we decide that microstructure effect for this frequency is negligible. paweł kliber 178 the figure 2 represents the activity signature plot for pkn orlen, but the graph is similar to the plots for liquid stock, future contract and currency prices. the plot resemble the case (17), when the process contains both jump and diffusion parts. the plot for index wig, shown on the figure 3, is typical for all indexes. in this case the process is continuous. figure 3. the activity signature plot for wig (vertical axis – exponents p , horizontal axis – the values of ˆ ( ; 2, )p h ) 3. estimating blumenthal-getoor index using threshold estimator the method of activity signatures allows us to identify the type of process that underlines prices, but usually does not allow estimating the value of blumenthal-getoor index. to estimate the values of this index we use threshold estimator proposed by aït-sahalia and jacod (2009). the estimator is given by the formula: ln ( , ) ln ( , )ˆ ( , , ) ln u h u k h k h k       , (18) where ,( )u h is counting function which counts the exceedances of the threshold:  ( ) 1 ( ), 1 i n x h h i hu      , (19) with (0,1 / 2) . estimator (18) uses two time scales, as activity signature method, but the latter method is based on limit properties of power variation, while the threshold estimator is based on different exceedance rates in different time scales. it allows us to estimate blumenthal-getoor index in the case when the process contains diffusion (continuous) part. it also allows for testing the accuracy of the estimation. the asymptotic standard error of the estimator equals: jumps activity and singularity spectra for instruments in the polish financial market 179 1 1 1 ln ( ,) (, )k hk u u h   . (20) we have calculated the estimator ˆ ( , , )k h  for all instruments in the sample. as in the original work of aït-sahalia and jacod (2009) we have taken 1 / 5  and 2k  . as for the threshold level  it was taken as seven times the estimated standard error of the continuous part of the process. the results are shown in table 1. table 1. estimators of blumenthal-getoor index instrument ˆ ( , , )k h  std. dev. ago 0.7382 0.0087 bre 1.0888 0.0074 kgh 1.5486 0.0052 pkn 1.5405 0.0085 wig 2.6208 0.0285 wig20 1.9684 0.0097 mwig40 2.2870 0.0185 fwig20 1.9280 0.0024 eur 1.9835 0.0097 usd 2.0491 0.0069 the results are only partially consistent with the ones obtained with activity signature method in the previous section. for the less-liquid stocks (ago and bre) the estimators of  are significantly lower than for the other instruments. they are however greater than 0. in case of stock indexes (wig, wig20 and mwig40) the estimators are high (close to 2 or even greater than 2), what is consistent with previous results, that trajectories of these instruments are continuous. liquid stocks have  between 1 and 2. as for the currencies (eur, usd) and futures contract (fwig20), the obtained values are close to 2, what stands in contradiction with the results from the previous section. the last column in the table 1 contains estimators’ errors. however they were calculated with asymptotic formula (20) and it is dubious if they give the true errors of estimators, especially if some values are greater than 2. 4. singularity spectra and jump activity the third method of estimating blumenthal-getoor index is based on singularity spectra of observed trajectories of prices. it is non-statistical methods. it is based on the fact that the trajectories of lévy processes with different  almost surely (i.e. with probability 1) reveals different types of continuity. let us first introduce the concept of singularity spectrum. paweł kliber 180 take any function :f r r . we say that the function is a -hölder continuous in the point 0t if there exists a polynomial ( )p t of order lower then a such that in some neighborhood of 0t : 0( ) ( ) a f t p t k t t   (21) for some 0k  . let  be a number such that for a  function f is a -hölder continuous at t and for all a  f is not a -hölder continuous at t . the  is called local hölder exponent of the function f at the point t and is denoted by ( )fh t . it is the measure of “regularity” of f in the neighborhood of t . the higher ( )fh t , the more regular the function is. for example the trajectories of wiener motion almost surely have local hölder exponent equal to 0.5 at each point. let ( )f  be the set of all points in which the function f has local hölder exponent equal to  :  ) :( ( )f ft h t    . (22) singularity spectrum it is the mapping which for all  returns the hausdorff dimension (see mallat, 2003; falconer, 2003) of the set ( )f  : h( ) dim ( )f fd    . (23) as was shown by jaffard (1999) the lévy processes with different blumenthal-getoor index almost surely have different singularity spectra. if the lévy process l does not contain diffusion part, then: 1 (dim ) for h l        (24) and ( )l    for 1 /  . the shape of singularity spectrum for such a process is show in the figure 4. if the process l contains diffusion part, then: 1 ( ) for dim 2 h l      , (25) (1di / 2)m 1h l  and ( )l    for 1 / 2  . for the wiener process (without jumps) (1di / 2)m 1h l  and ( )l    for 1 / 2  . jumps activity and singularity spectra for instruments in the polish financial market 181 figure 4. singularity spectrum for lévy process with blumenthal-getoor index  to calculate singularity spectra from discretely sampled data one uses so called “multifractal formalism”, introduced by frisch and parsi (1985) and developed by jaffard (1997a, 1997b). it can be shown that the function *fd , defined as: 0 * 1( ) lim inf ln ( ) ( ) ln q f h d q f x h f x dx h    (26) is the legendre transform4 of the singularity spectrum d , i.e.:  * ( ) inf ( )f f r d q q d      . (27) if the function )(fd  is convex (as it is generally assumed in the literature), then one can obtain fd by performing legendre transform on * fd . the function *fd can be estimated from sample moments. one has to calculate power variation (10) for multiple of sampling times h and then to use the regression: *ln ( , ) ( ) lnfv q h c qd h  . (28) this method however is not stable numerically and some better methods were proposed. most of them use wavelet transform. the review of them can be found in turiel, pérez-vincente, grazzini (2006) or oświęcimek (2005). in our research we have used method based on “modulus maxima” of wavelet coefficients, implemented in matlab package fraclab. the method was pro 4 lagendre transform is well-established method of convex analysis. see for example rockefaller (1970), ch. 26. α df(α) 1/β 1 paweł kliber 182 posed by jaffard (1997a) and very good description can be found in mallat (2003, ch. 6). figure 5. estimated singularity spectrum for pkn orlen the figure 5 shows the results for one stock (pkn orlen). the estimated singularity spectra for all other instrument revealed very similar pattern, so we omit them. one should also stress that the results bears a great resemblance to the results of cont and tankov (2004) for the stock from us market. the graph does not star at the origin, which is inconsistent with typical shape depicted on the figure 4. this however can be due to numerical inaccuracy. the main feature of the graph is the value at which the function reaches its peak, as it is the reciprocal of blumenthal-getoor index. for all instrument in the sample the peak values lie in the interval [0.6, 0.8], which means that the indexes  are between 1.2 and 1.8. conclusions the estimation of the blumenthal-getoor index is a complicated problem. we have used three different methods and as one can see the results are in many cases inconsistent. as for the less-liquid stocks, according to the activity signature method the price process is driven by compound poisson process, while the two other methods reveal positive value of blumenthal-getoor index. this inconsistency may be due to the fact of low liquidity of these stocks. there were days when the prices did not change for several hours. the estimators we have used take advantage of limit properties of price changes as time scale tends to 0. it is thus dubious if they gave correct estimators for such illiquid stocks. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 jumps activity and singularity spectra for instruments in the polish financial market 183 as for the liquid stocks all three estimators gave similar results. moreover the results are consistent with similar results for other markets (  at the level about 1.5). according to the first two methods (activity signature and threshold estimator) the indexes are continuous processes, while the third method (singularity spectrum) revealed 2  . probably in fact the former result holds, as the method of singularity spectrum is the most prone to numerical errors. the results for currencies and futures contract are ambiguous. this can be a result of active process of jumps with blumenthal-getoor index close to 2. as it was pointed out by zhang (2007) the jump process is then hardly distinguishable from continuous diffusion. references aït-sahalia, y., jacod, j. (2009), estimating the degree of activity of jumps in high frequency data, the annals of statistics 37, 2202–2244. barndorff-nielsen, o.e., shephard, n. (2002), power variation and time change, working paper, available in repec database. becry, e., muzy, j. f., arnédo, a. (1993), singularity spectrum of fractal signals from wavelet analysis: exact results, journal of statistical physics 70, 635–674. black, f., scholes, m. (1973), the pricing of options and corporate liabilities, journal of political economy 81, 637–654. cont, r. (2001), empirical properties of assets returns: stylized facts and statistical issues, qualitative finance 1, 223–236. cont, r., tankov, p. (2004), financial modelling with jump processes, chapman&hall, london, new york. falconer, k. (2003) fractal geometry, wiley. feller, w. (1971), an introduction to probability theory and its applications, vol. 2, wiley. frisch, u., parsi, g. (1985) fully developed turbulence and intermittency, in: giil, m. (ed.) turbulence and predictability in geophysical fluid dynamics and climate dynamics, north holland, amsterdam, 84–88. jaffard, s. (1997a), multifractal formalism for functions part i: results valid for all functions, siam journal of mathematical analysis 28, 944–970. jaffard, s. (1997b), multifractal formalism for functions part ii: self-similar functions, siam journal of mathematical analysis 28, 971–998. jaffard, s. (1999), the multifractal nature of lévy processes, probability theory and related fields 114, 207–227 jondeau, e., poon, s.-h., rockinger, m. (2007), financial modeling under non-gaussian distributions, springer. kallsen, j. (2000), optimal portfolios for exponential lévy processes, mathematical methods of operational research 51, 357–374. maldenbrot, b. b. (1963) the variation of certain speculative prices, journal of business 36, 394–419. malevergne, y., sornette, d. (2006), extreme financial risks, springer, berlin, new york. mallat, s. (2003), a wavelet tour of signal processing, elsevier, singapore. markowitz, h.m (1952) portfolio selection, journal of finance 7, 77–91. merton, r. c. (1973) theory of rational option pricing, bell journal of economics and management science 4, 141–183. oświęcimek, p. (2005), multifraktalne charakterystyki finansowych szeregów czasowych (multifractal characteristics of financial time series), doctor thesis, instytut fizyki jądrowej polskiej akademii nauk (nuclear physics institute of the polish academy of science). paweł kliber 184 rockafellar, r. t. (1970), convex analysis, princeton university press, princeton. sharpe, w. f. (1963), a simplified model for portfolio analysis, management science 9, 277–293. todorov, v., tauchen, g. (2009), activity signature function for high-frequency data analysis, journal of econometrics, preprint at http://ideas.repec.org (7.09.2011). turiel, a., pérez-vincente, c., grazzini, j. (2006), numerical methods for the estimation of multifractal singularity spectra on sampled data: a comparative study, journal of computational physics 216, 362–390. zhang, l., mykland, p. a., aït-sahalia y. (2005), a tale of two time scales: determining integrated volatility with noisy high-frequency data, journal of the american statistical association 100, 1394–1411. zhang, l. (2007), what you don’t know cannot hurt you: on the detection of small jumps, working paper at http://tigger.uic.edu/~lanzhang/ (7.09.2011). aktywność skoków i spectrum osobliwości dla instrumentów z polskiego rynku finansowego z a r y s t r e ś c i. w artykule podejmujemy próbę oszacowania aktywności skoków w procesach cen kilku instrumentów z polskiego rynku finansowego. jako miarę aktywności skoków przyjmujemy indeks β blumenthala-getoora dla procesów lévy’ego. pozwala nam to na rozróżnienie procesów charakteryzujących się rzadkimi i dużymi skokami i procesów o nieskończonej aktywności procesu skoków. aktywność skoków szacujemy trzema różnymi metodami. wykorzystujemy wykresy podpisu aktywności (activity signature plots) do zbadania typu procesu. następnie korzystamy z estymatora aït-sahalii i jacod, opartego na liczbie przekroczeń, do oszacowania wartości indeksu  . wreszcie korzystamy ze spektrum ciągłości oraz z odpowiednich twierdzeń na temat przebiegu tej funkcji dla procesów lévy’ego z różnymi wartościami indeksu  . s ł o w a k l u c z o w e: wykładnicze modele lévy’ego, indeks blumenthala-getoora, spektrum osobliwości. © 2012 nicolaus copernicus university press. all rights reserved. http://www.dem.umk.pl/dem dynamic econometric models vol. 12 ( 2012) 111−122 submitted october 25, 2012 issn accepted december 20, 2012 1234-3862 michał bernard pietrzak, natalia drzewoszewska, justyna wilk* the analysis of interregional migrations in poland in the period 2004–2010 using panel gravity model a b s t r a c t. the paper discusses the problem of migration in spatial and temporal perspective. the objective is to evaluate the intensity and direction of selected economic variables impact on the volume of interregional migration flows in poland in the period 2004–2010. the analysis was performed using panel gravity model with fixed effects. the socio-economic situation and especially the level of salaries determine migration directions in poland. significant movements keep occurring between economically stronger regions. they indicate tendencies towards obtaining positive net migration. k e y w o r d s: interregional migration, panel gravity model. j e l classification: c23, c32, j61, r11, r15 introduction migrations represent an integral part underlying the functioning of societies and economies. in the times of market economy internal migrations, which occur within the territory of a given country, are of particular importance. they regulate both, the size and structure of human resources as well as job market situation and the consumption of goods and services, etc. therefore they are of not just demographic dimension but also of economic one and have significant impact on regional development. in spite of the above, polish population is characterized by a relatively low territorial mobility, in particular with reference * correspondence to: michał bernard pietrzak, nicolaus copernicus university, ul. gagarina 13a, 87-100 toruń, e-mail: pietrzak@umk.pl, natalia drzewoszewska, nicolaus copernicus university, ul. gagarina 13a, 87-100 toruń, e-mail: drzewonata@gmail.com, justyna wilk, wroclaw university of economics, ul. nowowiejska 3, 58-500 jelenia góra, e-mail: juwil2002@interia.pl. michał pietrzak, natalia drzewoszewska, justyna wilk dynamic econometric models 12 (2012) 111–122 112 to long distance migrations. for this reason it is of crucial importance to research conditions responsible for interregional migration flows. in terms of relatively stable political and cultural terms, the strongest determinants of migration processes in poland are represented by economic and social motives, among others, the tendency towards improving the living conditions and standard. their origin is, to a great extent, influenced by socioeconomic situation in a given region, owing to a significant level of regional level diversification in poland. migrations represent a specific phenomenon, characterized by flows, since they occur in a territorial space. migration flows occur between two territorial units. therefore their conditions should be analyzed in the context of pushing and pulling factors influencing population migrations, as well as consider geographic distance. such approach to the discussed phenomenon is possible once gravity model is applied. questions related to gravity model implementation were discussed, among others, in the economic studies by: tinbergen (1962), chojnicki (1966), anderson (1979), grabiński et al., (1988), sen, smith (1995); frankel, stein, wei (1995), helliwell (1997), egger (2002), baltagi, egger, pfaffermayr (2003), anderson, van wincoop (2004), roy (2004); serlenga, shin (2007), faustino, leitão (2008), lesage, pace (2009), kabir, salim (2010). migrations are, at the same time, characterized by long-term processes which take many years to be carried out. their analysis in spatial perspective exclusively, e.g. covering just one year, does not allow for an overall evaluation of the occurred phenomenon. in such a situation one of the solutions is to apply an aggregated size of migration flows in relation to a given period. this approach, however, does not allow for the description of changes taking place in a given time. in such circumstances the best solution seems to use a panel model which facilitates the analysis of the studied phenomenon regarding its crosssectional and temporal aspects. additionally, it allows for considering individual effects with regard to territorial units covered by the study, as well as time effects referring to the years under analysis. therefore it facilitates data heterogeneity presentation which could not be accounted for by explanatory variables in the model. the paper discusses the proposal of an approach to be applied in modelling phenomena characterized by flows and analyzed in spatial and temporal perspective. such standpoint is based on the construction of a panel gravity model. the purpose of the paper is to apply this approach in the estimation of both intensity and directions, regarding the impact of economic aspects illustrating socio-economic situation in the regions, on the size of interregional migration flows in poland in the period 2004–20101. 1 “a region” in the investigation is interpreted as “a voivodship” which is a unit of territorial division in poland (nuts 2). interregional migrations means migration flows between voivodships. the analysis of interregional migrations in poland in the period 2004–2010… dynamic econometric models 12 (2012) 109–120 113 the size of migrations, based on the previous and current permanent residence, was accepted as a dependant variable. the set of explanatory variables included indicators responsible for the size of gdp, value of investment outlays, level of earnings and unemployment rate. the role of geographical distance was also considered. due to the fact that the intensity of internal migrations in poland is, to a great extent, influenced by economic prosperity, the study distinguishes two time spans, i.e. 2004–2007 and 2008–2010. such solution allowed for determining economic prosperity impact on potential reasons underlying interregional migrations. panel gravity model with fixed effects was constructed in order to measure the researched interdependencies. both, individual and temporal, effects were analyzed. haussman-taylor estimator was used to estimate of the structural parameters of the model. three research hypotheses were put forward within the framework of the carried out objective. the first assumes that economic factors, defining regional socio-economic situation, represent the determinants of interregional migration directions in poland. following the second hypothesis, during economic downturn in poland (in the period 2008–2010) the influence of analyzed factors on the level of interregional migration movements was smaller against the previous period of prosperity (2004–2007). in line with the third hypothesis it was assumed that regions featuring a relatively favourable socio-economic situation present a positive net migration. the first part of the paper discusses the construction of panel gravity model with fixed effects. the second focuses on the analysis scope and the considered model specifications. the final part presents the results of conducted research. 1. the construction of panel gravity model gravity models describe relations occurring between the size of flows in relation to a given category (e.g. economic) in territorial space – from the origin region to a destination one – and explanatory variables characteristic for both regions. the influence of explanatory variables on a dependent variable is considered in the context of push factors influencing flows from origin regions and pull factors attracting migrations to destination regions. additionally, gravity models allow for considering the role of a distance (e.g. geographical, temporal) between regional units (see chojnicki, 1966; grabiński et al., 1988; sen, smith, 1995). the dissemination of gravity models in economic studies resulted from research results published by tinbergen, who applied gravity model in international trade studies (see tinbergen, 1962). the typical gravity model formula (following logarithmic linearization2) takes the below form (see sen, smith, 1996; lesage, pace, 2009) 2 all variables are expressed in natural logarithms. michał pietrzak, natalia drzewoszewska, justyna wilk dynamic econometric models 12 (2012) 111–122 114 ,0 εdxxy +−++= γβββ ddoo (1) where: y – vector of flow values between regions, xo, xd – vectors of explanatory variable values respectively for origin and destination regions, d – vector representing distance between each pair of regions, 0β – constant, γββ ,, do –structural parameters of the model, ε – vector of random component. cross-sectional (spatial) and temporal data analysis applies panel models among which fixed effects models, the so called fe models, are responsible for macroeconomic phenomena. panel models present a major advantage over standard cross-sectional ones owing to their possibility of taking temporal changes into consideration. additionally, distinguishing individual and time effects from the fixed ones allows for data heterogeneity description, which is not accounted for by explanatory variables (see baltagi, 2005). in case when the research object is represented by a migration phenomenon analyzed not only in spatial, but also temporal perspective, the authors suggest to construct a panel gravity model with fixed effects which may take the form of the formula below: ,][][][][][ εzγβxβxααy +++++= ditdoitotiit (2) where: ][ity – vector of flow values between regions (i=1,2,...,n, i – object’s number3, t=1,2,...,t – number of time period), ][][ , itdito xx – matrixes of explanatory variables values respectively for origin and destination regions, z – matrix covering variables fixed in time, including distance between regions, ][iα – vector of individual effects, ][tα – vector of time effects, γββ ,, do – vectors of structural parameters of the model, ε – vector of the random component. 3 in case of panel gravity models, which analyze interregional migration flows, a pair of regions represents an object (unit). for n analyzed regions n2 objects are included in the study, i.e. pairs of regions between which migrations are observed. the analysis of interregional migrations in poland in the period 2004–2010… dynamic econometric models 12 (2012) 109–120 115 2. modelling of interregional migrations in poland the purpose of the study is to identify both the intensity and direction of the selected economic categories’ influence, which to some extent serve as an illustration of socio-economic situation, on interregional migration movements in poland in the years 2004–2010. within the analyzed period of time two sub-periods were distinguished. the first refers to the years 2004–2007, right after polish accession to the european union structures, representing the period of polish economy prosperity. these years were characterized by the significant intensity of domestic migrations. the second time span covers the years 2008–2010 and is analyzed in the context of global financial and economic crisis also experienced by polish economy. this period featured economic downturn and lower intensity, as well as stability, in migration flows within the country borders. panel gravity model with fixed effects, represented by the formula (2), was applied for the description of spatial-temporal migration aspects. the size of migration flows between regions was adopted as the dependant variable4. on the other hand, the selected economic indicators were accepted as explanatory variables, i.e. gdp per capita, the size of investment per resident, average monthly salary before tax and the registered unemployment rate. separate models were constructed for both time periods. due to the strong statistical correlation of explanatory variables5 the panel gravity models were performed separately for every analyzed economic variable6. additionally, in case of each model the impact of geographical distance on the size of migration movements was examined7. table 1 presents variables included in the study. the model assumes the possibility of individual effects occurrence for each analyzed pair of regions, as well as time effects for particular years8. three model specifications were considered and represented by formulas (3–5), i.e. panel gravity model fe: − with both individual and time effects (formula 3), − with individual effects only (formula 4), − with time effects only (formula 5). 4 the analyzed migration flows require their origin region i and their destination region j to be specified. each pair of regions is included in the panel twice to model the size of migration flows in both directions. the study does not cover internal regional migration flows. 5 distinguishing two periods of time did not influence, to a large extent, the decrease of statistical correlation level between explanatory variables. 6 it has to be considered that the regressions obtained represent gross regressions and apart from the effect of explanatory variables adopted in the model they also reflect impacts of variables correlated with them. 7 geographical distance between regional (voivodship) centroids was analyzed. 8 in case of the analyzed panel data a pair of regions represents a unit (240 pairs of regions), between which migration flows occur, while the time span is one year (2004, 2005,...,2010). michał pietrzak, natalia drzewoszewska, justyna wilk dynamic econometric models 12 (2012) 111–122 116 table 1. variables included in the analysis of interregional migrations variable definition accepted measure unit migrations size of interregional migration flows (permanent residence) accord-ing to previous and current residence address person gdp gross domestic product per capita 1000 pln per person investments investment outlays in enterprises per resident 1000 pln per person salaries average gross monthly salary in national economy 100 pln unemployment registered unemployment rate % distance geographical distance (distance between regional centroids) km the estimation stage, during panel gravity models application, is a problematic one. while analyzing the effect of distance on the size of migrations it has to be considered that this variable is fixed in time. the presence of such variable, or a set of variables fixed in time, excludes the ordinary estimation of fe model parameters with individual effects due to the occurring collinearity. the solution to the problem is the application of hausman-taylor estimator since it allows for estimations of all model parameters (see hausman, taylor, 1981). four separate models, referring to a selected explanatory variable9, were defined for each model specification and two analyzed time periods. in case of one explanatory variable and fixed in time distance between regions, panel gravity model represented by formula (2), depending on selected effects, is reduced to the following form: ,][][0][][][ εdxxααy +++++= γββ itdditotiit (3) ,][][0][][ εdxxαy ++++= γββ itdditoiit (4) ,][][0][][ εdxxαy ++++= γββ itdditotit (5) where: ][ity – vector of flow values between regions (i=1,2,...,n, i – object’s number, t=1,2,...,t – number of time period), ][][ , itdito xx – vectors of explanatory variable values respectively for origin and destination regions, ][iα , ][tα – vectors of individual and time effects, d – vector representing distance between each pair of regions, , ,o dβ β γ –structural parameters of the model, 9 finally 24 models were estimated, 12 for each analyzed period of time. adequate tests confirmed statistical significance of individual effects or periodical effects in case of each specification. the analysis of interregional migrations in poland in the period 2004–2010… dynamic econometric models 12 (2012) 109–120 117 ε – vector of the random component. prior to the estimation of subsequent fe panel gravity model the specifications’ significance, both jointly and separately, was analyzed for individual and time effects. the obtained test statistics indicated statistical significance of effects for all three specifications. having performed the estimation of the model specifications the statistical significance of the models' parameters were performed. the economic verification of the models with regard to interpretability of the obtained estimates was done as well. statistical insignificance of parameter estimations for the majority of accepted explanatory variables, or no possibility for correct interpretation of the analyzed economic phenomena impact, were characteristic for model specification with joint individual and time effects, as well as the specification with individual effects. therefore, considering individual effects provided an opportunity for capturing spatial heterogeneity but, at the same time, did not allow for impact measurement of the analyzed potential determinants on migration phenomenon10. it was only the choice of the final model specification, including time effects, which allowed for obtaining substantively interpretable parameter estimates for the selected explanatory variables. therefore the following part of the paper presents 8 models, out of 24 analyzed specifications, four in each subperiod. 3. determinants for interregional migrations in poland in the period 2004–2010 table 2 shows the results of estimation of panel gravity models with time effects for four explanatory variables. negative estimates of the distance parameter and its statistical significance for all models indicate the decreasing trends in migration intensity, proportionally to geographical distance increase between regions. this confirms unceasing timeliness of gravity model concept which assumes that geographical distance11 represents one of the basic factors determining the size of migration flows. in case of the remaining explanatory variables, i.e. gdp, investments, salaries and unemployment, for origin regions and destination ones parameters β1 and β2 proved to be statistically significant at 5% level of significance. therefore it may be agreed that all accepted variables describe, to some extent, the determinants of interregional migration flows in poland. 10 it presents an interesting situation and may result from strong correlation of individual effects with the analyzed explanatory variables. such correlation probably resulted in the occurrence of parameters insignificance for selected variables and changes in values or the sign of the obtained estimates of the parameters. 11 distance measure between regions in the gravity model of migration flows reflects costs of moving the place of residence. michał pietrzak, natalia drzewoszewska, justyna wilk dynamic econometric models 12 (2012) 111–122 118 table 2. the results of estimation of panel gravity models in the period 2004–2007 gdp investments parameters estimates p-value parameters estimates p-value α2004 1.385 0.02 α2004 12.646 0.0 α2005 1.165 0.04 α2005 12.388 0.0 α2006 0.948 0.1 α2006 12.140 0.0 α2007 0.596 0.3 α2007 11.686 0.0 β1 1.286 0.0 β1 0.695 0.0 β2 2.598 0.0 β2 1.444 0.0 γ -1.413 0.0 γ -1.445 0.0 r2 coefficient 68% r2 coefficient 66% salaries unemployment parameters estimates p-value parameters estimates p-value α2004 -47.015 0.0 α2004 18.695 0.0 α2005 -47.306 0.0 α2005 18.584 0.0 α2006 -47.633 0.0 α2006 18.393 0.0 α2007 -48.239 0.0 α2007 18.043 0.0 β1 2.845 0.0 β1 -0.375 0.0 β2 5.058 0.0 β2 -1.106 0.0 γ -1.500 0.0 γ -1.564 0.0 r2 coefficient 72% r2 coefficient 48% regarding the first three explanatory variables (gdp, investments and salaries) positive estimates of β1 and β2 parameters were obtained, however in case of unemployment the respective values were negative. this means that higher gdp, investments and salaries level and lower unemployment level act as a pull factors to the destination region, but also a push ones from the origin region. this observation provides grounds for concluding that the population from regions characterized by better socio-economic situation presents higher territorial mobility than residents of poorer regions. therefore it is highly likely that the most intensive migration flows occur between more developed regions. on the other hand, an unfavourable level of these determinants functions as the factor which slows migrations down. weak position of a given region is translated, to a great extent, into lowering local community living standards (including worse financial stability) and plays the crucial role in their migration potential reduction. the analysis of interregional migrations in poland in the period 2004–2010… dynamic econometric models 12 (2012) 109–120 119 the results of parameter estimations, referring to panel gravity models with time effects for the period 2008–2010 for four explanatory variables, are illustrated in table 3. table 3. the results of estimation for panel gravity models in the period 2008–2010 gdp investments parameters estimates p-value parameters estimates p-value α2008 0.395 0.6 α2008 11.377 0.0 α2009 0.217 0.8 α2009 11.569 0.0 α2010 0.251 0.7 α2010 11.604 0.0 β1 1.250 0.0 β1 0.565 0.0 β2 2.512 0.0 β2 1.204 0.0 γ -1.387 0.0 γ -1.404 0.0 r2 coefficient 69% r2 coefficient 59% salaries unemployment parameters estimates p-value parameters estimates p-value α2008 -49.789 0.0 α2008 18.412 0.0 α2009 -50.160 0.0 α2009 18.897 0.0 α2010 -50.455 0.0 α2010 18.984 0.0 β1 2.875 0.0 β1 -0.596 0.0 β2 5.098 0.0 β2 -1.429 0.0 γ -1.494 0.0 γ -1.482 0.0 r2 coefficient 71% r2 coefficient 55% models referring to the period 2008–2010 include all explanatory variables which are statistically significant, while the same signs of parameter estimates allow for identical interpretation of their influence directions as in the period 2004–2007. in this period it was also confirmed that migration intensity keeps decreasing along with increasing distance between regions, proved by γ parameter with negative sign. additionally, the statistical significance of parameters for the accepted economic variables allows for the first hypothesis verification that socio-economic situation in poland, which to some extent is represented by the adopted economic indicators, presents an important determinant of domestic interregional migration directions. it should also be observed that the impact intensity of adopted explanatory variables on migration phenomenon is diversified. variables can be arranged according to their impact on interregional movements. the size of explanatory michał pietrzak, natalia drzewoszewska, justyna wilk dynamic econometric models 12 (2012) 111–122 120 variable impact intensity results from the total (in absolute terms) value of estimates of the parameters (see tab. 4). the higher the total value the more intense migratory movements should be expected as the result of changes in the level of explanatory variables. the highest values of the parameter estimates regarding structural models, in case of both analyzed periods of time, referred to salaries and then subsequently to gdp. the sequence of variables by the significance level did not present radical changes in relation to economic prosperity transformations. however, in times of crisis the impact intensity of all variables (except distance) was reduced. most probably also other factors, not covered by the study, did play major role in the discussed period of time. table 4. the difference and the total value of parameter estimates regarding the origin and destination regions economic variable difference in values of parameter estimates (β2-β1) total value of parameter estimates |(β2+β1)| period 1 period 2 period 1 period 2 gdp 1.311723 1.262386 3.885185 3.762408 investments 0.749878 0.639443 2.140044 1.768789 salaries 2.213778 2.2235 7.902992 7.97324 unemployment -0.73106 -0.833313 1.48135 2.024317 therefore, the obtained results allow for concluding that in recent years higher salaries represented the strongest factor stimulating polish population to change their place of residence. this observation is of high probability since higher salary frequently constitutes not only the most important objective for a potential migrant, but just in itself allows for saving financial means indispensable for covering costs of moving to another place. the collected results suggest that the level of salaries is the most important determinant of interregional migration flows, which proves the crucial role of job market in exerting impact on migrations in poland. further analysis of the total value parameter estimates indicates that the second in line variable, influencing migration level most intensely, is gdp per capita. the impact of investment outlays against unemployment rate is stronger in the first period, while in the second it is changed and a reverse situation occurs. in accordance with the second research hypothesis economic factors, in the period of prosperity, have a stronger influence on migratory movements than in times of economic downturn. the power of these factors’ impact on migrations went up regarding values related to job market, i.e. the level of gross average monthly salary and the registered unemployment rate. however, in the period 2008–2010 the intensity of regional gdp per capita and the impact of investment outlays were reduced. this fact does not allow for an unequivocal verification of the second hypothesis. the results obtained suggest the need for more the analysis of interregional migrations in poland in the period 2004–2010… dynamic econometric models 12 (2012) 109–120 121 research covering other factors impact to be carried out, especially these referring to economic development and job market. interesting conclusions can also be drawn from the interpretation of the value of the parameters regarding push and pull factors, i.e. the difference in value of β2 and β1 (see tab. 4). owing to the fact that population migrations between regions are two-directional, positive parameter estimates for both origin and destination regions point to the migratory movements occurring simultaneously in both directions. the final balance of population flows is determined by the sign of the designated difference. positive difference indicates higher population movement towards regions featuring favourable values of explanatory variables. in case of interregional migration flows the positive difference illustrates the tendency towards obtaining positive net migration by regions enjoying favourable socio-economic situation. on the other hand, negative difference underlies tendencies towards more intense population outflow from regions showing less attractive values of explanatory variables. this allows for the third hypothesis verification following which the regions featuring the highest socioeconomic development present the tendency for positive internal net migration. conclusions the purpose of the paper was to analyze the selected economic variables’ impact on interregional migratory movements in poland in the period 2004– 2010. panel gravity model with time effects for the assessment of potential determinants’ impact intensity and direction was applied. the conducted research indicates the usefulness of such approach in interregional migration phenomenon modelling in poland. the results obtained in the investigation allowed for the first research hypothesis adoption according to which socio-economic situation illustrated, to some extent, by values of economic indicators represents an important determinant defining interregional migration directions in poland. the analysis of the underlying variables impact intensity in two analyzed periods of time did not allow for unequivocal verification of the second hypothesis which suggests a relatively higher influence of the discussed economic factors in the period of prosperity. it was only in case of gdp and unemployment that slightly higher values of parameter estimates were observed in the period of prosperity. however, it was noticed that the level of salaries represent the most important determinant of interregional population migration flows, which confirms the significant role of job market in relation to migrations. the obtained results also emphasized different, in time, nature of impact exerted by factors responsible for economic development, as well as these related to job market. the conducted analysis also provided grounds for the third research hypothesis verification stating that in regions characterized by the highest level of socio-economic development there is a tendency towards positive migration flow michał pietrzak, natalia drzewoszewska, justyna wilk dynamic econometric models 12 (2012) 111–122 122 balance. this is definitely one of the factors responsible for the divergence in the development of polish regions. the presented approach does not cover all options possible in relation to the phenomenon of migrations. among the problems still left open for consideration the following should be mentioned: an in-depth analysis of internal migration determinants in times of crises, analysis of factors influencing interregional migrations and also spatial relations which may occur in case of migrations, all of them to be studied using panel gravity model with spatial effects. references anderson, j. e. (1979), a theoretical foundation for the gravity model, american economic review, 69:1, 106–116. anderson, j. e., van wincoop, e. (2004), trade costs, journal of economic literature, 42(3), 691–751. baltagi, b. h., egger, p., pfaffermayr, m. (2003), a generalized design for bilateral trade flow models”, economics letters, 80, 391–397. baltagi, b. h. (2005), econometric analysis of panel data, john wiley&sons, england. chojnicki z., zastosowanie modeli grawitacji i potencjału w badaniach przestrzennoekonomicznych (application of gravity and potential models in spatial and economic research), pwn, warszawa, 1966. egger, p. (2002), an econometric view on the estimation of gravity models and the calculation of trade potentials, world economy, 25, 297–312. faustino, h., leitão, n. c. (2007), intra-industry trade: a static and dynamic panel data analysis, international advances in economic research, 13 (3), 313–333. frankel, j., stein, e., wei, s. j. (1995), trading blocs and the americas: the natural, the unnatural and the super-natural, journal of development economics, 47, 61–95. grabiński, t., malina, a, wydymus, s., zeliaś, a (1988)., metody statystyki międzynarodowej (international statistics methods), pwe, warszawa. hausman, j. a., taylor, w. e. (1981), panel data and unobservable individual effect, econometrica, vol. 49, no. 6, 1377–1398. helliwell, j. (1997), national borders, trade and migration, pacific economic review, 2, 165– 185. kabir, m., salim, r. (2010), “can gravity model explain bimstec’s trade?”, journal of economic integration, 25(1), 144–166. lee e., a theory of migration, demography, 3, 1966. lesage, j. p., pace, r. k. (2009), introduction to spatial econometrics, crc press, new york. roy, j. r. (2004), spatial interaction modeling: a regional science context. berlin: springerverlag. sen, a., smith, t. e. (1995), gravity models of spatial interaction behavior, springer, berlin heilderberg new york. serlenga, l., shin, y. (2007), gravity models of the intra-eu trade: application of the hausman-taylor estimation in heterogeneous panels with unobserved common time-specific factors, journal of applied econometrics, 22, 361–381. tinbergen, j. (1962), shaping the world economy, the twentieth century fund inc, new york. introduction 1. the construction of panel gravity model 2. modelling of interregional migrations in poland 3. determinants for interregional migrations in poland in the period 2004–2010 conclusions references dynamic econometric models © 2012 nicolaus copernicus university press. all rights reserved. http://www.dem.umk.pl/dem dynamic econometric models vol. 12 (2012) 89−103 submitted october 23, 2011 issn accepted october 29, 2012 1234-3862 dominik krężołek* non-classical measures of investment risk on the market of precious non-ferrous metals using the methodology of stable distributions† a b s t r a c t. the aim of this article is to present some non-classical risk measures which are commonly used in financial investments, including investments in assets from the market of precious non-ferrous metals. the time series of log-returns of gold, silver, platinum and palladium prices are considered. to properly asses the investment risk the measures based on value-at-risk methodology have been used (the var estimation approach based on values from the tail of the distribution). additionally, the measure comparing expected profits to expected losses from the opposite tails distribution has been shown – the rachev ratio. it was assumed that the log-returns of presented assets belong to the family of stable distributions. the results confirm the validity of the use of stable distributions to asses the risk on the precious non-ferrous metals market. k e y w o r d s: stable distributions, value-at-risk, expected shortfall, median shortfall, rachev ratio, precious metals. j e l classification: g11, c46. introduction contemporary financial markets represent very complex area both in terms of functional and investigating aspects. there are many ways to increase the value of invested capital, i.e. investments in securities, real estate, works of art, precious metals, etc. nevertheless, every possibility in investing money is related to uncertainty and risk. since 2007 the world economy is facing a financial crisis resulting mainly from the situation on the u.s. mortgage market. this situation has rapidly spread to other markets. the potential impact of the crisis * correspondence to: dominik krężołek, department of demography and economic statistics, university of economics in katowice, ul. bogucicka 14, building „d”, 40-226 katowice, poland, e-mail: dominik.krezolek@ue.katowice.pl † research supported by the grant number kbn: n n111 299838 dominik krężołek dynamic econometric models 12 (2012) 89–103 90 and the hedging methods against its extension have become the topic of fervent discussions both among politicians, practitioners and scientists. from the scientific point of view the sudden and unpredictable changes in the research area make inadequate that the current mathematical models used for describing analyzed reality. this raises the necessity of shifting the researcher’s attention from a classical to non-classical approach. similar necessity also applies to the analysis of investment and risk. the investment uncertainty can be considered as a derivative of decisions that have been made by a decision-maker and understood as the risky ones, with consequences in the future. this implies that the probability of occurrence of some particular event may be impossible to identify or be identified with some probability. thus, the significant difference between risk and uncertainty becomes clear: a risk can be clearly measured whereas an uncertainty is some kind of unmeasurable risk that refers to the investment. the purpose of this paper is to present some non-classical risk measures, widely used in risk analysis of financial markets, and presented therein with reference to the risk observed in the market of precious non-ferrous metals. the main hypothesis runs that the values of risk measures obtained for stable distributions are closer to the empirical ones if compared to the normal approach. the subject of the study are time series represented by the log-returns of the prices of gold, silver, platinum and palladium. the main reason for choosing this particular market is an increasing interest in investing in precious metals. deepening global crisis made investors look for alternatives to the traditional financial assets (stocks, bonds, etc.). the most popular precious metal, from the investing point of view, is gold. what investors gain are high availability and the possibility to hedge themselves in situations, where the financial system shows higher level of uncertainty. purchasing of gold is more secure than typical and popular investments, e.g. in exchange rates market. however, the investment in gold doesn’t mean investment only in gold bars. there are some alternative forms, such as golden coins, investment certificates or investing in units of funds connected to the companies actively acting on the gold market. as a result of still worsening situation on the biggest stock markets, majority of investors decide to allocate their money investing in assets that show the upward trend during crisis (e.g. as a hedge against increasing inflation). 1. methodology scientists have developed many advanced methods of measurement and analysis of risk. unfortunately, the more commonly used risk measures have the drawback of being unclear in terms of their theoretical underpinnings, their use and interpretations. therefore, the methodological and practical considerations play important role for anyone who is responsible for management of that kind of risk. the risk level is influenced by many factors, some of which can be completely independent. every theoretical model should accurately reflect the non-classical measures of investment risk on the market of precious… dynamic econometric models 12 (2012) 89–103 91 reality, and their construction is a very complex process. the methodological background for estimation and assessing risk, developed in the last century, requires to be modified due to the set of risk factors which are constantly varying over time. it results from the dynamic changes on the market, new possibilities of assets allocation and unpredictability of some unexpected events. such events are related to the higher risk level and may be impossible to forecast. this is a problem both for individual investors and institutions, which are exposed to huge financial loses or even bankruptcy. the literature on the subject recognizes a number of risk measures. in this paper the distinction between classical and non-classical measures is presented. as for classical ones all measures which represent a canon in risk analysis and are widely used in practice are considered here. most of these measures appeared together with the development of certain scientific theories and afterwards were modified according to the external factors which determine their use. moreover, the set of classical risk measures includes those based on commonly used probability distributions (especially the normal distribution). the set of non-classical risk measures encompasses all measures not included in the set of classical ones. the non-classical measures might be etymologically related to the interdisciplinary nature of science, deriving from the different scientific areas (physics, engineering, bio-medicine, etc.). however, because of their mathematical properties they are used in measuring risk on financial markets. due to strong assumption about the normality of distribution of stock returns in the set of classical measures, not met for the real data, some alternative measures based on different probability distributions have to be taken into account. the characteristics of empirical time series as high frequency of data, heteroscedasticity of variance, autocorrelation or fat tails of distributions are essential in risk analysis. the stable distributions, developed by b. mandelbrot in the 60s, have the wide application in this matter. distributions which belong to the class of stable ones, are described by the shape parameter allowing to model the asymmetry and fatness of the tail of distribution. it makes them useful in many scientific areas (from engineering, through physics to applications on financial markets (borak, härdle, weron, 2005). the main difficulty in application of stable distributions is that their probability distribution function is not clearly defined. to describe stable model the characteristic function is used (and so-called inverse fourier transform with respect to this function). if the random variable x has the cumulative distribution function ( )xf , then its characteristic function has the form (rachev, mittnik, 2000): ( ) ( )[ ] ( ) ( ),expexp ∫ +∞ ∞− == xdfitxitxetϕ (1) thus, for stable random variable the characteristic function is presented as: dominik krężołek dynamic econometric models 12 (2012) 89–103 92 ( ) ( ) ( ) , 1,ln 2 1 1, 2 1 ln        =    +− ≠            −− = α π βσµ α απ βσµ ϕ αα forttsignitti fortgtsignitti t (2) where ( )tsign is defined as ( )      <⇔− =⇔ >⇔ = 01 00 01 t t t tsign . the main characteristic of stable distribution is the shape parameter α allowing to measure the fatness of the tail of distribution. remaining parameters which describe the stable pdf are skewness parameter 1;1−∈β , scale parameter 0>σ and location parameter r∈µ . the practical use of stable distribution is related to complex estimation procedure of all parameters. in literature only three types of stable distributions are explicitly defined: normal distribution (where 2=α ), cauchy distribution (where 0,1 == βα ) and lévy distribution (where 1,2 1 −== βα ). the shape parameter plays significant role in probability distribution analysis. if it is less than 2, then the variance of the distribution is infinite and the location parameter is equal to the mean of distribution. in the case where the shape parameter is less than 1, both variance and mean are infinite (samorodnitsky, taqqu, 1994). the application of stable distributions is extremely justified if the fat tailed distributions are considered. in that case the probability that random variable takes values at the level significantly outlying from the center of the distribution is higher than in the normal case. hence, the risk measures which are used in risk analysis have to take into account such values. the most popular and the most widely used risk measure in this case is value-at-risk (var). several methods of estimating value-at-risk are used, but depend on many aspects, such as statistical assumptions related to the risk factors, dependency between these factors or the portfolio structure. in this paper the quantile based method for estimating var is used. in this method it is not necessary to assume the analytical form of function which describes the probability distribution of the risk. the historical data used in this method allows to estimate parameters which describe the real distribution of return. afterwards, the −α quantile of the distribution is estimated which allows to determine the var. in this method the most important problem is to select the proper probability distribution function, especially if the financial time series are analyzed. in addition, some tests of the fit of distribution have to be used to verify the convergence between empirical and theoretical distributions. non-classical measures of investment risk on the market of precious… dynamic econometric models 12 (2012) 89–103 93 2. non-classical risk measures the tail analysis of empirical distributions requires to define some class of measures which allow to asses risk related to the values significantly distant from the central part of the distribution. in this matter the quantile risk measures are considered. the idea of quantile risk measures is based on the value-at-risk (var) methodology, which is one of the most popular risk measures used in practice. its principal advantage is that the var conveys straightforward information about potential loss as a result of an investment. however, such information does not include cases where some extreme observation occurs. hence is not a good measure. artzner, delbean, eber and heat (artzner et al., 1997) have proposed the set of axioms which have to be met by a good risk measure. these axioms define a coherent risk measure. if ϑ is a coherent risk measure for a set of random variables y (defined on some probability space), such as +→ ry:ϑ , a measure ϑ have to be: sub-additive, for any ( ) ( ) ( )212121 :, xxxxyxx ϑϑϑ +≤+∈ positively homogeneous, for any 0≥λ and ( ) ( )xxyx λϑλϑ =∈ : monotonous, for any yxx ∈21 , if only ( ) ( )2121 : xxxx ϑϑ ≤≤ translation invariant, for any ( ) ( ) cxcxrcyx +=+∈∈ ϑϑ:, as was mentioned above, var is one of the most popular quantile measure used by practitioners. it gives an answer to a question of what is the maximum loss incurred by an investor (or institution) from an investment in some specified time horizon. this loss may occur with some probability α called tolerance level. maintaining previous assumptions and defining value-at-risk at the confidence level α , i.e. ( )xvarα , it is possible to calculate coherent risk measures determining expected value (in terms of mean and median) of an investment exceeding potential loss beyond var: the expected shortfall and median shortfall. these measures are calculated using formula: ( ) ( ) ( )[ ],xvarxxvarxexes ααα >−= (3) ( ) ( ) ( )[ ],xvarxxvarxmedianxms ααα >−= (4) these measures inform about the possible expected loss beyond the level represented by var. in other words, what follows from (3) and (4) is that for any given α there exists one-step-ahead prediction representing expected value (in terms of mean and median) of loss beyond the var. additionally, taking into account certain tails’ characteristics of empirical distributions it is necessary to compare if expected profits exceed expected losses. an interesting tool has been proposed by rachev – the rachev ratio ratior − (biglova, ortobelli, rachev, stoyanov, 2004). this measure is calculated using formula: dominik krężołek dynamic econometric models 12 (2012) 89–103 94 ( )( ) ( )( ) | | e x x var x r ratio e x x var x α β ≥ − − − = ≤ (5) if the random variable x represents profit or loss from a given investment, the rachev ratio −r ratio defines the ratio of expected profit above the var value (for a given −thα quantile) to expected losses below the var value (for a given −thβ quantile). if (in a special case) αβ −=1 , then the values of −r ratio are interpreted as follows: − 1r ratio− = – expected profits are equals to expected losses, − 1r ratio− > – expected profits exceed expected losses, − 1r ratio− < – expected profits are lower than expected losses. the risk assessment using −r ratio is based on maximizing its values. 3. empirical analysis the practical application of some non-classical risk measures is presented here in the relation to the market of precious non-ferrous metals, represented by the time series of 2902 daily log-returns of the prices of gold, silver, platinum and palladium in period of january 2000 – june 20111. as presented in table 1, there is a significant dispersion in prices of precious non-ferrous metals. table 1. descriptive statistics – prices gold silver platinum palladium mean 622.89 10.84 1 023.26 387.43 standard deviation 349.12 7.43 440.08 196.87 minimum 255.95 4.07 414.00 148.00 maximum 1 552.50 48.70 2 273.00 1 090.00 range 1 296.55 44.64 1 859.00 942.00 the highest dispersion level (in terms of range) is observable in prices of platinum and gold. although, if the coefficient of variation is taken into account, the highest dispersion is related to the prices of silver. the volatility of prices of gold and palladium is presented on figures 1 and 2 respectively. the prices of gold shows the behavior similar to the stock indices whereas the prices of palladium not. as is presented in figure 1, the price of gold has been increasing reasonably till february 2006. then some significant price changes have been observed. the last four years show that the price of gold hadn’t behave in a stable manner. 1 data from london metal exchange. non-classical measures of investment risk on the market of precious… dynamic econometric models 12 (2012) 89–103 95 figure 1. the volatility of price of gold from january 2000 to june 2011 the price of palladium has been changing without specifically defined direction. after first two years of growing the price has fallen dramatically returning to upward trend in first half of 2003. the year 2009 initiated the period of surge of prices that continues till now. in the next step the log-returns of corresponding prices have been calculated. figure 3 presents the volatility observed in time series of gold log-returns. figure 2. the volatility of price of palladium from january 2000 to june 2011 gold 0,00 600,00 1 200,00 1 800,00 1 501 1001 1501 2001 2501 pr ice [u sd ] palladium 0,00 400,00 800,00 1 200,00 1 501 1001 1501 2001 2501 pr ice [u sd ] dominik krężołek dynamic econometric models 12 (2012) 89–103 96 figure 3. volatility of log-return of gold as presented in figure 3, the period starting in the middle of 2005 reveals increased level of volatility (clustering and significant jumps in returns). the same or even more unstable behavior characterizes returns of palladium. clustering is observed within entire period, with higher intensity during last year (figure 4). figure 4. volatility of log-return of palladium log-return gold -0,1 -0,06 -0,02 0,02 0,06 0,1 log-return palladium -0,2 -0,1 0 0,1 0,2 non-classical measures of investment risk on the market of precious… dynamic econometric models 12 (2012) 89–103 97 one of the most important matter related to the risk analysis of financial markets is that, whether there exists a significant relation between assets. it is of great importance especially in portfolio analysis. to see that, table 2 presents the correlations between log-returns of metals analyzed, whereas table 3 shows the descriptive statistics. table 2. correlations gold silver platinum palladium gold 1.00000 0.56024 0.44953 0.37513 silver 0.56024 1.00000 0.45243 0.42328 platinum 0.44953 0.45243 1.00000 0.57408 palladium 0.37513 0.42328 0.57408 1.00000 the highest values of correlation are between pairs of returns of platinum and palladium and between gold and silver. the lowest values correspond to the relation between gold and palladium. it can be caused by a natural market properties of these precious metals. gold and silver are considered as investment precious metals while platinum and palladium are more related to industrial environment (especially the latter). table 3. descriptive statistics – returns gold silver platinum palladium mean 0.00058 0.00065 0.00048 0.00019 standard deviation ( nσ ) 0.01151 0.02107 0.01603 0.02392 minimum -0.07972 -0.18693 -0.17277 -0.17859 maximum 0.06841 0.18279 0.11129 0.16799 range 0.14813 0.36971 0.28406 0.34658 as presented in table 3, the highest value of expected return corresponds to the variables silver and gold. the comparable level reaches the return of platinum, whereas the return of palladium oscillates around zero. the dispersion analysis showed that the most risky (in terms of standard deviation) are investments in palladium and silver (in terms of range as well) while the safest one correspond to gold (figure 5). further analysis related directly to risk measurement requires to match the theoretical probability distribution to the real market data. classical risk analysis is based on the normal distribution, so this case has to be verified. to test the normality of log-returns of the selected precious metals the following goodnessof-fit methods have been used: jarque-bera test and anderson-darling test. the results of comparing empirical and normal distribution show that the assumption of normality has to be rejected (at the significant level of 0.01). therefore the empirical distributions do not belong to the family of normal ones. the rejection of normality requires the use of risk measures which are not based on this strong assumption and allow to use other statistical distributions. dominik krężołek dynamic econometric models 12 (2012) 89–103 98 figure 5. expected return vs. risk to solve this problem, the family of stable distribution is used. to estimate the parameters of stable distribution, the maximum likelihood methodology is applied and the results are presented in table 4. table 4. the parameters of stable distributions α β σ μ gold 1.63592 -0.04440 0.00620 0.00063 silver 1.60458 -0.03883 0.01074 0.00103 platinum 1.58989 -0.09054 0.00790 0.00061 palladium 1.54917 -0.01818 0.01192 0.00049 note: the parameters are significant at the level of 0.01. as was mentioned in section “methodology”, the parameter α plays the most important role. it is responsible for the thickness of tail of distribution. as results from calculations, each of log-returns distributions of analyzed assets is fat-tailed. the index of stability takes the highest value for the variable gold, and the lowest for the variable palladium. moreover, each of the distributions has a left-sided asymmetry which means that the left-side tail is heavier than the right-side one. if α takes values below 2, the location parameter is equal to the mean of distribution. therefore the more accurate information about the exex pected return v s. risk palladium silv er platinum gold 0,00000 0,00020 0,00040 0,00060 0,00080 0,00000 0,00500 0,01000 0,01500 0,02000 0,02500 0,03000 standard dev iation m ea n non-classical measures of investment risk on the market of precious… dynamic econometric models 12 (2012) 89–103 99 pected return of empirical data is provided rather by the value of location parameter than by the value of mean (which is more suitable if the assumption figure 6. fitting of normal distribution – platinum of normality is not rejected). the scale parameter σ can be interpreted as a dispersion parameter (just like the standard deviation in normal case) as the relation between location parameter of stable distribution and standard deviation of normal distribution is as follows: 2nσσ = . the fitting of normal and stable distributions to the empirical data of variable platinum is presented using histograms and qq-plots (figures 6–9). figure 7. fitting of stable distribution – platinum -0.15 -0.1 -0.05-1.38778 10-170.05 0.1 10 20 30 -0.15 -0.1 -0.05-1.38778 10-170.05 0.1 10 20 30 dominik krężołek dynamic econometric models 12 (2012) 89–103 100 the figures 6–7 show that stable distribution is better fitted to the empirical data of platinum than in the normal case. qq-plots confirm that in terms of quantile analysis. the more dotted line covers the solid one the better fit to the theoretical distribution. this property of qq-plot is rejected for normal case. figure 8. qq-plot of normal distribution – platinum figure 9. qq-plot of stable distribution – platinum further risk analysis is based on the non-classical measures arising from the methodology of value-at-risk. the concept of var is based on the estimation of quantile of arbitrary distribution. in the first step the value-at-risk for each variable is calculated. for calculation one-period forecast is used. the results presented in table 5. -0.04 -0.02 0.02 0.04 -0.06 -0.04 -0.02 0.02 0.04 -0.04 -0.02 0.02 0.04 -0.04 -0.02 0.02 0.04 non-classical measures of investment risk on the market of precious… dynamic econometric models 12 (2012) 89–103 101 table 5. value-at-risk estimates risk measure type of distribution quantile gold silver platinum palladium valueat-risk empirical distribution 0.01 -0.03298 -0.05946 -0.04992 -0.07466 0.05 -0.01727 -0.03093 -0.02266 -0.03747 normal distribution 0.01 -0.02414 -0.04929 -0.03684 -0.05663 0.05 -0.01815 -0.03344 -0.0262 -0.04074 stable distribution 0.01 -0.03638 -0.06713 -0.05238 -0.08346 0.05 -0.01651 -0.02931 -0.02218 -0.03453 table 5 presents value-at-risk estimates for quantiles 0.01 and 0.05 for three types of distributions. the results obtained for the stable model are similar to those obtained for the empirical distribution. however, comparing results of normal estimates, the differences are significant. for example, the var for gold at 0.01 confidence level for normal distribution is -0.02414 while for empirical and stable distribution is -0.03289 and -0.03638 respectively. therefore, taking into account the normal case, potential loss seems to be lower than it actually is (comparing to the empirical values). moreover, the results are corroboration of the fitting quality of stable models to the empirical data. table 6. expected shortfall estimates risk measure type of distribution quantile gold silver platinum palladium expected shortfall empirical distribution 0.01 -0.04349 -0.09007 -0.07171 -0.09979 0.05 -0.02707 -0.05209 -0.04001 -0.05930 normal distribution 0.01 -0.02715 -0.05739 -0.04209 -0.06523 0.05 -0.02207 -0.04352 -0.03286 -0.05051 stable distribution 0.01 -0.04429 -0.09507 -0.08766 -0.08540 0.05 -0.03563 -0.06547 -0.06267 -0.06288 similar results obtained for the expected shortfall (table 6) and median shortfall (table 7). expected loss exceeding var (in terms of mean and median) is similar for normal and stable distributions while it differs for the normal one. table 7. median shortfall estimates risk measure type of distribution quantile r-gold r-silver r-platinum r-palladium median shortfall empirical distribution 0.01 -0.04017 -0.07918 -0.06704 -0.09038 0.05 -0.02359 -0.04459 -0.03377 -0.05055 normal distribution 0.01 -0.02591 -0.05627 -0.03951 -0.06276 0.05 -0.02147 -0.04145 -0.03137 -0.05051 stable distribution 0.01 -0.04341 -0.07851 -0.07346 -0.09073 0.05 -0.02594 -0.03991 -0.03263 -0.04142 using values of var for a given pairs of quantiles, the results obtained for −r ratio are presented in table 8 and table 9. dominik krężołek dynamic econometric models 12 (2012) 89–103 102 table 8. r-ratio estimates (quantiles 0.99 vs. 0.01) risk measure type of distribution quantile r-gold r-silver r-platinum r-palladium r-ratio empirical distribution 0.99 vs.0.01 1.22435 1.12766 0.99684 1.00492 normal distribution 0.99 vs. 0.01 1.13321 1.19392 0.98224 1.02183 stable distribution 0.99 vs. 0.01 1.21334 1.14118 0.99733 1.01432 taking into account quantiles 0.99 and 0.01 the values of ratio obtained for stable distribution are closer to the values obtained from empirical distribution. this shows that the approach based on stable distribution is more adequate to the real behavior of stock returns. table 9. r-ratio estimates (quantiles 0.95 vs. 0.05) risk measure type of distribution quantile r-gold r-silver r-platinum r-palladium r-ratio empirical distribution 0.95 vs.0.05 1.01766 1.04638 1.00231 1.01233 normal distribution 0.95 vs. 0.05 1.12235 1.11321 0.99112 1.09121 stable distribution 0.95 vs. 0.05 1.02152 1.08199 0.99984 1.04523 the data presented in table 9 confirmed the results obtained for quantiles 0.99 and 0.01 – the advantage of using stable approach over the normal one. conclusions the application of non-classical risk measures plays very important role in financial market analysis. both scientists and practitioners confirm that the risk analysis has to be extended beyond the normal case, and this approach covers not only financial market but also other ones, e.g. non-ferrous metals market. this paper has presented the analysis of daily log-returns of gold, silver, platinum and palladium time series. as confirmed, the assumption of normality of distribution for each variables has been rejected and the stable models have been applied. the results show that stable distributions are better fitted to the empirical data than the normal ones, complying with leptokurtosis, heavy tails (the range of parameter α reflecting thickness of the tail is 1.55–1.64) and asymmetry. in the case of risk measurement the methodology of value-at-risk is the basis of calculating non-classical risk measures. both var and measures such as expected shortfall and median shortfall prove discrepancy between normal and empirical distributions. that is the reason why the other type of distributions should be taken into account. similar results for calculating coherent risk measures were obtained by trzpiot, krężołek (2009) for analyzing daily lognon-classical measures of investment risk on the market of precious… dynamic econometric models 12 (2012) 89–103 103 returns on polish financial market using stable models and by krężołek (2010) on precious non-ferrous metals market, but using geo-stable distributions (more precisely – asymmetric laplace distribution). the results, although straightforward in terms which class of theoretical distribution should be chosen, have to be interpreted very carefully. moreover, considering −r ratio the results obtained for stable distributions are closer to the values for empirical one comparing to the normal ones. references artzner, p., delbaen, f., eber, j. m., heath, d. (1997), thinking coherently, risk, 10, 68–71. biglova, a., ortobelli, s., rachev, s. t., stoyanov, s. (2004), different approaches to risk estimation in portfolio theory, journal of portfolio management, 31, 1, 106. borak, sz., härdle, w., weron, r. (2005), stable distributions, springer, berlin. hammoudeh, s., malik, f., mcaleer, m. (2011), risk management of precious metals, quarterly review of economics and finance, 51, 435–441. krężołek, d. (2010), kwantylowe oraz koherentne miary ryzyka – analiza empiryczna na rynku metali nieżelaznych z wykorzystaniem rodziny statystycznych rozkładów stabilnych (quantiles and coherent risk measures – an empirical analysis on the non-ferrous metals’ market based on the family of stable distributions), finanse, rynki finansowe i ubezpieczenia. skuteczne inwestowanie (finance, financial markets and insurance. effective investment), 29, 433–444. rachev, s., mittnik, s. (2000), stable paretian models in finance, series in financial economics and quantitative analysis, john wiley & sons ltd., england. samorodnitsky, g., taqqu, m. s. (1994), stable non-gaussian random processes. stochastic models with infinite variance, chapman & hall, new york, 1994. trzpiot, g. (2010), wielowymiarowe metody statystyczne w analizie ryzyka inwestycyjnego (multivariate statistical methods in investment risk analysis), wydawnictwo pwe, warszawa. trzpiot, g., krężołek, d. (2009), quantiles ratio risk measures for stable distributions models in finance (kwantylowe miary ryzyka z wykorzystaniem rozkładów stabilnych w finansach), studia ekonomiczne, 53 (economic studies, 53), zeszyty naukowe akademii ekonomicznej w katowicach, 109–120. nieklasyczne mierniki ryzyka inwestycyjnego na rynku szlachetnych metali nieżelaznych z wykorzystaniem metodologii rozkładów stabilnych z a r y s t r e ś c i. celem artykułu jest prezentacja wybranych nieklasycznych miar ryzyka, które mają szerokie praktyczne zastosowanie w przypadku inwestycji finansowych, w tym inwestycji w walory notowane na rynkach surowców (nieżelazne metale szlachetne). przedmiotem badania są szeregi czasowe reprezentowane przez stopy zwrotu cen złota, srebra, platyny oraz palladu. do oceny ryzyka inwestycyjnego wykorzystano miary wyznaczone w oparciu o metodologię valueat-risk stosując estymację opartą na wartościach z ogona rozkładu. dodatkowo do oceny ryzyka wykorzystano wskaźnik rachev’a. przyjęto założenie, że stopy zwrotu wykorzystanych zmiennych należą do rodziny rozkładów stabilnych. wyniki badania potwierdzają stosowność wykorzystania rozkładów stabilnych do oceny ryzyka na rynku metali nieżelaznych. s ł o w a k l u c z o w e: rozkłady stabilne, value-at-risk, expected shortfall, median shortfall, wskaźnik rachev’a, metale szlachetne introduction 1. methodology 2. non-classical risk measures 3. empirical analysis conclusions references © 2016 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2016.009 vol. 16 (2016) 133−144 submitted november 11, 2016 issn (online) 2450-7067 accepted december 20, 2016 issn (print) 1234-3862 beata szetela, grzegorz mentel, stanisław gędek * dependency analysis between bitcoin and selected global currencies a b s t r a c t. in this research we have tried to identify the relationship between the exchange rate for bitcoin to the leading currencies such as dollar, euro, british pound and chinese yuan and polish zloty as well. we have applied arma and garch models to model and to analyze the conditional mean and variance. the appliance of garch models have identified some dependency in explanation conditional variance between bitcoin and us dollar, euro and yuan, while arma analysis have shown no relations between bitcoin and other dependent variables. k e y w o r d s: arma, bitcoin, dependency, garch, variability. j e l classification: f31; c32. introduction progressive globalization and dynamic technological development has led to inequalities in sharing of resources and increase of distrust to governments, banks and other state and financial institutions. people began to look for new alternatives that would respond to the growing consumerism and surveillance. the creation of virtual currencies (vc) was such an answer. vc’s are developing in a dynamic way and are gaining more and more atten * correspondence to: beata szetela, rzeszow technical university, faculty of management, al. powstańców warszawy 8, 35-959 rzeszów, poland, e-mail: beata@prz.edu.pl; grzegorz mentel, rzeszow technical university, faculty of management, al. powstańców warszawy 8, 35-959 rzeszów, poland, e-mail: gmentel@prz.edu.pl; stanisław gędek, rzeszow technical university, faculty of management, al. powstańców warszawy 8, 35-959 rzeszów, poland, e-mail: gedeks@prz.edu.pl. beata szetela, grzegorz mentel, stanisław gędek dynamic econometric models 16 (2016) 133–144 134 tion. the experts from pricewaterhousecoopers in their report have admitted, that vc is a beginning of a new phase of technology-driven markets that have the potential to disrupt conventional market strategies, longstanding business practices, and established regulatory perspectives—all to the benefit of consumers and broader macroeconomic efficiency. [vc] carry groundbreaking potential to allow consumers access to a global payment system—anywhere, anytime. (pwc, 2015). following information available via: mapofcoins.com, currently there are ca. 600 different, active virtual currencies. among them is bitcoin – one of the most prominent examples, with the biggest capitalization (ca. 10 bn usd). it is the most popular and most widely recognized vc. many global companies are now accepting payments in bitcoin, e.g. wordpress.com, amazon, victoria’s secret, subway, bloomberg.com, sears, gap, apple app store, miscrosoft, dell, lot polish airlines, t-mobile polska, profident, etc. the father of bitcoin is considered satoshi nakamoto, who in 2008 published an article "bitcoin: a peer-to-peer electronic cash system", in which he described the concept of virtual, decentralized and independent means of payment. he has based his solution on the information protocol (block chain), which aims to eliminate all of the transactions factors, which are based on trust 1 . bitcoin supporters consider anonymity as its major advantage, as well as its broadest sense independence. therefore in this article we have used autoregressive models, such as arma and garch, to model and to analyze bitcoin’s conditional mean and variance in relation to other world currencies like us dollar (usd), euro (eur), british pound sterling (gbp) and chinese yuan (cny). the currencies were selected based on the transaction volume, in which the most of the transactions were performed. the results would give us an answer, whether bitcoin is impervious to external influences. such a feature would imply that this vc is impossible to control by third party agents, hence can be seen as a fully independent means of payment. 1. literature overview current achievements of scientists related to bitcoin, can be divided into four main streams of interest. considerations of a general and theoretical form for example: (dwyer, 2015), (dopierała and borodo, 2014), (liu et.al., 2015), (jagwani, 2015), (rogojanu and badea , 2015), etc. this group include 1 for more information about bitcoin and it’s technical details of creation, mining and functioning, please refer to (nakamoto, 2008), (nowakowski, 2013) and (nielsen, 2013). dependency analysis between bitcoin and selected global currencies dynamic econometric models 16 (2016) 133–144 135 a number of studies on the classification of bitcoin as a commodity or a currency. the scientists in their work roll extensive discussion regarding the intrinsic value of bitcoin, the future and the potential it can bring. the committee for payments and market infrastructure (cpmi) acting at the bank for international settlements in its publication "virtual currency” (cpmi, 2015), has stated, that bitcoin is similar in its concept to goods, such as gold, whose price is created by the power of supply and demand, with an except that it has no intrinsic value. (haubno-dyhrberg, 2016) raises other similarities between bitcoin and gold, e.g. limited amount and a similar way of its extraction ( "mining"). other similarities lists (weber, 2016), who points out that both gold and bitcoin supply is not controlled by any state or institution, and both products have the same means of exchange. the second group of papers, for example: (badev and chen, 2014), (taha, 2015), (luther and olson, 2015), (campbell, 2014) (tasca and de roure, 2014), (karama et.al., 2015) and others, is focusing mainly on issues relating to acquisition (mining), trade and broadly understood security. articles concerning regulatory and tax issues are covered by the third group, for example. (mandjee, 2014), (bryans, 2014), (plassaras, 2013), etc. this group also include the various types of reports and publications of banks, financial institutions and governments on bitcoin, i.e. european central bank (ecb, 2012), (ecb, 2015), (draghi, 2015); congressional research service acting on the needs of the us congress (murphy et.al. 2015), (congress, 2014); canadian central bank (gans and halaburda, 2013), (boc, 2014), (chiu and wong, 2015), and others. the last group focuses on the application of quantitative methods in the study of bitcoin, for example: (brandvold et.al., 2015) using the price discovering process studied which of the bitcoin’s trading platforms has the most impact on the bitcoin’s price and which one is following the general trend. the authors found that mtgox and btce are undisputed animators of the bitcoin’s prices. (haubno-dyhrberg, 2016a) and (haubno-dyhrberg, 2016b) have used garch to prove that bitcoin bears similarities to both the us dollar and gold, and that it can serve as an instrument to minimize the risk by people characterized by a strong risk aversion. (cheah and fry, 2015) showed that bitcoin like other cryptocurrency tend to generate bubbles, and they do not have fundamental value. (gronwald, 2015) using a generalized model of autoregressive conditional heteroscedasticity said that the fluctuations in bitcoin are characterized by sudden surges and extremes pricing, which is characteristic for immature markets. (szetela, 2016) has investigated bitcoin and dollar price variability using control charts. she has confirmed that price variability is strongly influenced by price jumps on the one hand, but on the other hand bitcoin’s beata szetela, grzegorz mentel, stanisław gędek dynamic econometric models 16 (2016) 133–144 136 price variability is tending to decrease. (macdonell, 2014) using arma model and lppl showed that the price of bitcoin depends on the cboe volatility index, which indicates the great potential of this speculative currency. (vockathaler, 2015) confirmed that fluctuations in the price of bitcoin are positively correlated with the amount of users btc and determined by the endogenous shocks of unknown source, origin and are not generated by the impact of specific variables, such as indexes s&p 500, gold rate against the us dollar (xau) and the shanghai stock exchange index (sse). according to (chu et.al., 2015) bitcoin’s price fluctuations are best characterized by generalized hyperbolic distribution. (bouoiyour and selmi, 2015) is their current research investigated the variability of bitcoin against the dollar in the period before and after the year 2015. they have found that despite the fact that the volatility of bitcoin in 2015 significantly decreased compared with the preceding period, still it cannot be said that the bitcoin can be regarded as mature currency. 3. methodology to analyze the relationship between given time series, we have used vectorautoregressive models formulated by sims in 1980.the analysis of the relationship between time series is complex and complicated. often, the dependent variables are modeled not only by a set of explanatory variables, but may also depend on their own historical observations and/or historical values of the independent variables. vector-autoregressive models (var) allow for simultaneous analysis of this type of relationship. an important advantage of these models is the lack of restrictions concerning the division of variables into endogenous and exogenous. this model also allows to analyze the bidirectional relation, that is when two variables interact with each other. this property is used e.g. (matuszewska and witkowska, 2006) to examine the interdependence between the exchange rate eur/usd and selected eight independent variables. var method is based on arma and garch models with all their generalizations and modifications. it is often use to model rates of return on various types of assets. arma allows the modeling of conditional mean, while the garch – conditional variance. in the literature, the combination of these two models to analyze the exchange rates applied, among others, (nakatsuma and tsurumi, 1999) (quaicoe et.al., 2015), (marreh et.al., 2014), (doman and doman, 2014), etc. arma (p, q) process is a combination of two types of processes, ie. an autoregressive of order p – ar (p) and the moving average of order q – ma (q). dependency analysis between bitcoin and selected global currencies dynamic econometric models 16 (2016) 133–144 137 following (montgomery et.al., 2008), the autoregressive process of an order p, can be written the formula: , (1) where: is a white noise; – parameters. the realization of a process at time t depends on p prior observations and the white noise at point t. if however the realization of a process at time t depends on q previous random terms, then such a process is a moving average of order q (pesaran, 2015). such dependency has the form: , (2) where: is a white noise; – parameters. the arma model is a combination of the autoregressive process of order p and a moving average process of order and can be written in the following form: , (3) the arch models (auto regressive conditional heteroskedasticity) were introduced to the literature by eagle in 1982. they are based on the assumption that variance of the process residuals is not constant over time. (mandelbrot, 1963) has proven, that in particular the financial data are affected by the presence of outliers and volatility clustering, which impacts the distortion of the distribution of the exanimated variables. arch models assume that the conditional variance of the error term at point t is dependent on the p previous error terms: , (4) where – variance, – error term, – parameters, – constant. the generalization of the arch model, a garch models 2 , was introduced in 1986 by boleslev, and has a form: , (5) where: is a white noise; – variance at time t; , – parameters. 2 for further details considered applied models please refer to hamilton (1994), montgomery et.al. (2008) and alberg et.al. (2008). beata szetela, grzegorz mentel, stanisław gędek dynamic econometric models 16 (2016) 133–144 138 vector autoregressive models can be applied to stationary time series. a time series is assumed to be strict stationarity, if the probability distribution function remains unchanged at each point of time, (markellos & mills, 2008): (6) the process can be described as stationary in a broader sense, if it has a constant mean and variance, and covariance of the observation depends solely on the distance between them. formally this conditions are described as follows: (7) (8) (9) in order to investigate, whether a considered time series follows a stationary process, a dickey fuller test is applied, which verifies a presence of a unit root under the null hypothesis ( ) versus an alternative hypothesis, which assumes process stationarity ( . 4. empirical results in our research we have taken into the consideration daily logarithmic rate of return for bitcoin to polish zloty (rbtc/pln) and compared it with the logarithmic rate of return for euro to zloty (reur/pln), us dollar to zloty (rusd/pln), british pound to zloty (rgbp/pln) and chinese yuan to zloty (rcny/pln). the mentioned currencies were chosen based on the volume of transactions performed on bitcoin 3 . we analyzed the period from january 2014 to june 2016. the data were collected from www.quandl.com and contained in total 641 full observations. the assumption of stationarity was verified by the dickey fuller unit root test (df). the test results presented in table 1, show that in all cases the df test was able to reject the null hypothesis of appearance of a unit root, proving stationarity for all of included variables. 3 following information available via: bitcoinity.org, the majority of transactions in terms of volume was performed in yuan, which accounted for approx. 84% of all transactions. the us dollar covered approximately 13% of all transactions performed on bitcoin, euro ca. 1%, british pound ca. 0.3% and polish zloty ca. 0.2%. dependency analysis between bitcoin and selected global currencies dynamic econometric models 16 (2016) 133–144 139 table 1. testing for the unit roots and stationarity results for the logarithmic rate of return for bitcoin to zloty (rbtc/pln), euro to zloty (reur/pln), dollar to zloty (rusd/pln), pound to zloty (rgbp/pln), daily observations from the period jan. 2014–june, 2016 dickey-fuller unit root tests variable zero mean p-value single mean p-value trend p-value rbtc/pln –17.54 <.0001 –17.53 <.0001 –17.74 <.0001 reur/pln –16.68 <.0001 –16.68 <.0001 –16.68 <.0001 rusd/pln –16.25 <.0001 –16.32 <.0001 –16.32 <.0001 rgbp/pln –15.73 <.0001 –15.75 <.0001 –15.75 <.0001 rcny/pln –16.42 <.0001 –16.44 <.0001 –16.44 <.0001 note: adf computes a test statistic for the time series with a zero mean, a single mean, which includes a constant term and a trend, which includes constant and a deterministic trend. for the adopted order of the model p and q, selected by the smallest value of corrected akaike inrofmation criterion, arma model parameters were estimated and the results are presented in table 2. none of the lagged dependent variables were statistically significant in explaining dependency between bitcoin and other currencies. the p-values are clearly above the 5% significance level. the results indicate, that no dependency exists between logarithmic rate of return of bitcoin to zloty and all other exchange rates in modeling conditional mean. table 2. results for the significance test for the estimated arma(1,0) model for the btc/pln as an dependent variable arma rusd/pln reur/pln rgbp/pln rcny/pln const 0.01410 0.02764 0.01374 0.02496 btc/pln (t–1) 0.04343 0.04758 0.04593 0.04629 rx/pln (t–1) 0.19033 –0.50527 0.42876 –0.09269 p rusd/pln reur/pln rgbp/pln rcny/pln const 0.9321 0.8671 0.9337 0.8800 btc/pln (t–1) 0.04015 0.2360 0.2516 0.2502 rx/pln (t–1) 0.26859 0.2721 0.1531 0.7314 std.err. rusd/pln reur/pln rgbp/pln rcny/pln const 0.16549 0.16506 0.16499 0.16532 btc/pln (t–1) 0.2798 0.04011 0.04003 0.04022 rx/pln (t–1) 0.4788 0.45963 0.29974 0.26988 note: btc/pln(t–1) is a lagged by one period btc/pln, rx/pln(t–1) – lagged by one period independent variable, i.e. rusd/pln, reur/pln, rgbp/pln, rcny/pln. we have applied the lagrange multiplier test to detect the arch effects in residuals. the highly significant p-value (<.0001) points at rejection of the null hypothesis, what indicates the existence of autocorrelation in residuals. to model the conditional variance, we have estimated forty different beata szetela, grzegorz mentel, stanisław gędek dynamic econometric models 16 (2016) 133–144 140 garch models. based on the smallest value of the information criterion, we have chosen an exponential garch(1,1), as a model best fitted to the data. the results of a significance test, presented in table 3 show, that all egarch parameters are statistically significant. moreover btc/pln logarithmic exchange rate is influenced by logarithmic exchange rate of usd/pln, eur/pln and cny/pln. table 3. results for the significance test for the estimated egarch(1,1) model for the btc/pln as an dependent variable egarch rusd/pln reur/pln rgbp/pln rcny/pln rx/pln 0.4053 0.9990 –0.1348 0.3742 earch0 0.2423 0.2472 0.2528 0.2477 earch1 0.2958 0.2994 0.3057 0.2952 egarch1 0.9245 0.9225 0.9208 0.9224 theta –0.0668 –0.0609 –0.0563 –0.0633 restrict –10.1788 1.2521 –9.7345 p rusd/pln reur/pln rgbp/pln rcny/pln rx/pln 0.0406 0.0026 0.5392 0.0746 earch0 <.0001 <.0001 <.0001 <.0001 earch1 <.0001 <.0001 <.0001 <.0001 egarch1 <.0001 <.0001 <.0001 <.0001 theta 0.2145 0.2570 0.3068 <.0001 restrict 0.0210 0.7469 0.0301 note: rx/pln–independent variable, i.e. rusd/pln, reur/pln, rgbp/pln, rcny/pln. to confirm the quality of the model fit we performed the bds test (table 4) and stability of the parameters were verified by the chow test (table 5). table 4. results for the bds test for the estimated egarch(1,1) model for the btc/pln as an dependent variable bds test eur/pln usd/pln gbp/pln cny/pln dimension bds pr > |bds| bds pr > |bds| bds pr > |bds| bds pr > |bds| 2 0.7035 0.4817 0.7654 0.4440 0.6031 0.5465 0.8044 0.4211 3 0.3521 0.7247 0.4092 0.6824 0.2908 0.7712 0.4463 0.6554 4 0.4129 0.6797 0.5354 0.5924 0.3920 0.6951 0.5691 0.5693 5 0.5807 0.5614 0.7242 0.4690 0.5796 0.5622 0.7769 0.4372 6 0.7871 0.4312 0.9528 0.3407 0.7898 0.4297 1.0111 0.3120 7 0.9275 0.3537 1.0873 0.2769 0.9216 0.3568 1.1630 0.2448 8 0.9925 0.3210 1.1470 0.2514 0.9824 0.3259 1.2311 0.2183 9 0.9798 0.3272 1.1308 0.2582 0.9781 0.3280 1.2127 0.2252 10 0.8498 0.3954 0.9882 0.3231 0.8494 0.3956 1.0701 0.2846 note: test prints the results for the correlation between residuals up to 10 lags. dependency analysis between bitcoin and selected global currencies dynamic econometric models 16 (2016) 133–144 141 the results of bds test indicate, that there exists no serial non-linear relationship between residual values, therefore it is assumed that all linear relationships have previously been removed from the model, what points at good model fit (brabazon and o’neill, 2008). the results of the chow test show that at the 5% significance level the test has failed to reject the null hypothesis, which assumes that the structural parameters of the estimated model are stable over time which provides a good fit to the data model. table 5. results of the chow testing the structural stability of the parameters chow test variable f value pr > f reur/pln 1.20 0.2733 rusd/pln 0.37 0.5415 rgbp/pln 0.51 0.4765 rcny/pln 0.18 0.6714 the above generated results lead to the conclusion that in terms of conditional mean bitcoin to polish zloty may be consider as independent from the impact of dollar, euro, pound and yuan. in term of conditional variance bitcoin seems to be dependent from usd, eur and cny, therefore it cannot be said, that bitcoin is fully independent currencies. conclusions one of the most important question which appear regarding bitcoin is its independency. this is one of its the main advantages, which is underlined strongly by its supporters. our research has focused on the analysis of certain relationships between bitcoin expressed in polish zloty and selected global currencies. in terms of conditional mean, modelled by arma process, we can say that bitcoin is independent from the influence of all of analyzed currencies. however bitcoin’s conditional variance, modelled by garch process, is influenced by the logarithmic rate of return of eur, usd and cny to pln and is independent from gbp/pln. additionally we have indicated exponential garch, as the most suitable to model bitcoin’s conditional variance. the results show that btc/pln is not fully independent from the external influences vc, thus it can be control by third party agents. such feature can be used, for example, by speculators to achieve abnormal gains. beata szetela, grzegorz mentel, stanisław gędek dynamic econometric models 16 (2016) 133–144 142 references alberg, d., shalit, h., yosef, r. (2008), estimating stock market volatility using asymmetric garch models, applied financial economics, 18, 1201–1208, doi: http://dx.doi.org/10.1080/09603100701604225. badev, a., chen, m. (2014), bitcoin: technical background and data analysis, finance and economics discussion series 2014–104, board of governors of the federal reserve system (u.s.). bank of canada (2014), decentralized e-money (bitcoin), backgrounders, http://www.bankofcanada.ca/wp-content/uploads/2014/04/decentralize-e-money.pdf. bouoiyour, j., selmi, r. (2015), bitcoin price: is it really that new round of volatility can be on way?, mpra paper 65580, university library of munich, germany. brandvold, m., molnár, p., vagstad, k., valstad, o. (2015), price discovery on bitcoin exchanges, journal of international financial markets, institutions & money, 36, 18–35, doi: http://dx.doi.org/10.1016/j.intfin.2015.02.010. brabazon, a., o’neill, m. (2008), natural computing in computational finance, springer, doi: http://dx.doi.org/10.1007/978-3-540-77477-8. bryans, d. (2014), bitcoin and money laundering: mining for an effective solution, indiana law journal, 89, 441, https://ssrn.com/abstract=2317990. campbell, h. r. (2014), bitcoin myths and facts, https://ssrn.com/abstract=2479670. cheah, e.-t., fry, j. (2015), speculative bubbles in bitcoin markets? an empirical investigation into the fundamental value of bitcoin. economics letters, 130, 32–36, doi: http://dx.doi.org/10.1016/j.econlet.2015.02.029. chiu, j., wong, t.-n. (2015), on the essentiality of e-money, staff working papers, bank of canada, 15–43, http://www.bankofcanada.ca/wp-content/uploads/2015/11/wp201543.pdf. chu, j., nadarajah, s., chan, s. (2015), statistical analysis of the exchange rate of bitcoin, plos one, 10(7), doi: http://dx.doi.org/10.1371/journal.pone.0133678. law library of congress (2014), regulation of bitcoin in selected jurisdictions. global legal research directorate staff. cpmi (2015), digital currencies. bank for international settlements, http://www.bis.org/cpmi/publ/d137.pdf. doman, m., doman, r. (2014), dynamic linkages in the pairs (gbp/eur, usd/eur) and (gbp/usd, eur/usd): how do they change during a day?, central european journal of economic modelling and econometrics, 6(1), 33–56. dopierała, ł., borodo, a. (2014), znaczenie waluty kryptograficznej bitcoin jako środka wymiany (the importance of cryptographic currency bitcoin as a medium of exchange), contemporary economy electronic scientific journal, 5(2), 1–12. draghi, m. (2015), letter from the president of the ecb to mrs. eva kaili concerning the schemes of virtual currency (vcs), eurpean central bank, http://www.ecb.europa.eu/pub/pdf/other/150421letter_buonanno_3.en.pdf. dwyer, g. (2015), the economics of bitcoin and similar private digital currencies, journal of financial stability, 17, 81–91, doi: http://dx.doi.org/10.1016/j.jfs.2014.11.006. ecb (2012), virtual currency schemes, frankfur am main: european central bank, https://www.ecb.europa.eu/pub/pdf/other/virtualcurrencyschemes201210en.pdf. ecb (2015), virtual currency schemes – a further analysis, frankfurt am main: european central bank, https://www.ecb.europa.eu/pub/pdf/other/virtualcurrencyschemesen.pdf, doi: http://dx.doi.org/10.2866/662172. dependency analysis between bitcoin and selected global currencies dynamic econometric models 16 (2016) 133–144 143 gans, j. s., halaburda, h. (2013), some economics of private digital currency, bank of canada working paper, 38, http://www.bankofcanada.ca/wp-content/uploads/2013/11/wp2013-38.pdf. gronwald, m. (2015), the economics of bitcoins – market characteristics and price jumps, cesifo area conference on macro, money and international finance, 5121. hamilton, j. d. (1994), time series analysis, princeton, princeton university press. haubno-dyhrberg, a. (2016a), bitcoin, gold and the dollar – a garch volatility analysis, finance research letters, 16, 85–92, doi: http://dx.doi.org/10.1016/j.frl.2015.10.008. haubno-dyhrberg, a. (2016b), hedging capabilities of bitcoin. is it the virtual gold? finance research letters, 16, 139–144, doi: http://dx.doi.org/10.1016/j.frl.2015.10.025. jagwani, b. (2015), bitcoins demystified. international journal of marketing, financial services & management research, 4(4), 29–35. karame, g. o., androulaki, e., roeschlin, m., gervais, a., čapkun, s. (2015), misbehavior in bitcoin: a study of double-spending and accountability, acm transactions on information and system security, 18(1), doi: http://dx.doi.org/10.1145/2732196. liu, j., kauffman, r., ma, d. (2015), competition, cooperation, and regulation: understanding the evolution of the mobile payments technology ecosystem, journal electronic commerce research and applications, 14(5), 372–391, doi: http://dx.doi.org/10.1016/j.elerap.2015.03.003. luther, w. j., olson , j. (2015), bitcoin is memory, journal of prices & markets, 3(3), 22–33, doi: http://dx.doi.org/10.2139/ssrn.2275730. macdonell, a. (2014, 4 1), popping the bitcoin bubble: an application of log-periodic power law modeling to digital currency, university of notre dame working paper. mandelbrot, b. (1963), the variation of certain speculative prices, the journal of business, 36(4), 394–419. mandjee, t. (2014), bitcoin, its legal classification and its regulatory framework, journal of business & securities law, 15(2), http://digitalcommons.law.msu.edu/jbsl/vol15/iss2/4. marreh, s., olubusoye, o., kihoro, j. (2014), modeling volatility in the gambian exchange rates: an arma-garch approach, international journal of economics and finance, 6(10), 118–128, doi: http://dx.doi.org/10.5539/ijef.v6n10p118. matuszewska, a., witkowska, d. (2006), analiza zmian kursu euro/dolar: model var i perceptron wielowarstwowy (analysis of change rate euro/dollar: var model and multilayer perceptron), zeszyty naukowe szkoły głównej gospodarstwa wiejskiego w warszawie. ekonomika i organizacja gospodarki żywnościowej (annals of warsaw agricultural university sggw-ar. economics and organization of food economy.), 241–250. mills, t. c., markellos, r. n. (2008), the econometric modelling of financial time series (third edition), cambridge university press. montgomery, d., jennings, c., kulahci, m. (2008), introduction to time series analysis and forecasting, john wiley & sons. inc. murphy, e. v., murphy, m., seitzinger, m. (2015), bitcoin: questions, answers, and analysis of legal issues. congressional research service, https://www.fas.org/sgp/crs/misc/r43339.pdf. nakatsuma, t., tsurumi, h. (1999), bayesian estimation of arma-garch model of weekly foreign exchange rates, asia-pacific financial markets, 6(1), 71–84, doi: http://dx.doi.org/10.1023/a:1010058509622. beata szetela, grzegorz mentel, stanisław gędek dynamic econometric models 16 (2016) 133–144 144 pesaran, m. h. (2015), time series and panel data econometrics, 1st edition, oxford university press. plassaras, n. a. (2013), regulating digital currencies: bringing bitcoin within the reach of imf, chicago journal of international law, 14, 377–408, http://chicagounbound.uchicago.edu/cjil/vol14/iss1/12. pricewaterhousecoopers (2015), money is no object: understanding the evolving cryptocurrency market, http://www.pwc.com/us/en/financial-services/publications/ assets/pwc-cryptocurrency-evolution.pdf. quaicoe, m., twenefour, f., baah, e., nortey, e. (2015), modeling variations in the cedi/dollar exchange rate in ghana: an autoregressive conditional heteroscedastic (arch) models, springerplus, 4(329), doi: http://dx.doi.org/10.1186/s40064-015-1118-0. rogojanu, a., badea, l. (2015), the issue of “true” money in front of the bitcoin's offensive, theoretical and applied economics, 22(2), 77–90. szetela, b. (2016), the use of control charts in the study of bitcoin's price variability, kounis l.d. (ed.) quality control and assurance – an ancient greek term re-mastered, intech. taha, a. s. (2015), bitcoin: perils of an unregulated global p2p currency (transcript of discussion), security protocols xxiii. 9379, 294–306, doi: http://dx.doi.org/10.1007/978-3-319-26096-9_30. tasca, p., de roure, c. (2014), bitcoin and the ppp puzzle, doi: http://dx.doi.org/10.2139/ssrn.2461588. vockathaler, b. (2015), the bitcoin boom: an in depth analysis of the price of bitcoins, major research paper university of ottawa, https://www.ruor.uottawa.ca/bitstream/ 10393/32888/1/vockathaler_brian_2015_researchpaper.pdf. weber, w. e. (2016), a bitcoin standard: lessons from the gold standard, bank of canada staff working paper, 2016–14, http://www.bankofcanada.ca/wp-content/uploads/ 2016/03/swp2016-14.pdf. analiza zależności pomiędzy bitcoinem a wybranymi walutami z a r y s t r e ś c i. w badaniach staraliśmy się przeanalizować i określić zależność pomiędzy kursem bitcoina do polskiego złotego, a innymi czołowymi walutami, takimi jak dolar, euro, funt brytyjski i chiński yuan. waluty zostały wybrane w oparciu o wielkość wolumenu transakcji do bitcoina. zastosowaliśmy modele arma do modelowania warunkowej średniej oraz modele garch do analizy warunkowej wariancji. wyniki nie wykazały związku pomiędzy logarytmiczną stopą zwrotu z bitcoina do złotówki, a pozostałymi kursami walut w zakresie warunkowej średniej. natomiast zastosowanie modeli gach wykazało pewną zależność pomiędzy bitcoinem a innymi walutami, w kontekście modelowania warunkowej wariancji. s ł o w a k l u c z o w e: arma, bitcoin, garch, var, zależność. microsoft word 10_bruzda_j.doc dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 joanna bruzda nicolaus copernicus university in toruń the haar wavelet transfer function model and its applications† a b s t r a c t. in the paper the haar wavelet transfer function models are suggested as a way to parsimoniously parametrise the impulse responses and construct models with parameters providing an insight into the frequency content of the relationships under scrutiny. besides, the models enable to verify hypotheses concerning changes of the regression parameters across dyadic scales (octave frequency bands). in the paper some theoretical properties of the models are investigated and an empirical illustration is provided. in the empirical study returns on wig are modelled with the help of returns on s&p 500. interestingly, besides the insight into the frequency content of the relationship, the empirical wavelet transfer function models also provided good forecasts. k e y w o r d s: wavelet transfer function model, haar wavelet, maximal overlap discrete wavelet transform. introduction there are two approaches to examine economic relationships with wavelets. in the first case, the processes under scrutiny are decomposed according to dyadic scales and the economic relationship is investigated for the separate octave frequency bands relying on dwtor modwt-based1 wavelet and scaling coefficient or, alternatively, dwtor modwt-based details and approximations (smooths). the second approach is more prediction-oriented and consists in replacing some or all of explanatory variables with their wavelet packet coefficients. the method was introduced in nason and sapatinas (2002) and applied to such problems as wind speed prediction (hunt, nason, 2001; nason, sapatinas, 2002), data segmentation (nason et al., 2001), modelling market shares of † the author acknowledges the financial support from the polish ministry of science and higher education under the grant no. n n111 285135. 1 the abbreviations dwt and modwt refer to the discrete wavelet transform and the maximal overlap (non-decimated) discrete wavelet transform accordingly. further we use also the acronym modwpt, which stands for the maximal overlap discrete wavelet packet transform. joanna bruzda 142 products with their relative prices (hunt, 2002) and constructing marketing mix models (michis, 2006). to overcome the problem of multicollinearity of wavelet packet coefficients form different decomposition levels, the coefficient vectors that show the maximum correlation with the dependent variable are usually used (nason, sapatinas, 2002) or the packet coefficients are replaced with their principal components (hunt, nason, 2001; hunt, 2002; michis, 2006). the approach suggested here resembles that of nason and sapatinas (2002), although in the construction of our wavelet transfer function model we put more emphasis on the interpretation of parameters and make use of the notion of the wavelet best basis. in what follows some theoretical properties of the suggested model are investigated and an empirical illustration is provided. in section 1 we introduce our haar wavelet transfer function model and examine spectral characteristics of the underlying bivariate process, while in section 2 the concept is confronted with some empirical data. in the empirical study returns on wig are modelled with the help of returns on s&p 500 and the wavelet as well as conventional transfer function models are used further for forecasting purposes. the last section offers brief conclusions. 1. the haar wavelet transfer function model let us consider modelling a response variable ty in terms of the present and previous values of an explanatory variable tx . we assume for the moment that the processes have the mean values of zero. we start with a construction utilising the haar wavelet and scaling coefficients and comment further on possible generalisations including the wavelet packet transfer function modelling. using the haar scaling and wavelet filters we have2: 1 2 x x x x t kt t t ktx v w w w        , where ][log 2 nk  and the modwt scaling and wavelet coefficients are obtained via the following recursive formulas: 1 1 1 1 1 1 2 1 1, 2 2 1 1, 2 1, 1,1, 2 1, 2 0.5( ); 0.5( ); 0.5( ); 0.5( ); 0.5( ); 0.5( ).k k x x t t t t t t x x x x x x t t t t t t x x x x x x kt k t kt k tk t k t v x x w x x v v v w v v v v v w v v                                      2 the level j haar scaling and wavelet filters are obtained via the formulas:  jj ljlg 221~  ,  jj ljlh 221 ~  , 12...,,0  jl , where )( and )( are the haar scaling and wavelet functions defined as )()( )1,0 xx  1 and )()()( )1,2/1)2/1,0 xxx   11 , respectively. the haar wavelet transfer function model and its applications 143 our proposal consists in using the following model to describing ty in terms of tx : 0 1 20 , 1 1, 2 2, , k x x x x t k t t t k k t ty v w w w                  , (1) where k ,,, 10  are nonnegative integers and it is assumed that t is strictly exogenous for the regressors in (1). the model enables to possibly parsimoniously parametrise an autoregressive distributed lag (adl) model, when the regression parameter and (or) the time delay is scale-dependent. furthermore, the model provides an insight into the frequency character of the relationship between tx and ty , being at the same time a simple forecasting instrument 3. let the bivariate process ),( tt yx defined via equation (1) be covariance stationary with an absolute summable autocovariance matrix. then, its crosscovariance function has the form:    0 1 1 2 2 1 1 2 1 0 0 1 3 1 1 2 0 2 2 1 2 1 0 2 ( ) 1 2 1 2 1 4 1 2 k k k k k k k xy t t t t j j t t t t t j t j j j k k t t j t j j j ex y e x x e x x x e x x x e x x x                                                                                                    1 1 2 1 0 0 1 1 1 0 1 3 2 2 2 0 2 2 1 2 1 0 2 2 ( ) 2 ( ) ( 1) 4 ( ) ( ) 2 ( ) ( ) , k k k k k x x x j x x j j k k x k x k j j j j j j j                                                                           while the cross-spectral density function is as follows: 3 applications of other types of causal filters to examine economic dependencies across frequency bands can be found in stawicki (1993) and ashley and verbrugge (2008). joanna bruzda 144 0 1 1 2 2 1 1 2 1 2 ( ) 2 2 ( 1) 0 1 0 1 3 2 ( ) 2 ( ) 2 0 2 2 1 2 1 2 ( ) 2 ( ) 0 2 ( ) ( ) 2 2 4 2 . k k k k k k i f j i f i fk xy x j i f j i f j j j i f j i f jk k j j s f s f e e e e e e e                                                                       (2) alternatively, (2) can be expressed as:  0 12 2 20 1 1( ) ( ) ( ) ( ) ( )ki f i f i fxy k k k xs f e g f e h f e h f s f               , where )( ~ fg j and )( ~ fh j denote the transfer functions of the level j scaling and wavelet filters. to see how the frequency characteristics of the bivariate process defined via (1) look like, let us start with the simple case of the first level decomposition: t x t x tt wvy     10 ,11,10 ~~ . (3) then, the cross-spectrum reduces to:  ][][)(5,0)( )1(221)1(220 1100     fifififixxy eeeefsfs and the amplitude spectrum is: .)2sin()](2sin[)2cos( 22 )()( 5,0 1010 2 1 2 0 2 1 2 0              ffffsfa xxy   in the case 10   the gain ( ) ( ) 0 ( ) xy xy x a f g f s f   is a monotonic function with values between || 0 and || 1 . it is easy to see that for all 0 and 1 the gain does not exceed  0,50 1 0 1max{| |,| |} (| | | |)     and its values at 0, 41 and 21 equal || 0 ,   0,52 2 0 1 0 1 0 1( ) 2 sin[ 2 ( )         and || 1 , respectively. the possibility to parsimoniously parametrise the impulse response function becomes more apparent, when further decomposition levels are considered, though the form of the theoretical amplitude spectrum of (1) is then fairly complicated, even in the ‘equal lag’ case. however, the values of the gain at 0, 41 and 21 always equal || 0 ,   0,52 2 1 2 1 2 2 1( ) 2 sin[ 2 ( )         and || 1 , respectively, and to a great extent, the beta coefficients in (6.1) reveal the shape of the gain function, especially in the case of identical lags. there are basically two problems with interpreting the beta coefficients in terms of the gain. first, the haar wavelet transfer function model and its applications 145 there is a substantial leakage associated with the haar wavelet and scaling filters. furthermore, if the lag parameters differ significantly across scales, the function becomes highly variable. figure 1 presents spectral characteristics of example bivariate processes defined via: t x t x t x tt wwvy     210 ,22,11,20 ~~~ . (4) 0 0.5 1 2 3 4 g; (a) 0 0.5 -10 -5 0 ; (a) 0 0.5 -4 -2 0 2 4 ; (a) 0 0.5 1 2 3 4 g; (b) 0 0.5 -0.5 0 0.5 ; (b) 0 0.5 -4 -2 0 2 4 ; (b) 0 0.5 0 2 4 g; (c) 0 0.5 -2 0 2 ; (c) 0 0.5 -4 -2 0 2 4 ; (c) 0 0.5 0 2 4 g; (d) 0 0.5 -10 0 10 ; (d) 0 0.5 -4 -2 0 2 4 ; (d) figure 1. spectral characteristics of the bivariate process (4). figure presents gains (left-hand column), phase spectra (middle) and time delays defined as f ff   2 )()(  (right-hand column), for the following four cases: (a) ,40  ,21  ,32  ,20  ,21  ,22  (b) ,20  ,31  ,42  ,00  ,01  ,02  (c) ,20  ,21  ,22  ,20  ,01  ,02  (d) ,20  ,21  ,42  ,00  ,21  02  . to draw the phase spectrum we used the unwrap matlab function, which converts increments greater in magnitude than or equal to  to their 2 complements. one generalisation of the approach presented here utilises the nondecimated haar wavelet packet transform coefficients. in order to better reflect the frequency character of the relationships under scrutiny we suggest to replace the lower level wavelet coefficients with the appropriate modwpt coefficients from a chosen decomposition level. the modwpt-enhanced model should joanna bruzda 146 enable to choose the best filters (as for their length and the frequency of oscillations captured) to describe the short term fluctuations. the best basis for the transform and the final variables left in the model can be chosen as to optimise the empirical model in terms of its parsimony (tests for equality and significance of parameters will be helpful), some goodness of fit measures and diagnostic tests. the haar wavelet packet filters, which produce the j-th level wavelet packet decomposition, are defined via lag polynomials of order j, whose all complex roots lie on the unit circle. for example, for j = 3 the non-decimated versions of the wavelet packet coefficients tnjw ,, ~ are obtained in the following way: 2 4 2 41 1 3,0, 3,1,8 8 2 4 2 41 1 3,2, 3,3,8 8 2 4 2 41 1 3,4, 3,5,8 8 21 3,6, 8 (1 )(1 )(1 ) , (1 )(1 )(1 ) , (1 )(1 )(1 ) , (1 )(1 )(1 ) , (1 )(1 )(1 ) , (1 )(1 )(1 ) , (1 )(1 t t t t t t t t t t t t t w l l l x w l l l x w l l l x w l l l x w l l l x w l l l x w l l                                   4 2 41 3,7, 8 )(1 ) , (1 )(1 )(1 ) .t t tl x w l l l x     the so-called sequency ordering instead of the natural ordering is applied to the coefficients above, i.e. the index n is associated with the frequency interval        11 2 1 , 2 jj nn . in the case of the usual wavelet decomposition at level j = 3 we would have four coefficients of the form: tw ,0,3 ~ , tw ,1,3 ~ , tt ww ,3,3,2,3 ~~  and tttt wwww ,7,3,6,3,5,3,4,3 ~~~~  . as we can see, within the modwpt framework the hypothesis about scale dependence of the regression coefficient is just one that can be tested. building the haar wavelet packet regression model for forecasting purposes we expect that the best basis will be different from that including all the k-level modwpt coefficients or that quite a big number of them will turn out to be insignificant. however, even if no reduction is possible, we still gain an interesting interpretation of the coefficients. turning to the specification step in building the haar wavelet transfer function models several remarks are at place. first, let us note that the regressors in model (1) are generally not pairwise orthogonal. though for the haar wavelet and scaling filters both the additive decomposition and the decomposition of variance hold, i.e. for the wavelet basis, for example, we have: xkt x t x ktt wwvx ~~~ 1   , 1var( ) var( ) var( ) var( ) x x x t kt t ktx v w w      , what implies also that for all decomposition levels j it holds: cov( , ) 0x xjt jtv w   , the wavelet coefficients themselves will be generally correlated. for example, it is easy to check that: the haar wavelet transfer function model and its applications 147 1 2 1 2cov( , ) 1 8[ (1) (3)] cov( , )t t x x t tw w k k w v       , where )(xk denotes the autocovariance function of tx . the crosscovariances can be even larger. this makes the identification of the model slightly more complicated. a good starting point in the procedure of building the haar wavelet model is as in the case of an ordinary transfer function model (see box et al., 2008, chapter xii), i.e. after differencing the series to achieve stationarity they are filtered with a prewhitening arma filter for the exogenous process. then, the cross-correlation function for the filtered series is computed. the shape of this function and results of significance tests of the cross-correlation coefficients will suggest orders of lag polynomials for a transfer function model and inform whether a haar wavelet model can be successful. the wavelet model offers a specific approach to a (relatively) parsimonious parametrisation of the impulse response function that can be applied instead of or next to the standard autoregressive structures. furthermore, the identification stage will give also the minimal time delay for the component series and will suggest the number of decomposition levels for the additive decomposition. however, it seems sensible to start with specifying the same time delays for all component series and then consider also other models, especially if the maximal values of crosscorrelations for component processes or an estimate of the phase spectrum point to the need to diversify these parameters. several tentative models can then be considered in further steps of the haar model building, which are exactly the same as in the case of standard transfer function models. in particular, the diagnostic checking stage includes also the inspection of the autocorrelation function of the residuals and the cross-correlation function involving the residuals and the input variable or its prewhitened version (see for details box et al., 2008, pp. 498–501). 2. an empirical example as an empirical illustration daily logarithmic returns on wig were modelled with the help of the returns on s&p 500. in this case the level j scaling and wavelet coefficient based on the haar wavelet are associated with j2 -day returns and daily increments of j -day returns, accordingly. the estimation period was 2008.04.01–2010.04.16 and included 534 daily quotations. both the johansen and engle-granger approaches to cointegration pointed to the lack of long-term relationships between logarithms of prices, so we turned to examining the daily logarithmic returns. as the returns on s&p 500, when accounting for the garch effect, did not show any signs of autocorrelation, before examining cross-correlation patterns the two series were only corrected for volatility clustering. garch models with student’s t conditional distribution were estijoanna bruzda 148 mated and the standardised residuals were used in the first step of the procedure of building the haar wavelet transfer function models. figure 2. estimated cross-correlation function for garch-filtered returns on wig and s&p with approximate two standard error bounds computed as n2 (xf and yf denote garch-filtered s&p and wig, accordingly) figure 2 presents the estimated cross-correlation function for the garch-adjusted returns, which shows a unidirectional character of the causal relationship and provides also a slight evidence for the presence of a longer lag distribution. initially, we considered up to seven decomposition levels and then used wald tests to examine equality of parameters in the wavelet models. in each case strict exogeneity of regressors was carefully investigated in order to enable a frequency characterisation of the relationship under scrutiny. however, for some of the most parsimonious representations of the data the p-values are sometimes still only slightly above 5%. several transfer function models were finally chosen. in each case the noise process was parametrised as a moving average with the least possible number of parameters. according to diagnostic checks the conditional normal distribution of innovations was eventually assumed. also some autoregressive specifications were examined, but the autoregressive terms turned out insignificant or produced worse models and forecasts. estimation outputs for the most interesting models are presented in tables 1–2. the tables include also the summary of goodness-of-fit evaluation and some of the diagnostic checks. the frequency characteristics themselves are presented in figure 3. for the high frequency components of the processes the cross-spectral measures for garch-filtered series gave somewhat better correspondence with the estimates in tables 1–2 than that for the original series. nevertheless, we decided to present estimates of the spectral characteristics for the original data as we noted that they correspond somewhat closer to the estimates of the long-term parameters in tables 1–2. finally, table 3 includes the haar wavelet transfer function model and its applications 149 a comparison of forecast accuracy of our models. predictions were made for the next five days using forecasts of the out-of-sample values of the regressors4. table 1. estimation results of transfer function models for logarithmic returns on wig variable coefficient standard error z-statistic p-value model i equation for the conditional mean s&p 0.312 0.019 16.49 0.0000 s&p(-1) 0.235 0.020 11.65 0.0000 s&p(-2) 0.073 0.020 3.640 0.0003 ma(2) -0.091 0.049 -1.846 0.0649 ma(6) -0.124 0.044 -2.807 0.0050 equation for the conditional variance c 1.49e-06 1.25e-06 1.195 0.2320 resid(-1)^2 0.068 0.019 3.671 0.0002 garch(-1) 0.926 0.019 48.08 0.0000 adj. r2 = 30.74%; aic = -5.7620; sc = -5.6976; q = 2.79 (0.43); arch = 1.71 (0.42); jb = 2.78 (0.25) model ii equation for the conditional mean s&p 0.309 0.020 15.45 0.0000 s&p(-1) 0.229 0.020 11.52 0.0000 s&p(-2) 0.070 0.021 3.290 0.0010 s&p(-5) 0.043 0.020 2.128 0.0333 s&p(-9) 0.061 0.021 2.951 0.0032 ma(2) -0.088 0.050 -1.768 0.0770 ma(6) -0.137 0.045 -3.066 0.0022 equation for the conditional variance c 1.33e-06 1.28e-06 1.041 0.2979 resid(-1)^2 0.069 0.021 3.272 0.0011 garch(-1) 0.926 0.022 42.02 0.0000 adj. r2 = 32.25%; aic = -5.7659; sc = -5.6846; q = 3.74 (0.29); arch = 2.45 (0.29); jb = 1.99 (0.37) model iii equation for the conditional mean w1 0.077 0.026 2.914 0.0036 w2+w3+w4 0.488 0.034 14.36 0.0000 v4 0.890 0.074 11.97 0.0000 ma(2) -0.130 0.049 -2.623 0.0087 ma(6) -0.154 0.045 -3.459 0.0005 equation for the conditional variance c 1.27e-06 1.23e-06 1.029 0.3036 resid(-1)^2 0.070 0.019 3.593 0.0003 garch(-1) 0.926 0.020 45.50 0.0000 adj. r2 = 31.35%; aic = -5.7622; sc = -5.6966; q = 5.18 (0.16); arch = 2.75 (0.25); jb = 2.42 (0.30) 4 a more precise evaluation of the forecast ability of our haar wavelet transfer function models for wig (as well as some other variables) can be found in bruzda (2011). joanna bruzda 150 table 1. continued variable coefficient standard error z-statistic p-value model iv equation for the conditional mean w1 0.084 0.027 3.140 0.0017 w2+w3 0.471 0.041 11.36 0.0000 v3 0.724 0.058 12.43 0.0000 ma(2) -0.111 0.049 -2.252 0.0243 ma(6) -0.142 0.044 -3.220 0.0013 equation for the conditional variance c 1.60e-06 1.35e-06 1.182 0.2372 resid(-1)^2 0.071 0.020 3.596 0.0003 garch(-1) 0.922 0.021 43.16 0.0000 adj. r2 = 30.49%; aic = -5.7527; sc = -5.6878; q = 3.56 (0.31); arch = 2.18 (0.34); jb = 3.14 (0.21) model v equation for the conditional mean v3 0.720 0.058 12.51 0.0000 w2+w3 0.480 0.040 11.86 0.0000 p6+p7 0.161 0.039 4.176 0.0000 ma(2) -0.135 0.044 -3.084 0.0020 ma(6) -0.104 0.049 -2.127 0.0334 equation for the conditional variance c 1.53e-06 1.34e-06 1.143 0.2532 resid(-1)^2 0.072 0.019 3.711 0.0002 garch(-1) 0.922 0.021 44.10 0.0000 adj. r2 = 31.22%; aic = -5.7648; sc = -5.6999; q = 3.50 (0.32); arch = 1.89 (0.39); jb = 1.94 (0.38) note: q – ljung-box statistic for standardised residuals and 5 lags; arch – arch lm test statistic for 2 lags; jb – jarque-bera normality test; p-values in brackets; two best values of the adjusted r2 coefficient and the information criteria are in bold; v – scaling coefficients, w – wavelet coefficients, p – wavelet packet coefficients in sequency ordering neither the ordinary transfer function models nor the haar wavelet models uniformly dominated in the model building part of our analysis. however, the wavelet models produced the best forecasts of wig and have comparable properties to the former models in terms of the fit and diagnostic checking. the best wavelet forecasts were obtained with the simplest wavelet models, while the wavelet packet-enhanced specifications resulted in a lower aic criterion, while still providing good forecasts in terms of the rmse. gain x:s&p y:wig f g a in 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50 coherence x:s&p y:wig f c o h e re n ce 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50 phase spectrum x:s&p y:wig f p h a se s p e ct ru m -1,2 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 -1,2 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50 figure 3. gain, coherence and phase spectrum for wig and s&p; estimates obtained via smoothing the cross-periodogram table 2. estimation results of the unreduced wavelet transfer function model for logarithmic returns on wig variable coefficient standard error z-statistic p-value equation for the conditional mean w1 0.101 0.028 3.570 0.0004 w2 0.496 0.054 9.235 0.0000 w3 0.505 0.082 6.194 0.0000 w4 0.459 0.101 4.540 0.0000 w5 0.981 0.121 8.122 0.0000 w6 0.862 0.171 5.034 0.0000 w7 0.993 0.256 3.888 0.0001 v7 0.850 0.174 4.886 0.0000 ma(2) -0.188 0.058 -3.219 0.0013 ma(6) -0.128 0.052 -2.464 0.0137 equation for the conditional variance c 9.11e-07 1.34e-06 0.679 0.4973 resid(-1)^2 0.080 0.024 3.320 0.0009 garch(-1) 0.917 0.024 38.73 0.0000 adj. r2 = 29.81%; aic = -5.6382; sc = -5.5100; q = 3.87 (0.42); arch = 2.25 (0.33); jb = 4.05 (0.13) note: see note below table 1. table 3. evaluation of forecast accuracy i ii iii iv v models for wig rmse 14.512 15.002 14.543 14.397 14.451 mae 13.546 14.133 13.449 13.363 13.588 note: rmse – root mean squared error; mae – mean absolute error; the mean forecast errors are multiplied by 10000; two best results according to each criterion are in bold. joanna bruzda 152 conclusions one of the most interesting feature of the haar wavelet transfer function models suggested here is the interpretation of their parameters. estimates of parameters in the haar regressions correspond to the absolute values of the gain function. although they do not provide precise values of the gain, they are able to recapture the shape of this function and to characterise the frequency content of a bivariate relationship. this interesting information is provided at a relatively low computational cost, as the computational complexity of the modwt is the same as that of the well known fast fourier transform. besides, the models can easily be used to verify hypothesis about changes of regression coefficients across scales. it is also worth emphasizing that forecasting with the haar wavelet transfer function models is no more complicated than in the case of standard transfer function models. furthermore, they do not require long time series and can be easily generalised to include deterministic components and multiple exogenous variables. the empirical analysis presented in the paper confirms that the haar wavelet transfer function model can be quite successful in describing economic relationships and in forecasting economic variables. the approach provides an interesting insight into the frequency character of the relationships under scrutiny, being at the same time simple and parsimonious in parameters. the causal filters applied here can also serve the purpose of band-pass filtering exogenous variables, when the causal relationship takes place in a constrained frequency range. an example of such an empirical model with an application to forecasting can be found in bruzda (2011). references ashley, r., verbrugge, r. j. (2008), frequency dependence in regression model coefficients: an alternative approach for modeling nonlinear dynamic relationships in time series, econometric reviews, 28, 4–20. box, g. e. p., jenkins, g. m., reinsel, g. c. (2008), time series analysis. forecasting and control, 4th edition, wiley, new jersey. bruzda, j. (2011), wavelet analysis of economic processes, monograph in preparation. hunt, k., nason, g. p. (2001), wind speed modelling and short-term prediction using wavelets, wind engineering, 25, 55–61. hunt, k. (2002), wavelet methods for transfer function modelling, phd thesis, university of bristol. michis, a. a. (2006), increasing marketing accuracy. wavelet based forecasting techniques, esomar congress 2006 research paper. nason, g. p., sapatinas, t. (2002), wavelet packet transfer function modelling of nonstationary time series, statistics and computing, 12, 45–56. nason, g. p., sapatinas, t., sawczenko, a. (2001), wavelet packet modeling of infant sleep state using heart rate data, sankhyā b, 63, 199–217. percival, d. b., walden, a. t. (2000), wavelet methods for time series analysis, cambridge university press, cambridge. the haar wavelet transfer function model and its applications 153 stawicki, j. (1993), metody filtracji w modelowaniu procesów ekonomicznych, (filtration methods in modelling economic processes), wydawnictwo umk, toruń. falkowy model funkcji transferowej oparty na falce haara i jego zastosowania z a r y s t r e ś c i. w artykule proponuje się falkowy model funkcji transferowej oparty na falce haara jako metodę konstrukcji modeli funkcji transferowej pozwalającą na oszczędną parametryzację odpowiedzi impulsowych oraz dostarczającą parametrów, które mają ciekawą interpretację częstotliwościową, dając wgląd w kształt funkcji przyrostu i spektrum fazowego procesu dwuwymiarowego. ponadto pozwalają one na weryfikację hipotez dotyczących zmian współczynnika regresji w zależności od diadycznej skali czasu. w artykule analizuje się teoretyczne własności takich modeli i ilustruje w przykładzie empirycznym dotyczącym modelowania stóp zwrotu z indeksu wig w zależności od stóp zwrotu z s&p 500. interesujące jest, iż poza ciekawymi interpretacjami parametrów oszacowane falkowe modele funkcji transferowej dostarczyły także dobrych prognoz. s ł o w a k l u c z o w e: falkowy model funkcji transferowej, falka haara, niezdziesiątkowana dyskretna transformata falkowa. microsoft word 05_doman_doman_.docx dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 małgorzata doman poznań university of economics ryszard doman adam mickiewicz university in poznań the impact of the exchange rate dynamics on the dependencies in global stock market† a b s t r a c t. the paper addresses the question of how the exchange rate dynamics affects the analysis of linkages between national stock markets. we consider two ways of tackling the problem. the first one consists in denominating the analyzed quotations in the same currency. the second deals with a direct introducing the exchange rate into a model. our analysis is based on the daily return series on selected stock indices from the period 1995-2010. we model the dependence structure using dynamic copulas. this allows us to separate the dynamics of dependence from the volatility dynamics. k e y w o r d s: stock market, stock index, linkages, denomination, exchange rate, copula. introduction the knowledge about linkages between stock markets is of importance in risk management and building investment strategies. moreover, it is crucial for understanding the nature of global financial market. it is thus quite natural that there exist many papers dealing with this problem. most of them belong to the contagion literature. the most popular approach here is to denominate the indices (or other stock market quotations) in local currencies (eun and shin, 1989; koutmos, 1992; theodossiou and lee, 1993; wong et al. 2004). the next popular choice is denomination in the us dollar (e.g. karolyi and stulz, 1996; rodriguez, 2007)). there exist analyses performed both in a local currency and the us dollar (e.g. lee et al., 2001). chen and poon (2007) use local currency for indices in the case of developed markets and for emerging market they use us dollar denominated indices. veiga and mcaleer (2004) remarked that the † this work was financed from the polish science budget resources in the years 2010-2013 as the research project n n111 035139. małgorzata doman, ryszard doman 74 use of the us dollar as a common currency is a complicating factor. this is because in such situation the us market is always included in the empirical analysis. changes in the us dollar are largely influenced by changes in us fundamentals, which also drive financial returns. thus, it is likely that some of the co-movements observed among returns in different markets expressed in a common currency are caused by changes in the fundamentals driving the us dollar exchange rate. however, the findings by veiga and mcaleer (2004) based on quite extensive analysis of the sensitivity of spillover effects on denomination show that the denomination has no significant impact on the results. in the paper, we ask how introducing the exchange rate dynamics influences the dynamics of linkages between stock indices. we consider dependencies between the elements of each pair of indices from the triple: the s&p500, the dax and the wig20 (the main index of the warsaw stock exchange). in addition, we investigate the linkages between the dax and the nikkei225. the analysis of linkages is performed by means of a dcc-copula model. we estimate dynamic copula correlations between the daily returns on the indices denominated in local currencies and in chosen alternative currencies. the aim of the presented investigation is to analyze the sensitivity of the dynamic copula correlation estimates to the denomination of the indices in alternative currencies. in the case of the s&p500 and the dax, the considered currencies are the us dollar and the euro. the analysis for the s&p500 (or the dax) and the wig20 includes denomination in the us dollar, the euro and the polish zloty. for the pair dax and nikkei225, the denominations in the us dollar, the japanese yen and the euro are included. moreover, for each of the considered pairs of the indices we calculate the dynamic copula correlations based on a three-dimensional dcc-copula model estimated jointly for the indices denominated in local currencies and the corresponding exchange rate (usd/eur for sp500-dax, usd/pln for sp500-wig20, eur/pln for dax-wig20, and eur/jpy for dax-nikkei225). 1. dcc-copula models modeling the dependencies between financial returns is a difficult task because of special properties of these series. typical return series usually exhibit conditional heteroskedasticity, different types of asymmetries and structural breaks which strongly influence estimation results for models of the dependence structure. moreover, the dynamics of dependencies significantly changes in time. for example, it is well documented in many studies that dependence between returns on different assets is usually stronger in bear markets than in bull markets (ang and bekaert, 2002; ang and chen, 2002; patton, 2004). this example of asymmetric dependence in financial markets is of great importance for portfolio choice and risk management. the main problem connected with this phenomenon is, however, that from the theoretical point of view the mentioned the impact of the exchange rate dynamics on the dependencies… 75 asymmetry cannot be produced by a statistical model for the returns that assumes an elliptical multivariate conditional distribution, and thus applying the linear correlation is not justified. an alternative concept that allows for modeling the dependence in a general situation is copula. roughly speaking, a d-dimensional copula is a mapping ]1 ,0[]1 ,0[: dc from the unit hypercube into the unit interval which is a distribution function with standard uniform marginal distributions. assume that ),,( 1 dxxx  is a d-dimensional random vector with joint distribution f and marginal distributions if , di ,,1  . then, by a theorem by sklar (1959), f can be written as: ))(,),((),,( 111 ddd xfxfcxxf   . (1) the function c is unique if if are continuous. otherwise, c is uniquely given by: ))(,),((),,( )1(1 )1( 11 ddd ufuffuuc   , (2) for ]1 ,0[iu , where })( :inf{)( 1 iii uxfxuf   . in that case, c is called the copula of f or of x. since the marginals and the dependence structure can be separated, it makes sense to interpret c as the dependence structure of the vector x. we refer to patton (2009) and references therein for an overview of financial time series applications of copulas. there one can also find more information about advantages and limitations of copula-based modeling. the simplest copula is defined by dd uuuuc    11 ),,( , and it corresponds to independence of marginal distributions. the next two important examples are ),,min(),,( 11 dd uuuuc    , and, in the two-dimensional case, )0 ,1max(),(  jiji uuuuc . the first corresponds to comonotonicity or perfect dependence (one variable can be transformed almost surely into another by means of an increasing map), and the second, to countermonotonicity or perfect negative dependence of the variables ix and jx (one variable can be transformed almost surely into another by means of a decreasing map). in the empirical part of this paper we will use the student t copula. it is defined as follows: ))(,),((),,( 11 1 ,1, d d d st ututtuuc    rr , (3) where r,t denotes the d-dimensional student’s t distribution with  degrees of freedom and correlation matrix r, and t stands for 1-dimensional student’s t distribution with  degrees of freedom. in the bivariate case we will use the notation tc  , where  stands for correlation coefficient. małgorzata doman, ryszard doman 76 the density associated to an absolutely continuous copula c is a function c defined by: d d d d uu uuc uuc       1 1 1 ),,( ),,( . (4) for an absolutely continuous random vector, the copula density c is related to its joint density function h by the following canonical representation: )()())(,),((),( 11111 ddddd xfxfxfxfcxxf   , (5) where dff ,,1  are the marginal distributions, and dff ,,1  are the marginal density functions. in the case of non-elliptical distributions, measures of dependence that are more appropriate than the linear correlation coefficient are provided by two important copula-based tools known as kendall’s tau and spearman’s rho (embrechts et al., 2002). since the dynamics of kendall’s tau can be easily derived for the results presented in this paper, we recall suitable definitions. if ),( yx is a random vector and )~,~( yx is an independent copy of ),( yx then kendall’s tau for ),( yx is defined as: }.0)~)(~{(}0)~)(~{(),(  yyxxpyyxxpyx (6) thus kendall’s tau for ),( yx is the probability of concordance minus the probability of discordance. if ),( yx is a vector of continuous random variables with copula c, then: 1),(d),(4),( 2]1,0[   vucvucyx . (7) for the student t copula tc  , , kendall’s tau equals )arcsin( 2   . a very important concept connected with copula, relevant to dependence in extreme values, is tail dependence (nelsen, 2006). if x and y are random variables with distribution functions f and g then the coefficient of upper tail dependence is defined as follows: ))(|)((lim 11 1 qfxqgyp qu     , (8) provided a limit ]1,0[u exists. analogously, the coefficient of lower tail dependence is defined as: ))(|)((lim 11 0 qfxqgyp ql     , (9) provided that a limit ]1,0[l exists. if ]1,0(u ( ]1,0(l ), then x and y are said to exhibit upper (lower) tail dependence. upper (lower) tail dependence the impact of the exchange rate dynamics on the dependencies… 77 quantifies the likelihood to observe a large (low) value of y given a large (low) value of x. the coefficients of tail dependence depend only on the copula c of x and y: q qqc ql ),( lim 0  , q qqc qu ),(ˆ lim 0  (10) where )1,1(1),(ˆ vucvuvuc  . for the student t copula stc  , , the coefficients of upper and lower dependence are both equal to  )1/()1)(1(2 1  t (see mcneil et al., 2005). introduced by patton (2004), the notion of conditional copula allows to apply copulas to modeling the joint distribution of tr conditional on information set 1t , where ),,( ,,1  tdtt rr r is a d-dimensional vector of financial returns. in this paper we consider the following general conditional copula model: )|( ~|,),|( ~| 1,1,1,11,1   ttdttdtttt frfr  , (11) )|(~| 11   tttt fr , (12) )|)|(,),|(()|( 11,,1,1,11   tttdtdttttttt rfrfcrf  , (13) where the set t includes the up to time t information on the returns on both considered financial assets, and tc is the conditional copula linking the marginal conditional distributions. further, we assume that: ttt yμr  , )|( 1 ttt e rμ , (14) tititiy ,,,  , )|var( 1, 2 ,  ttiti r , (15) ),,1 ,0(_ ~, iiti tskewiid  , (16) where ),,1 ,0(_ tskew denotes the standardized skewed student t distribution with 2 degrees of freedom, and skewness coefficient 0 (lambert and laurent 2001). to the marginal return series tir , , d, i ,1  , we fit armagarch models with skewed student’s t distributions for the 1-dimensional innovations. when modeling the joint conditional distribution, the evolution of the conditional copula tc has to be specified. usually (patton, 2004, 2006), the functional form of the conditional copula is fixed, but its parameters evolve through time. in this paper, we follow that approach and apply the dcc model proposed by engle (2002), extended to student’s t copulas. thus in our dcc-t-copula model we assume that the conditional copula tc is a student t copula t t c r, such that:       2121 diagdiag  tttt qqqr , (17) małgorzata doman, ryszard doman 78 111 ~~ )1(   tttt quuqq  , (18) where 0 , 0 , 1  , )(~ , 1 , titi utu   , )( ,,, tititi rfu  , d, i ,1  , and q is the unconditional covariance matrix of tu~ . 2. the data in the paper we present results of analysis concerning the dependencies between the daily returns for four pairs of indices: s&p500-dax, s&p500wig20, dax-wig20, and dax-nikkei225. we have chosen three indices representing stock markets of main economies from different parts of the world. the reason for the choice of the wig20 index is connected with the fact that the polish stock market is significantly influenced by the us financial markets and, on the other hand, there exist strong linkages between the polish and german economies. as it was mentioned in introduction, the very common approach in stock market linkages analysis is to investigate the indices of developed markets in local currency, and those from emerging markets – denominated in an alternative currency (mostly in the us dollar). we denominate the considered indices in their local currencies, in the usd, and in the euro. moreover, the dependencies involving the wig20 are analyzed for the indices denominated in the polish zloty, and those involving the nikkei225 – in the japanese yen. thus our dataset contains the quotations of the considered indices and the exchange rates eur/usd, usd/pln, eur/pln usd/jpy and eur/jpy. the quotation series were obtained from the service stooq. the period under scrutiny is from january 3, 1995 to december 11, 2009. table 1. descriptive statistics of the analyzed return series index mean maximum minimum stand. dev. skewness kurtosis s&p500 0.0238 10.957 -9.4695 1.289 -0.1783 10.9195 s&p500 in eur 0.0190 9.5946 -8.7688 1.4663 -0.2152 7.0311 s&p500 in pln 0.0282 10.054 -9.1886 1.4320 -0.0718 8.1544 dax 0.0276 10.797 -9.791 1.5877 -0.0635 10.9230 dax in usd 0.0325 13.5020 -9,4710 1.6741 0.0517 8.5868 dax in jpy dax in pln wig20 0.0302 13.709 -14.161 1.9544 -0.1548 6.7066 wig20 in usd 0.0262 14.995 -19.463 2.2717 -0.2531 8.2124 wig20 in eur 0.0212 16.368 -17.481 2.3220 -0.1743 8.6857 since the patterns of non-trading days in national stock markets differ, for the purpose of modeling dependencies the dates of observations for each pair of indices were checked and observations not corresponding to ones in the other index quotation series were removed. the time series under scrutiny are percentage logarithmic daily returns calculated by the formula: the impact of the exchange rate dynamics on the dependencies… 79 )ln(ln 100 ttt ppr  , (19) where tp denotes the closing index value on day t . the descriptive statistics of the analyzed return series are presented in table 1. in tables 2–5 we show in-sample estimates of the unconditional correlations. table 2. s&p500 and dax. estimates of the unconditional correlation of the returns s&p500 s&p500 in eur dax 0.5590 0.5309 dax in usd 0.5227 ---- table 3. s&p500 and wig20. estimates of the unconditional correlation of the returns s&p500 s&p500 in eur s&p500 in pln wig20 0.2682 ----0.1212 wig20 in usd 0.2824 ---- wig20 in eur ----0.2749 table 4. s&p500 and wig20. estimates of the unconditional correlation of the returns dax dax in usd dax in pln wig20 0.4494 ----0.3598 wig20 in usd ----0.5042 wig20 in eur 0.4887 table 5. dax and nikkei225. estimates of the unconditional correlation of the returns nikkei nikkei in usd nikkei in eur dax 0.2993 ----0.2225 dax in usd ----0.2616 dax in jpy 0,2853 3. empirical analysis of the stock market linkages the course of presented analysis is as follows. we investigate the dependencies between the returns for four pairs of indices: s&p500-dax, s&p500wig20, dax-wig20, dax-nikkei225. in each case we estimate the dynamic copula correlations by means of the dcc-t-copula model described in section 3. each pair of indices is considered in a local currency, in the us dollar, and in the euro. for s&p500-wig20 and dax-wig20 we additionally take into account denomination in the polish zloty, and for dax-nikkei225 – in japanese yen. moreover, we estimate jointly the dynamic copula correlations for triples of returns: s&p500-dax-eur/usd, s&p500-wig20-usd/pln, daxwig20-eur/pln and dax-nikkei225-eur/jpy. the advantage of copula models we apply here is that they allow to separate the dependence dynamics from the volatility dynamics. the feedback between these two features causes many problems in traditional analyses based on multimałgorzata doman, ryszard doman 80 variate volatility models. in our approach the volatility dynamics is captured by means of garch models and then the dependence structure is modeled. it means that the dcc-t-copula models are estimated using a two-step maximum likelihood approach. the first step includes fitting a garch model to each return series (laurent, 2009). the types of fitted models differ depending on currency used to denominate an index (table 4–5). next, the garch standardized residuals are transformed by means of their theoretical cumulative distribution functions to obtain the series of data uniformly distributed on [0,1]. in the second step the dcc-t-copula models are fitted to the transformed series. thus, we follow the method of inference functions for margins (joe and xu, 1996). the first observation coming from tables 6–7 is that the conditional mean an volatility dynamics is sensitive to denomination. the return series under scrutiny are long and include some crisis periods so the fitted garch models are mostly asymmetric and with a skewed student t as an error distribution. the dcc-t-copula parameter estimates are presented in tables 8–11. the results indicate that the dynamics of dependencies shows a high level of persistence in each considered case. table 6. s&p500 and nikkei225. types of fitted arma-garch models return series s&p500 s&p500 in eur s&p500 in pln nikkei225 nikkei225 in usd nikkei225 in eur arma (1,1) (0,2) (1,1) (0,0) (0,1) (0,0) garch gjrgarch(1,2) gjrgarch(1,2) garch(1,1) gjr(1,2) gjr(1,2) figarch(1,1) error distribution skewed student skewed student skewed student skewed student skewed student skewed student table 7. dax and wig20. types of fitted arma-garch models return series dax dax in usd wig20 wig20 in usd wig20 in eur arma (2,2) (1,0) (0,1) (2,0) (0,0) garch fiaparch(1,1) garch(1,1) fiaparch(1,1) gjr-garch(1,2) figarch(1,1) error distribution skewed student skewed student student student student table 8. s&p500-dax. parameter estimates for the fitted dcc-t-copula model (standard errors in parentheses) s&p500 and dax parameter in local currencies in eur in usd s&p500-dax -usd/eur  0.0146 (0.003 0.0204 (0.006) 0.0185 (0.003) 0.0175 (0.003)  0.9841 (0.004) 0.9750 (0.008) 0.9805 (0.004) 0.9789 (0.004)  14.4538 (3.819) 13.1894 (3.582) 15.5688 (4.665) 12.5504 (1.843) the impact of the exchange rate dynamics on the dependencies… 81 table 9. s&p500-wig20. parameter estimates for the fitted dcc-t-copula model (standard errors in parentheses) s&p500 and wig20 parameter in local currencies in eur in usd in pln s&p500-wig20 -usd/pln  0.0075 (0.008) 0.0109 (0.004) 0.0102 (0.004) 0.0072 (0.002) 0.0125 (0.003)  0.9900 (0.016) 0.9836 (0.007) 0.9878 (0.005) 0.9871 (0.005) 0.9821 (0.005)  14.6137 (3.710) 16.1879 (4.616) 14.9618 (3.911) 11.3950 (2.286) 16.6111 (3.004) table 10. dax-wig20. parameter estimates for the fitted dcc-t-copula model (standard errors in parentheses) dax and wig20 parameter in local currencies in eur in usd in pln dax-wig20 -eur/pln  0.0125 (0.005) 0.0181 (0.008) 0.0141 (0.003) 0.0109 (0.003) 0.0110 (0.002)  0.9875 (0.007) 0.9793 (0.011) 0.9852 (0.004) 0.9865 (0.003) 0.9863 (0.004)  12.017 (2.468) 12.154 (2.555) 14.516 (3.640) 14.372 (3.435) 17.213 (3.199) table 11. dax-nikkei225. parameter estimates for the fitted dcc-t-copula model (standard errors in parentheses) dax and nikkei225 parameter in local currencies in eur in usd in jpy dax-nikkei -eur/jpy  0.0150 (0.018) 0.0091 (0.004) 0.0080 (0.003) 0.0066 (0.003) 0.0114 (0.002)  0.9339 (0.142) 0.9889 (0.006) 0.9900 (0.004) 0.9864 (0.006) 0.9861 (0.003)  15.029 (4.870) 21.162 (7.524) 23.087 (7.556) 17.4376 (6.412) 13.248 (2.118) figure 1 shows a comparison of the dynamic copula correlations for the pair s&p500-dax obtained in all the considered cases. the dynamics of the correlations is quite strong. the strongest dependencies are observed in the years 2001-2004 and 2008-2009. the values of the correlations calculated for the indices denominated in local currencies, in the euro, and modeled jointly with the exchange rate eur/usd are quite close each other. only in the case of denomination in the us dollar the correlation estimates are clearly lower. the mean levels of the estimated dynamic copula correlations (table 12) are significantly different and the highest mean is obtained in the case of the dependencies between the indices and the exchange rate eur/pln modeled jointly. the null hypothesis about equality of the means was tested using the model confidence małgorzata doman, ryszard doman 82 set (mcs) procedure (hansen et al., 2003, 2011; hansen and lunde, 2007) applied to the set of the dynamic copula correlations series. figure 1. s&p500 and dax. dynamic copula correlations from dcc-t-copula model figure 2. s&p500 and wig20. dynamic copula correlations from dcc-t-copula model the estimates of dynamic copula correlations obtained for the pair s&p500wig20 are much lower but show similar pattern as in the previous case – the dynamics of the conditional copula correlations is strong but it does not depend significantly on the choice of currency. the only exception concerns the clearly weaker dependencies in the case of the indices denominated in the polish zloty. the difference is more visible after poland joining the eu. the testing procedure, the same as in the previous considered case, indicate that the mean levels ‐0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1 201 401 601 801 1001 1201 1401 1601 1801 2001 2201 2401 2601 2801 3001 3201 3401 3601 in local currencies in eur in usd jointly with eurusd ‐0,2 ‐0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 1 201 401 601 801 1001 1201 1401 1601 1801 2001 2201 2401 2601 2801 3001 3201 3401 3601 in local currencies in usd in eur in pln jointly with usdpln the impact of the exchange rate dynamics on the dependencies… 83 of the estimated dynamic copula correlations (table 12) are significantly different. table 12. means of the dynamic copula correlation estimates for the pairs s&p500dax and s&p500-wig20 s&p500 and dax wig20 in local currencies 0.4975 0.2434 in usd 0.4339 0.2345 in eur 0.4903 0.2711 modeled jointly with the exchange rate 0.4992 0.2455 in pln 0.1325 table 13. means of the dynamic copula correlation estimates for the pairs dax-wig20 and dax-nikkei225 dax and wig20 nikkei225 in local currencies 0.4185 0.2957 in usd 0.4432 0.2913 in eur 0.4436 0.2601 modeled jointly with the exchange rate 0.4195 0.3015 in pln 0.3386 ----- in jpy -----0.2571 figure 3. dax and wig20. dynamic copula correlations from dcc-t-copula models the estimates of dynamic copula correlations obtained for the pair daxwig20 are presented in figure 3. in general, the differences between the estimates are not very high and the dynamics in all cases is similar. however, once again, we can observe the impact of poland’s eu joining on the conditional correlations calculated for the indices denominated in the polish zloty. mean levels of the conditional correlation estimates are presented in table 13. the testing mcs procedure indicates that the mean levels of the conditional correla‐0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 201 401 601 801 1001 1201 1401 1601 1801 2001 2201 2401 2601 2801 3001 3201 3401 3601 in usd in pln jointly with eurpln in eur in local currencies małgorzata doman, ryszard doman 84 tions are statistically undistinguishable in the case of denomination in the eur and in the usd and this mean level is the highest one. figure 4. dax and nikkei225. dynamic copula correlations from dcc-t-copula models the dynamics of the conditional correlations between the returns on the dax and the nikkei 225 shows slightly different patterns (figure 4). denominating the indices in the yen results in the lowest values of the correlation estimates. the most interesting thing one can see in figure 4 is that from the beginning of the financial crisis 2007-2009 the plots of the conditional correlations estimates start to disperse. starting from that point, the dependencies measured for the indices denominated in the eur are the weakest. conclusions the aim of the presented research was to examine how the dynamics of linkages between stock markets changes when the exchange rate dynamics is introduced into the model. we considered dependencies between the s&p500 index and two european indices – the dax and the wig20, and for the pairs dax-wig20 and dax-nikkei225. to analyze the stock indices linkages we used dcc-t-copula models. the advantage of the applied approach is that it allows to separate the dynamics of linkages from the volatility dynamics. the presented results are slightly ambiguous but generally show that the impact of denomination or introducing the exchange rate directly into the model for dependencies is rather weak. however, as it was expected, some significant changes in the dynamics of the conditional dependence are observed for indices denominated in a common currency when events strongly influencing the considered exchange rates dynamics are present. ‐0,2 ‐0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 1 201 401 601 801 1001 1201 1401 1601 1801 2001 2201 2401 2601 2801 3001 3201 3401 in usd in eur in jpy jointly with eurjpy in local currencies the impact of the exchange rate dynamics on the dependencies… 85 the question about the proper way of analyzing the dependencies remains still open. the problem seems to be less important in the case of indices denominated in major currencies, i.e. the usd or the eur, and much more significant in the case of indices expressed in other (local) currencies. references ang, a., bekaert, g. (2002), international asset allocation with regime shifts, review of financial studies, 15, 1137–1187. ang, a., chen, j. (2002), asymmetric correlations of equity portfolios, journal of financial economics, 63, 443–494. chen, s., poon, s.-h. (2007), modelling international stock market contagion using copula and risk appetite, mbs working paper series, ssrn: http://ssrn.com/abstract=1024288 engle, r. f. (2002), dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models, journal of business and economic statistics, 20, 339–350. eun, c. s., shim, s. (1989), international transmission of stock market movements, journal of financial and quantitative analysis, 24, 241–256. embrechts, p., mcneil, a., straumann, d. (2002), correlation and dependence in risk management: properties and pitfalls. in: dempster m. (ed.), risk management: value at risk and beyond, cambridge university press, cambridge, 176–223. hansen, p. r., lunde, a., nason, j. m. (2003), choosing the best volatility models: the model confidence set approach, oxford bulletin of economics and statistics, 65, 839–861. hansen, p. r., lunde, a. (2010), mulcom 2.00, an oxtm software package for multiple comparisons, http://mit.econ.au.dk/vip_htm/alunde/mulcom/mulcom.htm hansen, p. r., lunde, a., nason, j. m. (2011), the model confidence set, econometrica, 79, 453–497. joe, h., xu, j. j. (1996), the estimation method of inference functions for margins for multivariate models, technical report no. 166, department of statistics, university of british columbia. karolyi, g. a., stulz, r. m. (1996), why do markets move together? an investigation of u.s.– japan stock market comovements, journal of finance, 51, 951–986. koutmos, g. (1992), asymmetric volatility and risk return tradeoff in foreign stock markets, journal of multinational financial management, 2, 27–43. laurent, s. (2009), estimating and forecasting arch models using g@rchtm6, timberlake consultants ltd, london. lambert, p., laurent, s. (2001), modelling financial time series using garch-type models with a skewed student distribution for the innovations, institut de statistique, université catholique de louvain, discussion paper 0125. lee, b., rui, o. m., wang, s. s. (2001), information transmission between nasdaq and asian second board market, journal of banking & finance, 28, 1637–1670. mcneil, a. j., frey, a., embrechts, p. (2005), quantitative risk management, princeton university press, princeton. nelsen, r. b. (2006), an introduction to copulas, springer science+business media, inc., new york. patton, a. j. (2004), on the out-of-sample importance of skewness and asymmetric dependence for asset allocation, journal of financial econometrics, 2, 130–168. patton, a. j. (2006), modelling asymmetric exchange rate dependence, international economic review, 47, 527–556. patton, a. j. (2009), copula-based models for financial time series. in: andersen t. g., davies r. a., kreiss j. p., mikosch t. (eds.), handbook of financial time series, springer, berlin, 767–785. małgorzata doman, ryszard doman 86 rodriguez, j. c. (2007), measuring financial contagion: a copula approach, journal of empirical finance, 14, 401–423. sklar, a. (1959), fonctions de répartition à n dimensions et leurs marges, publications de l’institut statistique de l’université de paris, 8, 229–231. theodossiou, p., lee, u. (1993), mean and volatility spillovers across major national stock markets: further empirical evidence, journal of financial research, 16, 327–350. wong, w.-k, penm, j., terrel, r. d., lim, k. (2004), the relationship between stock markets of major developed countries and asian emerging markets, journal of applied mathematics and decision sciences, 8, 201–218. veiga, b., mcaleer, m. (2004), testing the sensitivity of spillover effects across financial markets. in: pahl-wostl, c., schmidt, s., rizzoli, a. e., jakeman, a. j. (eds.), complexity and integrated resources management: transactions of the international conference on environmental modelling and software, osnabrueck, germany, iemss, manno, switzerland, 1523–1529. wpływ dynamiki kursów walutowych na zależności na globalnym rynku akcji z a r y s t r e ś c i. analiza powiązań pomiędzy narodowymi rynkami akcji jest zwykle oparta na modelach opisujących zależności pomiędzy stopami zwrotu z akcji lub indeksów. przy tym w niektórych badaniach wykorzystuje się notowania w walutach lokalnych, a w innych – notowania denominowane w tej samej walucie (zwykle w dolarze amerykańskim). w artykule zajmujemy się badaniem, jak uwzględnienie dynamiki kursów walutowych w modelu powiązań dla giełdowych stóp zwrotu wpływa na opis zależności. stosujemy i porównujemy dwa podejścia. pierwsze polega na denominowaniu rozważanych notowań w tej samej walucie, a drugie sprowadza się do bezpośredniego wprowadzenia kursu walutowego do modelu struktury zależności. prezentowana analiza jest oparta na szeregach stóp zwrotu z okresu 1995-2010. w celu opisu struktury zależności stosujemy dynamiczne modele kopuli. podejście takie pozwala nam na oddzielenie dynamiki zależności od dynamiki zmienności notowań. s ł o w a k l u c z o w e: rynek akcji, indeks giełdowy, powiązania, denominacja, kursy walutowe, kopula. © 2017 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2017.001 vol. 17 (2017) 5−18 submitted december 28, 2016 issn (online) 2450-7067 accepted october 11, 2017 issn (print) 1234-3862 tomasz schabek * , henrique castro “sell not only in may” seasonal effects on stock markets a b s t r a c t. described in literature stock market anomaly still remains unexplained. in long time series regressions and wide geographical spread research, “halloween effect” is significant on 19 amongst 73 markets, but also on 11 amongst 23 with long time series data. the data shows that abnormal returns could be realized also in strategies starting in october, november and december. we conclude that even with control of weather (sun hours), behavioral (sentiment index, number of ipos) and macroeconomic (industrial production) factors, the effect persists. k e y w o r d s: seasonal anomaly; behavioral factor; halloween indicator; january effect; sell in may. j e l classification: g15; q47; g10; g14. introduction seasonal anomalies are widely discussed in financial literature because of their unknown nature and relative simplicity in application as market strategies. “halloween effect” as one of them is subject of many articles. it was analyzed and tested on broad range of markets in bauman and jacobsen (2002). briefly speaking we can describe it as anomaly that is derived from old market saying: “sell in may and go away”. halloween indicator is another name to similar strategy that is to have long position on the market * correspondence to: tomasz schabek, university of lodz, faculty of economic and sociology, 3/5 polskiej organizacji wojskowej street, 90-255 lodz, poland, e-mail: tschabek@uni.lodz.pl; henrique castro, school of economics, business and accounting of the university of são paulo, são paulo, brazil, e-mail: hcastro@usp.br. https://orcid.org/0000-0002-4362-8864 tomasz schabek, henrique castro dynamic econometric models 17 (2017) 5–18 6 from october 31 till april 30 each year – described in o'higgins and downes (1990). although it has its beginnings only in market saying, numerous of studies has proven that it is still profitable and valid investment strategy. in our paper we focus on finding possible explanations of the anomaly. we use set of the variables representing fundamental and behavioral factors that could influence market participant’s behavior. we verify hypothesis about the existence of similar effects for all strategies starting in winter and finishing in summer months. weather factors has been broadly discussed as the possible reason of the anomaly, we address also this issue. our research brings an important contribution to existing literature because we test halloween effect in the context of behavioral variables. it allows to directly check if changes in investor sentiment are responsible for seasonal fluctuations. we used sentiment measure proposed by baker and wurgler (2007) which is solely based on market factors. we also analyzed the role of consumer confidence and industrial production. secondly we directly test influence of daylight in seasonal anomalies based strategies to confirm or reject the hypothesis of its influence on halloween effect. thirdly we confirm the existence of similar seasonal effects not only in end of october – end of april period but also from end of september (november) till end of march (may). 1. literature review numerous of studies: andrade, chhaochharia and fuerst (2013), jacobsen and others (2005), jacobsen and visaltanachoti (2009), zarour (2007), lean (2011) have confirmed the existence and significance of sell in may or halloween effects. researchers tried to explain the anomaly in many different ways. bouman and jacobsen (2002) in their research control for risk, january effect, changes in interest rates and volume and stated that even though the anomaly persists, they suggested summer holidays as the explanation of the anomaly (liquidity needs or changed risk aversion during vacations). summer holiday as a reason of the anomaly is supported by hong and yu (2009) which concluded that trading activity drops in summer months, also recent findings of kaustia and rantapuska (2016) seems to be in favor of holiday’s hypothesis. kamstra, kramer and levi (2003) postulated seasonal affective disorder (sad) as the main reason of halloween anomaly, their research however was heavily criticized in kelly, meschke (2010), jacobsen and marquering (2008, 2009). maberly and pierce (2004) questioned the existence of halloween effect as caused by outliers in october 1987 and august 1998 but haggard and witte (2010) acquire the results using robust “sell not only in may”. seasonal effect on stock markets dynamic econometric models 17 (2017) 7–18 7 estimation technique and confirms significant halloween anomaly. jacobsen and others (2005) checked if halloween effect is dependent on size, book to market or divided yield and conclude that it is not the case. gerlach (2007) found that returns from october to december are higher because of macroeconomic announcements but gugten (2010) reports that even with control of this factor halloween anomaly still exists. lucey and zhao (2007) postulated that january effect is the reason, but those results were questioned by haggard and witte (2010), because of short sub periods used in lucey and zhao (2007) tests. doeswijk (2008) hypothesizes that investors behavior is driven by optimism cycles. he stated that size and value factor do not influence “sell in may” effect, but behavioral factor – initial returns from ipos has explanatory power. in study of jacobsen and zhang (2012a) they use sample of 108 markets and very long time series (21 of those markets have more than 1000 monthly observations). they provide thorough and detailed analysis of persistence and significance of the halloween anomaly. their findings confirm significant and strengthening halloween effect. we rely on their studies therefore we will not replicate them in detailed way as jacobsen and zhang (2012a) to reaffirm existence of halloween effect, instead we will focus on verification of possible explanations of the effect. in recent paper jacobsen and zhang (2012) analyze 317 years of united kingdom stock prices data. such long time data series allowed testing halloween effect in 100and 50years subsample intervals. they report significant 0,56% higher winter returns than summer returns. may effect is present in all of their 100and 50years periods. 2. data in our study we divide used data into the following groups: a) stock markets monthly rates of return. all rates of return are calculated using last stock market session of the month closing price (in local currency). closing prices are acquired from routers database. the longest period covered is: jan 1964 to jun 2012 1 . b) us market sentiment is measured by baker and wurgler (2007) monthly sentiment index and acquired directly from wurgler web-page 2 . it covers the period of july 1965 to dec 2010. 1 different markets are covered with different periods as the effect of the fact that some analyzed market have shorter time series of required data. 2 http://www.stern.nyu.edu/~jwurgler/ tomasz schabek, henrique castro dynamic econometric models 17 (2017) 5–18 8 c) number of initial public offering (ipo) is calculated based on data obtained from reuters one database. the longest period starts from jan 1970 and finishes in jun 2012. d) industrial production – gathered from world bank global economic monitor (gem) database and oecd. covers period between jan 1991 to jun 2012 depending on country. e) consumer confidence indicator (cci) – acquired from oecd database. the longest period covered is: jan 1964 to jun 2012. f) sun hours – data comes from world meteorological organization, hong kong observatory and national meteorological institutes of analyzed countries. we analyze 73 indexes that cover 68 3 countries and 2 non-country indexes (msci world and crb commodity index). the selection of countries was driven by maximum possible geographical coverage and data availability. 3. methodology first objective of this study is to find presence of “sell in may and go away” (sim) and halloween (hal) effects in a group of emerging and developed capital markets. for this purpose we used methodology similar to bouman and jacobsen (2002). our second motive is to check hypothesis of significant role of daylight, u.s. market sentiment, number of ipos, and cci as possible causes of this effect. we also examine investing strategies starting in months other than may. our tests imitate strategy that could be utilized by investors on stock markets. this approach is very similar to buy-and-hold except that it keeps long position for six months of each year and during the rest of the year it takes form of passive “out of the market” strategy. we are examining if the starting month of the strategy plays any role and creates similar to sim anomalies and if yes, could it be explained by set of nonfundamental variables. this is the third motive of our study – to check if behavioral variables as postulated in doeswijk (2008) are responsible for sim/hal and similar effects. to identify sim and other month’s seasonal effects we run following time series regressions (1) or (2) for each market: , (1) if strategy include january, or: , (2) 3 us and chinese markets are represented by accordingly three and two indexes. “sell not only in may”. seasonal effect on stock markets dynamic econometric models 17 (2017) 7–18 9 if strategy does not include january. where: is the monthly rate of return calculated for i-th market index (from 1 to 73), is the dummy variable equal 1 when month t falling into the period of 6-months strategy initiated in the last day of month mon (except jan) and finishing in last day of month mon + 6 (except jan) and 0 otherwise. is the dummy variable equal 1 when (t=jan), 0 otherwise. is the dummy variable equal 1 when month t falling into the period of 6-months strategy initiated in the last day of month mon and finishing in last day of month mon + 6 and 0 otherwise; t symbolize time and mon symbolize months (jan, feb, …, dec). for instance if we analyze strategy starting in last day of april (mon = apr) regression (1) takes the following form: and means that dummy variable is equal 1 for months may till oct and 0 otherwise 4 . in the case of strategy starting in last day of october (mon = oct) regression (2) takes form of: and means that dummy variable is equal 1 for months nov, dec, feb, mar, apr and 0 otherwise. dummy variable is equal 1 for jan and 0 otherwise. in this part of the research we estimated 876 regressions (73 markets times 12 different starting months for our strategies). in these regressions we used estimation method introduced by huber (1973) (function rlm() in r program) 5 . we run regressions (1) and (2) with controlling (when possible – i.e. on 30 markets due to unavailability of data for other markets) for growth rate of industrial production as macroeconomic measure of business cycle and consumer confidence index – expressing consumer confidence that can be proxy of investor sentiment (as in e.g. qiu, welch, 2006). after identification of seasonal effects we moved to examining possible explanations of them. for ten markets with longest available ipo history (us, uk, japan, canada, australia, china, hong kong, india, south korea and taiwan) we can run time series regressions where number of ipos, u.s. market sentiment, macroeconomic variable controlling for cyclical changes (industrial production) and dummy variable representing each month, describe rates of return of selected market indices. in this approach we assume ad hoc that 4 we analyze monthly rates of return, but strategy is named after the level of index in given month so when strategy starts in the last day of month mon, the first rate of return is for month mon+1. 5 relying on jacobsen and zhang (2012a) where coefficients obtained from garch and ols models are similar we did not run garch regressions. tomasz schabek, henrique castro dynamic econometric models 17 (2017) 5–18 10 sentiment of u.s. investors affects other markets because of large capital flows and primary role of american stock exchange. there is no stock market related 6 sentiment index data for other than us countries publically available. we also used the proxy of stock market sentiment for other markets – consumer confidence indicator, delivered by oecd. we could not conduct this analysis for all the markets in our sample due to not enough ipo observation and lack of sentiment data, therefore we chose only ten mentioned above markets. ten regressions for each market used in this part of analysis take form of equation (3): , (3) where: is the monthly rate of return calculated for j-th biggest market index (from 1 to 10) in month t, is the dummy variable that takes value 1 when month of measured rate of return is equal respectively k = jan, feb, mar, … nov, otherwise 0. symbolizes number of ipos in month t for market j, is the baker-wurgler sentiment index in month t, is percentage change in industrial production in month t, for country j. as mentioned above we run similar to (3) regressions but with consumer confidence indicator (cci) for each country. symbolic representation of this regression is following: , (4) where: all symbols as previously except which is cci for j-th country. both (3) and (4) equations were tested for autocorrelation and possible arch effects – when found we applied accordingly ma and/or arch models, after that statistical properties of models was test again and this procedure was repeated till elimination of autocorrelation or arch effects. results of those estimations are presented in tables 1 and 2. next step in exploring possible reasons of seasonal effects is to include in regressions weather variable (sunlight). in this part we tested returns directly 6 like in baker and wurgler sentiment index, that is based exclusively on market data. “sell not only in may”. seasonal effect on stock markets dynamic econometric models 17 (2017) 7–18 11 from described previously 6-month strategies. in cross sectional regressions we regressed average returns from those strategies in the following manner 7 : , (5) 8 where: is average rate of return for six month strategies starting at the end of month m and at the end of month n – six month later in sequence for each of i-indexes; ( ) is average number of sunlight hours (standard deviation) for six month strategies starting at the end of month m and at the end of month n – six month later; is the dummy variable with value 1 for the second month of pair (m, n); symbol m and n marks pairs of months [(jan, jul), (feb, aug), … (jun, dec)] with six months “distant” from each other. we used pairs of data vectors to avoid multicollinearity due to the fact that we are using averages that in other regression settings would cause overlapping problem. we also wanted to compare averages between cold and hot halves of the year across the countries. we repeated calculations described in equation (5) but only for markets on which we found presence of sim effect. 9 4. results the outcomes of our analysis generally support existence of sell in may and halloween effect but also seasonal effects in other months. we can observe that 10 estimated coefficients from (1) or (2) are mostly negative for strategies starting at the end of northern hemisphere spring months (mar, apr, may) and mostly positive for strategies beginning in last days of autumn months (sep, oct, nov). numbers of markets that can be characterized by sim effect vary from 19 to 31 (11 to 21) when we use 0.10 (0.05) p-values for judging the significance of . comparing to the total number of ana 7 we cannot use longitudinal (time series cross sectional) regressions due to lack of publically available time series data for numbers of sun hours in each month of the particular year. 8 we also run this regression with additional part, where is average number of ipos but because of only few countries having those date appropriately frequent i.e. without too many “zeros”, we do not report those low power regressions, even if they have similar results. 9 we also run regressions with control of latitude of cities where analyzed stock exchanges are located; those results are similar and not presented here, but available upon request. 10 exact can be provided upon request. tomasz schabek, henrique castro dynamic econometric models 17 (2017) 5–18 12 lyzed indices (73) those numbers seem to be small, but what is worth noticing out of 10 stock exchanges which total capitalization corresponds to over 70% of world federation of exchanges (wfe) global capitalization, seven have significant sim effect 11 . seasonal effects are valid for most important and with the longest data series stock market indices. we did not test presence of sim effect in particular sub periods (due to relative short data samples) but according to the newest research of jacobsen and zhang (2012, 2012a) we can conclude that sim effect is persistent over time. our results with respect to the emerging markets with relatively short data series are similar to mentioned in jacobsen and zhang (2012a) paper and we conclude that even if sim effect is not present for all selected markets it can be due to lack of long time series data for them. one scratch on surface of hypothesis proving persistence of sim effect is significantly lower number 12 of sim and hal effects when we run our tests and start our samples just after bouman and jacobsen publication in 2002. similar conclusion could be found in jacobsen and zhang paper: only three markets 13 are characterized by significant “halloween effect” in period 2001–2011. again those results could mean the disappearing sim and hal anomalies after making them publicly known in scientific papers or it just could be caused by relatively short time series. other additional and not less interesting question is about the presence and persistence of january effect and its share in halloween effect. our results suggest that january effect plays less important role in observed seasonal anomalies. for six-months strategies that include january, only in eight to eleven (depending on the months used in estimates) regressions given by equation (1) parameter is significant with p-value of 0.05. another test conducted in this research checks if – in regressions based on time series data – seasonal effects still exist after controlling for number of ipos, level of us sentiment index and changes of industrial production. in time series regression analysis we need to acquire reasonable amount of data. while both descriptive variables: us sentiment index (or consumer confidence index) and industrial production have continued time series, the number of ipo for most of the markets is characterized by incomparably higher number of “zero” data due to not as often as in the largest markets occurrence of ipos. we choose the following approach to solve this prob 11 members of world federation of exchanges (wfe) are the biggest and most important stock exchanges in the world; data from wfe, december 2012. 12 results can be provided upon request. 13 table 5 in jacobsen and zhang (2012a). “sell not only in may”. seasonal effect on stock markets dynamic econometric models 17 (2017) 7–18 13 lem: in regressions (3) and (4) we include only those markets that meet two following conditions: 1. has at least 50 observations 2. has 95% or more of non-zero ipo data (i.e. number of non-zero ipo months – counted from given month to end of analyzed period – divided by the number of months in analyzed period is equal or greater then 0.95) this simple algorithm provides us ten indices that can be characterized by frequent ipos for longer period of time. outcomes of equation (3) and (4) presented in tables 1 and 2 are similar. we can find there that statistically significant seasonal effects – represented by coefficients related to particular months, exist. tables 1 and 2 shows january effect existing in korea, uk, hong kong and india 14 . most of the statistically significant seasonal anomalies, with negative coefficients, fall in the periods between april-september (or similarly: may–october) and it causes the cumulative rate of returns of 6-months investing strategies performing better, when started in autumn months. even with behavioral and macroeconomic control variables we still observe seasonal effects in all (excluding taiwan) markets. significantly lower rates in winter months means falling prices of stock indexes that leads to lower cumulative returns in six months strategies. results 15 of estimation of equation (1) and (2) show us that higher rates of returns occur not only for strategies lasting from end of october till end of april (hallowen effect) but the same is true for september – march and november – may. such coincidence with much different from each other seasons of the year could convince us that it is the weather factor that cause seasonal effects. indeed many authors like kamstra, kramer and levi (2003), cao and wei (2005), saunders (1993), hirshleifer and shumway (2003) support such thesis. from the other hand there is broad literature that criticize weather factor as cause of sim and hal effects, for instance: jacobsen and marquering (2008), and more recently kaustia and rantapuska (2016). the final step in our analysis is to examine postulated in literature role of sunlight in seasonal anomalies. to check this, we run cross-sectional regressions described in equation (5). we tested 6-months-strategies rates of returns, with risk (average standard deviation for time of the strategy) as control variable 16 . 14 january effect disappear after controlling for consumer confidence indicator in the case of korea and hong kong – see table 2. 15 results available upon request. 16 we also run these regressions with additional average ipo factor but, due to lack of frequent ipos in most countries we do not present these results. in case of behavioral variables, where we want to preserve the same source of data and methodology of calculations tomasz schabek, henrique castro dynamic econometric models 17 (2017) 5–18 14 table 1. coefficients of estimation of equation (3) – d, , , , for particular market variable australia canada china japan korea jan –0.01 –0.01 –0.04 –0.01 –0.06* feb –0.01 –0.02 0.01 –0.02 –0.02 mar –0.02 –0.03* –0.02 0.01 –0.01 apr –0.02 –0.02 –0.01 0.01 0.02 may 0 0 0.02 –0.02 –0.03 jun –0.03* –0.03** –0.06 –0.01 –0.04 jul –0.01 –0.01 –0.06 –0.02 –0.02 aug –0.02 –0.02 –0.04 –0.01 –0.03 sep –0.03** –0.04** –0.03 –0.03 –0.03 oct –0.01 –0.01 –0.07 –0.03* –0.06* nov –0.02 –0.02 –0.02 0 –0.01 –0.01* –0.01** 0.04** –0.02*** –0.01 indpr 0.72 –0.18 2.17*** 0.29 0.51* ipo 0 0 0 0 0.01** d 0.02 0.02 0.01 0.01 0.02 variable uk us hong kong india taiwan jan –0.03** –0.01 –0.04* –0.09** –0.04 feb –0.02 –0.02 0.01 –0.07* –0.01 mar –0.02 –0.02 –0.04** –0.05 –0.02 apr –0.01 –0.01 0 –0.01 –0.01 may –0.03** –0.01 0 –0.05 –0.01 jun –0.02 –0.03** –0.02 –0.06 –0.04 jul –0.01 –0.02 0 0 –0.01 aug –0.02 –0.02 –0.03 –0.03 –0.03 sep –0.03* –0.02 –0.02 0 –0.03 oct –0.01 –0.01 0 –0.07* –0.03 nov –0.01 –0.01 –0.01 –0.04 –0.02 –0.01* –0.01*** –0.02* –0.11*** –0.05** indpr 0.36 –0.72*** 1.79*** 0.69 0.69*** ipo 0 0 0 0 0 d 0.03** 0.02** 0.02 0.05 0.02 note: table contains estimations of coefficients from equation (3). significance levels: *** – 0.01, ** – 0.05, * – 0.10. data cover maximum possible period for each market. symbolizes number of ipos in month for market j, marks growth rate of industrial production for each market, symbolizes baker-wurgler sentiment index. we could not include cci factors in cross sectional regressions of 6 months strategies because cci is calculated by oecd in the way that six months moving averages have the same values. “sell not only in may”. seasonal effect on stock markets dynamic econometric models 17 (2017) 7–18 15 table 2. coefficients of estimation of equation (4) – , for particular market variable australia canada china japan korea jan –0.01 –0.01 –0.03 –0.01 –0.04 feb –0.01 –0.01 0.01 –0.01 –0.01 mar –0.03* –0.03** –0.02 0 0.01 apr –0.02 –0.02* –0.01 0 0.01 may –0.01 –0.01 0.01 –0.03* –0.03 jun –0.03** –0.04*** –0.07* –0.02 –0.03 jul –0.01 –0.01 –0.06 –0.02 0 aug –0.02 –0.02 –0.04 –0.02 –0.02 sep –0.04** –0.04*** –0.03 –0.03* –0.01 oct 0 –0.01 –0.06 –0.03* –0.05* nov –0.02 –0.02 –0.03 0 –0.01 0 0 0 0.01*** 0 indpr 0.44 –0.24 1.85*** 0.33* 0.56** ipo 0 0 0 –0.001* 0.01** g –0.36 –0.25 0.03 –0.88*** 0.27 variable uk us hong kong india taiwan jan –0.03** –0.01 –0.03 –0.07** –0.04 feb –0.02 –0.02 0 –0.06* –0.02 mar –0.02 –0.02 –0.04* –0.03 –0.04 apr –0.01 –0.01 0 –0.01 –0.02 may –0.03** –0.01 –0.01 –0.06* –0.03 jun –0.02* –0.03* –0.03 –0.05 –0.04 jul –0.01 –0.01 0 –0.02 –0.03 aug –0.02* –0.02* –0.03 –0.04 –0.05 sep –0.03** –0.02 –0.02 0.01 –0.03 oct –0.01 0 0 –0.05 –0.06 nov –0.01 –0.01 –0.02 –0.04 –0.02 0.01** 0 – – – indpr 0.19 –0.74*** 1.96*** 0.33 0.59*** ipo –0.0006** 0 0 0 0 g –0.35* –0.12 0.03 0.06** 0.04 note: table contains estimations of coefficients from equation (4). significance levels: *** – 0.01, ** – 0.05, * – 0.10. data cover maximum possible period for each market. symbolizes number of ipos in month t for market j, marks growth rate of industrial production for each market, symbolizes consumer confidence indicator for market j. for hong kong, india and taiwan cci data are not available. the results of these regressions prove that sunlight hours have no significant explanation power in the tested models 17 . even if we limit our sample to 17 results available upon request. tomasz schabek, henrique castro dynamic econometric models 17 (2017) 5–18 16 these markets on which sim effect is present, sunlight is significant only in one of our regressions. conclusions financial market puzzle described by bouman and jacobsen (2002) is still unsolved. controlling for macroeconomic conditions (industrial production), behavioral variables (consumer confidence index, number of ipos and baker–wurgler sentiment index) and weather factor (sun hours) did not cause disappearing of “sell in may” or “halloween” effects. we can conclude that answer for those anomalies should not be searched in sun cycles, industrial production changes or investors sentiment, as some studies suggest. existence of persisting for longer time anomalies is calling for new research, although after bauman and jacobsen publication the effect is fading – maybe because of exploitation of “sell in may” and “halloween” strategies or it is just short term disappearance of the anomaly. from the other hand more rigorous analysis shows that not only may but all spring/autumn months anomalies are present on many stock markets. we can conclude that old market saying is still able to deliver some positive rates of return to the investors who want to listen to it. references andrade, s. c., chhaochharia, v., fuerst, m. (2013), "sell in may and go away" just won't go away, financial analysts journal, 69(4), 94–105, doi: https://doi.org/10.2469/faj.v69.n4.4. baker, m., wurgler j. (2007), investor sentiment in the stock market, journal of economic perspectives, v.21, spring 2007, 129–157, doi: https://dx.doi.org/10.1257/jep.21.2.129. bouman, s., jacobsen b. (2002), the halloween indicator, "sell in may and go away": another puzzle, american economic review, 92, 1618–1635, doi: https://dx.doi.org/10.1257/000282802762024683. cao, m., wei, j. (2005), stock market returns: a note on temperature anomaly, journal of banking & finance, 29, 1559–1573, doi: https://doi.org/10.1016/j.jbankfin.2004.06.028. doeswijk, r. (2008), the optimism cycle: sell in may, economist (2008), 156, 175–200, doi: https://doi.org/10.1007/s10645-008-9088-z. dzhabarov, c., ziemba w. (2010), do seasonal anomalies still work, the journal of portfolio management, 36 (3), 93–104, doi: https://doi.org/10.3905/jpm.2010.36.3.093. gerlach, j. r. (2007), macroeconomic news and stock market calendar and weather anomalies, journal of financial research, xxx(2), 283–300, doi: https://doi.org/10.1111/j.1475-6803.2007.00214.x. https://doi.org/10.2469/faj.v69.n4.4 https://dx.doi.org/10.1257/jep.21.2.129 https://dx.doi.org/10.1257/000282802762024683 https://doi.org/10.1016/j.jbankfin.2004.06.028 https://doi.org/10.1007/s10645-008-9088-z https://doi.org/10.3905/jpm.2010.36.3.093 https://doi.org/10.1111/j.1475-6803.2007.00214.x “sell not only in may”. seasonal effect on stock markets dynamic econometric models 17 (2017) 7–18 17 gugten, t. v. d. (2010), stock market calendar anomalies and macroeconomic news announcements, working paper, erasmus university rotterdam, http://hdl.handle.net/2105/7758. gultekin, m. n., gultekin, n. b. (1983), stock market seasonality: international evidence, journal of financial economics, 12, 469–481, doi: https://doi.org/10.1016/0304-405x(83)90044-2. haggard, k. s., witte, h. d. (2010), the halloween effect: trick or treat? international review of financial analysis, 19(5), 379–387, doi: https://doi.org/10.1016/j.irfa.2010.10.001. hirshleifer, d., shumway t. (2003), good day sunshine: stock returns and the weather, the journal of finance, 58(3), 1009–1032 http://www.jstor.org/stable/3094570 hong, h., yu, j. (2009), gone fishin’: seasonality in trading activity and asset prices, journal of financial markets, 12(4), 672–702, doi: https://doi.org/10.1016/j.finmar.2009.06.001. huber, p. (1973), robust regression: asymptotics, conjectures and monte carlo, the annals of statistics, 1(5), 799–821. http://www.jstor.org/stable/2958283. jacobsen, b., mamun, a., visaltanachoti, n. (2005), seasonal, size and value anomalies, doi: http://dx.doi.org/10.2139/ssrn.784186. jacobsen, b., marquering, w. (2008), is it the weather? journal of banking & finance, 32, 526–540, doi: https://doi.org/10.1016/j.jbankfin.2007.08.004. jacobsen, b., marquering, w. (2009), is it the weather? response, journal of banking & finance, 33, 583–587, doi: https://doi.org/10.1016/j.jbankfin.2008.09.011. jacobsen, b., visaltanachoti, n. (2009), the halloween effect in us sectors, financial review, 44(3), 437–459, doi: https://doi.org/10.1111/j.1540-6288.2009.00224.x. jacobsen, b., zhang c.y. (2012a), the halloween indicator everywhere and all the time, http://ssrn.com/abstract=2154873. jacobsen, b., zhang, c.y. (2012), are monthly seasonals real? a three century perspective, review of finance, 17(5), 1743–1785, doi: https://doi.org/10.1093/rof/rfs035. kamstra, m. j., kramer, l. a., levi, m. (2003), winter blues: a sad stock market cycle, american economic review, 93(1), 324–343, doi: https://doi.org/10.1257/000282803321455322. kaustia, m., rantapuska, e. (2016), does mood affect trading behavior? journal of financial markets, 29, 1–26 doi: https://doi.org/10.1016/j.finmar.2015.08.001. keef, s. p., khaled, m. s. (2011), a review of the seasonal affective disorder hypothesis, journal of socio-economics, 40, 959–967, doi: https://doi.org/10.1016/j.socec.2011.08.012. kelly, p. j., meschke, f. (2010), sentiment and stock returns: the sad anomaly revisited, journal of banking & finance, 34(6), 1308–1326, doi: https://doi.org/10.1016/j.jbankfin.2009.11.027. lean, h. h. (2011), the halloween puzzle in selected asian stock markets, international journal of economics and management, 5(2), 216–225, http://www.scopus.com/inward/record.url?eid=2-s2.084869413435&partnerid=mn8toars. lucey, b. m, zhao, s. (2007), halloween or january? yet another puzzle, international review of financial analysis, 17, 1055–1069, doi: https://doi.org/10.1016/j.irfa.2006.03.003. maberly, e. d., pierce, r. p. (2004), stock market efficiency withstands another challenge: solving the "sell in may/ buy after halloween" puzzle, econ journal watch, 1, 29–46, https://econjwatch.org/158. https://doi.org/10.1016/0304-405x(83)90044-2 https://doi.org/10.1016/j.irfa.2010.10.001 https://doi.org/10.1016/j.finmar.2009.06.001 http://dx.doi.org/10.2139/ssrn.784186 https://doi.org/10.1016/j.jbankfin.2007.08.004 https://doi.org/10.1016/j.jbankfin.2008.09.011 https://doi.org/10.1111/j.1540-6288.2009.00224.x https://doi.org/10.1093/rof/rfs035 https://doi.org/10.1257/000282803321455322 https://doi.org/10.1016/j.finmar.2015.08.001 https://doi.org/10.1016/j.socec.2011.08.012 https://doi.org/10.1016/j.jbankfin.2009.11.027 https://doi.org/10.1016/j.irfa.2006.03.003 tomasz schabek, henrique castro dynamic econometric models 17 (2017) 5–18 18 o'higgins m., downes, j. (1990), beating the dow. a high-return-low-risk method for investing in industrial stocks with as little as $5000, harper collins, new york saunders jr., e.m., (1993), stock prices and wall street weather, the american economic review, 83(5), 1337–1345, http://www.jstor.org/stable/2117565. qiu l., welch i. ( 2006), investor sentiment measures, http://dx.doi.org/10.2139/ssrn.589641. zarour, b. a. (2007), the halloween effect anomaly: evidence from some arab countries equity markets. studies in business and economics, 13(1), 68–76, http://hdl.handle.net/10576/6858. “sell not only in may”. efekty sezonowe na rynkach akcji z a r y s t r e ś c i. opisywana w literaturze anomalia sezonowa (upraszczając: „sprzedaj w maju, kupuj w październiku”) nadal pozostaje niewyjaśniona. w przeprowadzonych regresjach, bazujących na długich szeregach czasowych oraz na licznej grupie indeksów giełdowych, efekt halloween jest istotny statystycznie w 19 spośród 73 badanych indeksów, ale także w 11 spośród 23 indeksów z najdłuższymi dostępnym szeregami czasowymi. wyniki badań wskazują, że ponadprzeciętne zyski mogą zostać zrealizowane również w strategiach zaczynających się w innych miesiącach jesiennych. stwierdzono, że także po włączeniu do testowanych modeli zmienny kontrolnych dotyczących pogody, sentymentu inwestorów, zmiennych makroekonomicznych – badana anomalia nadal istnieje. s ł o w a k l u c z o w e: anomalie sezonowe; czynniki behawioralne; efekt halloween; efekt stycznia. © 2016 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2016.007 vol. 16 (2016) 117−131 submitted november 30, 2016 issn (online) 2450-7067 accepted december 20, 2016 issn (print) 1234-3862 krzysztof kompa, dorota witkowska * performance of pension funds and stable growth open investment funds during the changes in the polish retirement system a b s t r a c t. the conditions of the pension funds (ofe) functioning were essentially changed in the years 2011–2014. the aim of the paper is to find out if these modifications influence the efficiency of the pension funds and to compare the performance of these funds to stable growth open investment funds (fio). the analysis is provided for selected funds in the years 2009–2015. we conclude that in the examined period, ofe performed better than fio, and the modifications of the rules for the pension funds caused the increase of risk and decrease of investment efficiency of these funds’ portfolios. k e y w o r d s: pension funds, stable growth open investment funds, investment efficiency, sharpe model, capm, sharpe, treynor and jensen ratios. j e l classification: g11; c12. introduction the pension system in communist poland was the defined benefit and pay-as-you-go (payg) scheme. however, the demographic changes, pension privileges concerning more and more occupational groups and economic sectors, together with the so-called early retirement regulations, which * correspondence to: krzysztof kompa, warsaw university of life sciences – sggw, department of econometrics and statistics, 166 nowoursynowska street, 02-787 warsaw, poland, e-mail: krzysztof_kompa@sggw.pl; dorota witkowska, university of lodz, department of finance and strategic management, 22/26 matejki street, 90-237 lodz, poland, dorota.witkowska@uni.lodz.pl krzysztof kompa, dorota witkowska dynamic econometric models 16 (2016) 117–131 118 went into force in late 70-ties 1 , caused the increase of the pension system costs. in 1981 the pension contribution was 25% of wages, in the years 1987–1989 it rose to 38%, and obtained the level of 45% in 1990 (see (wojciechowski 2011, podstawka 2005, p. 259)). the increasing deficit in the polish pension system enforced its transformation, which took place in 1999. the new pension system replaced defined benefit scheme by defined contribution one, it enriched the payg system by the mandatory capital-funded pillar, and introduced voluntary plans. new regulations were also to abolish sectoral and occupational privileges and early retirement programs. the reformed retirement system has been consisted in three pillars – social insurance institution (zus), open pension funds (ofe) and voluntary capital-funded system. the pension contribution (i.e. 19.52% of earnings) was divided between both mandatory pillars – zus 12.22% and ofe 7.3%. the first essential manipulation in the original pension reform was made in 2011 when the contribution to pension funds was diminished from 7.3% to 2.3%. the second and the most drastic regulation, consisted in shifting 51.5% of the assets, held by the pension funds, to the social insurance institution (including all debt securities issued and guaranteed by the state treasury). overhaul of the pension system also concerned changes in the ofes’ investment portfolio since private pension funds have no longer been allowed to invest in government bonds. these new law (from 2013) went into effect in february 2014. the third significant modification took place in 2014 and changed the character of pension funds which have been no longer obligatory. according to the new regulations, each employee has had four months every four years to decide whether 2.92 percent of their income goes to a chosen private fund or to zus. it means that after all mentioned above regulations the part of the pension contribution transferred to the pension funds decreased from 37.4% to 15% or even zero. there have been numerous studies concerning the efficiency of mutual and pension funds operating in poland. mutual funds’ performance is analyzed by karpio and żebrowska-suchodolska (2010, 2011, 2012), ostrowska (2003), perez (2012), witkowska (2009), witkowska et al. (2009) and zamojska (2012). whereas evaluation of pension funds investment activity is made by białek (2009), chybalski (2006, 2009), dybał (2008), karpio and żebrowska-suchodolska (2014, 2016), kompa and 1 early retirement regulations decreased the real pension age by several years since employees who fulfil certain conditions were allowed to obtain pension benefits earlier i.e. before statutory retirement age. performance of pension funds and stable growth open investment funds… dynamic econometric models 16 (2016) 117–131 119 wiśniewski (2015), kompa and witkowska (2015, 2016), marcinkiewicz (2009) and witkowska, kompa (2012). the mentioned research was provided for differently defined time spans, length of the samples, various frequency of measurement, taking into account bear and bull markets, and variety of efficiency measures. however, the investigation concerning the influence of political decisions to the performance of pension and investment funds market is rare. there is also lack of profound comparable research of the efficiency of pension and mutual funds. this paper is to fulfill that gap in literature. the aim of our research is to find out if mentioned above changes of the ofes’ functioning influence the efficiency of the pension funds and to compare the performance of these funds to stable growth investment funds (fio). the analysis is provided for selected funds applying statistical inference, sharpe and capital assets pricing (capm) models and classical investment efficiency ratios. 1. data and methods to answer the question about the consequences of manipulations in the pension system, the whole period of analysis, denoted by a, is divided into three pairs of sub-periods according to the moments when new regulations went into effect:  b, decreasing of the contribution transferred to ofe,  c, shifting of assets from ofe to zus and changes in the ofes’ portfolio composition,  d, waving the obligation of pension funds membership. empirical analysis of the investment efficiency is provided on the basis of daily observations from the years 2009–2015. the considered time span covers seven samples:  a from 1.01.2009 to 31.12.2015 (84 months),  b1 from 1.01.2009 to 30.04.2011, and b2 from 1.05.2011 to 31.08.2013 (28 months each sub-period),  c1 from 1.04.2012 to 31.01.2014, and c2 from 1.02.2014 to 31.12.2015 (22 and 23 months respectively),  d1 from 1.01.2013 to 30.06.2014, and d2 from 1.07.2014 to 31.12.2015 (18 months each). in our research we use daily logarithmic rate of returns evaluated from:  participation units of stable growth open investment funds and accounting units of pension funds, which are managed by the same six investment and pension funds companies, namely: allianz, aviva, nationale krzysztof kompa, dorota witkowska dynamic econometric models 16 (2016) 117–131 120 nederlanden, pekao, pko and pzu investment and pension funds companies 2 ,  warsaw stock exchange index – wig,  poland’s official treasury bonds index – tbsp.index, and  warsaw interbank offered rate – wibor. wig, tbsp and wibor (being the interest rate of 3 months loans) are used as benchmarks 3 in our analysis since they represent capital, treasury bond and money markets, respectively. wig is treated as the market index whereas tbsp and wibor – as risk free instruments. analysis of returns and risk generated by the investment portfolios constructed by selected funds is conducted using statistical interference (assuming the significance level 0.05). we will verify 4 the following null hypotheses concerning: rates of return risk sharpe model / capm e(rofe) = 0; e(rfio) = 0; e(rbenchmark) = 0  = 0;  = 0 e(rofe) = e(rfio) d2(rofe) = d2(rfio) ofe = fio e(rbefore) = e(rafter) d2(rbefore) = d2(rafter) before = after where, e(r) – expected returns, d 2 (r) – variance of returns, rofe, rfio – returns from ofe and fio respectively, ,  – parameters of sharpe model or capm, rbefore, rafter, before, after – returns from the portfolio and beta coefficients before and after the change went into effect, respectively. the comparison of the funds’ efficiency is provided applying sharpe, treynor and jensen ratios which are evaluated for all considered pension and mutual funds in all analyzed time spans, and for differently defined risk free instruments. 2. rates of return in the first step of our analysis we check out if daily rates of return are significantly positive or negative. it is visible (table 1) that expected returns from all considered pension funds’ investment are significantly positive in 2 the selection of mentioned above investment and pension funds companies is connected with the investigations previously provided by the authors separately for pension funds and investment funds. 3 it is worth mentioning that wig and wibor are used to calculate benchmarks, which are used to evaluate pension funds efficiency (dziennik ustaw rzeczypospolitej polskiej 2014, poz. 753). 4 description of all applied tests is presented in (tarczyński, witkowska and kompa, 2013, p. 18–19, 25, 72) performance of pension funds and stable growth open investment funds… dynamic econometric models 16 (2016) 117–131 121 the whole period of consideration (denoted by a), and for the periods b1 and c1, describing performance of pension funds before the essential changes of their functioning. it is also noticeable that in the periods c2 and d2, i.e. after the modifications went into effect, the average rates of return are negative, although the null hypothesis cannot be rejected for the significance level 0.05. table 1. test statistics verifying the hypothesis about expected returns from pension funds in considered periods h0: e(rofe) = 0 period a b1 b2 c1 c2 d1 d2 number of observations 1823 604 610 480 499 390 393 allianz 1.9306 2.7360 1.3129 2.4911 –0.1800 0.7590 –0.3116 aviva 1.7404 2.6509 1.2352 2.2974 –0.2728 0.7827 –0.4601 nationale nederlanden 1.7662 2.6364 1.3028 2.4438 –0.3501 0.8735 –0.5018 pekao 1.5230 2.7997 1.0987 2.2557 –0.4316 0.7572 –0.6005 pko 2.0540 2.9102 1.4263 2.6620 –0.1030 0.9630 –0.2776 pzu 1.7043 2.5413 1.0815 2.2353 –0.1397 0.8623 –0.3361 note: bold letters denote rejection of null hypothesis. table 2. test statistics verifying the hypothesis about expected returns from stable growth investment funds in considered periods h0: e(rfio) = 0 period a b1 b2 c1 c2 d1 d2 number of observations 1823 604 610 480 499 390 393 allianz 0.0849 0.5640 –0.1806 0.1617 –0.5957 –0.2008 –0.8648 aviva 2.0625 1.9847 0.7763 1.8784 0.0660 1.0115 –0.0550 nationale nederlanden 1.7755 1.8450 0.6528 1.6648 0.2879 0.6511 –0.2487 pekao 0.3032 1.6871 –1.0786 0.6526 –0.3112 0.1099 –0.5922 pko 2.1753 1.8920 0.8929 1.8669 0.5899 0.9977 0.1458 pzu 1.1236 1.5555 0.3250 1.3278 –0.6382 0.5266 –0.6100 note: bold letters denote rejection of null hypothesis. stable growth investment funds’ performance (table 2) seems to be worse than the pension funds since only aviva, nationale nederlanden and pko generated significantly positive returns in the periods a, b1 and c1. however, the comparison of returns, using cochran-cox test (table 3), shows that differences between returns obtained by both types of funds are not significant, except pekao in b2 when ofe performed better than mutual fund. analyzing returns from the benchmarks (table 4) one may notice that interest rate wibor generated positive returns in all periods, bond market performed well with significantly positive returns in all periods but d1 and d2, whereas expected returns from wig do not significantly differ from zero, except the period b1. krzysztof kompa, dorota witkowska dynamic econometric models 16 (2016) 117–131 122 table 3. test statistics for comparison of expected returns h0: e(rofe) = e(rfio) and risk h0: d 2 (rofe) = d 2 (rfio) for pension and investment funds period hypotheses allianz aviva nn pekao pko pzu a return 1.1902 –0.0116 0.1989 1.0688 0.2125 0.0000 variance 1.0935 1.1017 1.1702 1.0740 1.3329 1.5200 b1 return 0.8956 –0.0783 0.1499 0.1975 0.5473 –0.2761 variance 1.6486 1.4068 1.2048 1.4157 1.0283 1.9268 b2 return 0.8969 0.3501 0.3934 1.6668 0.3259 0.2907 variance 1.4784 1.0615 1.0968 1.3140 1.0511 2.0000 c1 return 1.9833 0.2024 0.6182 0.8171 0.7680 –0.2969 variance 1.1808 1.0581 1.2008 1.1005 1.2324 2.0000 c2 return 0.2589 –0.3431 –0.4286 –0.1533 –0.2260 0.4545 variance 2.0667 2.1169 2.1375 2.0351 2.4110 0.9857 d1 return 0.8041 0.0796 0.2887 0.5390 0.2098 0.0000 variance 1.6271 1.3958 1.4758 1.3195 1.7198 1.5306 d2 return 0.0885 –0.2676 –0.4538 –0.2788 –0.3561 0.2107 variance 2.0134 2.1991 2.1157 2.0944 2.3616 1.0462 note: nn is an abbreviation of nationale nederlanden. bold letters denote rejection of null hypothesis. shadowed cells denote situation when d 2 (rofe) <d 2 (rfio). table 4. test statistics verifying the hypothesis about expected returns from the benchmarks in considered periods h0: e(rbenchmark) = 0 periods a b1 b2 c1 c2 d1 d2 tbsp 5.4494 2.8694 3.7794 2.3963 2.8731 1.4714 1.6248 wig 1.0095 1.7045 –0.0816 1.0326 –0.4643 0.4855 –0.6640 wibor3m 135.2505 299.2855 132.8255 81.5094 109.2820 137.2229 116.3402 note: bold letters denote rejection of null hypothesis. table 5. cochran-cox test statistics comparing returns in the pairs of periods h0: e(rbefore) = e(rafter) funds type: ofe fio periods: b1:b2 c1:c2 d1:d2 b1:b2 c1:c2 d1:d2 allianz 1.0242 1.3305 0.7179 0.5409 0.5468 0.4707 aviva 1.0356 1.2945 0.8442 1.1207 1.2931 0.8161 nn 1.0311 1.5763 0.9342 0.9727 1.0382 0.6518 pekao 1.1848 1.3902 0.9399 1.9662 0.6945 0.4579 pko 1.0890 1.4153 0.8172 0.7175 1.3803 0.6347 pzu 1.1246 1.1066 0.7806 0.8989 0.9908 0.7972 note: nn is an abbreviation of nationale nederlanden. table 5 contains test statistics comparing expected returns generated by both types of funds in two sub-periods: before and after modification of the ofe environment i.e. conditions for their functioning. null hypothesis cannot be rejected in any case however returns for both pension and stable growth investment funds were always bigger before the change than after, performance of pension funds and stable growth open investment funds… dynamic econometric models 16 (2016) 117–131 123 and this difference was always bigger for the pension than for mutual funds. in table 6 comparison of risk is provided and one may notice that for all pension funds risk significantly increased in the periods c2 and d2 i.e. after the introduced by the government changes. for mutual funds we do not observe similar situation since the significant difference of risk between both comparing periods is visible only for aviva and nationale nederlanden in the sub-periods b together with aviva and pekao – in d but in both subperiods the risk before modification of ofe environment is bigger than after. table 6. fisher-snedecor test statistics comparing variances in the pairs of periods h0: d 2 (rbefore) = d 2 (rafter) returns ofe fio periods b1:b2 c1:c2 d1:d2 b1:b2 c1:c2 d1:d2 allianz 1.0153 3.6597 1.5669 1.1152 1.1070 1.0102 aviva 1.0448 3.9187 1.6562 1.3252 1.0003 1.2197 nn 1.1246 2.7625 1.6332 1.1563 1.0788 1.1200 pekao 1.0366 4.2110 1.6346 1.0483 1.1036 1.2319 pko 1.0429 3.1506 1.5867 0.7175 1.0869 1.0805 pzu 1.1452 4.3580 1.8796 1.0395 1.0606 1.1538 note: nn is an abbreviation of nationale nederlanden. bold letters denote rejection of null hypothesis. shadowed cells denote situation when d 2 (rbefore) < d 2 (rafter). pension funds’ portfolios are characterized by smaller risk than stable growth investment funds only in the sub-periods b, the period c1 for pekao and pzu, the period d1 for pzu and in the whole period a for allianz and pzu. although the null hypothesis is rejected only for pzu and allianz in all mentioned periods together with aviva, nationale nederlanden and pko in the period b1. in all sub-periods denoted as c and d risk of investments made by ofe is significantly bigger than by fio allianz, nationale nederlanden and pko, whereas for aviva and pekao this statement is true for c2, d1 and d2 (see table 3). comparison of variances in pairs of selected sub-periods (table 6) shows the significant increase of risk in all pension funds after the new regulations went into effect in 2014, and pekao also in 2011 whereas for tbsp and wibor only in 2011 (table 7). this conclusion is not true for stable growth investment funds and the rest of the benchmarks and sub-periods. it is also visible that significant changes in returns took place only in the money market (table 7). krzysztof kompa, dorota witkowska dynamic econometric models 16 (2016) 117–131 124 table 7. cochran-cox and fisher-snedecor test statistics comparing returns h0: e(rbefore) = e(rafter) and variances h0: d 2 (rbefore) = d 2 (rafter) expected returns variance periods b1:b2 c1:c2 d1:d2 b1:b2 c1:c2 d1:d2 tbsp –0.9878 –0.0836 0.0543 1.3576 1.2631 1.3371 wig 1.3593 1.0740 0.8047 1.4129 1.1331 1.2363 wibor3m –5.7610 33.2388 36.1675 5.6374 5.5480 1.6129 note: shadowed cells denote situation when d 2 (rbefore) < d 2 (rafter). bold letters denote rejection of h0. 3. single index and capital assets pricing models the models are estimated for each period of analysis using wig as the market index and two variants of risk free instruments i.e. tbsp and wibor in the capital assets pricing models (capm). therefore, there are 21 models for each fund, which parameters will be compared. beta coefficients in all models estimated for the pension funds are significantly positive (except the models estimated for ofe pzu in the periods: a, b1, c2, d1, and d2). value of parameter estimates increase from 0.2 in b1, by 0.3 in b2 and c1, 0.4 in d2 to 0.6 in c2 and d2. parameter estimates of beta and determination coefficients in the single index models and the capital assets pricing models with wibor as the risk free instrument are almost of the same values whereas using tbsp as the risk free instrument gives slightly different values of beta for all models estimated for the pension funds (the example in table 8). similar conclusions may be drawn for the stable growth investment funds (see table 9) however values of the determination coefficients are different in models describing both types of funds. table 8. estimates and determination coefficients of different pension funds models estimated for in the whole period allianz aviva nn pekao pko pzu beta coefficient sharpe 0.2976 0.3003 0.3119 0.3044 0.2897 0.0132 capm wibor 0.2976 0.3003 0.3119 0.3044 0.2897 0.0132 capm tbsp 0.2762 0.2785 0.2892 0.2827 0.2682 –0.0027 determination coefficient sharpe 0.5567 0.5538 0.5531 0.5189 0.5434 0.0009 capm wibor 0.5567 0.5539 0.5531 0.5190 0.5435 0.0009 capm tbsp 0.5182 0.5168 0.5201 0.4802 0.5029 0.0000 note: bold letters denote statistically significant parameters i.e. h0:  = 0; nn as in table 3. performance of pension funds and stable growth open investment funds… dynamic econometric models 16 (2016) 117–131 125 table 9. parameter estimates and determination coefficients of different models estimated for mutual funds in the whole period allianz aviva nn pekao pko pzu beta coefficient sharpe 0.3221 0.3379 0.3396 0.3396 0.2638 0.6389 capm wibor 0.3221 0.3379 0.3396 0.3396 0.2638 0.6389 capm tbsp 0.2977 0.3155 0.3165 0.3165 0.2409 0.6262 determination coefficient sharpe 0.5406 0.8632 0.9029 0.9029 0.8041 0.9406 capm wibor 0.5406 0.8632 0.9029 0.9029 0.8041 0.9406 capm tbsp 0.5115 0.8644 0.9435 0.9435 0.8103 0.9400 note: bold letters denote statistically significant parameters i.e. h0:  = 0; nn is an abbreviation of nationale nederlanden. in the next step we compare betas from the models estimated before and after the changes of the pension funds’ situation, introduced by the government 5 . it is visible that for all models estimated for the pension funds the risk significantly increased in the second considered periods (table 10). comparing beta coefficients estimated for the mutual funds the same results are obtained only for fio allianz whereas for other stable growth investment funds the results of the hypothesis verification are different (table 11). for aviva, pekao and pzu the risk significantly decreased in the period d2 in comparison to the period d1, also for the first considered pair of sub-periods fio aviva are characterized by the smaller risk after the modification of the system than before. compering a pair of sub-periods c1 and c2, we notice that there is only one case (pekao for the capm model with tbsp) when risk significantly decreased in the second compared period. table 10. student test statistics comparing betas estimated for the pension funds in the pairs of periods h0: before = after b1:b2 c1:c2 d1:d2 models sharpe capm tbsp sharpe capm tbsp sharpe capm tbsp allianz –4.70 –7.51 –33.46 –31.83 –12.89 –12.54 aviva –3.80 –6.71 –36.32 –36.51 –14.51 –14.62 nn –2.95 –5.81 –34.50 –34.90 –14.06 –14.55 pekao –5.35 –8.09 –37.60 –36.23 –14.08 –13.85 pko –4.45 –7.44 –35.79 –34.35 –13.31 –13.17 pzu –0.92 –3.16 2.02 3.22 –1.77 –1.00 note: bold letters denote rejection of null hypothesis. 5 in comparison parameter estimates from the capm using wibor as risk free instrument were omitted because they were of the same values as in the sharp models. krzysztof kompa, dorota witkowska dynamic econometric models 16 (2016) 117–131 126 table 11. student test statistics comparing betas estimated for the mutual funds in the pairs of periods h0: before = after b1:b2 c1:c2 d1:d2 models sharpe capm tbsp sharpe capm tbsp sharpe capm tbsp allianz –9.9973 –11.5962 –9.8910 –8.7862 –6.1679 –4.8829 aviva 13.9977 13.0450 –4.8720 –6.2323 2.0147 3.3168 nn –0.3876 –0.3003 –2.8112 –1.7663 –0.8466 –0.8954 pekao –8.0196 –8.9997 0.6125 2.2591 4.3964 4.3668 pko –11.0755 –11.7400 –14.4898 0.6200 –1.4324 –1.3214 pzu –18.4045 –20.9368 –89.8678 –85.1736 3.3144 3.9947 note: bold letters denote rejection of null hypothesis. we also compare betas from the models estimated for the pension and mutual funds (table 12), and it is visible that in the periods denoted as a, b1, b2 and c1, all pension funds except pko in a and allianz in c1 are characterized by bigger risk than mutual funds since betas are significantly bigger for ofe than for fio in the majority of cases. whereas in the sub-periods c2, d1 and d2 the situation is the opposite except pzu. table 12. student test statistics comparing betas estimated for the pension and mutual funds in all periods h0: ofe = fio a b1 b2 c1 c2 d1 d2 sharpe model allianz –3.5144 –3.0930 –16.4158 1.9904 51.7384 16.6141 41.4479 aviva –11.9270 –34.7256 –10.2976 –0.3780 47.4091 9.2631 43.6863 pko 8.3548 –5.9057 –12.4800 –0.5507 57.8852 18.6757 47.5616 nn –10.6034 –28.0973 –17.7427 –5.2973 57.7356 11.6558 47.2607 pekao –21.4350 –31.5047 –23.6178 –13.0494 53.8435 5.1663 47.3234 pzu –166.4176 –127.2500 –112.5233 –88.3038 –80.0261 –79.1466 –63.7252 capm tbsp allianz –3.1584 –2.8330 –16.8911 3.0855 75.3923 20.0192 64.5078 aviva –13.8325 –37.1812 –12.5294 1.1703 61.7681 15.9266 57.5696 pko 10.1111 –6.7755 –17.3235 0.8600 57.1452 25.1786 45.0649 nn –15.0291 –36.9608 –30.1785 –7.3074 78.8268 23.2502 66.6716 pekao –25.9866 –33.3984 –24.1046 –13.7403 33.1803 0.5113 28.3302 pzu –169.5121 –134.2389 –115.0464 –90.6541 –81.3673 –80.3593 –64.7672 note: bold letters denote rejection of null hypothesis. nn is an abbreviation of nationale nederlanden. 4. investment efficiency evaluation the last stage of our investigation consists in comparison of classical efficiency measures, which are evaluated for different portfolios and samples. it is worth reminding that applying so called reward-to-variability (sharpe) and reward-to-volatility (treynor) ratios, it is necessary to define performance of pension funds and stable growth open investment funds… dynamic econometric models 16 (2016) 117–131 127 the risk free instrument and the market index. since in our previous analysis we use two risk free instruments wibor and tbsp together with wig, which represents the capital market, the same instruments are used to evaluate the performance of considered pension and mutual funds. however, to evaluate treynor ratio we use betas from both i.e. single index and capital assets pricing models. in other words, for each pension or mutual fund we calculate two sharpe’s and jensen ratios, together with four treynor’s measures. tables 13 and 14 contain the average values of these efficiency measures. table 13. average values of the efficiency measures evaluated for pension funds period measure allianz aviva pko nn pekao pzu wig a sharpe 0.0100 0.0060 0.0125 0.0079 0.0025 0.0075 0.0096 treynor 0.0002 0.0001 0.0002 0.0001 0.0000 0.0057 0.0001 jensen 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 b1 sharpe 0.0716 0.0681 0.0768 0.0700 0.0731 0.0670 0.0584 treynor 0.0013 0.0012 0.0014 0.0010 0.0013 0.0114 0.0008 jensen 0.0001 0.0001 0.0001 0.0001 0.0001 0.0003 b2 sharpe –0.0006 –0.0042 0.0007 –0.0004 –0.0095 –0.0087 –0.0212 treynor 0.0000 –0.0001 0.0000 0.0000 –0.0002 –0.0013 –0.0002 jensen 0.0001 0.0000 0.0001 0.0001 0.0000 0.0000 c1 sharpe 0.0625 0.0535 0.0816 0.0758 0.0540 0.0519 0.0281 treynor 0.0007 0.0006 0.0011 0.0010 0.0007 0.0089 0.0003 jensen 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 c2 sharpe –0.0319 –0.0354 –0.0280 –0.0382 –0.0405 –0.0278 –0.0391 treynor –0.0003 –0.0004 –0.0003 –0.0004 –0.0004 0.0109 –0.0003 jensen 0.0000 0.0000 0.0000 0.0000 –0.0001 –0.0002 d1 sharpe 0.0127 0.0142 0.0232 –0.0024 0.0149 0.0187 0.0075 treynor 0.0002 0.0003 0.0003 0.0000 0.0002 0.0052 0.0001 jensen 0.0000 0.0000 0.0001 0.0001 0.0000 0.0001 d2 sharpe –0.0334 –0.0291 –0.0313 –0.0418 –0.0461 –0.0326 –0.0449 treynor –0.0003 –0.0004 –0.0003 –0.0004 –0.0005 –0.0072 –0.0004 jensen 0.0000 0.0000 0.0000 0.0000 0.0000 –0.0002 note: bold letters denote ratios evaluated for pension funds which are bigger than the ones calculated for wig. nn is an abbreviation of nationale nederlanden. it is visible that in the majority of cases (76%) pension funds performed better than capital market if the efficiency is measured by the averages of treynor and sharpe ratios. only nationale nederlanden ofe did not perform better in the periods a, c2 (only treynor measure), and d2. in fact, in c2 and d2 sub-periods also other pension funds show lower efficiency. applying jensen alpha, it is visible that in majority of cases values are nonnegative. although positive and statistically significant values of the parameter alpha in the capm (when wibor is the risk free instrument) are observed krzysztof kompa, dorota witkowska dynamic econometric models 16 (2016) 117–131 128 only for ofe allianz, nationale nederlanden and pko in the period c1, together with ofe pzu in the period b1 (see table 15). it is visible that in the majority of cases mutual funds performed worse than capital market. there are only 12 cases among 84 average of treynor or sharpe measure values (14%) when stable growth investment fund is more efficient than capital market represented by wig. the mutual funds which were better managed are fio aviva, pko and pzu. also jensen ratio is rarely positive for mutual funds and if the constant in the capital assets pricing models, estimated for investment funds, is statistically significant it is always negative for fio allianz and both risk free instruments in the whole period of analysis, fio nationale nederlanden, pekao, pko and pzu for tbsp in the whole period of analysis and in the sub-period b1 for nationale nedrelanden and tbsp, together with pekao regardless the risk free instrument (see table 15). table 14. average values of the efficiency measures evaluated for mutual funds period measure allianz aviva pko nn pekao pzu wig a sharpe –0.0298 0.0097 0.0034 0.0023 –0.0282 –0.0014 0.0096 treynor –0.0005 0.0001 0.0000 0.0000 –0.0003 0.0000 0.0001 jensen –0.0002 0.0000 0.0000 0.0000 –0.0002 –0.0001 b1 sharpe –0.0021 0.0522 0.0368 0.0442 –0.0743 0.0427 0.0584 treynor –0.0001 0.0008 0.0005 0.0005 –0.0010 0.0006 0.0008 jensen –0.0002 0.0000 0.0000 0.0000 –0.0001 –0.0001 b2 sharpe –0.0448 –0.0195 –0.0183 –0.0216 –0.0848 –0.0196 –0.0212 treynor –0.0006 –0.0003 –0.0002 –0.0003 –0.0010 –0.0099 –0.0002 jensen –0.0001 0.0000 0.0000 0.0000 –0.0003 0.0000 c1 sharpe –0.0534 0.0311 0.0271 0.0264 –0.0144 0.0420 0.0281 treynor –0.0006 0.0003 0.0003 0.0003 –0.0001 0.0004 0.0003 jensen –0.0002 0.0000 0.0001 0.0000 –0.0002 0.0001 c2 sharpe –0.0772 –0.0456 –0.0296 –0.0349 –0.0573 –0.0487 –0.0391 treynor –0.0007 –0.0005 –0.0003 –0.0003 –0.0005 –0.0004 –0.0003 jensen –0.0001 0.0000 0.0000 0.0000 –0.0001 –0.0001 d1 sharpe –0.0532 0.0143 –0.0187 –0.0033 –0.0256 0.0072 0.0075 treynor –0.0006 0.0002 –0.0002 0.0000 –0.0003 0.0001 0.0001 jensen –0.0002 0.0000 0.0000 0.0000 –0.0001 0.0000 d2 sharpe –0.0802 –0.0401 –0.0330 –0.0474 –0.0626 –0.0498 –0.0449 treynor –0.0008 –0.0004 –0.0003 –0.0004 –0.0006 –0.0004 –0.0004 jensen –0.0001 0.0000 0.0000 0.0000 –0.0001 0.0000 note: bold letters denote ratios evaluated for mutual funds which are bigger than the ones calculated for wig. nn is an abbreviation of nationale nederlanden. performance of pension funds and stable growth open investment funds… dynamic econometric models 16 (2016) 117–131 129 conclusion investigation is provided for the selected pension funds and stable growth open investment funds which have been operating in frame of the six investment and pension funds companies. during the period of analysis, i.e. the years 2009–2015, the essential changes of the pension funds functioning were introduced therefore, research is conducted for three distinguished pairs of sub-periods. these pairs: b (b1 and b2), c (c1 and c2), and d (d1 and d2) are characterized by similar numbers of observations to make the comparison of the considered pension and mutual funds’ performance possible. provided analysis let us conclude that changes in the operation of pension funds influence not only the pension funds market but the whole capital market in poland. the latter seems not to recover after the financial crisis 2007–2009 (see table 4) and its performance is worse in all periods after the modifications were introduced than before i.e. sharpe and treynor ratios are positive in the sub-periods b1, c1 and d1 and negative for b2, c2 and d2 (tables 13–14). also pension funds investment efficiency is worse after the changes went into effect, especially they cause the significant increase of the pension funds’ investment portfolios risk (see tables 6 and 10). table 15. parameter estimates of constant in capm for which h0:  = 0 is rejected fund ofe fio parameter period instrument parameter period instrument allianz 0.0002 c1 wibor –0.0001 a wibor –0.0002 a tbsp nationale nederlanden 0.0002 c1 wibor –0.0001 a tbsp –0.0001 b1 tbsp pekao –0.0001 a tbsp –0.0003 b1 wibor –0.0004 b1 tbsp pko 0.0002 c1 wibor –0.0001 a tbsp pzu 0.0003 b1 wibor –0.0001 a tbsp comparison of the pension and stable growth investment funds’ performance shows that the former were in analyzed periods more effective than the latter (compare tables 13 and 14). considered changes of the pension system have not effected mutual funds’ investment policy directly however the general situation of the polish capital market influences situation of fios. krzysztof kompa, dorota witkowska dynamic econometric models 16 (2016) 117–131 130 references analizy (2013), wspólny raport analiz online oraz izby zarządzających funduszami i aktywami z dnia 8 marca 2013, analizy online. białek j., konstrukcja miar efektywności otwartych funduszy emerytalnych, wydawnictwo uniwersytetu łódzkiego, łódź, 2009. chybalski, f. (2006), miary oceny działalności inwestycyjnej ofe, wiadomości statystyczne, 10, 22–35. chybalski, f. (2009) (red.), otwarte fundusze emerytalne w polsce. analiza działalności inwestycyjnej, finansów oraz decyzji członków, c.h. beck, warszawa. dybał, m. (2008), efektywność inwestycyjna funduszy emerytalnych, cedewu.pl, warszawa. karpio, a., żebrowska-suchodolska d. (2010), stabilność strategii inwestycyjnych fio akcji w latach 2003–2009, uniwersytet szczeciński, zeszyty naukowe nr 612 „finanse, rynki finansowe, ubezpieczenia”, 28, 409–418. karpio, a., żebrowska-suchodolska, d. (2011), stability of strategies of polish open-end investment funds investing in global markets during the financial crisis, acta scientiarum polonorum, oeconomia, wydawnictwo sggw w warszawie, 10(3), 51–59. karpio, a., żebrowska-suchodolska, d. (2014), ocena zarządzania portfelami otwartych funduszy inwestycyjnych z wykorzystaniem różnych miar efektywności inwestycyjnej, metody matematyczne i informatyczne w finansach i ubezpieczeniach, metody 2012, 207, 136–147. karpio a., żebrowska-suchodolska d. (2014a), the influence of pension funds on the polish capital market, metody ilościowe w badaniach ekonomicznych, wydawnictwo sggw, 15(1), 50–57. karpio, a., żebrowska-suchodolska, d. (2016), polish open-end pension funds performance and its persistency, acta scientiarum polonorum, oeconomia, wydawnictwo sggw, 15(1), 15–25. kompa, k., wiśniewski t. (2015), dywersyfikacja portfeli otwartych funduszy emerytalnych – kaprys czy konieczność? finanse, rynki finansowe, ubezpieczenia. zeszyty naukowe uniwersytetu szczecińskiego, 75, 245–255, doi: http://dx.doi.org/10.18276/frfu.2015.75-20. kompa, k., witkowska, d. (2014), pension funds in poland: efficiency analysis for years 1999–2013, dynamic econometric models, 14, 105–124, doi: http://dx.doi.org/10.12775/dem.2014.006. kompa, k., witkowska, d. (2015), pension system in poland: performance of pension funds, estudios de economia aplicada, 33(3), 965–984. kompa, k., witkowska, d. (2015), transformation of polish pension system: government manipulations instead of improvement, in: montero lorenzo, j.m., jiménez, j.m., salido, r.m. (eds.) sostenibilidad y suficiencia de los sistemas de pensiones (sustainability and sufficiency of pensions systems), anales de economía aplicada
 (xxix), 836–846, cuenca, asepelt, http://www.asepelt.org/modules.php?name=content&pa =showpage&pid=67. kompa, k., witkowska, d. (2016), efficiency of private pension funds in poland, aestimatio, the ieb international journal of finance, 12, 48-65, doi: http://dx.doi.org/10.5605/ieb.12.3. marcinkiewicz, e. (2009), analiza stóp zwrotu ofe w: chybalski f. (red.), otwarte fundusze emerytalne w polsce. analiza działalności inwestycyjnej, finansów oraz decyzji członków, c. h. beck, warszawa, 135–148. javascript:__dopostback('mytab$dgpublikacje$_ctl3$slinktyt_czasopisma','') javascript:__dopostback('mytab$dgpublikacje$_ctl3$slinktyt_czasopisma','') performance of pension funds and stable growth open investment funds… dynamic econometric models 16 (2016) 117–131 131 mikulec, a. (2004), ocena efektywności inwestowania otwartych funduszy emerytalnych, wiadomości statystyczne, 9, 26–39. ostrowska, e. (2003), efektywność funduszy inwestycyjnych na polskim rynku finansowym – wskaźniki sharpe’a, treynora i jensena, [w] prace naukowe akademii ekonomicznej we wrocławiu, inwestycje finansowe i ubezpieczenia – tendencje światowe a polski rynek, red. nauk. jajuga, k., ronka-chmielowiec, w., wydawnictwo akademii ekonomicznej we wrocławiu, wrocław. ostrowska, k. (2013), jakie opłaty w ofe, moja emerytura, serwis dziennika rzeczpospolita, www.rp.pl/artykul/975301-jakie-oplaty-w-ofe-.html#ap-1. perez, k. (2012a), efektywność funduszy inwestycyjnych. difin, warszawa. perez, k. (2012b), persystencja stóp zwrotu polskich funduszy inwestycyjnych, finanse, komitet nauk o finansach pan, 1(5), 81–113. podstawka, m. (2005), podstawy finansów teoria i praktyka, wyd. sggw, warszawa. tarczyński, w., witkowska, d. and kompa, k. (2013), współczynnik beta. teoria i praktyka, pielaszek research, warszawa, https://books.google.pl/books/about/wsp%c3%93%c5%81czynnik_beta_teoria_i_pr aktyka.html?id=znwzagaaqbaj&redir_esc=y. witkowska, d. (2009), efektywność wybranych funduszy akcyjnych w latach 2005–2007, ekonomika i organizacja gospodarki żywnościowej, zeszyty naukowe szkoły głównej gospodarstwa wiejskiego w warszawie, 74, 39–61. witkowska d., kompa k., grabska m. (2009), badanie informacyjnej efektywności rynku w formie silnej na przykładzie wybranych funduszy inwestycyjnych, metody ilościowe w badaniach ekonomicznych, warszawa, 10, 265–286. wojciechowski, w. (2011), reforma emerytalna: fakty i liczby zamiast błędów i mitów, liberte nr 22, 14.01.2011, http://liberte.pl/reforma-emerytalna-fakty-i-liczby-zamiastbledow-i-mitow. zamojska, a. (2012), efektywność funduszy inwestycyjnych w polsce. studium teoretycznoempiryczne, c. h. beck, warszawa. efektywność funduszy emerytalnych i otwartych funduszy inwestycyjnych stabilnego wzrostu w czasie zmian w polskim systemie emerytalnym z a r y s t r e ś c i. warunki funkcjonowania funduszy emerytalnych (ofe) zostały zasadniczo zmienione w latach 2011-2014. celem artykułu jest badanie, czy zmiany te wpływają na efektywność funduszy emerytalnych oraz porównanie efektywności tych funduszy ze efektywnością otwartych funduszy inwestycyjnych stabilnego wzrostu (fio). analizie poddano wybrane fundusze w latach 2009-2015. stwierdzono, że w badanym okresie ofe osiągały lepsze wyniki niż porównywane fio. zarazem modyfikacje zasad działania ofe spowodowały wzrost ryzyka i zmniejszenie efektywności inwestycyjnej portfeli tych funduszy. s ł o w a k l u c z o w e: fundusze emerytalne, otwarte fundusze inwestycyjne stabilnego wzrostu, efektywność inwestycyjna, model sharpe'a, capm, wskaźniki sharpe’a, treynora i jensena. microsoft word 01_szulc_e.docx dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 elżbieta szulc nicolaus copernicus university in toruń identification of the structures of spatial and spatiotemporal processes and a problem of data aggregation a b s t r a c t. the paper concerns the measurement of the dependence between economic spatial and also spatio-temporal processes at various levels of data aggregation. the considerations refer to the investigations confirming efficiency of the so-called quasi-congruent spatial model as a tool of the measurement of the dependence between economic processes. the main problem of the paper is to discuss such a description of the spatial and spatio-temporal connections, which would be adequate to express the autodependence of the investigated processes. the application of the economic distance, which characterizes the similarity of the regions on the ground of the values of the analyzed processes, is proposed. k e y w o r d s: quasi-congruent spatio-temporal model, autodependence, connectivity matrix, economic distance. introduction spatial models are estimated for the given spatial units, such as: countries, regions, provinces (voivodeships), census tracts and so on. data, usually spatially correlated, collected for these units create the established configurations which determine quantification of the spatial autodependence. configurations of the aggregated data simplify the connections of the smaller spatial units and then they may lead to the simplified description of some properties of the economic spatial processes and the dependence between them1. this is also the case of the spatio-temporal processes. in the paper attention is paid to the spatial aspect of the data aggregation. the question of the time aggregation is not taken into account. since the paper concerns the investigation of the spatial processes in time, some dynamic elements were introduced to the analysis. 1 see e.g. arbia (1988) for an early discussion of some of the issues. elżbieta szulc 6 the presented considerations refer to the previous investigations carried out on the basis of generated data (szulc et al., 2011) and empirical data for one year (szulc, 2011). at present the time range of the analysis has been extended to six years. in szulc et al. (2011) there was formulated and verified the thesis that finding out the value of the parameter which measures the actual influence of one spatial process on another spatial process needs to identify the properties and componential structures of the processes and to take them explicitly into account in the spatial regression model. formulation of the hypothesis mentioned above means the recommendation of construction of the quasi-congruent spatial model2 in which, thanks to taking into account the components describing the internal structure of the processes, the filtration of the processes takes place and the estimation of the dependence is more precise, as not disturbed by the so-called indirect influences. finally, on condition of the precise filtration, the regression coefficients in the quasicongruent spatial models estimated at the individual level should be the same (identical) as the ones estimated at the aggregated level. the hypothesis and its consequences may be extended to the case of the analysis of the spatio-temporal processes. in the quoted work there was shown, that the quasi-congruent spatial model was an effective tool of the measurement of the dependence between spatial processes (actual value of the regression coefficient), if the model description of the structures of the processes was correct and sufficient. the differences between the values of the regression coefficient obtained for the quasicongruent models before and after the aggregation of the data were dependent on the structures of the separate processes. for the processes only with the trend structure, regardless of the degrees of the trend and of the values of their parameters, the regression coefficients estimated at the various levels of data aggregation did not differ from one another significantly and oscillated around the actual value. on the contrary, in terms of the spatial autocorrelation, the value of the regression coefficient after the aggregation was overestimated, when y > x , and underestimated, when y < x 3. the regression coefficients in the spatial quasi-congruent models for the primary data and for the aggregated data with the spatial autocorrelation will differ in value first of all because the connectivity matrix in the aggregated model does not reflect the connections of the data before the aggregation. thus, the autodependence existing at the individual level cannot be precisely described at the aggregated level. moreover, the quantification of the spatial 2 the procedure of constructing such a model was described in the quoted work. it was also explained what the quasi-congruency of the spatial model meant. 3 y and x denote respectively the autoregression coefficients of the explaining and explanatory process. identification of the structures of spatial and spatio-temporal processes… 7 dependence by means of the standard matrixes of the connections, within the scheme of the structure of the spatial connections existing at the given level of data aggregation as well, may be inadequate. this is the situation, when the strength of the connection between the spatial units depends not only on the physical distance and the fact of being contiguous, but also on e.g. the economic similarity of the units. looking for the more precise measurement of the spatial autodependence with the help of other than the standard matrixes of connections, in szulc (2011) there was investigated the question of how the application of the certain matrix of the economic distance would influence the estimation of the dependence between the spatial processes. in the quoted investigation the difference between the parameters of the models for the data before aggregation and for the aggregated data slightly decreased, but it should not be treated as regularity. all the more it should not be expected that the difference will disappear because the proposed matrix is still based on the common border criterion. the present investigation is the continuation of the works quoted above. analogical analyses are carried out for the successive years in the period 2004– 2009. this approach allows to verify whether the observed regularities for 2007 (see szulc, 2011) are the same for other years. moreover, the spatio-temporal models are considered. in the investigation both the standard connectivity matrixes and the matrixes of the economic distance are used. the economic distance matrixes in the spatio-temporal models have the dynamic features in the sense that for each of the years another connectivity matrix is applied. 1. quantification of the spatial connections and the measurement of the spatial autodependence the basis of measurement of the spatial (auto)dependence is establishing the connections among spatial units. the spatial neighbours can be defined in a number of ways. the spatial connections are represented in the form of connectivity matrix w. assuming that there are n regions (spatial units), the matrix has as many rows and columns as there are the regions, i.e. n by n matrix w is considered. each row of the matrix contains non-zero elements in columns which correspond to the connected objects (the so-called neighbours), according to the received criterion. furthermore, the given object cannot be connected to itself, i.e. it cannot be a neighbour of itself, so wij = 0 for all i = j. thus, the diagonal elements of w are zeros. starting point in establishing the spatial connections is the binary matrix s, with elements:          ,if,0 if,1 inj inj sij (1) elżbieta szulc 8 where n(i) denotes the set of neighbours of spatial units i. the neighbours usually are established according to the common border criterion. then the rows in s are normalized, so that the row sums equal 1, as a result of dividing each entry on a row by the sum of the row values (the socalled row standardization to one). so, if  iddiagd , where    n j ij i s d 1 1 , i = 1, 2, …, n, then w = ds and 1 1   n j ijw . the weights wij which are established in this way signify that each j-th neighbour of the i-th spatial units is treated identically, and the greater the strength of its interactions with the neighbours is, the less neighbours it has. another is the case when weights wij are the functions of some properties of cells of the lattice, e.g. of the length of the common border, of the distance between the centres of the regions, or of other measures of similarity between the regions, e.g. of the so-called economic distance between them as well. various types of weights wij may be pointed out according to the established criteria (see haining, 2005, p. 83–84 ). as above, for all units i and j wij  0, when i  j, and wij = 0, when i = j. such weights create the generalized matrix of neighbourhood, which is row standardized to one by transforming its elements according to the formula     n j ij ij ij w w w 1 . in the paper the connections between the regions will be defined by using two approaches. the first one is traditional. in it the standard matrix of neighbourhood with the equal weights is used. the second one consists in that in the matrix of connections the economic distance (the essence of which is to establish similarity of the regions on the basis of the value of the analyzed economic process and the economic processes which determine its changeability) is taken into consideration. in the general form the economic distance between regions i and j is expressed as follows (see pietrzak, 2010, p. 75):                             ,for,0 for,|| 1 1 ... || 1 1 || 1 1 || 0 ,, 0 ,2,2 20 ,1,1 1 1 2 3 1 21 ji jixx k xx k xx k yy d n n k z ztnjztni n k z ztjzti k z ztjztiji ij   (2) identification of the structures of spatial and spatio-temporal processes… 9 where: yi, yj – values of the investigated spatial economic process at spatial location i and j respectively; xli, xlj (l = 1, 2, …, n) – values of explanatory processes, which determine changeability of the explained process at spatial locations as above; k1, k2, …, kn – constants denoting time lags for the considered processes; 1, 2, …, n+1 – normalizing constants. in this approach, the elements wij of the matrix of neighbourhood will be as follows:        . , 1 ji ji dw ijij if0, if (3) finally, as a result of row standardization to one the matrix of neighbourhood based on the economic distance is obtained. the more exact, because based on the economic criterion, measurement of the spatial autodependence should effect more precise filtration of the investigated processes (explained and explanatory), and as a consequence, lead to the appropriate estimation of the regression dependence. in other words, the more precise description of the spatial autodependence the more efficient filtration of the processes and the measurement of the dependence between the processes takes place at the level of the data which are more and more cleaned from the autodependence. the pure dependence between the processes would be the same, apart from the level of the data aggregation. in the approach proposed the effect of spatial aggregation of the data as the change of the regression parameter in the spatial model will be not eliminated, because the aggregated matrix of neighbourhood has still referred to another one than before aggregation of the data configuration. 2. empirical example the dependence between the unemployment rate and investment outlays of enterprises (in pln) in poland in years 2004–2009 across poviats (districts) and sub-regions was investigated. the data are from the internet source: http://www.stat.gov.pl. the conception of the spatial and spatio-temporal quasicongruent model was used. the trend-autoregressive structures of the individual processes at the both levels of the data aggregation were investigated by constructing the appropriate fundamental models4. for describing the trend structure the models of the spatial and spatio-temporal trend were used. the autoregressive structure was defined in two versions, with the use of (1) the 4 the fundamental model denotes here the model describing the componential structure of the spatial and spatio-temporal process. the considerations are accompanied by the foundation, that in economic processes at least two components are potentially present: the trend and autoregressive connections. elżbieta szulc 10 standard matrix with the equal weights – variant i and (2) the matrix of the economic distance (different weights) – variant ii. then the quasi-congruent models describing the dependence between the investigated processes were built and the obtained results were compared. for the investigated processes the following symbols were taken:  pty – spatial process of unemployment (unemployment rate in the region at spatial location p = [p1, p2] in established year t),  ptx – spatial process of investments (investment outlays per capita in the region at spatial location and time – as above),  ty ,p – spatio-temporal process of unemployment,  tx ,p – spatio-temporal process of investments. investigation of the dependence in years – variant i investigation of the dependence between the unemployment rate and the investment outlays per capita across poviats in the successive years allows to ascertain the following: 1. the models of unemployment and investments contained spatial trends of the first degree in all the years and spatial autodependence of the first order, except for 2004 and 2005, where the spatial autodependence in investments appeared statistically insignificant. 2. the quasi-congruent models in their full version took into consideration the trend structure and the autoregressive structure of the separate processes. thus, they were the models of the following form:          ppwppwp ttttt xxyppy   20111000 .(4) 3. after reduction of the statistically insignificant components, for all the years the models with analogous structures were obtained, i.e. the models of the form:        pppwp tttt xyppy   20111000 , (5) except for 2007, where the spatial trend was also reduced. the theoretical notation of the model for 2007 is following:        pppwp 444004   xyy . (6) in the investigation of the dependence between the unemployment and the investment outlays per capita across sub-regions in the successive years the following was established: 1. the fundamental models of unemployment and investments did not contain any significant spatial trend, but they had autoregressive components. only in 2004 and 2009 the spatial autoregression in investments appeared statistically insignificant. identification of the structures of spatial and spatio-temporal processes… 11 2. in the face of settlement 1. the full quasi-congruent models for 2005–2008 took the form:          ppwppwp ttttt xxyy   00 , (7) and for 2004 and 2009:        pppwp tttt xyy   00 . (8) 3. the reduction of the statistically insignificant components led to the models of the form (8), for all the years. table 1 presents specification of the values of the chosen parameters of the reduced quasi-congruent models for poviats and sub-regions. the comparison of the values leads to the statement, that in the successive years the regression coefficients estimated at the level of poviats are, as regards the absolute value, smaller than the ones obtained for sub-regions. in turn, the coefficients of autoregression for poviats are bigger than for sub-regions. table 1. coefficients of regression and autoregression in the models estimated at the level of poviats and sub-regions – variant i level of aggregation years 2004 2005 2006 2007 2008 2009 coefficients of regression  poviats -1.6371 -1.6489 -1.3799 -0.8802 -0.6379 -0.5991 sub-regions -2.5394 -2.5007 -1.9284 -1.4969 -1.2207 -1.3018 coefficients of autoregression  poviats 0.6833 0.6747 0.6724 0.6851 0.6632 0.6381 sub-regions 0.5112 0.5139 0.4353 0.3370 0.3544 0.4321 note: all the calculations for the investigation presented in the paper were done with the use of r-cran. investigation of the dependence in years – variant ii the influence of investment outlays on unemployment rate was also investigated by using the matrix of economic distance to define spatial autodependence in the considered processes. the matrix with differentiated weights, taking into account similarity of poviats/sub-regions established on the ground of the values of unemployment and investment outlays in the connected regions (i.e. in poviats, and then in sub-regions) was used. thus, the economic distance was defined as follows:       ,for0, for,|||| 21 ji jixxyy d jijiij  (9) where 1 = 2 = 0.9 5. in this case, for poviats, it was established, that: 5 the value has been established on the ground of the separated investigations. elżbieta szulc 12 1. the fundamental models of unemployment and investments identified at the level of poviats, for all the years, contained spatial trends of the 1st degree and the autodependence of the 1st order. it is the analogous result as that one obtained in variant i. 2. also, likewise in variant i of the analysis the full quasi-congruent models contained spatial trends, spatially lagged unemployment, current investments and spatially lagged investments (see formula (4)). 3. after reduction of the statistically insignificant components, for 2004–2007 the spatial autoregressive regression models were obtained (see, formula (8)). for 2008 from the full model only the spatial trend was eliminated, yet for 2009 the full model did not need any component to be reduced. the analysis of the data at the level of sub-regions led to the following settlements: 1. the unemployment and investments in all the years were described with the help of the pure autoregressive spatial models (without trend). 2. it led to the quite simple structure of the full quasi-congruent models, i.e.:          ppwppwp ttttt xxyy   00 . (10) 3. for all the years there were obtained the reduced models of the form:        pppwp tttt xyy   00 . (11) table 2. coefficients of regression and autoregression in the models estimated at the level of poviats and sub-regions – variant ii level of aggregation years 2004 2005 2006 2007 2008 2009 coefficients of regression  poviats -1.2893 -1.3225 -1.1025 -0.7173 -0.4401 -0.4075 sub-regions -2.3358 -2.2648 -1.7279 -1.3171 -1.0979 -1.1736 coefficients of autoregression  poviats 0.8158 0.8026 0.7954 0.7727 0.7506 0.7110 sub-regions 0.6170 0.5938 0.5506 0.5046 0.4829 0.5472 table 2 presents specification of the values of the chosen parameters of the reduced quasi-congruent models for poviats and sub-regions. comparing the results of the analysis for poviats and sub-regions, which are presented in table 2, it should be stated that in all the years the regression coefficients estimated at the level of poviats, are as regards the absolute value smaller than their estimates at the level of sub-regions. however, the coefficients of autoregression in the considered years are always bigger for poviats than for sub-regions. identification of the structures of spatial and spatio-temporal processes… 13 spatio-temporal models – variant i constructing the spatio-temporal models the analogous structure of components as in the case of the spatial models was established. in the analysis the deterministic spatio-temporal trends were taken into consideration. identification of the spatio-temporal autodependence was done with the use of the matrix of the form:              6 2 1 w00 0w0 00w w     , (12) where 621 ... www  – standard matrixes of the spatial connections, the same for all the considered years. with specification of the fundamental models it was confirmed, that both the unemployment rate and the investment outlays showed the spatio-temporal trends of the 1st degree and spatial autodependence of the 1st order. the components were observed as well at the level of poviats as at the level of subregions and they were taken into account in the quasi-congruent spatio-temporal models constructed afterwards. since the full models contained insignificant variables they were eliminated and in this way the reduced models were obtained. table 3 contains the characteristics of the quasi-congruent models obtained in the investigation at the level of poviats, yet table 4 presents the analogous specification for sub-regions. the reduced models do not contain the spatially lagged investments. this means that the unemployment in the given poviat does not depend on the investment outlays in the neighbouring poviats. the current unemployment in the given poviat is influenced by the current investment outlays in the same poviat (each thousand pln spent causes the unemployment rate to decrease on the average by 0.93 percentage point) and by the level of unemployment in the neighbouring poviats (the change of the unemployment rate in the given poviat by about 0.69 percentage point is connected with the one percentage change of the unemployment rate in the neighbouring poviats). likewise, the unemployment in the given sub-region does not depend on the investment outlays in the neighbouring sub-regions. the current unemployment rate in the given sub-region is influenced by the current investment outlays in this sub-region (each thousand pln spent causes the unemployment rate to decrease by about 1.74 percentage point) and by unemployment rate in the neighbouring sub-regions (the increase of the unemployment rate in the neighbouring sub-regions by one percentage point is connected with the increase of the unemployment rate in a given sub-region by about 0.39 percentage point). elżbieta szulc 14 the coefficient which measures the influence of the investment outlays on the unemployment rate in a sub-region differs from the analogous coefficient which is estimated at the level of poviats. the main reason of the difference observed is the spatial autocorrelation in unemployment and also in investments. the value of coefficient  for sub-regions is bigger than for poviats. in turn, autoregression coefficient , which measures the connections among the unemployment rates in the neighbouring areas at the level of subregions is visibly smaller than the analogous parameter estimated for poviats. the results are analogous to the ones obtained previously for the spatial models constructed for the successive years. they are also convergent with the results of the previous analyses, pointed out in the introduction, which were obtained on the basis of the generated data (especially, when y > x, the coefficient of regression, estimated on the basis of the aggregated data, is overestimated). table 3. characteristics of quasi-congruent models for poviats – variant i full model:          ttxtxtytppty ,,,,, 00120101100000 ppwppwp   parameters estimates of parameters standard deviations statistics z pr(>|z|) 000 100 010 001   7.6305 -0.1892 0.3472 -0.3433 -0.9315 -0.0332 0.6915 0.0652 0.0664 0.0665 0.0504 0.1085 11.0347 -2.9025 5.2255 -5.1612 -18.4788 -0.3060 0.0000 0.0037 0.0000 0.0000 0.0000 0.7596  = 0.6872 test lr: 967.48; p-value: 0.0000 wald statistic: 1409.3; p-value: 0.0000 aic: 13592 (aic for lm: 14557) residual autocorrelation test lm: 4.2911; p-value: 0.0383; moran statistic: -0.8026; p-value: 0.2111 reduced model:        ttxtytppty ,,,, 00120101100000 pppwp   parameters estimates of parameters standard deviations statistics z pr(>|z|) 000 100 010 001  7.5202 -0.1816 0.3464 -0.3472 -0.9334 0.6092 0.0613 0.0664 0.0647 0.0499 12.3439 -2.9644 5.2171 -5.3681 -18.6968 0.0000 0.0030 0.0000 0.0000 0.0000  = 0.6890 test lr: 1090.4; p-value: 0.0000 wald statistic: 1538; p-value: 0.0000 aic: 13590 (aic for lm: 14679) residual autocorrelation test lm: 3.7569; p-value: 0.0526 test moran statistic: -0.8867; p-value: 0.1876 identification of the structures of spatial and spatio-temporal processes… 15 table 4. characteristics of quasi-congruent models for sub-regions – variant i full model:          ttxtxtytppty ,,,,, 00120101100000 ppwppwp   parameters estimates of parameters standard deviations statistics z pr(>|z|) 000 100 010 001   17.2312 -0.0459 0.0482 -0.5223 -1.6817 -0.6643 2.0420 0.0156 0.0151 0.1628 0.1431 0.3645 8.4384 -2.9479 3.2023 -3.2082 -11.7497 -1.8225 0.0000 0.0032 0.0014 0.0013 0.0000 0.0684  = 0.3541 test lr: 30.415; p-value: 0.0000 wald statistic: 35.928; p-value: 0.0000 aic: 2279.5 (aic for lm: 2308) residual autocorrelation test lm: 6.2037; p-value: 0.0127 moran statistic: 0.6251; p-value: 0.2660 reduced model:        ttxtytppty ,,,, 00120101100000 pppwp   parameters estimates of parameters standard deviations statistics z pr(>|z|) 000 100 010 001  14.8309 -0.0321 0.0522 -0.6181 -1.7460 1.5208 0.0139 0.0149 0.1565 0.1365 9.7522 -2.3164 3.4839 -3.9504 -12.7951 0.0000 0.0205 0.0005 0.0000 0.0000  = 0.3883 test lr: 40.419; p-value: 0.0000 wald statistic: 50.843; p-value: 0.0000 aic: 2281 (aic for lm: 2319.5) residual autocorrelation test lm: 0.2836; p-value: 0.5943 moran statistic: -0.1386; p-value: 0.4449 spatio-temporal models – variant ii in the discussed variant for the purpose of quantification of the spatiotemporal connections there was used the matrix of the form:                    6 2 1 w00 0 0w0 00w w     , (13) where: 621 ...   www – matrixes of spatial connections, taking into account the economic distance between regions, different for the successive years. elżbieta szulc 16 various specifications of the fundamental models of unemployment and investments and the models of the dependence between the investigated processes, which resulted from them, were used. tables 5–6 present the results of the investigation, in which the time lags of the spatial dependence are not taken into account. tables 7–8, on the contrary, refer to the investigation, in which such the lags were considered. the structure of the full models results from the structures of the individual processes described by the fundamental models. in the first specification the fundamental models with the spatio-temporal trend and with the spatial autoregression were established. in the full model identified at the level of poviats all the components were statistically significant. other than in variant i, the measurement of spatial (auto)dependence caused the spatially lagged investments had not been removed from the model. it denotes some connection of the current unemployment rate in a given poviat with the investment outlays in the neighbouring poviats. in turn, in the analysis at the level of sub-regions the reduced model without the spatially lagged investments was obtained. like in variant i of the investigation, the difference in the estimation of the coefficient measuring the influence of investments on unemployment at the level of poviats and sub-regions, was observed. for poviats its value amounted to circa -0.69, while for sub-regions to circa -1.52. also the coefficient of autoregression, as in variant i, is bigger for poviats (about 0.76) than for subregions (about 0.53). table 5. characteristics of quasi-congruent models for poviats – variant ii a) full model:          ttxtxtytppty ,,,,, 00120101100000 ppwppwp    parameters estimates of parameters standard deviations statistics z pr(>|z|) 000 100 010 001   5.9703 -0.1403 0.2307 -0.1419 -0.6937 -0.4661 0.5611 0.0548 0.0548 0.0537 0.0453 0.1219 10.6413 -2.5583 4.2073 -2.6408 -15.3140 -3.8231 0.0000 0.0105 0.0000 0.0083 0.0000 0.0001  = 0.7595 test lr: 1467.6; p-value: 0.0000 wald statistic: 2912.2; p-value: 0.0000 aic: 12943 (aic for lm: 1.4408) residual autocorrelation test lm: 68.725; p-value: 0.0000 moran statistic: -2.9692; p-value: 0.0010 in the more extended specification of the structures of the investigated processes the time lags of the spatial autodependence were additionally taken into account. the characteristics of the full quasi-congruent model and of the identification of the structures of spatial and spatio-temporal processes… 17 reduced model for poviats are presented in table 7, whereas for sub-regions – in table 8. table 6. characteristics of quasi-congruent models for sub-regions – variant ii a) full model:          ttxtxtytppty ,,,,, 00120101100000 ppwppwp    parameters estimates of parameters standard deviations statistics z pr(>|z|) 000 100 010 001   12.3451 -0.0240 0.0315 -0.3442 -1.4485 -0.4057 1.7497 0.0143 0.0138 0.1466 0.1495 0.3925 7.0555 -1.6530 2.2901 -2.3488 -9.6917 -1.0337 0.0000 0.0983 0.0220 0.0188 0.0000 0.3013  = 0.5178 test lr: 77.168; p-value: 0.0000 wald statistic: 117.56; p-value: 0.0000 aic: 2235.4 (aic for lm: 2310.5) residual autocorrelation test lm: 37.529; p-value: 0.0000 moran statistic: -0.7447; p-value: 0.2282 reduced model:        ttxtytppty ,,,, 00120101100000 pppwp    parameters estimates of parameters standard deviations statistics z pr(>|z|) 000 100 010 001  11.1540 -0.0161 0.0343 -0.4002 -1.5245 1.3366 0.0128 0.0135 0.1362 0.1266 8.3450 -1.2578 2.5450 -2.9374 -12.0426 0.0000 0.2085 0.0109 0.0033 0.0000  = 0.5315 test lr: 86.996; p-value: 0.0000 wald statistic: 137.67; p-value: 0.0000 aic: 2234.5 (aic for lm: 2319.5) residual autocorrelation test lm: 14.549; p-value: 0.0001 moran statistic: -1.1656; p-value: 0.1219 in the full quasi-congruent model of unemployment for poviats the following were taken into consideration: spatio-temporal trend, current unemployment rate in the neighbouring poviats, unemployment rate from the previous period in the neighbouring poviats, current investment outlays in the given poviat and in the neighbouring poviats and also investment outlays in the neighbouring poviats from the previous period. most of the mentioned components were statistically significant. only the spatial trend appeared insignificant. after elimination of it the reduced model was obtained. starting from the analogous structure of the full model for sub-regions, as a result of eliminating the statistically insignificant components the reduced elżbieta szulc 18 model was obtained. in the model there are the following components: current unemployment in the neighbouring sub-regions, current investment outlays in a given sub-region, current and time lagged investment outlays in the neighbouring sub-regions. coefficient of regression () estimated at the level of poviats amounted to circa -0.63, and at the level of sub-regions – to circa -1.21. the coefficients of autoregression amounted to circa 0.45 and 0.33 respectively. table 7. characteristics of quasi-congruent models for poviats – variant ii b) full model:              ttxtxtx tytytppty ,1,,, 1,,, 00120101100000 ppwpwp pwpwp       parameters estimates of parameters standard deviations statistics z pr(>|z|) 000 100 010 001     -1.6307 0.1506 -0.0589 0.4975 0.4882 -0.6319 -0.7915 0.9709 0.8154 0.0632 0.0621 0.0794 0.0253 0.0461 0.1780 0.1823 -1.9998 2.3803 -0.9496 6.2677 19.2694 -13.7052 -4.4470 5.3247 0.0455 0.0173 0.3423 0.0000 0.0000 0.0000 0.0000 0.0000  = 0.4466 test lr: 189.85; p-value: 0.0000 wald statistic: 351.93; p-value: 0.0000 aic: 10456 (aic for lm: 10643) residual autocorrelation test lm: 3.4071; p-value: 0.0649 moran statistic: 0.7442; p-value: 0.2284 reduced model:              ttxtxtx tytytty ,1,,, 1,,, 001000 ppwpwp pwpwp       parameters estimates of parameters standard deviations statistics z pr(>|z|) 000 001     -0.2661 0.4718 0.4622 -0.6332 -0.8660 0.8658 0.5758 0.0775 0.0233 0.0462 0.1757 0.1784 -0.4622 6.0847 19.8427 -13.7195 -4.9275 4.8538 0.6439 0.0000 0.0000 0.0000 0.0000 0.0000  = 0.4483 test lr: 191.29; p-value: 0.0000 wald statistic: 359.94; p-value: 0.0000 aic: 10458 (aic for lm: 10648) residual autocorrelation test lm: 7.136; p-value: 0.0076 moran statistic: 1.0981; p-value: 0.1361 identification of the structures of spatial and spatio-temporal processes… 19 table 8. characteristics of quasi-congruent models for sub-regions – variant ii b) full model:              ttxtxtx tytytppty ,1,,, 1,,, 00120101100000 ppwpwp pwpwp       parameters estimates of parameters standard deviations statistics z pr(>|z|) 000 100 010 001     4.5273 0.0047 -0.0006 0.2194 0.4392 -1.1887 -1.6994 1.7360 2.8085 0.0165 0.0165 0.2483 0.0837 0.1569 0.5483 0.5679 1.6120 0.2861 -0.0371 0.8838 5.2446 -7.5757 -3.0992 3.0568 0.1070 0.7748 0.9704 0.3768 0.0000 0.0000 0.0019 0.0022  = 0.3148 test lr: 14.607; p-value: 0.0001 wald statistic: 26.774; p-value: 0.0000 aic: 1820.7 (aic for lm: 1833.3) residual autocorrelation test lm: 0.2322; p-value: 0.6299 moran statistic: 0.2175; p-value: 0.4139 reduced model:              ttxtx txtytyty ,1,, ,1,,, 000 ppwpw ppwpwp       parameters estimates of parameters standard deviations statistics z pr(>|z|) 000     6.3379 0.3855 -1.2131 -1.7387 1.8079 1.4305 0.0566 0.1491 0.5353 0.5220 4.4306 6.8075 -8.1375 -3.2479 3.4631 0.0000 0.0000 0.0000 0.0012 0.0005  = 0.3256 test lr: 16.022; p-value: 0.0000 wald statistic: 29.457; p-value: 0.0000 aic: 1815.6 (aic for lm: 1829.6) residual autocorrelation test lm: 1.1578; p-value: 0.2819 moran statistic: 0.4223; p-value: 0.3364 conclusions in the modelling of the dependence of spatial processes and also spatiotemporal processes the principle of congruency of the structures of the separate processes should be applied. limited efficiency of the spatial and spatiotemporal quasi-congruent models as the tools of discovering the real dependence between the processes (in particular the differences in estimating the dependence at different levels of data aggregation observed) result, first of elżbieta szulc 20 all, from imperfection of the measurement of autodependence in the investigated processes. references arbia, g. (1988), spatial data configuration in statistical analysis of regional economic and related problems, kluwer academic press, dordrecht. haining, r. (2005), spatial data analysis. theory and practice, cambridge university press, 3th ed., cambridge. pietrzak, m. b. (2010), dwuetapowa procedura budowy przestrzennej macierzy wag z uwzględnieniem odległości ekonomicznej (two-stage procedure of building a spatial weight matrix with the consideration of economic distance), oeconomia copernicana, 1, wydawnictwo umk, toruń, 65–78. szulc, e., müller-frączek, i., pietrzak, m. b. (2011), modelowanie zależności między ekonomicznymi procesami przestrzennymi a poziom agregacji danych (modeling of dependence between economic spatial processes and the level of data aggregation), in suchecka j. (ed.), ekonometria przestrzenna i regionalne analizy ekonomiczne (spatial econometrics and regional economic analyses), folia oeconomica, wydawnictwo uniwersytetu łódzkiego, 327–344. szulc, e. (2011), ekonometryczna identyfikacja struktur procesów przestrzennych wobec problemu agregacji danych (econometric identification of the structures of spatial processes and a problem of data aggregation), acta universitatis nicolai copernici, ekonomia xlii – nauki humanistyczno-społeczne, z. 392, toruń, in press. identyfikacja struktur procesów przestrzennych i przestrzennoczasowych wobec problemu agregacji danych z a r y s t r e ś c i. artykuł dotyczy odkrywania zależności między ekonomicznymi procesami przestrzennymi a także przestrzenno-czasowymi, gdy są one mierzone na różnych poziomach agregacji danych. rozważania nawiązują do badań, potwierdzających efektywność tzw. quasizgodnego modelu przestrzennego jako narzędzia pomiaru rzeczywistych zależności między procesami. wykorzystuje się koncepcję modelu quasi-zgodnego także w odniesieniu do procesów przestrzenno-czasowych. poszukuje się opisu powiązań przestrzennych i przestrzenno-czasowych adekwatnego do wyrażenia autozależności w badanych procesach. proponuje się wykorzystanie odległości ekonomicznej, mierzącej podobieństwo regionów na podstawie wartości analizowanych procesów. s ł o w a k l u c z o w e: przestrzenny i przestrzenno-czasowy model quasi-zgodny, autozależności, macierz sąsiedztwa, odległość ekonomiczna. d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2016.002 vol. 16 (2016) 21−35 submitted october 20, 2016 issn (online) 2450-7067 accepted december 10, 2016 issn (print) 1234-3862 marcin fałdziński, magdalena osińska * volatility estimators in econometric analysis of risk transfer on capital markets  a b s t r a c t. the purpose of the research is to compare the performance of different volatility measures while used in testing for causality in risk between several emerging and mature capital markets. the following volatility estimators are considered: parkinson, garman-klass, rogers-satchell, garman-klass-yang-zhang and yang-zhang and the ar-garch(1,1)-t model. additionally, the extreme value theory is also applied. several emerging capital markets are checked for being the source of the risk for both emerging and developed markets. the group of emerging markets includes the most intensively growing economies in the world. the final results are such as the number of relationships between the markets is considerably lower when the methods taken from the extreme value theory are used. k e y w o r d s: causality in risk, extreme value theory, growing emerging economies, risk transfer, volatility j e l classification: g15; q47. introduction the purpose of the research is to compare the performance of different volatility measures while used in testing for causality in risk between several emerging and mature capital markets. the problem considered in the report * correspondence to: marcin fałdziński, nicolaus copernicus university, faculty of economic sciences and management, 13a gagarina street, 87-100 toruń, poland, e-mail: marf@umk.pl; magdalena osińska, nicolaus copernicus university, faculty of economic sciences and management, 13a gagarina street, 87-100 toruń, e-mail: emo@umk.pl.  financial support of the national science centre in poland (umo2015/17/b/hs4/01000) is gratefully acknowledged. marcin fałdziński, magdalena osińska dynamic econometric models 16 (2016) 21–35 22 is rather complex from methodological perspective because it includes: a comparison of several estimators of volatility such as parkinson (1980), garman and klass (1980), rogers and satchell (1991), garman, klass, yang and zhang (1991) and yang and zhang (2000) while value at risk is calculated, a comparison of the mentioned estimators when extreme value theory (mcneil and frey, 2000; fałdziński, 2014) was added and testing for causality in risk using hong et al. (2009) procedure as well as candelon, joëts and tokpavi (2013) procedure. the garch(1,1) model with t-student error distribution is considered as the benchmark for all the comparisons. the wide empirical analysis is also provided in the paper. the two groups of markets represented by main indices are considered, i.e. emerging ones, such as: brazil (bovespa), russia (rts, micex), china (sse), india (bse), turkey (xu 100), indonesia (jci) and mexico (ipc) and mature ones, such as: usa (s&p 500, great britain (ftse 100), germany (dax), france (cac 40), japan (nikkei 225), switzerland (ssmi), hong kong (hsi), south korea (kospi) and australia (aor). the group of emerging markets includes the most likely intense growth economies that determine the state of the market, capital flows and global relationships. we try to establish the source and the effect of risk in most important capital markets in the era of globalization as well as to determine the most likely time periods for risk transferring. this paper develops and continues the research reported in our previous publications (fałdziński et al., 2012, osińska et al., 2012), in which garch-pot methodology has been applied. in this paper we not only compare different volatility measures but also use them for causality in risk testing and find them useful in certain cases. these findings are of methodological and practical nature. the paper can be included into spillover analysis, which can be examined in many ways. one of the results of spillover effect can be contagion. contagion is defined as a significant increase in market co-movement after a shock to one country. the paper by forbes and rigobon (2002) defines and illustrates this problem while a wide survey of methods of its analysis can be found in burzała (2014). in our publication we demonstrate that thanks to the extreme value theory only big shocks on financial markets, that may or may not cause contagion, are considered. risk transfer from one market to another, examined in this report, can be considered as an incentive for contagion but it is not a sufficient condition. 1. the methodology in our previous research (fałdziński et al., 2012, osińska et al. 2012) we applied granger causality in risk definition that was formulated by hong volatility estimators in econometric analysis of risk transfer on capital markets dynamic econometric models 16 (2016) 21–35 23 (2001) and testing idea that was derived by hong (2001) and further modified by hong et al. (2009). it was based on spectral representation of time series. in this paper candelon et al. (2013) test is applied. it differs from hong’s test in two ways. firstly, a multivariate linear regression is used to calculate the lr-type test and secondly, the breaks in value at risk (var) at different probability levels are considered. according to the definition of granger causality in risk 1 2 { , } t t y y is a bivariate not necessarily stationary stochastic time series and   1lt lt l ta a i  1, 2l  is the var at level (0,1)  for lt y predicted using the information set    ( 1) ( 2) 11 , ,l t l t ll ti y y y   available at time 1t  . lta satisfies   , 1|lt lt l tp y a i   . we define  lt lt ltv i y a  1, 2l  which denotes the var break indicator. the break indicator takes on the value of 1 when var is exceeded by loss and takes on the value of 0 otherwise. let assume that 1 { ,..., } m a   is the set of m different probability levels. next, we consider a vector , , 1 , ( ) [ ( ),..., ( )] 1, 2 i t i t i t m z a v v i   comprising of m different variables at time t respective to the assumed set of probability levels. in the case of the granger non-causality the null hypothesis is: 0 1, 1 1, 1, 1 : ( ) | ( ) | t t t t h e z a i e z a i          , (1) where 1, 1, { ( ), } t s i z a s t  and 1, 2, { ( ), ( ), } t s s i z a z a s t  with the alternative 1 1, 1 1, 1, 1 : ( ) | ( ) | t t t t h e z a i e z a i          . (2) the null hypothesis says that the process  2ty does not granger-cause the process  1ty in risk at the set of different levels  with respect to 1ti  . candelon et al. (2013) have shown that the test statistic can be formulated using multivariate linear regression of the form 1, 0 1 2, 1 2, 1 ( ) ( ) ... ( ) t t l t l t z a z a z a           (3) where 0  is the (m,1) dimensions vector of constants, , 1,..., s s l  are the ( , )m m dimensions matrices of parameters and 1t  is ( ,1)m dimensions residual process. marcin fałdziński, magdalena osińska dynamic econometric models 16 (2016) 21–35 24 the null hypothesis corresponds to the situation when 0 1 : ... 0 l h     which is fulfilled for 1, 0 2 ( ) t t z a    . the multivariate test statistic is defined as follows:      ' '2 2 1 1( 1) log loglr t ml              (4) where t is the number of observations of time series m is the number of different probability levels assumed, l is the number of lags in the regression. it informs about the time delay since the beginning till the end of the risk transfer. the test statistic follows 2  distribution with 2 lm degrees of freedom. due to the parameter uncertainty dufour (2006) proposes a monte carlo method to obtain p-values. in order to check the hypothesis of spillovers in financial markets different volatility measures have been used. these measures determine the empirical results and therefore are worth comparing. they do not affect the characteristics of the candelon et al. test because it operates on breaks of var which can be defined at different levels. to estimate value at risk the following methods have been applied: 1. volatility estimators such as: a) parkinson (1980) (p) 2 1 1 ln 4 ln(2) n i p i i h l               , b) garman and klass (1980) (gk) 2 2 1 1 ln (2 ln(2) 1) ln 2 n i i gk i i i h c l o                              , c) rogers and satchell (1991) (rs) 1 ln ln ln ln n i i i i rs i i i i i h h l l c o c o                             , d) garman, klass, yang and zhang (1991) (gkyz) 2 2 2 1 1 1 ln ln (2 ln(2)) ln 2 n i i i gkyz i i i i o h c c l o                                             e) yang and zhang (2000) (yz) 2 2 2 (1 ) yz overnight volatility open to close rs k k       volatility estimators in econometric analysis of risk transfer on capital markets dynamic econometric models 16 (2016) 21–35 25 0,34 1 1,34 1 k n n     , 2 2 1 1 ln ln 1 n i i opent to close volatility i i i c c n o o                      , 2 2 1 1 1 1 ln ln 1 n i i overnight volatility i i i o o n c c                        , where n is the number of days taken into estimation, i h is the highest price, i l is the lowest price, i o is the open price and i c is the close price at day i . 2. conditional volatility models (ar(p)-garch(1,1) and ar(p)tarch(1,1) both with student error distribution (zakoian, 1994) 3. conditional volatility models with the extreme value theory (ar(p)garch(1,1) with student error distribution and peaks over threshold (pot) approach (mcneil and frey, 2000; fałdziński, 2014). 4. volatility estimators described in 1 with peaks over threshold (pot) approach. it is worth mentioning that using the extreme value theory represented by peaks over threshold (pot) enables identifying shocks (extreme changes) in some financial markets that affect other markets. thus finding the break in var when pot approach is applied is a strong argument for the spillover effect. the peaks over threshold method was described in fałdziński (2014). to explain it briefly let us assume that a given sequence of i.i.d. observations 1 , , n x x comes from unknown distribution function f , where we are interested in excesses over a high threshold value u . conditional excess distribution function (cedf) u f is defined as  ( ) | ,uf y p x u y x u    0 f y x u   , where x is a random variable, u is a given threshold, and y x u  is the excess (mcneil and frey, 2000). the distribution u f can be written as: ( ) ( ) ( ) ( ) ( ) 1 ( ) 1 ( ) u f u y f u f x f u f y f u f u        (5) the realizations of the random variable x lie between 0 and u , therefore the estimation of f in this interval generally poses no problems. according to the pickands-balkema-de haan (pickands (1975), balkema, de haan (1974)) theorem, for x u , we can use the tail estimate marcin fałdziński, magdalena osińska dynamic econometric models 16 (2016) 21–35 26   , ,ˆ ( ) 1 ( ) ( ) ( )n nf x f u g x f u     , where , , ( )g x   is the generalized pareto distribution (gdp), to approximate the distribution function ( )f x . it can be shown that ˆ ( )f x is also generalized pareto distribution, with the same shape parameter  , but with scale and location parameters, correspondingly equal:  1 ( )nf u     and   1 ( ) 1 /nf u           . thus, the pot estimator of p x is obtained by inverting the formula for ˆ ( )f x then substituting unknown parameters of the gpd by estimates ˆ ˆ( , )  , we get: ˆ 1 ˆ ˆ, , ˆ( ) 1ˆˆ ( ) 1 ˆ1 ( ) 1 ( ) n p u n n p f u p x f p g u f u f u                               . (6) if u n is the number of exceedances of the threshold u and n is the total number of realizations that we have from the distribution f , value-at-risk in the peaks over threshold method equals: ˆ ˆ ( ) 1 ˆ u n var u n                  , (7) where  is a tolerance level. 2. characteristics of the data according to world economic outlook released in 20161 the potential for economic growth in china is projected to decrease from 7.3 in 2014 to 6.0 in 2017 although it will still remain a very important country. the most prospective growth is projected in india: from 7.3 in 2014 to 7.5 in 2017 and in mexico: from 2.3 in 2014 to 2.9 in 2017. other countries like russia and brazil are expected to lose their growth rate and reach negative values. as concerns turkey, its growth rate in 2015 was quite high. it amounted to 4% with decreasing perspective. there are also other very fast developing emerging economies like kenya or nigeria but we excluded them from the study because of relative smaller liquidity in financial markets. another very fast developing economy is indonesia, which yearly growth rate in 2015 was 5.5%. among these countries there is a competition to be not only the best 1 http://www.imf.org/external/pubs/ft/weo/2016/update/01/pdf/0116.pdf http://www.imf.org/external/pubs/ft/weo/2016/update/01/pdf/0116.pdf volatility estimators in econometric analysis of risk transfer on capital markets dynamic econometric models 16 (2016) 21–35 27 emerging economy in the world but also a very important investment market. in sequent years one may observe and predict development of new important economic areas. in fig. 1 the annual growth rate of mentioned economies in 2006–2015 has been shown. figure 1. annual growth rate in emerging economies over 2006–2015 [in %] 2 the dynamics of chinese economy is still dominating although it started to decline after 2012. the opposite tendency can be noticed for india. indonesia’s growth seems to be stable over the decade. mexico, turkey, brazil and russia suffered hard from the recession in 2009, but mexico seems to be the most promising for the future. the data suggest that so called “bric group” that was considered a new economic body at the beginning of xxi century is no longer the case and other developing countries try to move from peripheries to the center. on the opposite side – developed financial markets are represented by traditional markets such as the usa, the great britain, germany, france, switzerland and australia that was completed by relatively new but mature markets from far east asia such as hong kong and south korea. in the paper we took into account the linkages between stock markets from different continents so north and latin americas, asia, australia and europe are 2 source: based on international monetary fund data. -10 -5 0 5 10 15 20 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 china india indonesia turkey russia mexico brazil marcin fałdziński, magdalena osińska dynamic econometric models 16 (2016) 21–35 28 represented. such a selection does not cover all possible linkages but allows answering the question of direction of capital transfers in both periods: the bullish and the bearish markets. big (and often negative) shocks in the financial market are usually perceived negatively by all groups of investors, market makers and supervisors. they may be due to: huge market uncertainty, policy changes, unpredicted information, speculative attacks, and transfers from other markets. sometime many causes may act simultaneously. some of them may cause extreme changes in values of losses (and/or profits). in general the process of globalization caused that the financial markets seem to act in the same way; they are linked. it is interesting that little attention is paid to the big and positive changes in financial markets. however in the literature one can find several individual cases of little linkages between different markets. for example china during asian crisis 1997–1998 was an example of completely separated market that was analyzed by lardy (1998). on the other hand, when markets are related it can be expected to transfer from one market to another like in the period 2007–2009 between usa and europe. risk can be generated locally or take the specific form like it was in 1997 between japan and usa (see: peek and rosengren, 1997). to answer the question of risk transfer between emerging and mature markets we used daily data from the period 03.01.2010–02.01.2015 (t=1260 observations). the log returns has been used for calculations in the form:  1100 ln( ) ln( )t t tr p p  , while testing for granger causality in risk we have used different lags 5,10,15, 25l  and we obtained p-values using monte carlo simulations (dufour, 2006), having assuming 1000 repetitions. in figure 2 the comparison of var breaks’ computed with different volatility estimators basing on dax returns is shown. one can notice that the highest values are indicated by garman, klass, yang and zhang estimator. in figure 3 the results of the latter are compared with the ar(1)-garch(1,1) with pot model. in the cases of shocks the ar(1)-garch(1,1) with pot seem to perform better. volatility estimators in econometric analysis of risk transfer on capital markets dynamic econometric models 16 (2016) 21–35 29 figure 2. vars computed with different volatility estimators basing on dax returns figure 3. vars computed with garch-pot method and garman-klass-yangzhang volatility estimator basing on dax returns -6 -4 -2 0 2 4 6 1 0 -1 5 -2 0 1 0 1 1 -1 5 -2 0 1 0 1 2 -1 5 -2 0 1 0 1 -1 5 -2 0 1 1 2 -1 5 -2 0 1 1 3 -1 5 -2 0 1 1 4 -1 5 -2 0 1 1 5 -1 5 -2 0 1 1 6 -1 5 -2 0 1 1 7 -1 5 -2 0 1 1 8 -1 5 -2 0 1 1 9 -1 5 -2 0 1 1 1 0 -1 5 -2 0 1 1 1 1 -1 5 -2 0 1 1 1 2 -1 5 -2 0 1 1 1 -1 5 -2 0 1 2 2 -1 5 -2 0 1 2 3 -1 5 -2 0 1 2 4 -1 5 -2 0 1 2 5 -1 5 -2 0 1 2 6 -1 5 -2 0 1 2 7 -1 5 -2 0 1 2 8 -1 5 -2 0 1 2 9 -1 5 -2 0 1 2 1 0 -1 5 -2 0 1 2 1 1 -1 5 -2 0 1 2 1 2 -1 5 -2 0 1 2 dax var (parkinson) var (garman-klass) var (rogers-sachell) var (garman-klass-yang-zhang) var (yang-zhang) -8 -6 -4 -2 0 2 4 6 8 1 -4 -2 0 1 0 3 -4 -2 0 1 0 5 -4 -2 0 1 0 7 -4 -2 0 1 0 9 -4 -2 0 1 0 1 1 -4 -2 0 1 0 1 -4 -2 0 1 1 3 -4 -2 0 1 1 5 -4 -2 0 1 1 7 -4 -2 0 1 1 9 -4 -2 0 1 1 1 1 -4 -2 0 1 1 1 -4 -2 0 1 2 3 -4 -2 0 1 2 5 -4 -2 0 1 2 7 -4 -2 0 1 2 9 -4 -2 0 1 2 1 1 -4 -2 0 1 2 dax var (garch-pot) var (garman-klass-yang-zhang) marcin fałdziński, magdalena osińska dynamic econometric models 16 (2016) 21–35 30 a precise comparison of different estimators of volatility was presented in table 1. we compared the analyzed volatility estimators with ar-garch, ar-garch-pot and garman-klass-yang-zhang with pot using seven different loss functions (described below the table). the preferred model is ar-garch-pot which was indicated in 4 cases on 7. twice the garmanklass-yang-zhang with pot was indicated and once the rogers-satchell estimator. table 1. mean value of the loss functions for var(0.95) method qps i qps ii qps iii rlf flf lf olf ar-garch 0.0988 5.0516 0.0360 0.0778 0.1669 0.1168 7.4261 ar-garchpot 0.0940 4.8277 0.0337 0.0730 0.1647 0.1094 7.7466 parkinson 0.1619 4.6296 0.0698 0.1028 0.1719 0.1642 5.7280 rogers-satchell 0.1770 6.0812 0.1020 0.1218 0.1877 0.1916 5.4615 garman-klass 0.1738 5.5389 0.0893 0.1159 0.1822 0.1834 5.4851 garman-klass yang-zhang 0.1323 4.5242 0.0499 0.0852 0.1638 0.1341 6.5845 yang-zhang 0.1979 6.7888 0.1331 0.1386 0.1991 0.2180 5.0100 garman-klassyang-zhang pot 0.1303 4.3008 0.0500 0.0812 0.1615 0.1283 6.8112 note: qps i means quadratic probability score function with binary loss function (lopez, 1998), qps ii means quadratic probability score function with size-adjusted loss function (lopez, 1998), qps iii means quadratic probability score function with size loss function (blanco and ihle, 1998), rlf means regulatory loss function (sarma et al., 2003), flf means firm's loss function (sarma et al., 2003) with opportunity cost of capital equals 0.05, lf means loss function (angelidis and degiannakis, 2006) and olf means overestimation loss function (fałdziński, 2011). the lowest (best) values of measures are in bold. 3. causality in risk between emerging and developed markets in this section we show the results of hong et al. (2009) and candelon et al. (2013) tests for granger causality in risk when emerging markets are indicated to be a source of risk transfer. having in mind the results shown in table 1 the following methods of volatility analysis were used: ar(p)garch(1,1)-pot with t-distribution and garman-klass-yang-zhang volatility estimator. the results are presented in tables 2–5. volatility estimators in econometric analysis of risk transfer on capital markets dynamic econometric models 16 (2016) 21–35 31 table 2. granger causality in risk for long position where ar-garch-pot with t-distribution was applied (5 up to 25 lags, hong et al. test) bovesp a → aor, bse, cac40, dax, hsi, jci, kospi, micex, nikkei225, rts, sse, ssmi, xu100 bse aor, jci, kospi, micex, nasdaq, sp500 ipc aor, dax, hsi, kospi, micex, nikkei225, rts, s&p 500 jci kospi micex aor, bse, hsi, kospi, nasdaq, nikkei225, xu100 rts aor, bse, hsi, kospi, nasdaq, nikkei225 sse ftse100, kospi, nasdaq xu100 hsi, kospi note: “→” shows direction of causality. table 3. granger causality in risk for long position where garman-klass-yangzhang volatility estimator was applied (5 up to 25 lags, hong et al. test) bovespa → bse, ftse100, jci, ssmi bse aor, bovespa, cac40, dax, ftse100, hsi, ipc, kospi, nasdaq, s&p 500, ssmi ipc aor, bovespa, hsi jci aor, bovespa, bse, cac40, dax, ftse100, hsi, ipc, kospi, micex, nasdaq, rts, s&p 500, ssmi, xu100 micex aor, bovespa, bse, dax, nasdaq, s&p 500 rts aor, bovespa, bse, dax, ftse100, nasdaq, s&p 500 sse bovespa, dax, ftse100, micex, nasdaq, rts, s&p 500 xu100 aor, bovespa, bse, cac40, dax, ftse 100, jci, kospi, nasdaq, rts, s&p 500, ssmi note: “→” shows direction of causality. table 4. granger causality in risk for long position where ar-garch-pot with t-distribution was applied (5 up to 25 lags, candelon et al. test) bovespa → aor, bse, dax, hsi, ipc, kospi, nikkei 225 bse ipc, kospi ipc bse, cac 40, ftse 100, ipc, kospi, micex, nikkei 225, rts, sse jci ftse 100, kospi micex cac40, ftse 100, jci, micex, sse rts bse, cac40, ftse 100, ipc, jci, micex, sse sse cac40, jci, micex xu 100 cac40, ftse 100, jci, micex note: “→” shows direction of causality. marcin fałdziński, magdalena osińska dynamic econometric models 16 (2016) 21–35 32 table 5. granger causality in risk for long position where garman-klass-yangzhang volatility estimator was applied (5 up to 25 lags, candelon et al. test) bovespa → aor, bse, cac40, dax, ftse100, hsi, ipc, jci, kospi, nasdaq, nikkei225, sse, ssmi bse aor, cac40, dax, hsi, ipc, kospi nasdaq, nikkei225, sse ipc aor, bse, cac40, dax, hsi, jci, kospi, nasdaq, nikkei225, sse, s&p 500, ssmi jci bovespa, bse, cac40, dax, ipc, kospi, nasdaq, nikkei225, sse micex bse, cac40, dax, hsi, ipc, kospi, nasdaq, nikkei225, s&p 500, ssmi rts bovespa, aor, bse, cac40, dax, ftse100, hsi, ipc, kospi, nasdaq, nikkei225, s&p 500, ssmi sse bovespa, bse, cac40, dax, hsi, ipc, kospi, nasdaq, nikkei225 xu100 bse, cac40, dax, hsi, ipc, kospi, nasdaq, nikkei225, s&p 500, ssmi note: “→” shows direction of causality. in the tables 2–5 the results of testing for granger causality in risk for long positions (losses) are presented. computing results for short positions (profits) we can observe lower number of relationships between the markets. it may suggest that taking into account profits the markets are more independent (they do not share profits) while in the case of losses otherwise situation takes place. the remained results are available on request. in general, we can say that the garman-klass-yang-zhang volatility estimator indicates the granger causality in risk more frequently than the ar-garch-pot method. this was intuitively expected, because the latter method takes into account the extreme observations while the volatility estimators includes all observations corrected by the high, low minimum and maximum values. the results show that there is granger causality in risk between emerging capital markets and highly developed ones. some capital markets absorb risk more often than others. we can delineate the markets which absorb the risk (risktakers) most frequently when the risk transfer is from emerging markets: aor, cac 40, ftse 100, hsi, nikkei 225 and kospi in the case of the garch-pot method. in the case of the volatility estimators the group of the risk-takers is: aor, bovespa, bse, cac 40, dax, nikkei 225, s&p 500 and nasdaq. the latter group is larger which is not surprising due the fact that volatility estimators fit better to the ‘average values’ of the time series. the overlapping of two methods of estimating value-at-risk and testing for granger causality in risk is rather easily visible. the difference between hong et all test and candelon et al. test is such that in case of argarch-pot model the results are the same in 13 cases only. the causal impact of micex, rts, sse and xu 100 on other markets was found to be volatility estimators in econometric analysis of risk transfer on capital markets dynamic econometric models 16 (2016) 21–35 33 quite different while the impact of bovespa, bse ipc and jpc can be considered as similar. conclusions in the paper we extended our previous investigations using quite sophisticated research methods and we concentrated on huge magnitude of changes (extreme values). our findings should be considered when systemic risk in the global economy is analyzed. they are linked with the problem of spillover in the sense that risk transfer from one market to another can be considered as an incentive for extending negative trends (contagion) but it is not a sufficient condition. in the paper we analyze the linkages between capital markets located in both emerging and developed economies. the difference between emerging and mature markets lays in different types of institutions like financial supervision, possibility of quoting the instruments from abroad, the number and type of listed instruments and, what is probably most important, in market liquidity. the question whether less liquid market can ‘produce’ more risk due to the lack of many alternatives within the market and more loosely rules is very important in the era of globalization. the answer can have many practical implications including financial regulations. in our research it was indicated that basing on volatility estimators it is possible to find more “causal” relationships than basing on ar-garchpot methods. it is due to the fact that volatility estimators are better fitted to the average volatility values than the methods based on the extreme value theory. we can observe risk transfer from emerging markets to the highly developed ones, so that the research helped to find the explanation of the problem put in the introduction. the markets which absorb risk transfers most frequently are: s&p500, cac40, nikkei225, nasdaq and ftse100. the empirical results of hong et al test and candelon et al test are slightly different that results from different tolerance levels allowing in both testing procedures. references angelidis, t., degiannakis, s. (2006), backtesting var models: an expected shortfall approach, working papers no 701, university of crete, athens university of economics and business, http://econpapers.repec.org/paper/crtwpaper/0701.htm (01.10.2016) balkema, a. a., de haan, l. (1974), residual life time at great age, annals of probability, vol.2, no. 5, 792–804, doi: http://dx.doi.org/10.1214/aop/1176996548. blanco, c., ihle, g. (1998), how good is your var? using backtesting to assess system performance, financial engineering news, august, 1–2. marcin fałdziński, magdalena osińska dynamic econometric models 16 (2016) 21–35 34 burzała, m. (2014), wybrane metody badania efektów zarażania na rynku kapitałowym, wyd. uniwersytetu ekonomicznego w poznaniu, poznań, candelon, b., joëts, m., tokpavi, s. (2013), testing for granger causality in distribution tails: an application to oil markets integration, economic modelling, 31, 276–285, doi: http://dx.doi.org/10.1016/j.econmod.2012.11.049. dowd, k. (2004), measuring market risk, john wiley & sons, west sussex, doi: http://dx.doi.org/10.1002/9781118673485. dufour, j.-m. (2006), monte carlo tests with nuisance parameters: a general approach to finite sample inference and nonstandard asymptotics, journal of econometrics, 27 (2), 443–477, doi: http://dx.doi.org/10.1016/j.jeconom.2005.06.007. fałdziński, m. (2011), on the empirical importance of the spectral risk measure with extreme value theory approach. financial markets principles of modelling forecasting and decision-making, findecon, lodz, 73–86, fałdziński, m. (2014), teoria wartości ekstremalnych w ekonometrii finansowej, wydawnictwo umk, toruń, fałdziński, m., osińska, m., zdanowicz, t. (2012), detecting risk transfer in financial markets using different risk measures, central european journal of economic modelling and econometrics, vol. 4, issue 1, 45–64, forbes, k. j., rigobon, r. (2002), no contagion, only interdependence: measuring stock market comovements, the journal of finance, 57(5), 2223–2261, doi: http://dx.doi.org/10.1111/0022-1082.00494. garman, m.b., klass, m.j. (1980), on the estimation of security price volatilities from historical data, journal of business 53, 67–78, hong, y. (2001), a test for volatility spillover with applications to exchange rates, journal of econometrics, 103(1–2), 183–224, doi: http://dx.doi.org/10.1016/s0304-4076(01)00043-4. hong, y., liu, y., wang, s. (2009), granger causality in risk and detection of extreme risk spillover between financial markets, journal of econometrics, 150(2), 271–287, doi: http://dx.doi.org/10.1016/j.jeconom.2008.12.013. lardy, n. (1998), china and the asian contagion, foreign affairs, 77, 78–88, doi: http://dx.doi.org/10.2307/20048967. lopez, j.a. (1998), regulatory evaluation of value-at-risk models, federal reserve bank of new york economic policy review, october, 119–124, doi: http://dx.doi.org/10.21314/jor.1999.005. mcneil, j.a., frey, f. (2000), estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach, journal of empirical finance, 7, 271–300, doi: http://dx.doi.org/10.1016/s0927-5398(00)00012-8. osińska, m., fałdziński, m., zdanowicz, t. (2012), econometric analysis of the risk transfer in capital markets. the case of china, argumenta oeconomica, 2(29), 139–164, parkinson, m. (1980), the extreme value method for estimating the variance of the rate of return, journal of business 53, 61–65, doi: http://dx.doi.org/10.1086/296071. peek, j., rosengre, e.s. (1997), the international transmission of financial shocks: the case of japan, the american economic review, 87, 495–505, doi: http://dx.doi.org/10.2139/ssrn.36583. pickands, j. (1975), statistical inference using extreme order statistics, annals of statistics, 3(1), 119–131, rogers, l.c.g., satchell s.e. (1991), estimating variance from high, low and closing prices, annals of applied probability 1, 504–512, doi: http://dx.doi.org/10.1214/aoap/1177005835. http://dx.doi.org/10.1016/j.econmod.2012.11.049 volatility estimators in econometric analysis of risk transfer on capital markets dynamic econometric models 16 (2016) 21–35 35 sarma, m., thomas, s., shah, a. (2003), selection of value-at-risk models, journal of forecasting, 22, 337–358, doi: http://dx.doi.org/10.1002/for.868. yang, d., zhang, q. (2000), drift independent volatility estimation based on high, low, open and close prices, journal of business 73, 477–492, doi: http://dx.doi.org/10.1086/209650. zakoian, j.-m. (1994), threshold heteroscedastic models, journal of economic dynamics and control, 18 (5), 931–955. estymatory zmienności w ekonometrycznej analizie transferu ryzyka na rynkach kapitałowych z a r y s t r e ś c i. celem badania jest porównanie wykorzystania różnych estymatorów zmienności, zastosowanych do testowania przyczynowości w ryzyku, między kilkoma wybranymi rynkami wschodzącymi i rozwiniętymi. w pracy uwzględniono następujące estymatory zmienności: parkinsona, garmana i klassa, rogersa i satchella, garmana, klassa, yanga i zhanga, yanga i zhanga oraz model garch(1,1)-t. dodatkowo wykorzystano narzędzia teorii wartości ekstremalnych. kilka wybranych rynków wschodzących zostało przebadanych, czy są źródłem ryzyka dla rynków rozwijających i rozwiniętych. wyniki pokazują, że przyczynowość w ryzyku występuje rzadziej w przypadku modeli z wykorzystaniem teorii wartości ekstremalnych. s ł o w a k l u c z o w e: przyczynowość w ryzyku, teoria wartości ekstremalnych, rozwój rynków wschodzących, transfer ryzyka, zmienność. microsoft word 08_kliber a.docx dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 agata kliber poznan university of economics sovereign cds instruments in central europe – linkages and interdependence† a b s t r a c t. in the article, linkages among sovereign cds instruments in central europe are investigated. special attention is paid to the change of causality patterns during the hungarian and greek crises. the results of the research reveal that the expectations do play a role in determining the prices of the contracts, as well as that there exist regional causality relationships between the instruments. the strength of causality between the volatilities of polish – hungarian and czechhungarian cds prices weakened during the hungarian crisis, while the volatilities of the three time series reacted rapidly and strongly to the greek one. this suggest that the european events should play more important role in determining the dynamics of the contracts than the problems of the country of the weakest fundamentals in the region. k e y w o r d s: multivariate volatility, credit default swap, contagion, sunspot, central europe. introduction sovereign cds contracts attracted special attention of the researchers starting from the outbreak of the recent financial crisis. beforehand, most of the research concentrated on the corporate cds contracts and analysed interrelations of the cds spreads with other financial instruments. to name a few: hull, predescu and white (2004) analysed relationships between the cds spreads, bond yields and credit rating announcements over the period 1998-2002. benkert (2004) searched for the factors explaining the dynamics of the cds premia, finding option-implied volatility especially significant in explaining their variability. some studies concentrated on the interrelations between the premia of the corporate cds contracts. such an example can be the paper by coudert and gex (2008) in which the authors analyse possible contagion of the † the work is supported by the polish ministry of science and higher education through the project nn 112372340. the author would like to thank the members of the project, especially jacek wallusch and michał adam for their help, cooperation and fruitful comments on the article, as well as the two anonymous referees for their useful and inspiring remarks. agata kliber 112 crisis experienced by general motors and ford in may 2005 to the whole cds market. jorion and zhang (2007) show how information of credit event of a given company is transferred among the other companies through the cds market in usa over the period 2001–2004. most recently, more and more attention is paid to the interrelations between the sovereign cds instruments. an example can be the work of calice, chen and williams (2011) in which the authors analyse the interrelations between the cds contract and bonds markets in the eurozone. alter and schüler (2011) investigated interdependence of the sovereign and banks’ cds series of chosen european countries (france, germany, italy, ireland, netherlands, portugal and spain). alfonso, furceri and gomes (2011) carried an event-study analysis to check the reaction of sovereign bond yields and cds spreads to the rating agency announcements, finding among all that the reaction of the cds spreads to the negative news increased after the lehman brothers’ bankruptcy. the dataset used by the authors included emu and non-emu countries, among all czech republic, hungary and poland, and covers the period from january 1995 to october 2010. the aim of this paper is to investigate possible linkages and interdependence among central european sovereign cds instruments of 5 year maturity during the hectic period of recent financial crisis. the author wants to verify whether the price of the sovereign cds instruments is determined mainly by fundamentals or whether the premium is influenced by expectations (sunspots). the author first analyses volatilities of the cds spreads to find any common patterns of their dynamics. next, the non-causality test of cheung and ng (1996) is run to verify the possible moments of causality and to relate them to the crisis events. eventually, to check the strength of causality, the author estimates the multivariate stochastic volatility model with granger-causality of yu and meyer (2006). the results of the research show that the price of the instruments is indeed influenced by expectations (e.g. a significant reaction of volatilities to the greek crisis was found). regional linkages (measured by grangercausality relations) do exist, but in the analysed period they played less important role (reaction of volatilities of polish and czech instruments to the hungarian problems was not comparable to their reaction to the greek crisis). the finding that the negative events influence significantly the price of the cds is consistent with the previous finding, cited above. 1. cds instruments – an overview credit default swap (in short: cds) is an instrument that gives the holder right to sell a bond for its face value in the event of a default by the issuer (hull, 2008). this credit derivative constitutes a form of insurance which protects the buyer of the cds in the case of a loan default by particular company. the company is known as the reference entity and a default by the company is sovereign cds instruments in central europe – linkages and interdependence 113 known as a credit event. the buyer of the cds makes periodic payments to the seller until the end of life of the cds or until a credit event occurs. the total amount paid per year as a percent of the notional principal to buy the protection is called the cds spread (hull, 2008). many different companies and countries are reference entities for the cds contracts. special kinds of cds contracts are the sovereign ones, the underlying instruments of which are government bonds. spreads of such cds instruments can be also used as indicators of the credit-worthiness of a given country (hull, 2008). usually, each negative (and positive) piece of information about financial situation of a given country is reflected in the cds spread. it is noticeable that negative information contributes to the growth of the cds premia, while the positive one – to its lowering. the more liquid the market, the more perceptible are these movements. the sovereign debt market has attracted considerable attention since september 2008, while before the crisis trading concentrated on private sector instruments (fontana, scheicher, 2010). 2. cds instruments in central europe let us take a look at the figure 1. it presents the dynamics of the cds spreads of 5 year maturity quoted for czech republic, hungary and poland over the period march 2008 – march 2011. first of all, we can notice that the spreads for hungary are the highest. next, we can observe some similarities in the behaviour of all the three spreads. there was a significant jump in the middle of october 2008. according to the financial stability report (nbp, 2009) “the rise in cds premium on the polish government debt was largely connected with the global tendency to assess credit risk very prudently and the negative impact of the situation in the region on perception of investment risk in poland”. moreover, the probability of default of the polish government implied by the prices of cds was much higher than other ratings suggested. the rating implied by the cds prices was bbb, while moody’s assessment was a2 and s&p’s and fitch’s: a-. on contrary, the rating for hungary was lowered several times in 2008 and 2009. for example, on october 15th the rating for hungary was changed by s&p from a2 to a2 with negative outlook. the rating was lowered again on november 17th, 2008 and on march 30th, 2009. the changes in ratings appeared rather seldom and were not as sharp as the dynamics of the cds prices could suggest. in the second half of 2009 the situation stabilised. since the beginning of 2010 the cds premia in the three countries decreased, falling to the lowest levels since september 2008. however, in may 2010 the rise in risk aversion connected with greek insolvency affected also the central european cds market. thus, we could observe the rise in the cds premia to the level recorded in the second quarter of 2009. on may 9th, the european stabilisation mechanism has agata kliber 114 been announced and shortly afterwards the risk assessment of central european bonds improved (nbp, 2010). however, in june it rose again. the highest growth was observed in hungary. this growth of the cds premium in hungary can be also explained by the worsening economic situation of the country. already on june 3rd, a leader of the ruling fidesz party said that the country's finances were in much worse condition than previously expected. also the vice chairman of fidesz party stated that there was only a slim chance to avoid greek scenario (according to: www.napi.hu). in consequence, the five-year credit default swaps on hungary rose by 58 basis points to 320 basis points, which is reflected in figure 1. on june 12th, moody changed its rating for hungary from baa1 to baa3 with negative outlook. again, on july 17th the eu and imf suspended a review of hungary’s funding program (which had been set up in 2008 to save the country from financial problems) and said that the country must have taken action to meet targets for cutting its budget deficits. the suspension meant that hungary would not have access to the remaining funds in its loan package. this event could have contributed to the increase of the cds price in the second half of july. eventually, the pick of the cds price in november can be explained by the reforms of the pension system announced by the hungarian government. figure 1. corporate 5-year cds spreads in bps for czech republic, poland and hungary 2.1. data description as already said, the analysed sample covered the period from march 2008 to march 2011 and consisted of 796 observations. the data was provided by bloomberg. obviously, the spreads were not stationary, so we run all our tests on the changes of the prices ( t t tp p  ). we did not take the logarithms of the increments to avoid problems with interpretation of such transformed series (the prices of the cds contracts are expressed in basis points). sovereign cds instruments in central europe – linkages and interdependence 115 table 1 presents the descriptive statistics of the increments of the daily changes of the prices for czech republic, hungary and poland. as expected, the highest deviation is observed in case of the hungarian instruments, while the lowest – in case of the czech ones. table 1. descriptive statistics of the price changes of the central european cds variable min value mean value max value std. deviation dcds_cz -57.503 0.038 54.998 6.693 dcds_hu -83.332 0.156 98.033 14.294 dcds_pl -46.654 0.064 47.333 8.154 date 10.03.2008 28.03.2011 3. non-causality analysis – tests based upon the garch-type models in order to test whether expectations play a role in cds pricing, we tested for the volatility interactions in the cds prices. detecting volatility spillovers in the data is the sign of contagion. the type of contagion which is expectationdriven is called sunspot – see e.g. (keister, 2006). the occurrence of the sunspot is connected with the period of volatility growth – thus we first estimate univariate volatility models to test for possible moments of contagion. 3.1. volatility dynamics of the cds prices – a univariate garch analysis to begin with, we first fit univariate garch models to the data and searched for the moments of higher volatility. as stated before, the moments of abnormal volatility growth can indicate the sunspot event. thus, we fit the garch models to each of the series. since in each case the sum of the parameters  and  in estimated garch(1,1) models exceeded 1, we decided for igarch models with student distribution. in case of hungarian and polish cds it was also necessary to account for arma effect in the conditional mean equation, as well as to assume student distribution. each of the fitted model was successfully tested against autocorrelation in mean (box-pierce test) as well as arch effect (engle test) and stability of the parameters (nyblom test). all computations presented in this paragraph were performed using oxmetrics 6 with package g@rch. tables 2– 4 present the estimated parameters, while the figures 2 to 4 the estimates of volatilities of each time series. table 2. estimates of arma(0,0)-igarch model for czech cds coefficient estimated value std. error p-value  0.248 0.181  0.175 0.043 0.000  0.825 note: all parameters appeared to be stable, according to the nyblom stability test. agata kliber 116 table 3. estimates of arma(1,0)-igarch(1,1) model for polish cds coefficient estimated value std. error p-value ar(1) 0.136 0.040 0.001  0.344 0.195  0.188 0.029 0.000  0.812 degrees of freedom 4.867 0.632 note: all parameters appeared to be stable, according to the nyblom stability test. table 4. estimates of arma(0,1)-igarch(1,1) model for hungarian cds coefficient estimated value std. error p-value ma(1) 0.163 0.036 0.000  3.616 1.937  0.280 0.088 0.002  0.720 degrees of freedom 3.881 0.460 note: all parameters appeared to be stable, according to the nyblom stability test. we plot the obtained estimates of conditional variances in figures 2–4. we can observe that the moments of volatility growth in the three countries overlap. the first one started in september 2008 and was most probably connected with the worldwide financial crisis. the second one started in may 2010 and may have two causes. either it was the reaction for the greek crisis or the echo of the turmoil in hungary. it is worth noting that the second pick of volatility was much smaller in the case of czech cds price than the one observed for polish and hungarian cds price. what is important, during the “second turmoil” we observe the picks in volatility on may 11th in the case of the three time series. however, in the case of hungary, this pick is relatively small as compared to the one observed on june 7th, which may have been caused by the statements of the leaders of the ruling fidesz party about the very poor economic situation of the country. the third pick appeared on july 21st and was most probably the consequence of the suspension of the funding program for hungary. figure 2. estimates of conditional variance for czech cds sovereign cds instruments in central europe – linkages and interdependence 117 figure 3. estimates of conditional variance for polish cds figure 4. estimates of conditional variance of hungarian cds 3.2. non-causality test of cheung and ng based upon the obtained estimates of the volatility of the cds contracts, we computed the statistics of the non-causality test of cheung and ng (1996) – see the appendix for details. the statistics are used to test the null hypothesis of no causality in variance. table 5 presents the results. in all the cases the null of no causality was rejected. however, the highest value took the statistics obtained for hungarian and polish cds variances. table 5. statistics of the non-causality in variance test of cheung and ng j=0,k= cz→pl pl→cz hu→pl pl→hu hu→cz cz-→hu 2 1 43.045 43.554 60.905 60.768 23.392 23.392 5.991 2 43.563 44.105 61.311 61.116 23.410 23.495 7.815 3 44.436 44.328 61.312 61.119 23.644 24.153 9.488 agata kliber 118 additionally, taking advantage of the statistics constructed for small samples, we computed the test for the moving window of 120 observations. let us assume that the relationship implied by the test is valid for the period of 120 days, ending on the day of the window end. we present the results in figures 5–10. the points of the second coordinate equal to 1 denote the moments in which we rejected the hypothesis of non-causality at lag 1. the pictures reveal some interesting patterns. the causality relationships are bilateral (there occurs the “feedback” according to the cheung and ng terminology). yet, in the case of poland and czech republic, the period of constant causality in variance begins in summer 2009, while in the case of poland and hungary – it ends in summer 2010. this means that there was a year period of feedback polandhungary, poland-czech republic, which ended in case of the poland-hungary relationships in the moment of the turbulence in hungary. what is interesting, the hungarian crisis did not affect to such extend the causality patterns with czech republic. although there appear moments of non-causality, as for example in november 2010, in general the null is rejected. however, the fact that in november 2010 the causality linkages weakened suggests that the pension crisis in hungary did not affect the way the market participants value the risk of investment in czech republic. figure 5. periods of causality in variance: poland czech republic figure 6. periods of causality in variance: czech republic poland figure 7. periods of causality in variance: czech republic – hungary sovereign cds instruments in central europe – linkages and interdependence 119 figure 8. periods of causality in variance: hungary czech republic figure 9. periods of causality in variance: poland hungary figure 10. periods of causality in variance: hungary-poland the results of the test suggest that the two moments of turbulence denoted by volatility growth were of different nature. while in the first phase of the crisis we could observe significant interrelations between volatility of polish and hungarian cds instruments, as well as between czech and hungarian ones, suggesting that the crisis may have been regional or that at least it was considered so by the investors, the situation changed during the second volatility pick. although in spring 2010 the null of non-causality was rejected, in summer and autumn – when the crisis in hungary intensified – the causality relations weakened significantly. this can suggest that the contagion did occur, but its source was outside the central europe. most probably, as suggested in the nbp’s financial stability reports, there was a global growth of risk perception as a reaction to the greek problems. however, the hungarian crisis did not infect the rest of the analysed central european countries. it seems that either the investors did not perceive the hungarian problems as contagious or that the cds prices did not reflect the market reality. the latter problem could occur if the market was not liquid. however, as presented in figure 11 the instruments are indeed traded intensively, and the most liquid ones are the hungarian cds contracts. agata kliber 120 figure 11. turnovers on the cds market quarterly data 4. analysis of granger causality in variance – a multivariate stochastic volatility model the test of cheung and ng allows us to test the direction of causality, not the strength of it. thus, we also estimated a model of yu and meyer (2006) – a multivariate stochastic volatility model with granger causality (see the appendix for details), allowing not only for detecting the causality patterns (in granger sense), but also the strength of them. first, we estimated a var equation for the polish and hungarian cds prices in order to account for any linear dependencies in the data. the results are presented in tables 6–7. order of equation was chosen using schwarz information criterion. in case of the hungarian cds price, it appears that it was influenced by the past values of this instrument price (lagged value of polish cds price is insignificant). similarly, the price of polish cds is also influenced by the past value of hungarian cds price. table 6. var equation for hungarian cds (polish and hungarian cds in the system) parameter name estimate std. error p-value cdshu(-1) 0.261 0.054 0 cdspl(-1) -0.014 0.094 0.878 const. 0.079 0.482 0.870 table 7. var equation for polish cds (polish and hungarian cds in the system) parameter name estimate std. error p-value cdspl(-1) -0.0230 0.054 0.864 cdshu(-1) 0.154 0.031 0 const. 0.047 0.276 0.864 the volatility model was estimated for the residuals from the var system, using the free software winbugs (version 1.4). since the program uses the sovereign cds instruments in central europe – linkages and interdependence 121 bayesian approach to the estimation, we assume that all the parameters are stochastic variables. to give the readers an approximation of the point estimates, we present the means and medians of the distributions, as well as the standard deviations. the results are presented in table 8. moreover, in the case of the parameters 12 and 21 , we present the obtained density functions (figure 11 and 12). table 8. results of the estimation of gc-msv model for hungarian (1) and polish (2) cds prices variable name mean standard deviation mc error 2.5% percentile median 97.5% percentile 1 4.063 0.368 0.025 3.340 4.038 4.858 2 3.100 0.395 0.027 2.332 3.077 3.964 1 0.697 0.061 0.004 0.566 0.702 0.803 2 0.877 0.043 0.003 0.785 0.879 0.954 12 0.098 0.044 0.003 0.021 0.096 0.192 21 0.247 0.058 0.004 0.146 0.241 0.373 1  0.674 0.061 0.004 0.554 0.674 0.796 2  0.431 0.053 0.005 0.330 0.429 0.537  0.752 0.019 0.001 0.714 0.753 0.789 thus, we conclude that there exist causality between the volatility of polish and hungarian cds contracts. based upon the obtained distribution we can conclude that the volatility of polish cds granger-causes the volatility of hungarian cds. since the result was obtained for the whole sample, we can expect that the direction of causality is constant over time. table 9. var equation for polish cds (polish and czech cds in the system) parameter name estimate std. error p-value const. 0.054 0.278 0.847 cdspl(-1) 0.044 0.047 0.356 cdspl(-2) -0.050 0.048 0.296 cdscz(-1) 0.241 0.057 0 cdscz(-2) 0.064 0.056 0.259 table 10. var equation for czech cds (polish and czech cds in the system) parameter name estimate std. error p-value const. 0.023 0.227 0.919 cdspl(-1) 0.314 0.039 0 cdspl(-2) 0.106 0.039 0.006 cdscz(-1) -0.147 0.047 0.002 cdscz(-2) -0.184 0.046 0 agata kliber 122 we followed the same procedure in order to verify the interactions between polish and czech, as well as czech and hungarian cds prices. the results are presented in tables 9–11. figure 12. density of the parameter 12 (causality from poland to hungary) figure 13. density of the parameter 21 (causality from hungary to poland) figure 14. density of the parameter 21 (causality from poland to czech republic) figure 15. density of the parameter 12 (causality from czech republic to poland) in case of the interactions between polish and czech cds prices we observe that polish cds price are influenced by the prices of czech cds from the previous day, while the prices of czech cds are influenced by the prices of both polish and hungarian cds from one and two days before. in case of the second moment dependency, it is again polish cds volatility that influences the volatility of the czech cds (parameter 21 ). the second “causality parameter” – 12 – takes smaller value and its standard deviation amounts to 60% of its mean value. thus, we conclude that the investors do not assess the risk of investment in poland based upon the worsening situation of other central europesovereign cds instruments in central europe – linkages and interdependence 123 an countries, as before. moreover, it is the volatility of polish cds that affects the volatility of the other central european ones the most. figures 14 and 15 present the obtained densities of the causality parameters. eventually, we investigated the causality patterns in case of hungarian and czech cds prices. the results are displayed in tables 12–14. figure 16. density of the parameter 21 (causality from czech republic to hungary) figure 17. density of the parameter 12 (causality from hungary to czech republic) table 11. results of the estimation of gc-msv model for polish (1) and czech (2) cds prices variable name mean standard deviation mc error 2.5% percentile median 97.5% percentile 1 3.150 0.365 0.027 2.474 3.150 3.942 2 2.196 0.421 0.031 1.419 2.190 3.097 1 0.844 0.080 0.006 0.666 0.856 0.969 2 0.558 0.125 0.010 0.283 0.562 0.767 12 0.106 0.065 0.005 0.003 0.096 0.250 21 0.499 0.151 0.012 0.251 0.493 0.828 1  0.422 0.073 0.006 0.307 0.412 0.590 2  0.574 0.095 0.007 0.388 0.572 0.766  0.622 0.026 0.001 0.568 0.622 0.669 based upon the estimated var model we can conclude that the price of czech cds depended on the past values of itself and hungarian cds price. in case of the causality in the second moments, we observe small values of the “causality coefficients”. although the value of 12 is more than two times higher than the one of 21 , in both cases the standard deviation of the obtained coefficients exceed 35% of their means. thus, we can expect bivariate causality on agata kliber 124 quite low level, a little higher in case of hungarian influence on the volatility of czech cds. the results are presented in figure 16 and 17. table 12. var equation for czech cds (czech and hungarian cds in the system) parameter name estimate std. error p-value const. 0.025 0.228 0.914 cdscz(-1) -0.095 0.043 0.027 cdshu(-1) 0.161 0.021 0 table 13. var equation for hungarian cds (czech and hungarian cds in the system) parameter name estimate std. error p-value const. 0.078 0.481 0.871 cdscz(-1) 0.143 0.091 0.114 cdshu(-1) 0.213 0.043 0 table 14. results of the estimation of gc-msv model for czech (1) and hungarian (2) cds prices variable name mean standard deviation mc error 2.5% percentile median 97.5% percentile 1 2.053 0.394 0.026 1.284 2.051 2.878 2 4.040 0.334 0.022 3.373 4.035 4.706 1 0.700 0.091 0.008 0.501 0.714 0.839 2 0.772 0.068 0.006 0.621 0.777 0.897 12 0.312 0.109 0.010 0.153 0.293 0.562 21 0.138 0.055 0.004 0.041 0.134 0.260 1  0.719 0.096 0.009 0.562 0.704 0.918 2  0.669 0.089 0.008 0.465 0.670 0.841  0.588 0.028 0.001 0.532 0.588 0.641 conclusions the aim of the research was to verify whether there appear significant interactions between volatilities of the central european cds prices, suggesting that their price can be influenced not only by fundamentals, but also by expectations (sunspot event). to verify this, first univariate volatility models were estimated to check for the periods of volatility growth. next, the cheung-ng test for noncausality in variance was run, using the rolling-over approach and the test version suitable for the small samples. the author was interested in revealing the possible sources of the volatility growth in case of the cds prices in the three countries, in spring 2010. usually the situation of volatility growth is associated with the occurrence of sunspot event. there may have been two sources of such contagion – either the global growth of risk aversion due to the greek crisis, or the local turmoil caused by the worsening economic situation of hungary. it appeared, however, that the moments in which the null hypothesis of nonsovereign cds instruments in central europe – linkages and interdependence 125 causality in the pairs including hungary was not rejected, overlapped with the moment of hungarian crisis in 2010. it was also observed that the periods of higher volatility in hungary did not have equivalents in polish and czech cds volatility. such situation can indicate that the investors do not expect the crisis outbreak in poland or czech republic when it happens in hungary and they do not assess the risk of investment in the whole central europe based upon the situation in the country of weaker fundamentals. eventually, yet anoher test for causality in variance was performed, through estimating bivariate stochastic volatility models with granger causality for each pair of instruments. the results are clear: there is a strong causality from polish cds volatility to the volatility of the rest cds prices. the increase in conditional volatilities of the instruments in spring 2010 can suggest that contagion did occur. however, according to the presented results, the source of it was outside the central europe. if there was a reaction to the turmoil in hungary, it was not reflected in the volatilities of polish and czech cds prices. thus, the reaction of the cds prices of poland and czech republic to the problems in hungary was not noticeable. however, the fact that the events of causality appeared during the problems in greece can suggest that expectations do play a role in cds pricing. since the central europe is not critically linked via fundamentals with the southern one, the contagion which occurred in may 2010 must be so called “expectations-driven”. what is more, the results of the research show also that there exist causality linkages in the second moments of the cds prices processes. significant reactions of volatilities of czech and hungarian cds prices to the changes of the volatility of polish cds price were found. the linkages are in fact bilateral, but the causality from poland is much stronger. on the other hand, the turnover on the hungarian cds market was the highest in the analysed period. the explanation of the result can be the following – the sovereign cds instruments are bought mainly for speculation. in the period over study hungary was the most risky country from the region, and investments in hungarian swaps could allow for the highest earnings. thus, the investors were interested in trading these particular instruments. however, from the analysed markets, the polish one is the biggest and if an investor wants to sell his assets in a central european market as quickly as possible, the easiest way is to trade them on the polish stock exchange. this may explain the fact that – although the turnover on the hungarian cds market was the highest – the strongest causality in variance occurred from the polish one. this conclusion could explain why the volatility of polish and czech cds prices did not react to the hungarian problems. agata kliber 126 references alfonso, a., furceri, d., gomez, p. (2011), sovereign credit ratings and financial markets linkages: application to european data, working paper series 1347, european central bank: http://www.ecb.europa.eu/pub/pdf/scpwps/ecbwp1347.pdf (10.01.2012). alter, a., schüler, y.s. (2011) credit spread interdependencies of european sates and banks during the financial crisis, university of konstanz, working paper: http://www.clevelandfed.org/research/seminars/2011/alter2.pdf (10.01.2012). bollerslev, t. (1990), modelling the coherence in short-run nominal exchange rates: a multivariate generalized arch model, review of economics and statistics, 20, 498–505. benkert, c. (2004), explaining credit default swap premia, the journal of futures market, vol. 24, no. 1, 71 – 92. calice, g., chen, j., williams, j. (2011), liquidity spillovers in sovereign bonds and cds markets: an analysis of the eurozone sovereign debt crisis, journal of economic behaviour and organization (in press). cheung, y. w., ng, l. k. (1996), a causality-in-variance test and its application to financial market prices, journal of econometrics, 72, 33–48. coudert, v., gex, m. (2008), contagion in the credit default swap market: the case of the gm and ford crisis in 2005, cepii working paper, 14, 1 – 67: http://www.cepii.fr/ /anglaisgraph/workpap/pdf/2008/wp2008-14.pdf (10.01.2012). engle, r. f. (2002), dynamic conditional correlation: a simple class of multivariate generalised autoregressive conditional heteroskedasticity models, journal of business and economic statistics 20, 339–350. fontana, a., scheicher, m. (2010), an analysis of euro area sovereign cds and their relation with government bonds european central bank working paper series no 1271: http://www.ecb.int/pub/pdf/scpwps/ecbwp1271.pdf (25.10.2011). hull, j. (2008), options, futures and other derivatives (7th edition), prentice hall, upper saddle river, 2008. hull, j., predescu m., white, a. (2004), the relationship between credit default swap spreads, bond yields, and credit rating announcements, journal of banking and finance, vol. 28, no. 11, 2789 – 2811. jorion, p., zhang, g. (2007), good and bad credit contagion: evidence from the credit default swaps, journal of financial economics, 84, 860 – 883. keister , t. (2006), expectations and contagion in self-fulfilling currency attacks, staff reports 249, federal reserve bank of new york. national bank of poland (2009), financial stability report, june 2009, warsaw, 2009. national bank of poland (2010), financial stability report, june 2010, warsaw, 2010. yu, j., meyer, r. (2006), multivariate stochastic volatility models: bayesian estimation and model comparison, econometric reviews, 25, 361–384. www.napi.hu (25.10.2011). appendix non-causality test of cheung and ng let us consider two stationary and ergodic time series processes: tx and ty , as well as two information sets defined by: 1{ , 0}t ti x j  and { , , 0}.t t j t jj x y j   ty is said to cause 1tx  in variance if sovereign cds instruments in central europe – linkages and interdependence 127 2 2 1 , 1 1 , 1[( ) | ] [( ) | ],t x t t t x t te x i e x j       where , 1x t  is the conditional mean of 1tx  (conditioned on ti ). feedback in variance occurs when x causes y, and y causes x. let us also suppose that: 1/ 2 , , 1/ 2 , , , . t x t x t t t y t y t t x h y h         in the model above, ,z t denotes conditional mean, ,z th conditional variance, while t and t are white noise processes with null mean. let tu and tv denote squares of standardised residuals: 22 ,, 2 2 , , ( )( ) , . t y tt x t t t t t x t y t yx u v h h         let , ( )u vr k denote cross-correlation between u and v, for the k-th lag: , , , , ( ) ( ) , ( (0) (0)) u v u v u u v v c k r k c c  where , ( )u vc k denotes covariance between u and v at lag k. since the processes u and v are independent: , , ( ) 0 1 0 , , . 0 0 1( ) u v u v t r k n k j t r j                 cheung and ng (1996) proposed the following test to verify the causality in variance. first, we construct the following statistics: 2 , ( ), k u v i j s t r i    which is distributed according to 2 distribution with (k-j+1) degrees of freedom. if the sample is small, the following, corrected version of the statistics is applied: 2 , ( ), k i u v i j s t r i    where 2 . | | i t t i     agata kliber 128 multivariate stochastic volatility model with granger causality of yu and meyer the model has the following form: , ~ ( , ),t t t t iid y ω ε ε 0 σ 1 2 2 2 1 ,( ) ~ (0, ( , )),t t t t iid diag       h μ φ h μ η η where: 1 2( , ) 'y yy denotes the mean-adjusted time series of data, t is a diagonal matrix ( (exp( )), 2 t t h diagω 1 , 1t            σ 11 12 21 22 = .           φ naturally, 1 2 1, 2, 1, 2, 1, 2,( , ) ', ( , ) ', ( , ) ', ( , ) '.t t t t t t t t th h        μ h ε η in the model it is assumed that the (mean-adjusted) returns and volatilities are cross-dependent, as well as that the volatility of the first asset can be granger-caused by the volatility of the second one, and that the volatility of the second asset can be caused by the volatility of the first one. in order to investigate the direction of volatility spillovers we estimate the model and check the estimates of the parameters 12 and 21 . zależności i powiązania między instrumentami cds na dług rządowy w gospodarkach europy środkowej z a r y s t r e ś c i. artykuł przedstawia badanie współzależności między procesami cen instrumentów cds (credit default swap) w europie środkowej. autorka analizuje zależności między zmiennościami cen instrumentów w okresie kryzysowym starając się znaleźć odpowiedź na pytanie, czy ich cena determinowana jest w znacznym stopniu przez zależności regionalne i oczekiwania inwestorów (zjawisko zarażania sterowane oczekiwaniami, ang. sunspot). wyniki przeprowadzonego badania sugerują, że ceny instrumentów cds są w znacznym stopniu sterowane oczekiwaniami (gwałtowna reakcja na kryzys grecki), jak również że zależności regionalne – mimo że istnieją – mają mniejszy wpływ na cenę kontraktów, niż wydarzenia ogólnoeuropejskie. s ł o w a k l u c z o w e: wielowymiarowe modele zmienności, cds, europa środkowa, przenoszenie zmienności, sunspot. microsoft word 11_bruzda_bejger.docx dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 sylwester bejger, joanna bruzda† nicolaus copernicus university in toruń detection of collusion equilibrium in an industry with application of wavelet analysis a b s t r a c t. in the present paper an attempt was made to verify the possibilities of the use of a marker of structural changes of market price variance in the detection of trade collusion between business players. we used the theoretical model of strategic behaviour of trade players with the assumption of exogenous and time-constant cartel quota (market shares), which justifies the application of a marker for business with specific parameters. the paper contains empirical employment of a marker for a sequence of average lysine price on the usa market in 1990–1996. wavelet analysis was applied, for the first time in this context, as the econometric method for the detection of structural changes in the variance. k e y w o r d s: explicit and tacit collusion, supergame with a fixed structure of market shares, price variance, wavelet analysis introduction in the paper bejger (2010), on the basis of well-known lysine1 cartel (1990–1996 period) a theoretical model of strategic behaviour of industry players was constructed as a standard supergame model with a cournot type stage game with additional assumption of exogenous and time-constant cartel quota (market shares). for business branches bound by specific parameters, e.g. lysine manufacturers2 in the above mentioned period, the model may indicate certain characteristic structural disorders in the variance of the market price, resulting from the likelihood of the development of price war phase, caused by † the author acknowledges the financial support from the polish ministry of science and higher education under the grant no. n n111 285135. 1 lysine is an α-amino acid that the human body cannot synthesize. lysine production for animal feed is a major global industry, as lysine is an important additive to animal feed because it is a limiting amino acid when optimizing the growth of certain animals such as pigs and chickens for the production of meat. 2 for a detailed description of lysine conspiracy see connor (2001). sylwester bejger, joanna bruzda 156 a player with no intention of maintaining or establishing collusion due to a too low predicted or factual share on the market or a certain market price stiffness in the collusion phase in the periods of market contraction. present paper includes an attempt of the application of wavelet analysis with the aim of an empirical verification of the correctness of theoretical findings. the verification is made possible thanks to the well known history of strategic behaviour of the players in the business. the following chapter will introduce the summary of the to-date empirical research with the use of variance change marker. chapter 2 contains the description of research methodology. chapter 3 contains empirical work (in a case-study flavour). in chapter 4 the results of the study were presented. chapter 5 summarizes the whole of the research. 1. to-date empirical research with application of variance change marker assuming that the theoretical model accurately describes strategic behaviour in the industry of lysine manufacturers, the following research hypotheses could be put forward:  regime changes of the variance of average lysine prices are possible due to a probability of the occurrence of price war phases,  in the collusion phase the variance should be lower than in competition phase. these two specific patterns in variance process could distinguish collusion from competition and are usually named markers of collusion (harrington, 2005, p. 25). with the aim of detecting such variance disorders a wavelet analysis was applied. so far, such approach has not been used. basing on theoretical findings, one could put forward a hypothesis that, on average, in the collusion phase price variance is lower than in the competition phase. one should also expect regime variance change while switching from collusion phase to competition phase (price war). to-date papers connected with the detection of collusion on the basis of the detection of structural changes in the variance embraced the application of descriptive statistics methods in the comparison of variance levels in the phases of collusion and competition (abrantes-metz, froeb, geweke, taylor, 2006), the application of arch / garch specification in the process of market price including the additional 0-1 variable describing the phases of collusion and competition (bolotova, connor, miller, 2008), as well as the application of markov switch model of ms(m)(ar(p))garch(p,q) type for the variance and/or average (constant) of the price process (bejger, 2009). the two latter articles are all the more interesting due to the fact that they refer to the cartel of lysine manufacturers. in the paper (abrantes-metz, froeb, geweke, taylor, 2006) the cartel of frozen fish suppliers was the research subject. the existence and functioning of the cartel was confirmed by the authorities. detection of collusion equilibrium in an industry with application of wavelet analysis 157 in the course of the research it was found that during collusion phase (before the discovery of the cartel) market price variance (offered by colluders at bargains) was significantly lower than after the fall of the cartel. the hypothesis that justified the research was the supergame model with sppe equilibrium. in the article (bolotova, connor, miller, 2008) arch / garch specification was used with the aim of examining the disturbances in the variance and the average of the lysine market price process during the phases of competition and collusion. the influence was described through 0-1 variables for the phases of cartel and competition. a statistically significantly lower average variance was reached in the periods of collusion than in the phases of competition. however, it should be underlined that the marking off of phases of both types (with the aim of 01variable specification) took place on the basis of a posteriori evidence from the proceedings in the case of the cartel. in the same article a cartel of citric acid producers was studied, with no particular findings as to the lower variance in the collusion phase. the theoretical basis for the marker was sppe-type equilibrium. in the paper (bejger, 2009) markov switching model was employed with the aim of detecting moments of change in the level of variance and the average of lysine price process. the results confirmed the existence of two variance regimes and a high likelihood of the process staying within the lower variance regime, especially during the second collusion phase. the study seems of importance in view of the absence of assumptions as to the moments of switch. the detection of those moments partly confirmed the history of cartel activity known from the trial evidence. 2. wavelet analysis methodology assuming that the theoretical model accurately describes strategic behaviour in the business of lysine manufacturers, one could put forward the following research hypotheses:  regime changes of the variance of average lysine prices are possible due to different phases of price war,  in the collusion phase the variance should be lower than in the competition phase. with the aim of detecting such variance disorders a wavelet analysis was applied. so far, such approach has not been used. comparing it to variance marker-based econometric methods of collusion detection applied before (mentioned in chapter 1), it can be noticed that the wavelet analysis makes it possible to utilize simple methods of statistical inference, allows the preliminary, graphic, estimation of the changes in variance, and as a non-parametric method is not burdened with model specification error. in addition, it allows for the indication of the scales in case of which a change occurs; this however requires the access to long time sequences. sylwester bejger, joanna bruzda 158 wavelet analysis consists in the decomposition of the process into components constituting shifted and rescaled versions of the so-called mother wavelet, , which is a function with unit energy, fulfilling the so-called admissibility condition ( percival, walden, 2000, p. 4). let a vector be given in the form of ),,,( 110  nxxx x where jn 2 . for and (j – decomposition level, t – wavelet coefficient number) we define discrete wavelet transformation (dwt) of the vector x:      1 0 ,, n n tjntj nxw  , (1) where )(, tj are versions of the basis wavelet shifted by an integer number and rescaled on the dyadic scale ,...2,1,2 1   jjj , i.e.:  txx jjtj   22)( 2/,  . (2) in the case of the popular daubechies wavelets for a given j wavelet coefficients tjw , are proportional to differences (of various orders) of weighted averages over the scale j . for the stochastic process tx a time-varying wavelet variance is defined in the following way: )var( 2 1 )( , 2 tj j jt w   . (3) under the assumption that the quantity above does not depend directly on time 3, we obtain a decomposition of the variance according to the scale in the form (percival, walden, 2000, pp. 296-298):        1 1 2 , )()var( 1 2 1 )var( j j jtj j t wx  . (4) wavelet variance on the level j corresponding to the scale 12  jj , )( 2 j , informs about the changeability of fluctuation in cycles contained approximately in the bracket j2 – 12 j . in the estimation of wavelet variance and wavelet correlation in practice, dwt is replaced by its modification in the form of modwt (maximal overlap discrete wavelet transform)4, which does not require handling long ranges being 3 the assumption is fulfilled also for non-stationary processes on condition that the processes are integrated of order d, while the applied wavelet filter is sufficient to eliminate the nonstationarity, i.e. it is a daubechies filter (daublet, symlet or coiflet) of an appropriate length (see percival, walden, 2000). 4 modwt is preferably pronounced ‘mod wt’ – modified wt see: (percival, walden, 2000, p. 159). other names for this kind of transformation are non-decimated wavelet transformation, continuous-discrete wavelet transformation or french-derived term algorithme à trous. )( jj ,,2,1  12,,1,0   jjt  detection of collusion equilibrium in an industry with application of wavelet analysis 159 the subsequent powers of the number 2, provides a more effective estimator )(2 j and has invariance properties due to shifts in time. (percival, walden, 2000, pp. 308-310; gençay and others, 2002, p. 135). the estimator of wavelet variance is expressed by means on the following formula:     1 1 2 , 2 ~ ~ 1 )(~ n lt tj j j j w n  , (5) where are modwt coefficients, is the length of the wavelet filter for the scale (l is the length of the basic wavelet filter), whereas is the number of coefficients distorted by the extrapolation at the ends of the sample. (1–)-per cent confidence interval for )(2 j can be approximated in the following way: 5,0 , ~ 2 ~ )0(ˆ )(~ 2          j jw j n f  , (6) where 2  is the (1–/2) quantile of the normal distribution, whereas )0(ˆ ,~ jwf is an estimate of the spectral density function for squares of wavelet coefficients for scale j at 0. 2.1. testing homogeneity of variance the attractiveness of wavelet coefficients in testing volatility changes results from their two vital properties. firstly, wavelet coefficients are closely tied to the changes at various scales and moments in time, and thus they carry information about the variability of the process. secondly, (conventional) wavelet transformation provides coefficients which can be treated as approximately noncorrelated both in the case of short and long-memory processes, which significantly simplifies the inference procedures (comp. whitcher, 1998, ch. 4.1.2; percival, walden, 2000, p. 351). meanwhile, in the estimation of the switch moment the employment of modwt is proposed on the account of eliminating subsampling effects, which entails higher precision of an estimate. the approach put forward in the phd dissertation of whitcher (1998) is presented below, and the direct application of the inclán and tiao (1994) method to wavelet coefficients is proposed. it is obligatory to stress that the methods of wavelet detection of changes in variance are discussed primarily in the context of longmemory processes, but – because the property of the approximate decorrelation is valid also for the short-memory processes – it is possible to apply the method in the case of the latter processes as well. tjw , ~ 1)1)(12(  ll jj j 1 ~  jj lnn sylwester bejger, joanna bruzda 160 let }{ ,tjw be wavelet coefficients of the jth decomposition level. we are now interested in testing the following hypothesis: )var()var()var(:h 12/,1,,0   j jj njljlj www  , where jl is the number of boundary coefficients of the wavelet transform at the jth decomposition level (dwt coefficients, the value of which is influenced by the extrapolation method at the ends of the sample)5. furthermore, we assume that the wavelet decorrelation is effective what means that the coefficients }{ ,tjw form a second-order gaussian white noise. moreover, we assume that the length of the wavelet filter is sufficient for the elimination of deterministic components, i.e. 0)e( , tjw . the statistics in the test against the alternative hypothesis in the form: )var()var()var()var(:h 12/,1,,,1   j j njkjkjlj wwww  , where k is the unknown location of the variance change, is based on the normalized cumulated sum of squares (cusum for squares):       12/ 2 , 2 , j j j n lt tj k lt tj k w w w , (7) and have the forms: a. (inclán, tiao, 1994) j j jj j j n lk k lk n wit       1 12,, 2 max  ; (8) b. (whitcher, 1998)   ddw jn ,max 2 , (9) where:  knlk nlk wd j j j j      1 1 22/,, max  ,   1 22/,, max      j j j j n lk k nlk wd  , whereas j j j lnn  2/ is the number of coefficients distorted by the extrapolation method at the ends of the sample. in the case where jn amounts to at least 128, the following distribution for the test statistics is used (see: inclán, tiao 1994, p. 923; whitcher 1998, p. 60). small sample critical values (9) can be found in the works of whitcher (1998), whitcher and others (2002). 5 for the applied wavelets those values amount to 0 in the case of the haar wavelet whereas in the case of d4 wavelet – 1 for the first decomposition level and 2 for higher levels – compare (percival, walden, 2000, p. 136). detection of collusion equilibrium in an industry with application of wavelet analysis 161 2.2. estimating the location of variance change k ~ , for which the appropriate expressions in test statistics reach their maximum as mentioned earlier, it is necessary to base the estimation of the location of variance change on the coefficients tjw , ~ of the non-decimated wavelet transformation. it is then that the index values, indicates the moment of change. obviously, in such statistics the lower and upper summation limits must be modified accordingly, in the way so that all non-boundary coefficients are taken into account. and thus, the statistic in inclán and tiao version now takes the form: 1 2 1,,12 1 max      j j j j ln lk k nlk ln wit  , (10) whereas the expressions in statistics (9) transform into:  klnlknlk wd jjj      2 2,,1 max  ,  1 1, , 2 max ,j j j k l k n l k l n d w         then the normalized cumulated sum of squares is in the form:       1 1 2 , 1 2 , ~ ~ n lt tj k lt tj k j j w w w . (11) finally, we receive the moment of change in real time, by applying additional corrections due to a phase shift6. in the case in which there is a possibility of the occurrence of multiple changes in variance, it is proposed (inclán, tiao, 1994) to apply a procedure described as the algorithm of iterated cumulative sums of squares, icss, based on the method of binary segmentation. the application of such an algorithm in the context of wavelet detection was proposed by whitcher et al. (2000). the course of the algorithm can be depicted in the following way:  having a succession of non-boundary dwt coefficients: 12/,1,, ,,,  jjj njljlj www  we assume 12/, 21  j j ntlt and compute the value of the statistic it or w for the range ],[ 21 tt . if the statistic points to the occurrence of change, we mark the moment k , for which its value was determined and we pass on to the next stage.  we determine the value of the test statistic to the left of k , i.e. in the range ]1,[ 1  kt and to the right of k , i.e. in the interval ],1[ 2tk   . we keep 6 in the case of the filters applied further in the empirical part, the evaluation of the moment of change in real time, with the assumption that the time is numbered beginning with 1, is obtained as follows: in the case of the haar wavelet without any modification of k ~ , while in the case of d4 wavelet – by subtracting 1 from k ~ . sylwester bejger, joanna bruzda 162 finding subsequent points by following the same procedure – we appropriately divide the intervals and determine the values of statistics in the smaller ranges of coefficients. the procedure comes to an end when the rejection of the zero hypothesis does not occur.  we arrange the detected points in the ascending order nkkk ,,, 21  while additionally assuming jlk 0 and 12/1  j n nk . for each j = 1, …, n we carry out a test in the interval between the points of change adjacent to jk , i.e. in the area between 1jk and 1jk . if a potential change-point is not detected again, we reject jk from the considered set. the procedure is continued on a new set of points and we ultimately finish when there are no reductions in the number of detected changes. with the aim of selecting the appropriate manner of implementation for the presented methods, it is worth discussing their statistical properties. firstly, wavelets with smaller support (shorter wavelet filters) have better localization properties – out of the simulation analyses presented in whitcher’s phd dissertation (1998) we can infer that the localization of the moment of switch is burdened with a bigger mistake for longer wavelet filters. secondly, wavelet coefficients on the second decomposition level require higher snr (signal to noise ratios), in order to attain the same level of accuracy as on the first level, both in terms of the detection of change and its localization – thus it is vital to make use of primarily first levels of decomposition. other conclusions of the simulation analyses are as follows: the estimation of the moment of change is lightly biased towards the centre of the test (inclán, tiao, 1994; whitcher, 1998), in the test of homogeneity of the wavelet variance, the haar, d4 and la8 wavelets provide similar rejection rates in tests for one or two changes in a wide range of value for the parameter d of fractional integration (0,05–0,45), especially if small sample critical values were used and the test was carried out on low (1–2) decomposition levels (whitcher, 1998; whitcher and others, 2002). obviously, daubechies wavelets (see daubechies, 1992; percival, walden, 2000) with longer support allow to analyse non-stationary processes of a more complicated structure but – simultaneously – render a smaller number of useable coefficients. in the case of the tested sequence we do not analyse higher levels of decomposition also for the reason that it would only contain 19 coefficients (so there is a slim chance of as many as two moments of change being discovered), moreover, wishing to carry out a j-level wavelet analysis without having to write in any additional observations we should be equipped with a series of length which is a multiple of j2 (the length of our series – 78 – does not allow to take the analysis to higher levels without writing in additional observations). summing up, two wavelets will be applied in the analysis: the haar wavelet and detection of collusion equilibrium in an industry with application of wavelet analysis 163 d4, yielding filters of the length of 2 and 4 respectively7. in the testing for homogeneity of variance we will use the first decomposition level. 3. empirical study in the first stage of the analysis graphs were made of the modwt coefficients and the wavelet variance in its rolling version. figure 1 was then marked with modwt coefficients obtained by means of the haar wavelet on 4 levels of decomposition, together with scaling coefficients from the fourth level as well as the original time line. the values are shifted on the time axis in the way so as to cancel the phase shift. figure 1. wavelet, scaling coefficients and the time series. the vertical lines delineate the boundary regions 7 the d4 wavelet cannot be written in closed form, while the haar wavelet is given as: )()()( )2/1,0)1,2/1 xxx   11 . a detailed presentation of the level j haar and d4 wavelet filters can be found in percival and walden (2000), ch. iv. 10 20 30 40 50 60 70 0.7 0.8 0.9 1 1.1 1.2 1.3 x t-1w 1 t-2w 2 t-2v 2 10 20 30 40 50 60 70 0.7 0.8 0.9 1 1.1 1.2 1.3 x t-1w 1 t-2w 2 t-4w 3 t-4v 3 10 20 30 40 50 60 70 0.7 0.8 0.9 1 1.1 1.2 1.3 x t-1w 1 t-2w 2 t-4w 3 t-8w 4 t-8v 4 sylwester bejger, joanna bruzda 164 figures 2 and 3 (next page) depict the wavelet variance in the rolling (local) version, computed with small portions of wavelet coefficients, shifted on the time axis, together with 95-percent confidence intervals, estimated on the basis of modwt. the calculations omit all boundary coefficients (i.e. the coefficients the value of which is influence by the extrapolation method at the ends of the series). two moments can be seen: one located around observation 28 and the other further off. figure 2. rolling wavelet variances with with 95-percent confidence intervals – haar wavelet, window length 30 20 4 0 60 0 0. 5 1 1. 5 x 10 -3 sca le 1 wavelet variances 20 40 60 0 0.005 0. 01 0.015 scale 2 2 0 40 60 0 0.02 0.04 0.06 0.08 sca le 4 figure 3. rolling wavelet variances with with 95-percent confidence intervals – d4 wavelet, window length 30 the next stage consisted in testing of structural changes and variances by means of wavelet methods and included a direct application of the inclán and tiao method to dwt coefficients and testing according to whitcher’s wavelet method. the results of both tests are summed up in tables 1 and 2. the occurrence of the subsequent point of change was also verified with the application of wavelet d4. this time coefficients with numbers from the interval [1, 20] were evaluated. appropriate statistics were equal: 0.4106 (it), 0.6290 (w), pointing to the lack of another switch within the variance. the d4 wavelet indicates two moments of change: we infer about the first one at the low significance level, i.e. empirical significance level remains below 10%, yet the inference about the presence of the latter is already made at 1% level. the observations are made in accordance with the course of rolling wavelet variances. 20 40 60 0 1 2 3 4 x 10 -3 scale 1 wavelet variances 20 40 60 0 0.005 0.01 0.015 scale 2 20 40 60 0 0.02 0.04 0.06 0.08 scale 4 20 40 60 0 0.1 0.2 0.3 0.4 scale 8 detection of collusion equilibrium in an industry with application of wavelet analysis 165 table 1. testing for change in variance – the haar wavelet method and stage of analysis statistics no. of the dwt coefficient (kj) location of the change in variance with the help of the modwt coefficients it first change-point 1.4144** 22 (k1) 45 w – first change-point 1.3458** 22 (k1) 45 it – second change-point searched to the left from the first 0.4627 10 38 w – second change-point searched to the left from the first 0.7356 18 38 it – second change-point searched to the right from the first 0.5335 27 61 w – second change-point searched to the right from the first 0.8494 34 61 note: it – inclán and tiao test; w –whitcher test; asymptotic critical values: 1.224 (10%), 1.358 (5%), 1.628 (1%), small-sample critical values (whitcher, 1998): n = 16 – 1.135 (10%), 1.265 (5%), 1.508 (1%), n = 32 – 1.157 (10%), 1.293 (5%), 1.553 (1%); small-sample critical values (inclán, tiao, 1994): n = 100 – 1.14 (10%), 1.27 (5%), 1.52 (1%); numeration of the dwt coefficients: 0, 1,…, 38; location of variance change in real time; ‘**’ denotes rejection of the homogeneity of variance at the 5% significance level. table 2. testing for change in variance – the d4 wavelet method and stage of analysis statistics no. of the dwt coefficient (kj) location of the change in variance with the help of the modwt coefficients it first change-point 1.1297 23 (k2) 42 w – first change-point 1.0583 23 42 it – second change-point searched to the left from the first 1.2241* 20 (k1) 28 w – second change-point searched to the left from the first 1.5256*** 20 (k1) 28 it – second change-point searched to the right from the first 1.0331 34 63 w – second change-point searched to the right from the first 0.6940 34 63 note: it – inclán and tiao test; w –whitcher test; asymptotic critical values: 1.224 (10%), 1.358 (5%), 1.628 (1%), small-sample critical values (whitcher, 1998): n = 16 – 1.135 (10%), 1.265 (5%), 1.508 (1%), n = 32 – 1.157 (10%), 1.293 (5%), 1.553 (1%); small-sample critical values (inclán, tiao, 1994): n = 100 – 1.14 (10%), 1.27 (5%), 1.52 (1%); numeration of the dwt coefficients: 0, 1,…, 38; location of variance change in real time; designation of the coefficients indicating a change in variance is in an increasing order; ‘*’, ‘***’ denote rejection of the homogeneity of variance at the 10%, 1% significance level, respectively. as far as the moments of switch are concerned, certain ambiguities appear. the first change-point is in the interval 42–46 (rather closer to 45–46, as the haar sylwester bejger, joanna bruzda 166 wavelet has better localization properties, and moreover – as it was already mentioned – the evaluation of the moment of change is burdened with error towards the centre of the sample), while the other one – as it results from the test with the use of d4 wavelet as well as the charts – around obser vations 28–29. 4. interpretation of results in order to better illustrate the received results, the series of average lysine prices was presented in figure 4, together with observation numbers. figure 4. average price of lysine, observation numbers on upper horizontal axis. data from connor, (2000), appendix a, table a2 as results from the study we carried out, the moments of variance change are identified around observation 28 – the increase of the variance – and around observation 45 – the fall of the process’ variance. this piece of statistical information is confirmed by the analysis of charts 1 and 2. basing on theoretical motivation of the marker of collusion it is mandatory to make the following statements: a) until april 1992 no significant change of the variance regime can be found, therefore the period cannot be considered a factual phase of the price war, b) from april 1992 until august 1993 we can talk about a significant phase of the growth of the process variance, which points to the price war. it is the period of the announcement by adm (precisely april 1992) of the proposition to establish the ‘amino acids manufacturer society’, until august 1993, which is around two months before the irvine meeting (where representatives of two main players, amd and ajinomoto had confidentially agreed cartel quotas of lysine supply, as they further witnessed), 1 7 13 19 25 31 37 43 49 55 61 67 73 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 ja n -9 0 m a y9 0 s e p -9 0 ja n -9 1 m a y9 1 s e p -9 1 ja n -9 2 m a y9 2 s e p -9 2 ja n -9 3 m a y9 3 s e p -9 3 ja n -9 4 m a y9 4 s e p -9 4 ja n -9 5 m a y9 5 s e p -9 5 ja n -9 6 m a y9 6 average price of lysine (usd/pound) detection of collusion equilibrium in an industry with application of wavelet analysis 167 c) from september 1993, therefore exactly a month before the irvine meeting, a meaningful drop of the process variance occurs, which may indicate the establishment of collusion and the willingness to maintain it. there might be a certain mistake in pinpointing the moment of change or perhaps the players had gone into informal agreements before the time they indicated in their testimonies. comparing the results of the wavelet analysis with familiar facts from the cartel history (see: connor, 2000), it is necessary to notice that the analysis pointed out significant changes in the process variance. the discovered moments of change are closely tied to vital facts from the cartel history, therefore we can assume that they indicate the change in the players’ price behaviour. if we accept the proposed game theory model (bejger, 2010) as the theoretical base for that behaviour, changes in the variance correspond to phases of the price war and collusion. in confrontation with the to-date analyses of the workings of the cartel, the present empirical study leads to the following differences in the history evaluation:  price war phase is delayed by around 12 months,  first phase of collusion is not clearly distinguished. on the one hand, those differences may be caused by a small sample series, which disables wavelet analysis at higher decomposition levels, on the other they may point to a history of lysine industry collusion that is much different from the one generally accepted. conclusions taking as an theoretical motivation the model of strategic behaviour of the players in an industry with the assumption of time-fixed cartel quotas (market shares) an attempt of explaining the behaviour of the players in the industry of lysine manufacturers in the years 1990 – 1996 was made. the model provides theoretical support to the method of the detection of collusion based on the analysis of the variance of the market price process, as well. for empirical evaluation the marker of variance change was applied for a series of average lysine prices on the usa market in the studied period. wavelet analysis was employed, for the first time in such a context, as the econometric method of the detection of structural changes in the variance. the proposed method of detection proved to be very useful, indicating the significant changes of the variance regime and precisely detecting the moments of such changes, closely connected to the key dates in the history of the cartel. while enumerating the advantages of wavelet analysis in the proposed application, it is essential to mention:  parsimony of specification – as a non-parametric method it is not burdened with specification error of the econometric model, sylwester bejger, joanna bruzda 168  simplicity of application – the work indispensable to apply the method to the data is minimal. therefore, the method is quick in application,  precise indication of the moment of variance change, without any assumptions as to their localization. the method is thus absolutely objective,  possibility of preliminary, graphic assessment of the variance changes by means of modwt charts as well as rolling wavelet variances. among the drawbacks to wavelet analysis we can indicate relatively high requirements as to the length of the observation series and the lack of the direct link of the method to the structure of equilibrium strategy (as it takes place in e.g. the application of markov switching model of ms-ar-garch type). therefore, it can be assessed that wavelet analysis in the proposed application may serve as a preliminary detector of the variance changes because its application is cheap and quick with accurate findings. wherever it is theoretically justified other methods may then be applied with the aim of further verification of the hypothesis of collusion in an industry. our work has a case –study character so our conclusions should be verified in further research. next steps in the study will be the modification of the theoretical model for the strategies with different penal codes and – on the empirical plane – an attempt to apply the wavelet analysis of variance for other price time series in industries susceptible to explicit or tacit collusion. references abrantes-metz, r., froeb, l., geweke, j., taylor, c. (2006), a variance screen for collusion, international journal of industrial organization 24, 467–486 athey, s., bagwell, k., sanchirico, c. (2004), collusion and price rigidity, review of economic studies, 71, 317–349. bejger, s. (2010), collusion and seasonality of market price – a case of fixed market shares, business and economic horizons, vol 2, 48–59. bejger, s. (2009), econometric tools for collusion detection, aunc ekonomia xxxix, 125–132. bolotova, y., connor, j. m., miller, d. j. (2008), the impact of collusion on price behavior: empirical results from two recent cases, international journal of industrial organization 26, 1290–1307. connor, j. (2001), our customers are our enemies”: the lysine cartel of 1992–1995, review of industrial organization, 18, 5–21. connor, j. (2000), archer daniels midland: price-fixer to the world, staff paper no. 00-11, department of agricultural economics, purdue university, west lafayette, in. daubechies, i. (1992), ten lectures on wavelets, siam, philadelphia. gençay, r. f., selçuk, f., whitcher, b. (2002), an introduction to wavelets and other filtering methods in finance and economics, academic press, san diego. harrington, j. e. (2005), detecting cartels, working paper, john hopkins university. inclán, c., tiao, g. c. (1994), use of cumulative sums of squares for retrospective detection of changes of variance, journal of the american statistical association, 89, 913–923. percival, d. b., walden, a. t. (2000), wavelet methods for time series analysis, cambridge university press, cambridge. west, k. d., cho, d. (1995), the predictive ability of several models of exchange rate volatility, journal of econometrics, 69, 367–391. detection of collusion equilibrium in an industry with application of wavelet analysis 169 whitcher, b. (1998), assessing nonstationary time series using wavelets, phd thesis, university of washington. whitcher, b., byers, s. d., guttorp, p., percival, d. b. (2002), testing for homogeneity of variance in time series: long memory, wavelets and the nile river, water resources research, 38 , 1054–1070. whitcher, b., guttorp, p., percival, d. b. (2000), multiscale detection and location of multiple variance changes in the presence of long memory, journal of statistical computation and simulation, 68, 65–88. detekcja równowagi zmowy w branży z wykorzystaniem analizy falkowej z a r y s t r e ś c i. artykuł zawiera empiryczne zastosowanie markera zmian strukturalnych wariancji procesu ceny rynkowej dla szeregu cen średnich lysiny na rynku usa w latach 1990– 1996. jako metodę ekonometryczną detekcji zmian strukturalnych w wariancji zastosowano, po raz pierwszy w tym kontekście, analizę falkową. metoda ta ma w omawianym zakresie aplikacji istotne zalety, takie jak oszczędność w zakresie wymaganych danych statystycznych oraz bardzo dobre własności lokalizacyjne w dziedzinie czasu. w pracy wykorzystano model teoretyczny zachowań strategicznych graczy w branży motywujący zastosowanie wymienionego markera. s ł o w a k l u c z o w e: zmowa jawna i milcząca, supergra ze stała strukturą udziałów w rynku, lysina, wariancja ceny, analiza falkowa. 01_blangiewicz_milobedzki.pdf © 2012 nicolaus copernicus university press. all rights reserved. http://www.dem.umk.pl/dem dynamic econometric models vol. 12 (2012) 5−17 submitted may 6, 2012 issn accepted october 8, 2012 1234-3862 maria blangiewicz, paweł miłobędzki* the expectations hypothesis of the term structure of libor us dollar interest rates† a b s t r a c t. using the monthly sampled data on libor us dollar interest rates and maturities ranging from 1 to 12 months from 1995 to 2009 we provide with a number of tests of the expectations hypothesis based on a 3-variable var allowing for a time-varying term premium. we find some evidence against the expectations hypothesis. the term premia appear to vary in time and the yield spread has a good predictive power, however the long rates under-react to current information about future short rates. unexpected changes in holding period returns to large extent depend upon revisions to forecasts about future short rates and to small extent upon revisions to future term premia. k e y w o r d s: term structure of interest rates, expectations hypothesis, term premium, libor, var. j e l classification: e43. introduction the expectations hypothesis (eh) of the term structure of interest rates credited to fisher (1886, 1930) and lutz (1940) states that the expected oneperiod holding period return on a bond that has n periods to maturity (long bond) equals to the return on one-period (short) bond increased by the term premium. if valid it has two important implications: the yield on a long bond (long rate) equals to the average of expected yields on the short bond (short rates) over the life of the long bond plus the rolling-over term premium, and the * correspondence to: paweł miłobędzki, department of econometrics, faculty of management, university of gdańsk, ul. armii krajowej 101, 81-824 sopot, poland. tel/fax: +48 585231408, e-mail: milobedp@wzr.ug.edu.pl † the earlier version of this paper was presented at the dynamic econometric models conference held at the nicolaus copernicus university, toruń, september 7-9, 2011. the research was founded by the polish ministry of science and higher education under the grant n n111292238 the term structure of libor interest rates (struktura terminowa stóp libor). maria blangiewicz, paweł miłobędzki dynamic econometric models 12 (2012) 5–17 6 actual yield spread between the long and the short rate is an optimal predictor of the next period’s change in the long rate as well as future changes in the short rate. the early tests of the eh invented by campbell and shiller (1991) examine the ability of the yield spread to predict future changes in the short and the long rates. embedded in either a single equation or var setting they provide with a very limited support for the eh when performed on the us and the other data. the long rates appear to move in the opposite direction to that predicted by theory. the short rates move in the correct direction, however the yield spread is their poor predictor at the shorter end of maturity spectrum (see campbell, shiller, 1991; hardouvelis, 1994; gerlach, smets, 1997, among many others). the empirical failure of the eh is explained in a number of ways. it is usually accounted for the existence of a time-varying term premium which is assumed constant in traditional tests. the other explanations include a small sample bias of the eh tests remaining severe in large samples, the over-reaction of long rates to current short rates as well as the asset pricing anomaly disappearing once it is widely recognized to the public (tzavalis, wickens, 1997; bekaert et al., 1997; garganas, hall, 2011; bulkley et al., 2011). it is also stressed that the predictive power of the yield spread depends upon monetary policies implemented by the central bank being much stronger at the times of monetary targeting than interest rates smoothing (see mankiw, miron, 1986; mccallum, 2005, among many others). in this paper we report on that whether the libor us dollar interest rates behave according to the eh. we assume that the term premium vary over time and nest the analysis within a 3-variable var of tzavalis and wickens (1997). we estimate it on the monthly sampled data from 1995 to 2009. in doing so we use maturities ranging from 1 to 12 months. the data come from thomson reuters1. to test for the time-varying term premium, the ability of the yield spread to predict future changes in the short rate and the link between the current yield spread and that predicted from the var we set restrictions on its parameters and statistics. we provide with some evidence against the eh. the results reported in the paper complement those of hurn et al. (1995) and miłobędzki (2010) who analyzed the libor interest rates in sterling and using the var methodology found much support for the eh at the whole maturity spectrum. the remainder of the paper proceeds as follows. section 1 introduces the eh of the term structure of interest rates and shows its implications. section 2 reviews the var based tests of the eh allowing for the time-varying term premium. section 3 discusses our empirical findings. the last section briefly concludes. 1 the data are supplied under the agreement between thomson reuters poland and the university of gdańsk. the expectations hypothesis of the term structure of libor us dollar interest rates dynamic econometric models 12 (2012) 5–17 7 1. eh of the term structure of interest rates and its implications the eh of the term structure of interest rates may be formally stated as: ( ) ( ) ( ) ( ) ( )1 1 1 1ln ln n n n n t t t t t t te h e p p r θ − + +  = − = +  , (1) where ( )ntp is the price at time t of pure discount bond with face value of $1 and n periods to maturity, ( )1tr is the certain (riskless) one-period interest rate, and ( )ntθ is a term premium which compensates for the risk of investing in long bonds. the term premium is admitted to vary in time but presumed to be a stationary random variable. variants of the eh include the pure ( ( ) 0ntθ = , peh), constant ( ( )nt constθ = , ceh) and liquidity preference versions ( ( ) ( )1n n t tθ θ −> > ( )2 tθ> , lpeh) (for all t and n ). the term premium, ( )n tθ , according to eq. (1), is reflected by the expected excess one-period holding period return, ( ) ( )1 1 n t t te h r+ − . to demonstrate implications of the eh for the interest rates the following is usually undertaken (campbell, shiller, 1991; cuthbertson, 1996; tzavalis, wickens, 1997; cuthbertson, nitzsche, 2003). firstly, a continuous compounding is assumed, i.e. ( ) ( )ln n nt tp nr= − , where ( )n tr is the spot yield on the long bond. then some manipulations of eq. (1) result in: ( ) ( ) ( ) ( )1 1 0 1 nn n t t t i ti r n e r − += = + θ∑ , (2) where ( ) ( ) ( )1 0 1 nn n i t t t ii n e θ − − += θ = ∑ . subtracting ( )1tr from both sides of eq. (2) and rearranging terms yield: ( ) ( ) ( ) ( )1 1 1 1 nn n t t t i ti s e i n r − += = − ∆ +θ∑ . (3) eq. (3) shows that the observed yield spread should equal to the sum of the optimal forecast of future changes in the short rate, ( ) ( )1 1 1 1 n t t ii e i n r − += − ∆∑ , and the average of term premia expectations, ( )ntθ (rolling-over term premium). thus it can be concluded that at time t no other information apart from that contained in both variables should help predict future changes in the short rate. the immediate consequence of the latter is twofold: ( )nts should granger cause ( )1t ir +∆ , and in the case term premium ( )n tθ is not time-varying the expected excess one-period holding period return is constant and should not depend upon its past values as well as past values of the actual spread and changes in the future short rate. maria blangiewicz, paweł miłobędzki dynamic econometric models 12 (2012) 5–17 8 last but not least, substituting eq. (2) into the unanticipated change (‘surprise’) in the one-period holding period return, ( ) ( ) ( )1 1 1 n n n t t t teh h e h+ + += − , gives (tzavalis, wickens, 1997): ( ) ( ) ( ) ( ) ( ) ( ) ( )1 11 11 1 1 1 11 1 n nn n i n t t t t i t t t i t ti i eh e e r e e er eθ θ − − − + + + + + + += =  = − − − − = − + ∑ ∑ , (4) where ( ) ( ) ( )11 11 1 1 n t t t t ii er e e r − + + += = − ∑ exhibits the ‘news’ about future short rates and ( ) ( ) ( )11 1 1 nn n i t t t t ii e e eθ θ − − + + += = − ∑ exhibits the ‘news’ about future term premia. hence, unanticipated change in the one-period holding period return must be due to either a revision to expectations about future short rates or a revision to expectations about future term premia. 2. three-variable var based tests of the eh the var based tests of the eh solving for the time-varying term premium hinge on the extended 2-variable var of campbell and shiller (1991) in which the yield spread, ( )nts , and the change in the short rate, ( )1 tr∆ , are supplemented by the excess one-period holding period return, ( ) ( )11 n t th r −− . such a var of order p with vector ( ) ( ) ( ) ( )1 1* 1 n n t t t t tz s r h r − ′ = ∆ −  containing stationary variables is stacked into companion form as a first order var (see tzavalis, wickens, 1997; cuthbertson, bredin, 2001; cuthbertson, nitzsche, 2003; blangiewicz, miłobędzki, 2008): 1 ,t t tz az u−= + (5) where a is a square ( )3 3p p× matrix of coefficients, tz is a ( )3 1p × vector of regressors like ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 11 1 1 1 n n n n n n t t t p t t p t t t p t pz s s r r h r h r− + − + − − + − ′ = ∆ ∆ − −    , and tu is a ( )3 1p × vector of errors. variables included in the var can be picked up from the system using ( )3 1p × selection vectors 1e ′ , 2e ′ and 3e ′ with unity in the first, second and third row, respectively, and zeros elsewhere so that ( ) 1nt ts e z′= , ( )1 2t tr e z′∆ = and ( ) ( )1 1 3 n t t th r e z− ′− = . their predictions from the var can be computed throughout the chain rule of forecasting as: ( )| kt k t te z z a z+ = . (6) the tests of interest verify whether the excess one-period holding period return is not time-varying, what the sources of ‘surprise’ in its performance are (if there are any), as well as whether long rates properly react to current information about future short rates. construction of the appropriate test statistics is based on the assumption that predictions from the var system are adequate. the expectations hypothesis of the term structure of libor us dollar interest rates dynamic econometric models 12 (2012) 5–17 9 the prediction of the expected excess one-period holding period return from the var is ( ) ( )11 13 3 n t t t t te h r e z e a z+ +′ ′− = = which in the case of time-invariant term premium should equal to some constant. in terms of the var with demeaned variables it requires a set consisted of 3 p linear restrictions be such that 3 0e a′ = . this is tested with the use of a wald test. under the null the relevant test statistics is distributed as 2χ variable with 3 p degrees of freedom. the prediction of the yield spread from the var (‘theoretical spread’) is (cuthbertson et al., 2000): ( ) ( ) ( )1* 1 1 1 2 nn t t t i ti s e i n r e z − += ′= − ∆ = λ∑ , (7) where ( )( )( ) ( )1 11 na i n i a i a i a− − λ = − − − −  . it should track the actual spread, ( ) 1nt ts e z′= , provided expectations about the future term premia, ( )n i t t ie θ − + , are constant over time. in such circumstances it is expected that ( ) ( )* n n t ts s= , which implies the following set of var metrics: 1' 2 ' 0e e− λ = , (8) ( ) ( )*2 2 1n nt tvr s sσ σ   = =    . (9) ( ) ( )* , 1n nt tcorr s sρ  = =  , (10) where vr and ρ stand for a variance ratio and a correlation coefficient, respectively. the set of nonlinear cross-equation restrictions from eq. (8) can be tested for with the use of a wald test. the relevant test statistics is: 1 ( ) ( )ˆ( ) ( )aa f a f a w f a f a a a − ∂ ∂ ′= × σ × ′∂ ∂  , (11) where ( ) 1' 2 ' 0f a e e= − λ = and ˆ aaς is either the standard or the eicker-white heteroscedasticity consistent variance-covariance matrix of var parameters estimator. under the null (and standard properties of error term tu ) it is distributed as the 2χ variable with 3 p degrees of freedom. to proceed with the metrics contained in eq. (9) and (10) it is worth noting that under the peh the series of theoretical and actual spread should move together. a high degree of co-movement indicates that variation in the spread is mainly due to rationally expected changes in future short rates with no or only minor variation in the premia (engsted, 1996). the validity of peh can be informally deduced plotting ( )* nts versus ( )n ts , while it can be more formally evaluated using the above two metrics. since vrβ ρ= is the ols estimator maria blangiewicz, paweł miłobędzki dynamic econometric models 12 (2012) 5–17 10 of the slope in the regression of actual spread onto theoretical spread, which should also be unity, both the numerator and denominator should be close to unity or one of them must be approximately the inverse of the other. thus the rejection of 1β = is to be accounted for either the over-reaction (underreaction) hypothesis or the presence of the time-varying term premium. if ( ) 1vr < > and 1ρ ≈ , then the slope would be more (less) than unity and the actual spread is more (less) volatile than the theoretical spread, the optimal predictor of future short rates. hence, although there is a strong relationship between ( )* nts and ( )n ts , the long rate is over-reacting (under-reacting) to current information about future short rates. in the case neither are close to unity, the actual spread behaves differently from the theoretical spread and the overreaction (under-reaction) could be the consequence of a time-varying term premium (campbell, shiller, 1991; hardouvelis, 1994). bekaert et al. (1997), bekeart and hodrick (2001) and garganas and hall (2001) show that a 2-variable var based tests of the eh with the exception of ( ) ( )* ,n nt tcorr s s   are biased in small samples in the case the short rate is persistent. the wald test tends to over-reject the null, while the variance bound ratio favours it too often. the bias in these tests increases with the degree of short rate persistence. the rejection of the eh may be also due to noise traders. their excessive activity relative to that of smart money traders increases interest rates volatility which results in a downward bias of all var metrics (cuthbertson et al., 1996). we are now to asses what a portion of ‘surprise’ in the one-period holding period return, ( ) ( ) ( )1 1 1 n n n t t t teh h e h+ + += − , is due to the ‘news’ about future short rates, ( )1 1ter + , and the ‘news’ about future term premia, ( ) 1 n teθ + . such a decomposition is based on residuals from the var system. to see this note that the error from the second var equation, 2, 1 12t tu e u+ +′= , represents the ‘surprise’ in the future change of the short rate, while the error from its third equation, 3, 1 3t tu e u+ ′= – the ‘surprise’ in the excess one-period holding period return. since (see tzavalis, wickens, 1997): ( ) ( ) ( ) ( ) ( ) ( ) ( )1 11 1 1 11 1 1 11 1 11 n n i t t t t t t t t ji i j er e e r e e n r r − − + + + + += = =  = − = − − + ∆ =  ∑ ∑ ∑ ( ) ( )1 11 1 1 n i t t t ji j e e r − + += = − ∆∑ ∑ , (12) the ‘surprise’ in the term premia can be calculated from: ( ) ( ) ( )1 1 1 1 n n t t te er ehθ + + +=− − = (13) ( ) ( ) ( ) ( )( )2 2 1 3, 12 1 2 3 1 n t te n i n a n a n n a u u− + + ′ − + − + − + + − − −  , using the appropriate var residuals. the first term on the right hand side the expectations hypothesis of the term structure of libor us dollar interest rates dynamic econometric models 12 (2012) 5–17 11 of eq. (13) stands for the weighted sum of the ‘surprises’ in future short rates so that matrices sa exhibit the degree of persistence in the ‘news’ about future short rates ( 1, 2, , 2s n= − ). suppose further that a revision to expectations about future term premia are negligible ( ( )1 0 n teθ + ≈ ). this yields ( ) ( )1 1 1 n t teh er+ +≈ − , and the following metrics also apply: ( ) ( )12 2 1 1 1 n t ter ehσ σ+ +    ≈    , (14) ( ) ( )1 1 1, 1 n t tcorr er eh+ +  ≈ −  . (15) in addition, from eq. (1) and (4) we obtain: ( ) ( ) ( ) ( ) ( )1 1 1 1 1 n n n t t t t th r er eθ θ+ + +− = − − . (16) hence we can conclude that ( )21 r− of the one-period holding period return equation in the var system indicates a proportion of the excess one period holding period return that is due to variation in the ‘news’ about future short rates. 3. empirical results since interest rates are believed to be integrated of order one variables the use of the var based tests in the applied work is limited to cases in which all variables in the var system (actual yield spread, change in the short rate, expost excess one-period holding period return) are stationary2. this is to be empirically confirmed, however. hence the analysis sets off with testing for (non) stationarity of the individual us dollar libors and the variables entering the var. for testing purposes we employ the df-gls and kpss tests (see elliot et al., 1996; kwiatkowski et al., 1992). their results (available to readers upon a request) prove that the variables in question are integrated of order zero. the results from the var models are stacked in table 1 (see appendix). var order p for each maturity is set with the use of schwarz information criterion but occasionally increased to remove autocorrelation in residuals3. the 2 in such circumstances a 3-variable var of tzavalis and wickens (1997) implies a vector error correction model with the yield spreads and excess one-period holding period return being the co-integrating vectors; see appendix c in king and kurmann (2002) for details regarding a 2-variable var of campbel and shiller (1991). 3 there is some unclear picture of autocorrelation for the yield spread and the change in short rate equations (for 4n = and 12n = , respectively). the estimates of the breusch-godfrey test statistics used to test for no-autocorrelation of up to the 12-th order are just equal to the critical value of the f variable with 12 and ( )3 1 12t p− + − degrees of freedom, while the estimates of the ljung-box test statistics for these maturities are far away from the critical value of the 2χ variable with 12 degrees of freedom at the conventional 5 per cent significance level. we are not able maria blangiewicz, paweł miłobędzki dynamic econometric models 12 (2012) 5–17 12 first two equations in the system have from a relatively moderate to large explanatory power as reflected by their coefficient of determination 2r estimates. nevertheless quite a lot of unexplained variation in the ex-post excess one-period holding period return equation is left to be attributed to a revision to the expectations about future short rates and future term premia (estimates of 21 r− in the third equation range from 0.58 to 0.81). in addition, the estimates of granger non-causality test statistics prove the ability of the yield spread to predict future changes in the one-month us dollar libor. table 2 (see appendix) reports the results of testing for the eh using the restrictions set on the var parameters and other metrics. restriction 3 0e a′ = is rejected for all maturities so that we suspect the term premia are timevarying. turning now to the var metrics, a graph of the actual and theoretical spread for both 4n = and 12 show their rather poor correspondence over time with some visual evidence of under-reaction of the actual spread relative to the expected changes in future short rates (see fig. 1-2, right panels, appendix). the same somewhat poor correspondence is apparent when the first spread is scattered versus the latter (see fig. 1-2, left panels, appendix). empirical points on these panels are much dispersed along the straight 45-degree line indicating that for both maturities correlation between ( )nts and ( )* n ts may substantially differ from one. the null stating that ( ) ( )* n nt ts s= is also rejected at 5 per cent significance level for all interest rates but not for the 12-month us dollar libor (in this case it is rejected at 10 per cent significance level) which assures that the term premia are time-varying. a more formal measures of the relationship between the actual and theoretical spread to much extent support the under-reaction hypothesis. while for all maturities the vr estimates does not depart from unity by more than its 2 standard deviations (the relevant 95 per cent confidence interval obtained from the bootstrap covers unity in all cases apart from those of 8n = , 10 and 11), the correlation coefficient estimates are less than unity by more than its two standard deviations for all maturities except 12n = . given the result that a lot of unexplained variation in the ex-post excess one-period holding period return equation is due to a revision to the expectations about future short rates and future term premia (see estimates of 21 r− from the third equation of the var in table 1, appendix) we are to evaluate the size of their contribution to the overall effect. formally, the estimates of ( ) ( )11 1, n t ter ehρ + +   1≈ − and for all maturities they do not differ from to remove autocorrelation without over-parametrizing the system. hence, for these two maturities some caution should be retained when further predictions about the theoretical spread as well as predictions based upon all var metrics employing the change in short rate are made. the expectations hypothesis of the term structure of libor us dollar interest rates dynamic econometric models 12 (2012) 5–17 13 minus unity by more than its 2 standard deviations, and those of ( ) ( )12 2 1 1 n t ter ehσ σ+ +       are close to but above unity and slightly differ from that by more than its 2 standard deviations (its 95 per cent confidence interval from the bootstrap does not cover unity for all maturities except the 11-month us dollar libor). this indicates that a portion of ‘surprise’ in the one-period holding period return due to ‘news’ about future term premia for all n is not negligible, however small. taking into account the above findings we can conclude that the evidence we have gathered against the eh in the london interbank market is strong enough to reject it due to under-reaction of long rates to current information about the future short rate. on the other hand, as campbell and shiller (1991) have argued, rejection of the cross-equation parameter restrictions is not a final argument against the eh on economic grounds as long as the theoretical spread closely tracks the actual spread. out of all var metrics we primarily trust correlation coefficient ρ which properties are not much distorted by the short rate persistence and for that its estimates substantially depart from unity for all maturities. this is supported by the variance ratio metrics for some rates at the longer end of maturity spectrum. the rejection of the eh due to under-reaction could be totally erroneous, however. if agents use the var methodology for forecasting purposes, when forming expectations about the future short rates are expected to utilize information on a more frequent basis (minute-to-minute, hourly, daily; see cuthbertson et al., 1996). in addition, large banks can meet their longer-term needs for monies at a lower cost outside london. information regarding interest rates of both origins is not exhibited in our data set so that the predictions we have obtained from the var system might be heavily biased. in particular, using the theoretical spread we could substantially underestimate agents’ expectations about the future short rates. our predictions could also poorly track the true expectations. conclusions in this study using the monthly sampled data on libor us dollar interest rates from 1995 to 2009 and a wide range of maturities we find a rather conclusive evidence against the eh. however the term premia appear to vary in time and the yield spread has a good predictive power, the long rates under-react to current information about the future short rates. unexpected changes in the holding period returns to a large extent depend upon revisions to forecasts about the future short rates and to a small extent upon revisions to the future term premia. the results reported in this paper are in some respects in contrast to those of hurn et. al. (1995), engsted (1996), cuthbertson (1996), cuthbertson et al. (1996), blangiewicz and miłobędzki (2009, 2010) and miłobędzki (2010) who maria blangiewicz, paweł miłobędzki dynamic econometric models 12 (2012) 5–17 14 analyzed the term structure of interest rates at the danish, polish and the uk money markets with the use of either a 2 or 3-variable var and thus provide with the nearest comparison to our work. the main difference revealed in their work is that of a time-invariant term premium which is consistent with the peh (hurn et al., 1995; cuthbertson, 1996; miłobędzki, 2010 – for pound sterling, engsted, 1996 – for danish kroner; blangiewicz, miłobędzki, 2010 – for polish zloty, for all or some maturities), and the main similarity – a strong predictive power of the yield spread (all authors except from engsted, 1996, for danish kroner during the period of interest rates smoothing). appendix table 1. summary statistics for var ( ) ( )1* (1)1 n t t t t tz s r h r − ′ = ∆ −  n p autocorrelation r2 granger lm(12) a) ljung-box(12) b) noncaus ality (n) ts (1) tr∆ ( ) ( )n 1 t t 1h r −− (n) ts (1) tr∆ ( ) ( )n 1 t t 1h r −− (n) ts (1) tr∆ ( ) ( )n 1 t t 1h r −− 3 9 1.43 0.96 1.07 7.00 6.63 7.39 0.69 0.45 0.28 135.69 (0.15) (0.49) (0.38) (0.87) (0.88) (0.83) (0.00) 4 20 1.79 1.56 1.42 3.92 3.68 3.78 0.79 0.53 0.42 96.89 (0.05) (0.11) (0.16) (0.99) (0.99) (0.99) (0.00) 5 7 1.13 0.98 0.94 5.73 9.69 7.35 0.71 0.36 0.20 101.96 (0.34) (0.47) (0.51) (0.93) (0.64) (0.83) (0.00) 6 22 1.76 1.29 1.57 2.58 3.73 3.93 0.8 0.53 0.23 88.09 (0.06) (0.23) (0.10) (0.99) (0.99) (0.99) (0.00) 7 12 1.72 1.18 0.99 3.64 3.98 3.57 0.76 0.41 0.22 70.66 (0.07) (0.30) (0.46) (0.99) (0.98) (0.99) (0.00) 8 18 0.99 1.36 1.33 1.63 3.92 3.86 0.80 0.49 0.38 76.60 (0.46) (0.187) (0.21) (1.000) (0.99) (0.99) (0.00) 9 16 0.85 1.76 1.70 1.97 4.97 3.47 0.80 0.48 0.27 54.61 (0.60) (0.06) (0.07) (0.99) (0.96) (0.99) (0.00) 10 18 0.86 1.08 1.15 2.24 3.41 3.39 0.81 0.46 0.35 52.90 (0.59) (0.38) (0.33) (0.99) (0.99) (0.99) (0.00) 11 17 1.07 1.13 1.02 1.84 3.56 2.32 0.79 0.42 0.30 48.80 (0.39) (0.34) (0.43) (1.00) (0.99) (0.99) (0.00) 12 12 0.68 1.78 1.57 1.226 4.35 3.21 0.78 0.37 0.19 54.18 (0.78) (0.05) (0.10) (1.00) (0.98) (0.99) (0.00) note: a) estimates of the breusch-godfrey [ljung-box] test statistics for autocorrelation of order 12 under the null of no-autocorrelation distributed as ( )12, 3 1 12f t p− + −   [ ( ) 2 12χ ], t – number of observations; relevant p-values in brackets under the estimates; b) estimates of the wald test statistics for granger noncausality from ( )nts to ( )1 tr∆ under the null distributed as ( )2 pχ variable; relevant p-values in brackets under the estimates. the expectations hypothesis of the term structure of libor us dollar interest rates dynamic econometric models 12 (2012) 5–17 15 table 2. var restrictions and other metrics, variance decomposition n excess one period holding period return not time varying actual (n)st and theoretical *(n)st spread news about future short rates and one period returns *(n) (n)s =st t 2 2 *(n) t (n) t s s  σ    σ    ρ  *(n) (n) t ts ,s b) ( ) ( ) 12 t 1 n2 t 1 er eh + +  σ    σ   ( ) ( ) + +  ρ  1 n t 1 t 1er ,eh b) e3 a=0′ a) vr ci vr ci 3 w(27)=90.37 w(27)=93.57 0.91 0.61 0.71 1.28 1.12 -0.99 (0.00) (0.00) (0.26) 1.62 (0.10) (0.08) 1.45 (0.00) 4 w(60)=138.15 w(60)=182.80 1.22 0.69 0.60 1.31 1.04 -0.98 (0.00) (0.00) (0.41) 2.28 (0.10) (0.14) 1.60 (0.01) 5 w(21)=60.33 w(21)=50.64 0.90 0.51 0.72 1.33 1.03 -0.98 (0.00) (0.00) (0.31) 1.73 (0.12) (0.15) 1.63 (0.01) 6 w(66)=278.42 w(66)=161.53 1.28 0.76 0.58 1.64 1.20 -0.99 (0.00) (0.00) (0.47) 2.60 (0.12) (0.23) 2.10 (0.01) 7 w(36)=114.86 w(36)=73.69 1.14 0.55 0.70 1.57 1.10 -0.98 (0.00) (0.00) (0.47) 2.36 (0.13) (0.24) 2.03 (0.02) 8 w(54)=177.26 w(54)=135.30 1.82 1.09 0.52 1.50 1.02 -0.96 (0.00) (0.00) (0.75) 3.96 (0.09) (0.29) 2.00 (0.03) 9 w(48)=161.83 w(48)=103.48 1.42 0.67 0.64 1.58 1.04 -0.99 (0.00) (0.00) (0.61) 3.03 (0.13) (0.27) 2.10 (0.02) 10 w(54)=139.17 w(54)=96.06 1.99 1.48 0.51 1.53 1.04 -0.99 (0.00) (0.00) (0.90) 4.80 (0.08) (0.26) 2.08 (0.02) 11 w(51)=89.64 w(51)=123.33 2.14 1.36 0.50 1.41 0.96 -0.99 (0.01) (0.00) (1.46) 6.83 (0.11) (0.24) 1.92 (0.01) 12 w(36)=52.23 w(36)=49.85 1.41 0.60 0.74 1.70 1.03 -0.99 (0.04) (0.06) (0.69) 3.24 (0.15) (0.35) 2.36 (0.01) note: a) relevant p-values in brackets under the wald test statistics estimates; b) ρ – linear correlation coefficient. relevant standard errors from the bootstrap under the variance ratio vr and correlation coefficient ρ estimates. ci – 95 per cent confidence interval from the bootstrap. -1 -.5 0 .5 1 th eo re tic al -.5 0 .5 1 actual -1 -.5 0 .5 1 0 100 200 300 actual __ theoretical figure 1. actual and theoretical spread (4 vs. 1-month us dollar libor) maria blangiewicz, paweł miłobędzki dynamic econometric models 12 (2012) 5–17 16 -2 -1 0 1 2 th eo re tic al -1 -.5 0 .5 1 1.5 actual -2 -1 0 1 2 0 100 200 300 actual __ theoretical figure 2. actual and theoretical spread (12 vs. 1-month us dollar libor) references bekaert, g., hodrick, r. j. (2001), expectations hypotheses tests, journal of finance, 56, 1357– 1394. bekaert, g., hodrick, r. j., marshall, d. (1997), on biases in tests of the expectations hypothesis of the term structure of interest rates, journal of financial economics, 44, 309–348. blangiewicz, m., miłobędzki, p. (2009), the rational expectations hypothesis of the term structure at the polish interbank market, przegląd statystyczny, 1, 23–39. blangiewicz, m., miłobędzki, p. (2010), the term structure of interest rates at the polish interbank market. a var approach, in milo w., wdowiński p. (eds.), forecasting financial markets. theory and applications, wydawnictwo uniwersytetu łódzkiego, łódź, 197– 209. bulkley, g.,harris, r. d. f., nawosah, v. (2011), revisiting the expectations hypothesis of the term structure of interest rates, journal of banking & finance, 35, 1202–1212. campbell, j. y., shiller, r. j. (1987), cointegration and tests of present value models, journal of political economy, 95, 1062–1088. campbell, j. y., shiller, r. j. (1991), yield spreads and interest rates movements: a bird’s eye view, review of economic studies, 58, 495–514. cuthbertson, k. (1996), the expectations hypothesis of the term structure: the uk interbank market, economic journal, 106, 578–592. cuthbertson k., hayes s., nitzsche d. (1996), the behaviour of certificate of deposit rates in the uk, oxford economic papers, 48, 397–414. cuthbertson, k., hayes, s., nitzsche, d. (2000), are german money market rates well behaved?, journal of economic dynamics & control, 24, 347–360. cuthbertson, k., bredin, d. (2001), risk premia and long rates in ireland, journal of forecasting, 20, 391–403. cuthbertson, k., nitzsche, d. (2003), long rates, risk premia and over-reaction hypothesis, economic modelling, 20, 417–435. elliot, g., rothenberg, t. j., stock, j. h. (1996), efficient tests for an autoregressive unit root, econometrica, 64, 813–836. engsted, t. (1996), the predictive power of the money market term structure, international journal of forecasting, 12, 289–295. engsted, t., tanggaard, c. (1995), the predictive power of yield spreads for future interest rates: evidence from the danish term structure, scandinavian journal of economics, 97, 145–159. the expectations hypothesis of the term structure of libor us dollar interest rates dynamic econometric models 12 (2012) 5–17 17 fisher, i. (1886), appreciation and interest, publications of the american economic association, 11, 1–98. fisher, i. (1930), the theory of interest, macmillan, london. garganas, e., hall, s. g. (2011), the small sample properties of tests of the expectations hypothesis: a monte carlo investigation, international journal of finance and economics, 16, 152–171 hardouvelis, g. a. (1994), the term structure spread and future changes in long and short rates in the g7 countries – is there a puzzle?, journal of monetary economics, 33, 255– 283. hurn, a. s., moody, t., muscatelli, v. a. (1995), term structure of interest rates in the london interbank market, oxford economic papers, 47, 418–436. king, r. g., kurmann, a. (2002), expectations and the term structure of interest rates: evidence and implications, federal reserve bank of richmond quarterly, 88 (4), 49–95. kwiatkowski, d., phillips, p. c. b., schmidt, p., shin, y. (1992), testing the null hypothesis of stationarity against the alternative of a unit root, journal of econometrics, 54, 159–178. lutz, f. a. (1940), the structure of interest rates, quarterly journal of economics, 55, 36–63. mankiw, n. g., miron, g. (1986), the changing behaviour of the term structure of interest rates, quarterly journal of economics, 101, 211–228. mccallum, b. t. (2005), monetary policy and the term structure of interest rates, federal reserve bank of richmond economic quarterly, 91, 1–21. miłobędzki, p. (2010), the term structure of libor sterling rates, prace naukowe uniwersytetu ekonomicznego we wrocławiu, 138 (5), 88–102. tzavalis, e. (2003), the term premium and the puzzles of the expectations hypothesis of the term structure, economic modelling, 21, 73–93. tzavalis, e., wickens, m. (1998), a re-examination of the rational expectations hypothesis of the term structure: reconciling the evidence from long-run and short-run tests, international journal of finance and economics, 3, 229–239. hipoteza oczekiwań struktury terminowej stóp procentowych libor dla dolara usa z a r y s t r e ś c i. w artykule przedstawia się wyniki testów hipotezy oczekiwań struktury terminowej stóp libor dla dolara usa opartych na 3-wymiarowym modelu var. model ten oszacowano na podstawie miesięcznych szeregów czasowych stóp procentowych z lat 1995-2009 i zapadalności od 1 do 12 miesięcy. zaleziono kilka przesłanek świadczących przeciwko tej hipotezie. chociaż premie płynności okazały się być zmiennymi w czasie, a spredy stóp procentowych – mieć silne własności prognostyczne, niemniej stopy długie w niedostateczny sposób reagowały na bieżące informacje odnośnie do przyszłych stóp krótkich. niespodziewane zmiany w okresowych stopach zatrzymania były w dużej mierze spowodowane rewizjami prognoz przyszłych stóp krótkich, a tylko w skromnej mierze rewizjami prognoz przyszłych premii płynności. s ł o w a k l u c z o w e: struktura terminowa stóp procentowych, hipoteza oczekiwań, premia płynności, libor, var. introduction 1. eh of the term structure of interest rates and its implications 2. three-variable var based tests of the eh 3. empirical results conclusions appendix references © 2016 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2016.005 vol. 16 (2016) 49−64 submitted november 30, 2016 issn (online) 2450-7067 accepted december 17, 2016 issn (print) 1234-3862 jerzy różański, paweł sekuła * determinants of foreign direct investment in developed and emerging markets a b s t r a c t. we analyzed fdi determinants for 26 developed economies and 25 emerging markets. the analysis was conducted using a panel regression model for the period 1996– –2014 as well as macroeconomic and institutional variables. growth dynamics, increasing welfare, and the size of the market positively influence fdi. among institutional variables, government stability index and the rule of law index exert positive impact upon fdi. misgivings with respect to the quality of democracy and corruption do not undermine fdi inflow. k e y w o r d s: developed economies, emerging markets, foreign direct investment, institutional determinants, panel data. j e l classification: f21. 1. introduction increasing importance of foreign direct investment (fdi) in modern global economy is connected with globalisation that has facilitated the flows of capital, goods, and services among individual countries. it enhances the role of fdi as a factor that boosts the dynamics of economic growth of coun * correspondence to: jerzy różański, university of łódź, faculty of management, department of finance and strategic management, 90-237 łódź, matejki street 22/26, poland mail: almera@uni.lodz.pl; paweł sekuła university of łódź, faculty of management, department of finance and strategic management, 90-237 łódź, matejki street 22/26, poland mail: pasek@uni.lodz.pl. jerzy różański, paweł sekuła dynamic econometric models 16 (2016) 49–64 50 tries. on the other hand, for many companies foreign investment has become the vehicle of expansion and profit multiplication. the goal of the paper is to identify factors that impact the scale of fdi in the host countries, considering not just the major economic indicators of the host country but also qualitative factors. we also analysed differences between factors that motivate to invest in developed economies and in developing countries, which often offer high growth potential. to this end, we used panel model to more accurately estimate the impact of individual factors upon the size of foreign direct investment in both groups of host countries and changes in the area of foreign direct investment in these countries. 2. literature review fdi is considered to be one of the key drivers of economic growth in many countries, hence the analysis of its determinants has been the subject of various studies (review in assuncao et al., 2011). views on important factors that motivate investors to make fdi have evolved rather substantially. conducted studies took account of many different conditions in micro and macroeconomic scale. faeth stresses that heckscher-ohlin (1933) and macdougall-kemp (1960,1964) models were among the first concepts to explain fdi and they pointed to determinants such as high profitability in foreign markets, lower cost of labour and foreign exchange risk (faeth, 2009). vernon (1966) analysed corporate propensity to get involved into fdi from the point of view of a product life cycle. he arrived at a conclusion that manufacturers’ requirements as to the skills of the labour force and technological innovation diminish with time. that is why enterprises in the growth stage invest in developed countries because demand grows quickly and the production can be sold relatively effortlessly while in the stage of maturity of a product, production is transferred to developing countries. at that point the market is saturated and the product is no more innovative, which forces out the reduction of costs. caves (1971) highlights imperfect competition as fdi determinant. foreign direct investment and adequate product differentiation produce more benefits than exports and licensing. dunning’s approach, i.e. eclectic paradigm (or oli), combines internationalisation theory and traditional trade theories. according to him, geographical distribution of international production is determined by three components: ownership advantages – o, location advantages – l, internalisation advantages – i. ownership advantages bring benefits connected with the ownership and control over manufacturing, patents and technology. location advantages may give access to a protected market, lower costs of production and transportation, determinants of foreign direct investment in developed and emerging markets dynamic econometric models 16 (2016) 49–64 51 favourable tax system, lower business risk. advantage of internalisation reduces transaction costs (dunning, 1988, 2000). subsequent concepts referred to as the „new theory of trade” analysed, inter alia, market size, transport cost, barriers to entry, and factor endowments (markusen, venables, 1998, 2000). theoretical models also emerged to study political variables (review in assuncao et al., 2011). until the end of the 1990s studies on the determinants of foreign investment were dominated by analyses that referred to the classical investment model. the impact of the size of the market and its growth rate, tariff related arrangements or the depth of integration were analysed rather commonly. for example, root and ahmed (1978) pointed to favourable tax rates as incentives for industrial investment. however, they stressed their volatile impact due to fears of their withdrawal by the host country. schneider and frey (1985) analysed fdi for a group of eighty emerging markets and identified inflation rate and salaries and wages as important determinants. high inflation rate and deficit of the balance of payments adversely affect the inflow of foreign investment because they might be indicative of the lack of economic stability and restrict free movement of capital. smaller distance from developed markets, gnp per capita, and gnp growth rate also had positive impact upon fdi. lucas (1993) analysed fdi determinants for countries of east and southeast asia. he stressed the sensitivity of foreign investment to costs of production and pointed to higher impact of salaries and wages than that of capital cost as well as higher impact of demand in export markets than in the domestic market. when analysing fdi determinants, wang and swain (1997) studied factors that attracted foreign capital to hungary and china. they found out that fdi inflow is determined by the size of the market, cost of capital, and political stability. in the case of china, foreign exchange rate and labour cost were also vital. as of 2000 increasingly more studies have been considering not only macroeconomic but also institutional factors that describe the quality of state organisation and functioning. biswas (2002) provided evidence for positive relationship between the quality of infrastructure and fdi inflow. based on an integrated index that takes account of bureaucracy quality, corruption, and risk of expropriation he demonstrated positive impact of institutional quality upon investment. botrić and škuflić (2006) while analysing developing countries from south-east europe used the number of the internet connections as a measure of infrastructure development and concluded that the relationship between infrastructure and fdi is negative. the study provided evidence for positive impact of low deficit of the balance of payments, private sector development, and the gdp. relationship with salaries and wages was negative, jerzy różański, paweł sekuła dynamic econometric models 16 (2016) 49–64 52 which the authors explained with increased fdi inflows into the service sector where salaries in countries included in the study were higher. asiedu (2006) studied 22 countries of sub-saharan africa and noticed positive impact of the size of the market, openness of the economy, quality of infrastructure and human capital, and the quality of institutional performance of the state on fdi. inflation rate and corruption index had negative impact on foreign investment. azman-saini et al. (2010) analysed the impact of economic freedom on fdi in 85 countries. they concluded that the inflow of fdi is closely linked to economic freedom. kinda (2010) examined the impact of investment climate upon fdi inflows analysing 77 developing countries. he provided evidence that infrastructural problems of the host country, financial restrictions and institutional issues are obstacles to fdi inflows. vijayakumar et al. (2010) explored fdi determinants in brics countries. in their studies they demonstrated positive impact of the gdp, salaries and wages and the quality of infrastructure on investment. weak and unstable foreign exchange rate turned out to have negative impact upon fdi. doytch and eren (2012) studied determinants of foreign investment in eastern europe and central asia across sectors. they provided evidence, inter alia, that human capital and quality of democracy have positive impact upon fdi. on top of that, they claimed that the inflow of investment to the service sector is driven by the level of education of the labour force while cheap labour and natural resources attract fdi to agriculture and manufacturing sectors. the role of fdi and its significant increase initiated a series of studies designed to identify key determinants of investment inflow. in practice, however, consensus over their results has been hard to achieve and identified key fdi determinants are often manifestly different. moreover, many studies focus on specific regions and there are fewer studies that would cover bigger groups of countries. 3. foreign direct investment inflows, 1996–2014 globalisation has contributed to enhanced international capital transfers and to the change in the structure of their allocation. years covered by the analysis, 1996–2014, are marked with significant fluctuations in fdi inflows. in the examined period, fdi inflows substantially increased, especially in developing countries. for the developed economies the growth fdi amounted to 111%, while for the emerging markets it reached 332%. the structure of fdi allocation also clearly evolved. in 1996, fdi inflows to emerging markets accounted for 64% of fdi inflows to developed economies, while in 2014 fdi inflows to emerging markets amounted to usd 653 determinants of foreign direct investment in developed and emerging markets dynamic econometric models 16 (2016) 49–64 53 bn and were by usd 154 bn higher than fdi inflows to developed economies. in developed economies we could observe significant fluctuations in fdi inflows caused by changes in global economic situation, especially following the downturns in 2000 and in 2008. emerging markets exhibited relatively stable increasing tendency in fdi inflows, which confirmed their increasingly prominent role in the world economy. figure 1. fdi inflows, by group of economies, 1996–2014 (millions of usd) source: own elaboration based on unctad fdi statistics. considering the above observations and existing studies we have formulated the following research hypotheses: h1: economic situation of the host country and its economic growth exert positive impact upon fdi; h2: quality of institutions in the host country measured with the worldwide governance indicators exerts positive impact upon fdi; h3: fdi determinants are different for developed economies and for emerging markets. 4. data and methodology research sample included 51 countries from asia, australia and oceania, europe, north and south americas. sample selection was determined by the economic status of a country and availability of data used as variables in the models. to divide the research sample into developed economies and emerging markets we used guidelines worked out by the international mone0 200000 400000 600000 800000 1000000 1200000 1400000 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 developed economies emerging markets jerzy różański, paweł sekuła dynamic econometric models 16 (2016) 49–64 54 tary fund (imf), morgan stanley capital international (msci), and bbva research. the group of developed economies was made up of 26 countries and the group of emerging markets consisted of 25 countries 1 . analysis was conducted for nineteen years 1996–2014. such a time horizon of the analysis resulted from the availability of institutional data published by the world bank. studies were conducted in three options: for all countries included in the analysis on a research sample of 814 observations and separately for developed economies and emerging markets. in the last two cases the research sample consisted of 416 and 398 observations, respectively. the first biennial cycle of publications of world governance indicators by the world bank, were based on panel and non-balanced data of crosssectional and time-based nature. in such a case, relations among variables can be studied using the classical ordinary least squares (ols) method. however, we need to bear in mind that the condition of the absence of individual effect must be met. hence, the research procedure included three stages. first, using the breusch-pagan test, we checked whether the introduction of individual effects could be justified. where no grounds were found to reject the null hypothesis, we assumed that a given panel model can be estimated using the classical ordinary least squares (ols). if test values were high (lm multiplier), we rejected the null hypothesis in favour of the alternative one and we added individual effects. in the next stage, we conducted hausman test to choose between fixed effects and random effects. high value of h statistics of the hausman test gave preference to fixed effects model while low value of the statistics suggested random effects model. the last stage consisted in estimation of the selected model. in the analysis we used the following panel regression model: 1 developed countries: australia, austria, belgium, canada, cyprus, denmark, finland, france, germany, greece, hong kong, iceland, ireland, israel, italy, japan, netherlands, new zealand, norway, portugal, singapore, spain, sweden, switzerland, united kingdom, united states; emerging markets: argentina, brazil, bulgaria, chile, china, colombia, czech republic, estonia, hungary, india, indonesia, republic of korea, latvia, malaysia, mexico, peru, philippines, poland, romania, russian federation, slovakia, slovenia, thailand, turkey, venezuela. determinants of foreign direct investment in developed and emerging markets dynamic econometric models 16 (2016) 49–64 55 all variables in the model are represented by individual data of the i-th country and the t-th year, is a total random error. fdi inflows are dependent variable in the study. in the analysis we used fdi inflows in individual countries measured annually in us dollars and published by unctad. there were eleven independent variables in the study. four of them represented the impact of macroeconomic factors on fdi: effective exchange rate indices, gdp growth, gdp per capita, inflation. one variable – population – referred to the size of the fdi host country. six variables described the quality of state organisation and functioning and were represented by the worldwide governance indicators: voice and accountability, political stability & absence of violence/terrorism, government effectiveness, regulatory quality, rule of law, control of corruption. effective exchange rate indices (eer) – in the study we used data published by the bank for international settlements. eer is calculated as a weighted geometrical mean of bilateral exchange rates adjusted for the consumer price index. the impact of the effective exchange rate on fdi is ambiguous. on the one hand, depreciation of the currency of the host country favours those who acquire assets in the host country. on the other hand, the strengthening of domestic currency boosts the purchasing power of the residents, which may also be positive. gdp growth (gdpgr) – we used data published by the imf. it represents annual percentage changes in gross domestic product at constant prices. gdp per capita (gdppc) – we also used the imf statistics. gross domestic product per capita is reported in current prices in us dollars. in our analysis we assumed that gdp growth and gdp per capita are two variables, which identify economic potential of the fdi host country and should be positively correlated with the level of fdi inflows. inflation (inf) – we used data published by the imf included in the annual consumer price index. we assumed that relatively high inflation rate – that has been maintained for several years – may be indicative of macroeconomic instability, which may adversely affect fdi. population (pop) – based on data published by the imf. our assumption was that the population of the host country reflects its size and potential and as such it should have positive effect upon fdi. institutional variables were represented by the worldwide governance indicators (wgi) published by the world bank. wgi consist of six aggregate indicators and measure various aspects of the functioning of the state. they are typically based on the opinions of businesses, individual citizens, jerzy różański, paweł sekuła dynamic econometric models 16 (2016) 49–64 56 and experts in individual countries. for the analysis we used wgi ranging from –2.5 to 2.5 points where higher values inform about stronger and better quality governance.  voice and accountability (vaa) index measures the quality of democracy, citizens’ impact upon government, freedom of association, freedom of speech and media.  political stability & absence of violence/terrorism (psavt) index measures governance stability and the probability of government getting destabilised by the use of violence.  government effectiveness (ge) index measures the quality of state civil service and its independence of political pressures, the quality of state infrastructure.  regulatory quality (rq) index assesses the capability of a government to pursue policy that would support and promote the growth of the private sector.  rule of law (rol) index informs about the quality of the judiciary and police, respect for ownership rights and order, crime rates.  control of corruption (coc) index evaluates corruption rate in a country in different areas. in our study we assumed that wgi growth reflecting higher quality of state and its functioning should positively impact fdi inflows. 5. descriptive statistics descriptive statistics were examined for two groups of countries: developed economies and emerging markets. it helped overview differences in statistics and confirm the thesis about obviously divergent statistics for developed economies and emerging markets. average fdi for developed economies was more than twice as high as the for emerging markets. clear differences were observed also in macroeconomic variables. average gdp growth dynamics for emerging markets was 3.94%, while for developed economies it amounted to 2.31%. significant differences were observed in wealth levels measured by gdp per capita. for developed economies average gdp per capita amounted to usd 36.4 k and for emerging markets ca. usd 7.4 k. we need to stress, however, differences in the size of population where median for developed economies was 9.90 million and for emerging markets 38.02 million. differences also manifested in levels of eer index; for developed economies the average exceeded 100 points meaning currencies were relatively strong contrary to the currencies of emerging markets. significant disproportions were determinants of foreign direct investment in developed and emerging markets dynamic econometric models 16 (2016) 49–64 57 reflected in institutional variables represented by wgi, especially political stability and control of corruption where for emerging markets average values were negative. we need to highlight deep differentiation in wgi indices within countries that belong to the emerging markets. table 1. descriptive statistics developed economies variables mean median std. dev. cv fdi 25,678.80 10,700.10 42,953.50 1.6727 eer 101.28 100.00 11.84 0.1169 gdpgr 2.31 2.43 2.81 1.2194 gdppc 36,384.00 33,540.00 15,537.00 0.4270 inf 2.08 2.02 1.71 0.8188 pop 34.503 9.906 60.451 1.7521 vaa 1.24 1.37 0.42 0.3345 psavt 0.84 0.99 0.59 0.7002 ge 1.61 1.71 0.43 0.2693 rq 1.45 1.54 0.36 0.2494 rol 1.52 1.64 0.41 0.2684 coc 1.67 1.81 0.62 0.3707 note: unit variables: fdi – million usd; pop – million; gdppc – usd; gdpr, inf – percentage point; eer – index point; coc, ge, psavt, rol, rq, vaa – index point, range <–2.5, 2.5>. source: own elaboration based on fdi – unctad; eer – bank for international settlements; gdpgr, gdppc, inf, pop – imf; vaa, psavt, ge, rq, rol, coc – world bank. table 2. descriptive statistics emerging markets variables mean median std. dev. cv fdi 11,218.10 4,864.64 18,779.30 1.6740 eer 94.93 95.70 23.79 0.2506 gdpgr 3.94 4.47 4.16 1.0576 gdppc 7,430.00 5,719.00 5,765.00 0.7759 inf 11.32 5.04 50.75 4.4845 pop 144.764 380.230 320.969 2.2172 vaa 0.25 0.36 0.68 2.7822 psavt –0.17 0.01 0.87 5.0726 ge 0.27 0.18 0.58 2.1287 rq 0.37 0.41 0.66 1.7736 rol 0.06 –0.01 0.71 12.2265 coc –0.04 –0.16 0.60 14.0014 note: unit variables: fdi – million usd; pop – million; gdppc – usd; gdpr, inf – percentage point; eer – index point; coc, ge, psavt, rol, rq, vaa – index point, range <–2.5, 2.5>. source: own elaboration based on fdi – unctad; eer – bank for international settlements; gdpgr, gdppc, inf, pop – imf; vaa, psavt, ge, rq, rol, coc – world bank. jerzy różański, paweł sekuła dynamic econometric models 16 (2016) 49–64 58 6. empirical results impact of analysed variables upon fdi was examined along three lines. to start with, research sample included all analysed countries and then the analysis was repeated for developed economies and emerging markets. in the case of the analysis of the total research sample composed of 51 countries, breusch-pagan and hausman tests suggested we should apply the fixed effects model. estimated values of independent variables for fdi are presented in table 3. out of analysed variables, seven had statistically significant impact upon the dependent variable, i.e. gdp growth (positive impact), gdp per capita (positive impact), population (positive impact) voice and accountability (positive impact), political stability & absence of violence/terrorism (positive impact), rule of law (positive impact), and control of corruption (negative impact). the accuracy of the model measured with adjusted r-square amounted to 68.89%. the first hypothesis (h1), which assumed a positive relationship between economic performance of a country and fdi inflows was confirmed by our analysis. gdpgr and gdppc, the two variables that describe the dynamics and tendency of economic growth, were found to be statistically significant. we also need to stress the importance of the size of a country as an important determinant of fdi inflows. for the second hypothesis (h2) conclusions are no longer so unambiguous. out of six examined wgi indices, four: vaa, psavt, rol, and coc were statistically significant. however, we need to bear in mind that the impact of vaa variable was different from what we assumed in the hypothesis. negative influence of the vaa index in the estimated model suggested increased fdi inflows to countries where the quality of democracy is lower and media freedom restricted. analogous situation was revealed for coc. corruption problems posed no barrier to fdi inflows. on the other hand, however, investors paid attention to political stability, absence of violence, and adequate quality of legal solutions. the same test was conducted for developed economies. independent variables for fdi were estimated using fixed effects model (table 4). the choice was dictated by breusch-pagan and hausman tests. five independent variables had statistically significant impact upon fdi: gdp per capita (positive impact), population (positive impact), voice and accountability (negative impact), political stability & absence of violence/terrorism (positive impact), control of corruption (negative impact). the accuracy of the model measured with adjusted r-square amounted to 65.29%. determinants of foreign direct investment in developed and emerging markets dynamic econometric models 16 (2016) 49–64 59 table 3. determinants of fdi inflows for the entire research sample variables coefficient stand. error t-student p value const –11,651.55 9,106.41 –1.2790 0.2011 eer –12.47 61.33 –0.2033 0.8390 gdpgr 463.43 222.57 2.0820 0.0377** gdppc 0.4177 0.0971 4.3000 <0.0000*** inf –37.84 98.96 –0.3823 0.7023 pop 257.99 47.68 5.4110 <0.0000*** vaa –18,702.00 5,850.50 –3.1970 0.0014*** psavt 10,288.70 3,334.76 3.0850 0.0021*** ge 4,764.52 5,458.58 0.8729 0.3830 rq 1,474.92 4,960.29 0.2973 0.7663 rol 13,657.10 6,956.65 1.9630 0.0500** coc –9,871.71 4,874.72 –2.0250 0.0432** r-square = 0.7123 adjusted r-square = 0.6889 f test = (61, 752) = 30.5193 (p < 0.00001) breusch-pagan test lm = 1,718.97 (p < 0.00001) hausman test h = 39.53 (p = 0.00004) note: significant variable at * p < 0.10, ** p < 0.05, *** p < 0.01. source: own elaboration based on fdi – unctad; eer – bank for international settlements; gdpgr, gdppc, inf, pop – imf; vaa, psavt, ge, rq, rol, coc – world bank. results of analysis for developed economies were close to those for the entire research sample although some differences were observed. gdppc index, which is one of macroeconomic parameters was statistically significant and exerted positive impact upon fdi but gdpgr index did not explain fdi in a statistically significant way. pop index was found to be statistically significant, however, at p below 10%. some differences were also observed for institutional indices. three of them were statistically significant: vaa, psavt, and coc, while rol turned out to be insignificant. results obtained for developed economies, similarly to the results for all of the research sample, did not confirm positive impact of the quality of state performance upon fdi inflows. although psavt index had positive effect upon fdi, vaa and coc indicators had negative impact meaning some aspects of high quality state performance are not fundamental for decisions on capital allocations. our last analysis focused on emerging markets. we started with breusch-pagan and hausman tests, which recommended fixed effects model, similarly to the earlier direction adopted in the study. estimated independent variables for fdi are included in table 5. results demonstrate that the constant and five variables had statistically significant impact upon the dependent variable: gdp growth (positive impact), gdp per capita (positive impact), population (positive impact), voice and accountability (negative imjerzy różański, paweł sekuła dynamic econometric models 16 (2016) 49–64 60 pact), and rule of law (positive impact). the accuracy of the model measured with adjusted r-square amounted to 82.68%. table 4. determinants of fdi inflows for developed economies variables coefficient stand. error t-student p value const –8,257.52 36,694.70 –0.2250 0.8221 eer –119.95 175.74 –0.6826 0.4953 gdpgr 769.40 536.30 1.4347 0.1522 gdppc 0.4175 0.1567 2.6647 0.0080*** inf 1,224.42 896.57 1.3657 0.1729 pop 914.47 478.24 1.9122 0.0566* vaa –27,594.90 14,541.00 –1.8977 0.0585* psavt 21,934.40 7,558.14 2.9021 0.0039*** ge 7,588.36 10,034.90 0.7562 0.4500 rq 11,677.00 10,524.30 1.1095 0.2679 rol 3,576.45 13,606.10 0.2629 0.7928 coc –14,363.70 8,582.37 –1.6736 0.0950* r-square = 0.6830 adjusted r-square = 0.6529 f test = (36, 379) = 22.6803 (p < 0.00001) breusch-pagan test lm = 80.27 (p < 0.00001) hausman test h = 33.66 (p = 0.00041) note: significant variable at * p < 0.10, ** p < 0.05, *** p < 0.01. source: own elaboration based on fdi – unctad; eer – bank for international settlements; gdpgr, gdppc, inf, pop – imf; vaa, psavt, ge, rq, rol, coc – world bank. the results of studies for emerging markets have turned out to be largely convergent with the results for the entire sample. the gdpgr and gdppc exerted statistically significant impact upon fdi, which allowed confirming the first hypothesis (h1) concerning positive impact of the host country economic performance upon fdi. also the size of the host country was important. yet, the assessment of the second hypothesis (h2) on the quality of state upon fdi was ambiguous. the vaa had negative influence upon fdi, while rol’s impact was positive. the effect of the other institutional variables was statistically insignificant. analysis conducted along all of the lines helped assess the third among formulated hypotheses (h3) about fdi determinants that are different for developed countries and for emerging markets. the above discussed descriptive statistics for both groups of economies revealed rather substantial differences in their investment profiles, as well as in the organisational and functional quality. panel analyses of fdi determinants confirmed some divergences between developed economies and emerging markets. the effect of gdppc, pop, and vaa on fdi was statistically significant. determinants of foreign direct investment in developed and emerging markets dynamic econometric models 16 (2016) 49–64 61 table 5. determinants of fdi inflows for emerging markets variables coefficient stand. error t-student p value const –30,398.40 4,239.41 –7.1704 <0.0000*** eer 48.11 31.87 1.5097 0.1319 gdpgr 308.19 117.09 2.6321 0.0089*** gdppc 0.7039 0.1466 4.7998 <0.0000*** inf –37.49 46.34 –0.8092 0.4190 pop 240.19 21.28 11.2874 <0.0000*** vaa –10,488.20 3,235.71 –3.2414 0.0013*** psavt 2,049.11 1,804.98 1.1353 0.2570 ge –4,003.34 3,848.81 –1.0402 0.2990 rq –2,534.04 2,897.53 –0.8746 0.3824 rol 14,662.40 4,579.66 3.2016 0.0015*** coc –4,048.78 3,327.49 –1.2168 0.2245 r-square = 0.8421 adjusted r-square = 0.8268 f test = (35. 362) = 55.1519 (p < 0.00001) breusch-pagan test lm = 249.14 (p < 0.00001) hausman test h =110.65 (p < 0,00001) note: significant variable at * p < 0.10, ** p < 0.05, *** p < 0.01. source: own elaboration based on fdi – unctad; eer – bank for international settlements; gdpgr, gdppc, inf, pop – imf; vaa, psavt, ge, rq, rol, coc – world bank. differences concerned the impact of one macroeconomic variable and three institutional variables. when it comes to developed countries, psavt and coc significantly influenced fdi, while gdpgr and rol were significant for the emerging markets. it is worth stressing that vaa adversely affected fdi in both groups of countries. according to the findings – over the period covered by the study and for the sample at hand – wealth measured with gdppc and the size of the market measured with pop were relevant fdi determinants for both developed and emerging economies. economic growth dynamics was an additional relevant variable for the emerging markets, which did not impact fdi inflows to developed countries. that would mean that investors involved in international capital allocations directed to emerging markets consider their scale, wealth, growth rate, while in developed economies they focus mainly on the size of the market and wealth. with respect to institutional variables, there were some differences in factors specific for developed economies and emerging markets. in developed countries political stability index was important for fdi inflows, while in the emerging markets similar role was played by rule of law index. it may mean that investors in emerging markets pay special attention to the quality of the judiciary, crime rates and ownership rights while in developed economies they are interested in political stability. in developed economies corruption indicator was also important but its impact was negative, meaning the bigger jerzy różański, paweł sekuła dynamic econometric models 16 (2016) 49–64 62 problems with corruption the bigger fdi inflows. negative fdi impact was also detected for vaa, both in developed economies and in emerging markets. we should conclude from the above that the lower quality of democracy, the less impact citizens have upon governments and the lower media freedom the higher fdi inflows. for quality variables there were some differences in answers for developed economies and emerging markets, but general conclusions about their ambiguous impact upon fdi were rather close. on the one hand, investors take account of the quality of judiciary, respect for ownership rights, political stability but on the other hand, they negate issues connected with the quality of democracy, individual freedoms or corruption levels. 7. conclusion our analysis belongs to the increasingly bigger stream of studies on fdi determinants. it covers the period from 1996 to 2014, hence it considers the latest dynamic increases in global fdi as well as rapid fluctuations caused by crises in 2000 and 2007. moreover, it compares fdi determinants for developed economies and emerging markets, which helps us analyse more in-depth the increasingly more prominent ranking of developing countries in attracting foreign investments. results of studies confirmed the first of our hypotheses (h1) about positive impact of economic performance of a country upon fdi inflow over the analyzed period. the gdp rate of growth, which reflects the dynamics of economic growth, exerted positive impact upon fdi. also citizens’ wealth (gdppc) and the size of the market measured with the size of the population were significant fdi determinants. on top of that, the studies explored the effect of institutional variables on fdi. to this end, we used six wgi indices published by the world bank. obtained results were rather ambiguous, which prevented us from the adoption of the second hypothesis (h2) on beneficial impact of high quality institutional performance of the host country upon fdi. political stability (psavt) and adopted legal regulations (rol) positively influenced fdi but the impact of indices reflecting the quality of democracy (vaa) and corruption (coc) was negative. the results suggested that investors when making fdi decisions consider civil freedoms, the freedom of media, democracy or corruption only to a limited extent. the analysis confirmed the third hypothesis (h3) about different sets of fdi determinants for developed and emerging economies. differences concerned the impact of one macroeconomic and three institutional variables. determinants of foreign direct investment in developed and emerging markets dynamic econometric models 16 (2016) 49–64 63 speaking of emerging markets, gdp growth dynamics and the quality of regulations exerted positive impact upon fdi; for developed countries we observed positive influence of government stability index (psavt) and negative impact of corruption index (coc). it would suggest that besides the size of the market, growth dynamics is an important fdi determinant for the emerging markets. with regard to institutional variables we realized that investors in developed countries are interested in government stability while in the emerging markets they take care of the rule of law, which includes the quality of the judiciary, crime levels and respect for ownership rights. moreover, in developed economies corruption problem did not undermine fdi inflows. summing up, we need to highlight certain limitations in drawing conclusions resulting from the study. the analysis covers a relatively short period of time and a set of eleven independent data. that is why we see the need to pursue further studies and to expand the scope of analysis. references asiedu, e. (2006), foreign direct investment in africa: the role of natural resources, market size, government policy, institutions and political instability, world economy, 29(1), 63–77, doi: http://dx.doi.org/10.1111/j.1467-9701.2006.00758.x. assuncao, s., forte, r., teixeira, a. a. c. (2011), location determinants of fdi: a literature review, fep working papers, universidade do porto, 433,1–26. azman-saini, w. n. w., baharumshah, a. z., law, s. h. (2010), foreign direct investment, economic freedom and economic growth: international evidence, economic modelling, 27(5), 1079–1089, doi: http://dx.doi.org/10.1016/j.econmod.2010.04.001. biswas, r. (2002), determinants of foreign direct investment, review of development economics, 6(3), 492–504, doi: http://dx.doi.org/10.1111/1467-9361.00169. botrić, v., škuflić, l. (2006), main determinants of foreign direct investment in the southeast european countries, transition studies review, 13(2), 359–377, doi: http://dx.doi.org/10.1007/s11300-006-0110-3. caves, r. (1971), international corporations: the industrial economics of foreign investment, economica, 38(149), 1–27, doi: http://dx.doi.org/10.2307/2551748. doytch, n., eren, m. (2012), institutional determinants of sectoral fdi in eastern european and central asian countries: the role of investment climate and democracy, emerging markets finance & trade, 48(4), 14–32. dunning, j. h. (1988), the eclectic paradigm of international production: a restatement and possible extensions, journal of international business studies, 19(1), 1–31, doi: http://dx.doi.org/10.1057/palgrave.jibs.8490372. dunning, j. h. (2000), the eclectic paradigm as an envelope for economic and business theories of mne activity, international business review, 9(2), 163–190, doi: http://dx.doi.org/10.1016/s0969-5931(99)00035-9. faeth, i. (2009), determinants of foreign direct investment – a tale of nine theoretical models, journal of economic surveys, 23(1), 165–196, doi: http://dx.doi.org/10.1111/j.1467-6419.2008.00560.x. jerzy różański, paweł sekuła dynamic econometric models 16 (2016) 49–64 64 kinda, t. (2010), investment climate and fdi in developing countries: firm-level evidence, world development, 38(4), 498–513, doi: http://dx.doi.org/10.1016/j.worlddev.2009.12.001. lucas, r. e. b. (1993), determinants of direct foreign investment: evidence from east and southeast asia, world development, 21(3), 391–406, doi: http://dx.doi.org/10.1016/0305-750x(93)90152-y. markusen, j. r., venables, a. j. (1998), multinational firms and the new trade theory, journal of international economics, 46(2), 183–203, doi: http://dx.doi.org/10.1016/s0022-1996(97)00052-4. markusen, j. r., venables, a. j. (2000), the theory of endowment, intra-industry, and multinational trade, journal of international economics, 52(2), 209–234, doi: http://dx.doi.org/10.1016/s0022-1996(99)00055-0. root, f. r., ahmed, a.a. (1978), the influence of policy instruments on manufacturing direct foreign investment in developing countries, journal of international business studies, 9(3), 81–93, doi: http://dx.doi.org/10.1057/palgrave.jibs.8490670. schneider, f., frey, b. s. (1985), economic and political determinants of foreign direct investment, world development, 13(2), 161–175, doi: http://dx.doi.org/10.1016/0305-750x(85)90002-6. vernon, r. (1966), international investment and international trade in the product cycle, quarterly journal of economics, 80(2), 190–207, doi: http://dx.doi.org/10.2307/1880689. vijayakumar, n., sridharan, p., rao, k. c. s. (2010), determinants of fdi in brics countries: a panel analysis, international journal of business science and applied management, 5(3), 1–13. wang, z., swain, n. j. (1997), determinants of inflow of foreign direct investment in hungary and china: time-series approach, journal of international development, 9(5), 695– 726, doi: http://dx.doi.org/10.1002/(sici)1099-1328(199707)9:5%3c695::aidjid294%3e3.0.co;2-n. determinanty bezpośrednich inwestycji zagranicznych na rynkach rozwiniętych i wschodzących z a r y s t r e ś c i. badane są determinanty fdi dla 26 państw rozwiniętych i 25 państw wschodzących. analizę przeprowadzono przy wykorzystaniu modelu regresji panelowej w okresie 1996–2014, dla zmiennych makroekonomicznych i instytucjonalnych. stwierdzono pozytywny wpływ dynamiki wzrostu gospodarki, poziomu zamożności i wielkości rynku na fdi. dla zmiennych instytucjonalnych odnotowano dodatni wpływ indeksu stabilności rządu i indeksu ładu prawnego na fdi. problemy z jakością demokracji i korupcją nie ograniczały napływu fdi. s ł o w a k l u c z o w e: gospodarki rozwinięte, rynki wschodzące, bezpośrednie inwestycje zagraniczne, determinanty instytucjonalne, dane panelowe. dynamic econometric models © 2012 nicolaus copernicus university press. all rights reserved. http://www.dem.umk.pl/dem dynamic econometric models vol. 12 (2012) 73−87 submitted october 22, 2011 issn accepted march 21, 2012 1234-3862 milda maria burzała* the probability of recession in poland based on the hamilton switching model and the logit model a b s t r a c t. in the article dating method for the four phases of economic activity is presented. comparison of probabilities of recession occurrence in poland based on the hamilton switching model and the logit model was conducted in the empirical research. the study shows the convergence of indications based both on the proposed dating method and on the hamilton model. in the presented version the hamilton model adequately describes the probability of occurrence of two decline phases. the logit model allows to obtain satisfactory results for the division on four phases of economic activity. however, in the domain of the polish economy, more research is needed in recognising the symptomatic properties of various macroeconomic indicators. the interest rate spread, used successfully in advanced marked economies, continues to alter its characteristics under polish economic conditions and is currently not the best possible indicator forecasting a recession. k e y w o r d s: switching model, logit model, dating of economic activity phases, probability of recession. j e l classification: e32, e37. introduction analysts often emphasise that many financial and economic indicators tend to behave differently during growth and decline. therefore, it is a wellgrounded assumption that the parameters of the models describing the formation of such values change. the switching models allow to test such an assumption. if the turning points, otherwise known as the moments of switching between periods of diversified behaviour of the variables, are known, then the segment model for the quantitative variable or the probability model for the selected variants of the qualitative variable, is estimated. if researchers cannot * correspondence to: milda burzala, department of econometrics, faculty of informatics and electronic economy, poznan university of economics, ul. towarowa 53, 61-896 poznań, poland, e-mail: m.burzala@ue.poznan.pl milda maria burzała dynamic econometric models 12 (2012) 73–87 74 agree upon a single method for establishing such turning points, then the markov-switching model, as proposed by hamilton, can be used. in the case of economic activity, the moments of switching depend, among others, on the accepted method of decomposition of time series of selected macroeconomic ratios. in research on the american market, data provided by the nber concerning the turning points for growth and decline phases in the u.s. economy is used as the point of reference. yet in many countries there is no established system to indicate the beginning and end of a recession. that is why it is worthwhile to analyse the convergence of indications resulting from various methods. zarnovitz and ozyildirim (2006) discuss the influence of the accepted method of decomposition on the variability of the course of a u.s. growth cycle. they compare the dating of turning points based on cycles of levels, trend deviations and smoothed growth cycle. the results obtained with the use of the pat method are very similar to the results obtained on the basis of the hodrickprescott filtering method, local linear trend as well as band-pass filtering method1. in poland, a comparative analysis of business cycles obtained using different methods, made under various assumptions, was described, among others, by skrzypczyńska (2011) and burzała (after revives, in press). this article presents a comparison of the indications for a recession phase based on the two models, with known and unknown switching points. section 1 presents the dating method of economic activity phases, which allows to determine the moments of switching between the phases of high and low economic activity. sections 2 and 3 describe the models (hamilton’s switching model and the logit model, respectively) which are used to estimate the probability of a recession. the research results and the comparison of indications based on the accepted method of dating of phases are described in section 4. section 5 is a summary of the research results. 1. dating of economic activity phases in a time of growth-based market economies, it is difficult to clearly determine which of the observed changes have resulted from long-term economic growth and which stem from economic fluctuations. despite the many research studies and upgraded decomposition methods, the division between ‘trend’ and ‘cycle’ has always been accepted as a convention and can be considered somewhat artificial. therefore, it would seem to make sense to employ an approach based on an analysis of the growth rates of a time series with the seasonal and random fluctuations removed, and with no decomposition into trend and eco 1 pat (phase average trend) is a 10-step procedure as described by boschan, ebanks (1978). it was applied by the nber to a large number of indicators with generally satisfactory results in terms of timing and conformity to aggregate growth cycles. the probability of recession in poland… dynamic econometric models 12 (2012) 73–87 75 nomic fluctuations. thus, the research is focused on economic activity in general, rather than on the course of a business cycle. in the empirical studies presented in this paper, the measure of economic activity were the annual indices of total industrial output sold pr_irt – as recorded on a monthly basis between january 1993 and march 2011 – with seasonal and random fluctuations removed. the selection of output indices mostly resulted from a larger frequency of quotations than from the gnp. this approach is compatible with the generally accepted growth-based definition of the business cycle as commonly assumed in empirical analyses (mintz, 1972). the dating rules for economic activity phases, as used in this paper, make use of shortand long-term changes in the output indices (burzała, 2005). the annual index pr_irt measures the change in a value prt with respect to the corresponding period of the preceding year prt-12 and constitutes the measure of changes ‘within a long period of time’. the monthly index pr_imt measures the change in a value prt with respect to the preceding month prt-1 and is a measure of changes ‘within a short period of time’. these indices represent the following respective dependencies: .100_,100_ 112 ⋅=⋅= −− t t t t t t pr pr impr pr pr irpr (1) the first index (pr_irt) is a reference series, as has already been mentioned. the proposal of dividing the set of all observations of the reference series into separable subsets (economic activity phases) has been based upon tests which were to determine whether the short-term growth rate implies long-term changes. depending on whether a given index is above or below 100 (which corresponds to a positive or negative growth rate), an observation is classified as characteristic of a given phase of the economy. this approach uses the rules of symbolic taxonomy, in which a phase (state) is described through a conjunction of values of selected indicators (gatnar, 1998). these rules allow to distinguish four phases of the economy (burzała, 2005 a,b): a) 100_ ≥ t impr and, simultaneously, 100_ ≥ t irpr – implied growth, which denotes high economic activity (conventionally called the prosperity phase and marked with the code w_im); b) 100_ ≥timpr and 100_ <tirpr – non-implied growth, which denotes a rise of the monthly index and most often occurs following a recession (code w_nim); c) 100_ <timpr and 100_ <tirpr – implied decline, which denotes an economic recession (code s_im); milda maria burzała dynamic econometric models 12 (2012) 73–87 76 d) 100_ <timpr and 100_ ≥tirpr – non-implied decline, which denotes a dropping monthly index – which is usually a pre-recession phase (code s_nim). the advantage of such a division is that two transitory phases have been identified, i.e. non-implied decline, which in itself is a warning against adverse change in business conditions, and non-implied growth, which indicates an improving environment. the growth and decline phases discerned in this paper may be combined for a less detailed differentiation (prosperity vs. recession). 2. hamilton’s switching model the assumption in this case is that the behaviour of certain macroeconomic indicators changes as a result of changed economic activity. however, a phase of economic activity is not directly observable, and thus it is difficult to establish which phase an economy is in at a given moment. in the terminology of switching models, a phase of activity is one of the possible states referred to as regimes. in the original hamilton model (1989), it was assumed that there were two possible regimes, s (s = 1, 2), corresponding to the condition of an economy (prosperity vs. recession). in each regime, indicator values are generated by two separate and independent processes. in the hamilton model, these were the conditional average processes, ar(4), for quarterly gnp changes. thus, in a general case: ,)( ...)()( 21 2211 tsptp ststst pt ttt y yyy εµφ µφµφµ +−+ +−+−=− − −− − −− (2) where: εt ~n(0,σ 2), µs denotes the expected value under the s regime. at present, models with switches resulting from the changing variance σ2, or the average and a variance are estimated particularly with regard to financial markets (doman, doman, 2009). such a model, entailing two regimes, was used in the research studies. consequently: ).1,0(~ ,/.../1 , 2222 11 1 iid h h t sptpstt ttst ptt t ξ σεασεα ξσε −− −− +++= = (3) the series of random variables si in the subsequent moments in time t (t=1,..., t) has the markov property, i.e. its value at the time moment t+1, i.e. st+1, depends only on the regime at the t moment, rather than on all the preceding regimes, which is formally formulated as: .)(,...),( 111 ijttttt pisjspksisjsp ======= −−+ (4) the probability of recession in poland… dynamic econometric models 12 (2012) 73–87 77 the probabilities pij denote the probability of an economy’s switching from regime i into regime j. these are unknown parameters that are estimated, where: ).22( ),21( ),12( ),11( 122 121 112 111 === === === === + + + + tt tt tt tt sspp sspp sspp sspp (5) based on conditional probabilities (4 and 5), it is possible to determine the unconditional probability p(st = i) of an economy remaining at the t moment under an ith regime. in the case of two regimes, the following is obtained: . 2 1 )2(, 2 1 )1( 2211 11 2211 22 pp p sp pp p sp tt −− − == −− − == (6) the inverse of probabilities p(st =i) is interpreted as the expected time of resumption of regime i: , )( 1 )( isp imt t = = (7) whereas the expected time of remaining under regime i is provided by means of the dependency: . 1 1 1 ijii i pp d = − = (8) the parameters of the model (pij, µs φi, σs 2) are estimated using the maximum likelihood method2. if at the t moment the process was under the st =j regime, then the conditional probability density function of the explained variable yt can be represented as ),( 1−ψ= ttt jsyf , where 1−ψt denotes the history of the process until the t-1 moment. even knowledge of the model’s parameters does not allow to establish which regime a given economy is under at the t moment. any suppositions on the actual regime may only be made by means of a conditional probability: , )(),( )(),( )( 2 1 11 11 ∑ = −− −− ψ=⋅ψ= ψ=⋅ψ= =ψ= i ttttt ttttt tt ispisyf jspjsyf jsp (9) 2 the switching models were estimated using the tsm programme. the relevant likelihood function is presented as part of the description of the programme (davidson, 2011). milda maria burzała dynamic econometric models 12 (2012) 73–87 78 where: ∑ −−− ψ==ψ= 2 1 111 ).()( ttijtt isppjsp (10) the maximised likelihood function is as follows: ∑ ∑ = = −− ψ=⋅ψ== t t j ttttt jspjsyfl 1 2 1 11 ).(),(log (11) it is not easy to estimate the model’s parameters. numerical problems result from the occurrence of local extrema of the logarithmic likelihood function. this is why normally two, up to three, regimes under which a process may be are distinguished. 3. logit model the dating method of economic activity phases, as proposed in section 2, allows to date the time periods relating to a recession. let us assume that this time the explained variable yt assumes the value of 1 at the t moment, if a recession occurs in an economy (s_im); otherwise, the value is 0. the explained variable thus defined is a qualitative binary variable, usually modelled using the logit model. in the model with a qualitative explained variable, the theoretical probability pt1 of occurrence of the first option of the variable at the t moment is defined by the cumulative distribution function (β′xt), i.e.: ),'(1 ttt fp ξ+= xβ (12) where xt is a vector of explanatory variables, β is the parameters vector and ξt is a random disturbance. the type of distribution assumed for the ξt random variable generates the type of model under consideration. in the logit model, it is assumed that probability pt1 is defined using the cumulative distribution function of a standardised logistic distribution ξt ∼ l(0, π 2/3). hence: ( ) . 1 1 1 ' ' 21 t t t t e dξ e e p tξ ξ t xβ xβ − ∞− − − + = + = ∫ (13) it can be proved that the logit model’s random component is heteroscedastic (jajuga, 1990). for this reason, to estimate the model’s parameters, the method with the utmost probability is most often used. the maximised logarithmic likelihood function is provided by means of the dependency: .lnln) 1 2 1 ∑ ∑ = = == t t j tjtj * pyll )(( ββ (14) the probability of recession in poland… dynamic econometric models 12 (2012) 73–87 79 as for the uses of models with the qualitative explained variable, the article by estrella, mishkin (1998) is of primary importance for measuring economic activity. these authors tested macroeconomic indicators that preceded the onset of an economic recession in the u.s. by using a binary probit model. as it turned out, the difference in interest rates applied to ten-year treasury notes and three-month treasury bills (the so-called spread) quite reliably indicates, with a one-year advance, an economic recession in the u.s. in the research studies carried out for the polish economy, apart from the interest rate spread (spread), the annual indices of money supply m1 (m1_ir), m2 (m2_ir), m3 (m3_ir) and the wig stock-exchange index (wig_ir) were tested. the symptoms of a recession were chosen according to research studies carried out by this author in an unpublished phd thesis (burzała, 2005a). it should be emphasised that any given set of leading indicators is not effective for all countries. a different set of indicators is used by nber or oecd researchers, and specialists from germany or japan use yet a different methodology. under polish economic conditions, the methods of testing the business cycle are still being improved and the list of quantitative indicators is being expanded. the idea of using the interest rate spread for forecasting economic activity is derived from the rational expectations hypothesis. in a market economy, a negative interest rate spread is one of the symptoms of a recession (short-term interest rates are higher than long-term interest rates)3. the growing interest in financial indicators stems from economists’ convictions that financial variables have an increasing influence on the real sphere of the economy, and thus the entire economic policy. the meaning of a monetary policy that is handled appropriately results from the function that regulates both the money supply and lending in order to maintain employment, price stability and economic growth. the importance of financial indicators was analysed by atta-mensah, tkacz (1998), estrella, hardouvelis (1991), stock, watson (1993), wright (2006), and nyberg (2009). the purpose of this article is not to determine the best set of indicators, but to check the properties of some of these indicators. 4. research results in order to illustrate the recession indications based on the dating method as proposed in section 1, the results obtained for the u.s. economy were compared with information from the nber (figure 1). it turned out that the time periods shown for the recession appear to show high convergence, which may prove that the proposed dating method is correct. it is worth emphasising that in dating the cycles, nber imposes additional restrictions for the duration of the decline and growth phase (a minimum of 6 months) and for maintaining the growths and declines around the turning points 3 in the countries of central and eastern europe (including poland), the interest rate spread does not fulfil the given assumptions during an economic transformation. milda maria burzała dynamic econometric models 12 (2012) 73–87 80 (for a period of 5 months). the method proposed in this paper does not take such conditions into consideration in this version, yet it can easily be extend. 0 0,2 0,4 0,6 0,8 1 19 59 m 01 19 62 m 01 19 65 m 01 19 68 m 01 19 71 m 01 19 74 m 01 19 77 m 01 19 80 m 01 19 83 m 01 19 86 m 01 19 89 m 01 19 92 m 01 19 95 m 01 19 98 m 01 20 01 m 01 20 04 m 01 20 07 m 01 20 10 m 01 80 85 90 95 100 105 110 115 nber_rec recession pr_ir figure 1. u.s. economy recession phases, january 1959 to march 2011, base on nber data fundamental research studies on polish economic activity were based on statistics from the international monetary fund (ceic). figure 2 shows the economic activity phases for poland. the bar chart highlights the recession phases. 0 0,2 0,4 0,6 0,8 1 19 92 m 01 19 93 m 01 19 94 m 01 19 95 m 01 19 96 m 01 19 97 m 01 19 98 m 01 19 99 m 01 20 00 m 01 20 01 m 01 20 02 m 01 20 03 m 01 20 04 m 01 20 05 m 01 20 06 m 01 20 07 m 01 20 08 m 01 20 09 m 01 20 10 m 01 20 11 m 01 80 85 90 95 100 105 110 115 120 125 s_im (recession) w_im s_nim w_nim figure 2. economic activity phases for poland, january 1993 to march 2011, based on ceic data the probability of recession in poland… dynamic econometric models 12 (2012) 73–87 81 the regularities that occur are worth consideration: a recession phase and a prosperity phase are always preceded by a transitory phase, which signalises a deterioration or improvement of the economic conditions. the economic activity phases determined as such were compared with the probability of occurrence of two regimes (prosperity and recession, by definition) based upon the hamilton model. the results hamilton presented referred to the differences between the gnp logarithms. the proposed pattern, once directly applied to the output indices in the polish economy, has not yielded satisfactory results. therefore, it was proposed that the explained variable under the hamilton model be the logarithm of the annual output index pr_irt (1), as recorded on a monthly basis. the parameters of the model as estimated are shown in table 1. in the construction of a dynamic model describing changes in output, a conclusion was applied which is used in the dating of economic activity phases. the tests that were carried out have shown that the model which takes into account the first and twelfth lag (a counterpart of the shortand long-term changes) gives the best results. table 1. parameters of hamilton’s switching model parameters (standard error) probability of transition φ1 = 0.952 (0.007) p 11 = 0.920 p 12 = 0.027 φ12 = 0.547 (0.032) p 21 =0.079 p 22 = 0.973 expected regime i duration µ1 = 317.204 (6.548) d 1 = 12.577 d 22 = 36.873 µ2 = 316.875 (6.545) unconditional probability σ1 = 0.736 (0.052) p(s t = 1) = 0.254 p(s t =2) = 0.746 σ2 = 0.175 (0.011) expected time of regime i resumption r2 = 0.9829 mt(1) = 3.932 mt(2) = 1.341 changes of output indices pr_irt, phases of recession s_im and the probability of the occurrence of regime 1 (rg1_smprobs) are shown in fig. 3. our analysis implies that regime 1 corresponds with decreased economic activity. the high probability of regime 1 covers not only the recession periods s_im but also non-implied decline periods s_nim. the non-implied decline phase usually precedes an economic recession. the period of june 2004 to may 2005 was one of significant turbulence in the polish economy and was connected with the eu accession boom. the period’s prosperity was disturbed by a decline as well as by non-implied growth. none of the observations, however, has been considered a time of economic recession. under the hamilton model a high probability of regime 1 occurring can be observed for this period. assuming that regime 1 was to appear, with the probability of its occurrence exceeding 0.5, then the accuracy of indications was evaluated. higher accuracy rates were obtained with the assumption that regime 1 corresponds to a combination of two decline phases (indications for both decline phases proved to be 90% accurate, with a total accuracy of 97%; as compared to the accuracy for the recession phase equal to 94%, with a total accuracy of 81%). it is worth noting that both regimes milda maria burzała dynamic econometric models 12 (2012) 73–87 82 differ insignificantly in terms of the expected value. the period of lower economic activity is characterised by definitely higher variance. both activity states are relatively durable (with the probability of remaining in the lower activity phase equal to 0.920, during a prosperity phase equal to 0.973), whereas the probabilities of transition between the states are small. the table also shows the unconditional probabilities which indicate a higher probability of prosperity occurring (0.746). prosperity is also characterised by a longer expected duration period (c.37 months) and a shorter time of return from the state of lower economic activity (c.1.3 months). 0 0,2 0,4 0,6 0,8 1 19 92 m 01 19 93 m 01 19 94 m 01 19 95 m 01 19 96 m 01 19 97 m 01 19 98 m 01 19 99 m 01 20 00 m 01 20 01 m 01 20 02 m 01 20 03 m 01 20 04 m 01 20 05 m 01 20 06 m 01 20 07 m 01 20 08 m 01 20 09 m 01 20 10 m 01 20 11 m 01 80 90 100 110 120 130 rg1_smprobs s_im (recession) w_im s_nim w_nim figure 3. hamilton model-based probability of regime 1, set against the background of the economic activity phases in poland the second research approach using the logit model assumed that the moments of switching to the recession phase are known. the qualitative explained variable yt assumes a value of 1 if the observation concerned an implied decline s_im, otherwise it takes on the value of 0. the model was used to estimate the probability of a recession occurring on the basis of values assumed by selected leading indicators. they were represented in the model by annual indices of change – the exception being the interest rate spread as the difference in the interest rate of the ten-year treasury notes and the three-month treasury bills. should the sample be balanced (the number of 1s being identical with the number of 0s for the explained variable), we assume that a recession is to occur if the probability of it occurring exceeds 0.5. for an unbalanced sample (which appeared in the research study), the limit value is usually assumed to be the share of 1s in the sample. measures of fit and the accuracy of estimations from the logit models are shown in table 2. the forecast of symptomatic variables (leading indicators) was determined based on the maximum value of mcfadden’s r2 (determination coefficient) and akaike’s information criterion. the probability of recession in poland… dynamic econometric models 12 (2012) 73–87 83 table 2. logit models and their verification m od el va ria bl e la g pa ra m et er e st im at e pva lu e m cf ad de n rsq ua re d ak ai ke in fo rm at io n cr ite rio n fr eq ue nc y of 1 s (e m pi ric al) ac cu ra cy , i n to ta l ac cu ra cy o f 1 s m1 spread 12 -0.345 1.40e-03 0.122 78.36 0.105 76% 54% const -2.771 1.88e-11 m1a spread 12 -0.340 1.89e-02 0.412 55.82 0.099 82% 85% pr_ir 1 -0.333 3.00e-04 const 30.710 6.00e-04 m2 m1_ir 7 -0.105 4.40e-03 0.083 97.98 0.105 65% 44% const 9.647 1.78e-02 m2a m1_ir 7 -0.06 1.72e-01 0.322 75.49 0.105 78% 88% pr_ir 1 -0.269 4.98e-05 const 32.07 1.00e-04 m3 m2_ir 4 0.077 3.31e-02 0.046 102.44 0.103 63% 56% const -11 9.00e-03 m3a m2_ir 4 0.023 6.23e-01 0.309 77.24 0.103 80% 100% pr_ir 1 -0.267 4.12e-05 const 22.481 1.76e-02 m4 m3_ir 4 0.069 6.08e-02 0.035 103.58 0.103 63% 63% const -10.08 1.85e-02 m4a m3_ir 4 0.026 5.63e-01 0.280 80.23 0.103 78% 100% pr_ir 1 -0.254 3.82e-05 const 20.914 1.91e-02 m5 wig_ir 2 -0.058 5.95e-06 0.312 79.91 0.083 80% 94% const 2.981 4.20e-03 m5a wig_ir 2 -0.082 4.20e-03 0.409 70.00 0.087 84% 94% pr_ir 1 0.017 8.71e-01 const 3.443 6.96e-01 m6 spread 12 -0.354 1.65e-02 0.462 53.59 0.099 88% 100% m3_ir 4 0.006 9.59e-01 wig_ir 2 -0.085 3.80e-03 const 3.811 7.85e-01 taking the individual indicators into account, the wig stock-exchange index appears to have the highest symptomatic properties. unfortunately, from a practical standpoint, a two-month forecast is too short. due to the length of the forecast period, the interest rate spread better fits the purpose. a higher accuracy of indications for the recession phase, given a four-month forecast, was obtained for annual changes in the index of money supply m3. in all cases the accuracy of indications was clearly improved if a lagging industrial output index was milda maria burzała dynamic econometric models 12 (2012) 73–87 84 added to the model. an attempt was also made in this research to combine the symptomatic properties of three leading indicators (i.e. index of money supply m3, the stock-exchange index and the interest rate spread). this latter model is characterised by the highest degree of measure of fit, the lowest akaike’s criterion value as well as the highest accuracy of indications within the sampling period. the probability of occurrence of a recession based on the m6 model is shown in figure 4. 0 0,2 0,4 0,6 0,8 1 19 92 m 01 19 92 m 12 19 93 m 11 19 94 m 10 19 95 m 09 19 96 m 08 19 97 m 07 19 98 m 06 19 99 m 05 20 00 m 04 20 01 m 03 20 02 m 02 20 03 m 01 20 03 m 12 20 04 m 11 20 05 m 10 20 06 m 09 20 07 m 08 20 08 m 07 20 09 m 06 20 10 m 05 80 90 100 110 120 130 logit m6 s_im (recession) w_im s_nim w_nim figure 4. probability of a recession based on the logit model m6 and the recession phases in poland summarising the results of the research – the duration of a recession was analysed based on the proposed dating method, the hamilton model and one of the logit models – the m6 (table 3). table 3. recession time periods beginning of recession end of recession s_im, s_nim s_im hamilton model logit model m6 s_im, s_nim s_im hamilton model logit model m6 1998-06 1998-04 1998-12 1998-12 2001-01 2001-05 2001-01 2001-02 2002-04 2002-04 2002-03 2002-04 2004-06 2003-10 2005-01 2005-02 2008-03 2008-09 2008-02 2008-08 2009-02 2009-02 2009-02 2009-09 the classic approach was additionally adopted in the comparisons, which states that every phase should last at least half a year (bry, boschan, 1971). this allowed us to exclude short periods of implied decline in the comparisons and means that the rise or drop of a measure must remain on both sides of the turning point for at least six months. for the hamilton model this was the rise/drop the probability of recession in poland… dynamic econometric models 12 (2012) 73–87 85 of the probability of recession above/below the limit value of 0.5. for the logit model, as a result of not balancing the statistic sample, the limit value was the empirical frequency of 1s occurring for the qualitative explained variable. in the proposed dating method it was the change in behaviour of the annual and monthly output indices. all of the analysed dating methods show significant convergence in terms of dating the end of recession (a 1-month difference). the only exception is the logit model, which shows the end of the last recession to be later. however, the dating of the beginning of recession by the hamilton model shows convergence with the results from the logit model in 2001 and the two phases of decline s_nim, s_im in 1998 and 2001. in the case of the last economic recession, the hamilton model and the combined phases s_nim and s_im indicate the beginning of recession in february/march 2008, while the logit model and the phase of implied decline s_im indicate it as late as in august/september. yule’s coefficient of association, based on chi-square statistics, which is a measure of interdependence of qualitative variables measured in a nominal scale, was used to quantitatively estimate the similarity of indications (it has a value from -1 to 1). the indications of the hamilton model demonstrate the highest level of association with the indications of two phases of decline s_nim and s_im – 0.55, however, the logit model indications demonstrate the highest level of association with the indications of implied decline s_im – 0.42. 5. conclusions the research conducted shows the convergence of indications based both on the proposed dating method and on the hamilton model. in the presented version the hamilton model adequately describes the probability of occurrence of two decline phases. the dating of phases as proposed in section 2 is simple in empirical applications and allows for a division into four phases of economic activity, which is beneficial as it distinguishes two transitory phases. the logit model allows to gain satisfactory results for a more detailed division of the recession phase s_im, based on the value of selected leading indices. however, in the domain of the polish economy, more research is needed in recognising the symptomatic properties of various macroeconomic indicators. the interest rate spread, used successfully in advanced marked economies, continues to alter its characteristics under polish economic conditions and is currently not the best possible indicator forecasting a recession. references amemiya, t. (1981), qualitative response models: a survey, journal of economic literature, 19, 481–536. atta-mensah, j., tkacz g. (1998), predicting canadian recessions using financial variables: a probit approach, department of monetary and financial analysis, bank of canada working paper 98−5, ottawa. milda maria burzała dynamic econometric models 12 (2012) 73–87 86 barczyk, r., kowalczyk, z. (1993), metody badania koniunktury gospodarczej (research methods of business condition), pwn, warszawa-poznań. bry, g. and boschan, c. (1971), cyclical analysis of times series: selected procedures and computer programs, technical paper 20, nber, new york. burzała, m. (2005a), prognozowanie aktywności gospodarczej polski przy wykorzystaniu zagnieżdżonych modeli logitowych, niepublikowana praca doktorska (polish economic activity forecasting using nested logit models – unpublished phd thesis). burzała, m. (2005b), kilka uwag na temat wyboru modelu dyskretnego do prognozowania aktywności gospodarczej polski (a few comments on choosing discrete model for economic activity forecasting), in jurek w., prace z ekonometrii finansowej (research in financial econometrics), 19–42, wydawnictwo ae, poznań. burzała, m., (2011), porównanie faz aktywności gospodarczej uzyskanych na podstawie filtru hodricka-prescotta, stóp wzrostu oraz przełącznikowego modelu hamiltona (comparision of the phases of economic activity generated from hodrick-prescot filter, smoothed growth rates and switching model of hamilton), in appenzeller d., matematyka na usługach ekonomii (mathematics in the service of economics), 65–77, wydawnictwo uep, poznań. boschan, c., ebanks, w.w. (1978), the phase-average trend: a new way of measuring growth, in: 1978 proceedings of the business and economic statistics section, american statistical association, washington, d.c. cieślak, m. (2001), (red.), prognozowanie gospodarcze. metody i zastosowania (economic forecasting. methods and implementations), pwn, warszawa. czerwiński, z. (2002), przyczynek do dyskusji nad problemem „dobrego” modelu ekonometrycznego (contribution to the discussion on the problem of “good” econometric model), in czerwiński z. (red.), moje zmagania z ekonomią (my wrestling with economy), 389411, wydawnictwo akademii ekonomicznej w poznaniu, poznań. davidson, j. (2011), time series modeling version 4.32, university of exeter. doman, m., doman, r. (2009), modelowanie zmienności i ryzyka. metody ekonometrii finansowej (volatility and risk modeling. methods of financial econometrics), oficyna, kraków. estrella, a., hardouvelis, g. a. (1991), the term structure as a predictor of real economic activity, journal of finance, 46, 555−576. estrella, a., mishkin, f.s. (1998), predicting u.s. recessions: financial variables as leading indicators, the review of economics and statistics, 80, 45−61. gatnar, e. (1998), symboliczne metody klasyfikacji danych (symbolic methods of data classification), pwn, warszawa. gruszczyński, m. (2001), modele i prognozy zmiennych jakościowych w finansach i bankowości (models and forecasts of qualitative variables in finances and banking), monografie i opracowania 490 (monographs and studies), sgh, warszawa. hamilton, j. d. (1989), a new approach to the economic analysis of nonstationary time series and the business cycle, econometrica, nr 57, 357–384. hamilton, j. d. (2005), regime-switching models, department of economics, 0508, university of california, san diego. jajuga, k. (1990), modele z dyskretną zmienną objaśnianą (models with discrete explained variable), in bartosiewicz st. (ed.), estymacja modeli ekonometrycznych (estimation of econometric models), we, warszawa. konopczak, k. (2009), analiza zbieżności cyklu koniunkturalnego gospodarki polskiej ze strefą euro na tle krajów europy środkowo-wschodniej oraz państw członkowskich strefy euro (convergence analysis of polish economy business cycle with euro area against the background of middle east european countries and euro area member states), in raport na temat pełnego uczestnictwa rzeczypospolitej polskiej w trzecim etapie unii gospodarczej i walutowej (report on the polish republic full participation in the third stage of economic and monetary union), nbp. the probability of recession in poland… dynamic econometric models 12 (2012) 73–87 87 mintz, i. (1972), dating american growth cycles, in zarnowotz, victor (ed.), 39−88. nyberg, h. (2009), dynamic probit models and financial variables in recession forecasting, journal of forecasting, 29(1−2), 215−230. skrzypczyńska, m. (2011), pomiar cyklu koniunkturalnego w polsce – analiza porównawcza (measurement business cykle in poland − comparative analysis), bank i kredyt, 42 (4), 31-54. stock, j., watson, j. (1993), a procedure for predicting recessions with leading indicators: econometric issues and recent experience, business cycles, indicators and forecasting, 95-156, nber. train, k.e. (2003), discrete choice methods with simulation, mit press, cambridge. wright, j. h. (2006), the yield curve and predicting recessions, technical report, division of monetary affairs, federal reserve board. zarnowitz v., ozyildirim a. (2006), time series decomposition and measurement of business cycles, trends and growth cycles, journal of monetary economics, 53, 1717−1739. federal reserve bank of new york, http://www.newyorkfed.org/research/capital_markets/ ycfaq.html (17.08.2011). prawdopodobieństwo kryzysu w polsce z modelu przełącznikowego hamiltona i modelu logitowego z a r y s t r e ś c i. w wielu krajach brakuje wypracowanego systemu oznaczania początku i końca kryzysu. w proponowanej metodzie periodyzacji każda z czterech faz aktywności gospodarczej opisywana jest przez koniunkcję wartości rocznych i miesięcznych indeksów produkcji przemysłowej. analitycy rynku zwracają szczególną uwagę na zróżnicowanie zachowania się większości wskaźników makroekonomicznych w czasie spadków i długookresowego wzrostu. w związku z tym uzasadnione jest założenie o zmieniających się parametrach modeli opisujących kształtowanie się tych wielkości. realizację takiego założenia umożliwiają zarówno modele przełącznikowe jak i modele logitowe. w badaniach empirycznych przeprowadzono porównanie prawdopodobieństwa wystąpienia kryzysu z obu modeli analiza wyników pokazuje duże podobieństwo wskazań z zaproponowanej metody periodyzacji i modelu hamiltona. model hamiltona w prezentowanej wersji dobrze opisuje prawdopodobieństwo wystąpienia dwóch faz spadkowych. model logitowy pozwala na uzyskanie zadawalających rezultatów dla podziału bardziej szczegółowego. na gruncie gospodarki polskiej należy jednak w dalszym ciągu prowadzić badania nad rozpoznaniem własności symptomatycznych różnych wskaźników makroekonomicznych. s ł o w a k l u c z o w e: model przełącznikowy, model logitowy, periodyzacja faz aktywności gospodarczej, prawdopodobieństwo kryzysu. introduction 1. dating of economic activity phases 2. hamilton’s switching model 3. logit model 4. research results 5. conclusions references © 2012 nicolaus copernicus university press. all rights reserved. http://www.dem.umk.pl/dem dynamic econometric models vol. 12 (2012) 19−32 submitted january 25, 2012 issn accepted october 8, 2012 1234-3862 dorota witkowska, krzysztof kompa, aleksandra matuszewska-janica* analysis of linkages between central and eastern european capital markets† a b s t r a c t. the aim of the research is analysis of shortand long-term international relations between stock exchanges in central and eastern europe. the analysis is provided in 3 stages. in the first step the order of the variables integration is examined. in the second stage short-run relationships for pairs of indexes are analyzed using granger causality test. in the last step longrun relationships for pairs of indexes are examined applying johansen cointegration method. k e y w o r d s: emerging markets, equity cee markets, cointegration, granger causality, longrun relationships, short-run relationships. j e l classification: g15. introduction the analysis of common stock market movements is important for effective portfolio diversification and a possible starting point to examine the functioning of the global financial system. therefore international market linkages has attracted investors and policy-makers’ attention. consequently, international equity market integration is a topic often discussed in literature, especially many researchers have investigated the short-term and long-term interrelationships among worldwide financial markets. the theory review, evidence and implications of international equity market integration are presented in (kearney, lucey, 2004; bailey, choi, 2005) among others. various aspects of equity markets relationships have been considered, including: * correspondence to: dorota witkowska, department of econometrics and statistics, warsaw university of life sciences, ul. nowoursynowska 166, 02-787 warszawa, poland, e-mail: dorota_witkowska@sggw.pl † scientific research with the financial support of the polish ministry of science and higher education no n n111 43 1837. dorota witkowska, krzysztof kompa, aleksandra matuszewska-janica dynamic econometric models 12 (2012) 19–32 20 − volatility spillovers across markets (e.g. engle, susmel, 1993; kearney, 2000; koutmos, booth, 1995; ng, 2000); − market correlation structures (e.g. koedijk et al., 2002; longin, solnik, 1995) and − financial crises contagion (e.g. claessens, forbes, 2001; rigobon, 1999). empirical investigations discussed in literature can be classified into 3 major classes due to following criteria: − regions and periods of provided analysis, − length of the return intervals, − methods of analysis. empirical analysis considering relations among mature markets has been provided since the end of the 20-th century (eun, shim 1989; hamao et al., 1990; kasa 1992; engle, susmel 1993; lin et al., 1994; longin, solnik 1995, 2001; koutmos, booth, 1995; kim, rogers, 1995; karolyi, stulz, 1996; choudhry, 1996; koutmos, 1996; serletis, booth et al., 1997; king, 1997; rigobon, 1999; witkowska, 1999; kearney, 2000; ng, 2000; claessens, forbes, 2001; macdonald, 2001; shachmurove, witkowska, 2001; forbes, rigobon, 2002; koedijk et al., 2002; serwa, bohl, 2005; sharkasi et al., 2004; kearney, lucey, 2004; baur, 2004; phylaktis, ravazzolo, 2005). while investigation of mutual market linkages for emerging markets has shorter history, especially consideration for post-communist countries. syriopoulos (2007) notices that despite the growing importance of the emerging central and eastern european stock markets (see fig. 1), the relevant body of research remains surprisingly limited. furthermore, the empirical findings on this topic appear rather ambiguous and contradictory. for emerging markets we should mention research provided for: − asean (janakiraman, lamba, 1998; gosh et al., 1999; masih, masih, 2001; siklos, ng, 2001), − middle and south americas (phylaktis, ravazollo, 2005; diamandis, 2009) and − central and eastern europe (voronkova, 2004; gilmore et al., 2008; syllignakis, kouretas, 2010). taking into account length of the investigated samples we notice that 10year or longer periods are very often considered, for instance (caporale, spagnolo, 2010; gilmore et al., 2008; sharkasi et al., 2004). however shorter periods are also used in comparable analysis as dubinskas, stunguriene (2010) who consider 2-years period or gilmore et al. (2008) who use rolling windows approach. the length of the returns interval is also crucial and influences the results of investigation. in fact different intervals are used, for instance daily and weekly returns are discussed by caporale, spagnolo (2010), monthly – in baur (2004), and 5-minutes intraday data – in hanousek et al. (2008) and hanousek, kočenda (2009). analysis of linkages between central and eastern european capital markets dynamic econometric models 12 (2012) 19–33 21 figure 1. cee equity markets capitalization as a percentage of fese equity markets capitalization investigations has been provided applying different methods of analysis, the most popular are: − correlation measures (as in: panton et al., 1976; watson, 1980; meric, meric 1989; bailey, stulz, 1989; fisher, palasvirta, 1990; longin, solnik, 1995), − causality analysis (for instance kwan et al., 1995; roca, 1999; huang et al., 2000; narayan et al., 2004; matuszewska-janica, 2010), − var models and cointegration analysis (eun, shim, 1989; kasa, 1992; richards, 1995; hassan, naka, 1996; choundry, 1997, gosh et al., 1999; witkowska, 1999; shachmurove, witkowska, 2001; masih, masih, 2001; siklos, ng, 2001; chen et al., 2002; pascual, 2003; yang et al., 2004; gilmore et al., 2008; kuçukcolak, 2008; matuszewska-janica, 2011), − garch models (baele, vennet, 2001; voronkova, 2004; li, majerowska, 2008), − taxonomic methods as kompa (2010). the aim of the research is identification of shortand long-term international relations between stock exchanges in central and eastern europe. the analysis is provided in 3 stages in which: 1. the order of the variables integration, 2. short-run relationships for pairs of indexes, using granger causality test and 3. long-run relationships for pairs of indexes, applying johansen cointegration method are investigated. 1. data description the research is provided for quotations of 14 indexes from the capital markets in central and eastern europe (cee) – table 1, from the period: january 2000 – november 2010. in our research we consider daily, weekly and monthly (for the last quotation in the week and month respectively) data. the observations are transformed into natural logarithms and logarithmic rates of return. dorota witkowska, krzysztof kompa, aleksandra matuszewska-janica dynamic econometric models 12 (2012) 19–32 22 table 1. analyzed indexes lp index type of index stock exchange (se) 1 atx price, blue-chip index vienna se 2 px price, blue-chip index prague se 3 pxglob price, broad index prague se 4 bux performance, blue-chip index budapest se 5 sbi20 price, broad index ljubljana se 6 sax total return, blue chip index bratislava se 7 bet price, blue-chip index bucharest se 8 sofix total return, broad index bulgarian se 9 omxbb performance, 32 companies from baltic market – benchmark omx group, baltic countries 10 omxt total return, all share index tallin se, omx group 11 omxr total return, all share index riga se, omx group 12 omxv total return, all share index vilnius se, omx group 13 wig performance, all share index warsaw se (wse) 14 wig20 price, blue-chip index warsaw se (wse) it is worth mentioning that for 2 indexes: sofix and sbi20 the data are available only from january 2001 till october 2010, therefore analysis is provided for 2 samples as it is shown in table 2 where time ranges, symbols of samples and numbers of observations are presented. missing data are completed by repeating the last observation (i.e. foregoing the lacking one). table 2. the considered periods, number of observations and notation of samples indexes date of first observation date of last observation frequency of data daily weekly monthly a b a b a b omxbb, omxt, omxr, omxv, atx, sax, bux, px, pxglob, bet, wig, wig20 2000-01-03 2010-11-05 pd1 2790 556 pw1 130 pm1 sofix, sbi20* 2000-12-29 2010-10-14 pd2 2515 501 pw2 118 pm2 note: a – symbol of samples, b – number of observations, * – quotation of sbi20 was stopped in october 2010. 2. results in the first step the order of the variables integration is identified applying augmented dickey-fuller (adf) test1. the results indicate, that all examined time series of indexes are nonstationary while all returns are stationary so indexes are i(1). 1 for technical details see e.g. maddala, kim (1998), elder, kennedy (2001). analysis of linkages between central and eastern european capital markets dynamic econometric models 12 (2012) 19–33 23 in the second step, short-run relationships between all indexes are examined employing granger causality test2. for two indexes (x and y) we denote the direction of granger causality by the arrow (i.e. x→y means that x causes changes in y, and y→x the opposite). causality analysis is provided for 182 mutual relations (for 14 indexes), considering from 1 to 10 lags for each investigated relation. hypotheses are verified at the significance level 0.05. table 3 contains the results presented as percentage of cases when the null hypothesis is rejected. more detailed results are presented in tables a1–a3 in the appendix. for daily data the greatest percentage of rejections is obtained for following relations: wig20→y (93%), bux→y (92%), wig→y (87%), x→sofix (93%) and x→sbi20 (82%). on another words, daily changes of wig20, wig and bux are the most often causes of changes in other investigated indexes, while sofix and sbi20 are the most sensitive indexes. the smallest number of h0 rejections is observed for following relations: sax→y (22%), omxr→y (23%), x→wig20 (18%) and x→wig (35%). thus the changes of sax and omxr influence other indexes very rarely while the less sensitive to changes of other indexes are wig and wig20. it can be explained by the fact that warsaw stock exchange is the biggest market in cee region and it reacts due to the world biggest markets changes. table 3. results of the granger causality test – percentage of rejection h0 relation % rejections of the h0 relation % rejections of the h0 pd1/pd2 pw1/pw2 pm1/pm2 pd1/pd2 pw1/pw2 pm1/pm2 atx→y 78% 55% 59% x→atx 66% 45% 2% bet→y 48% 71% 23% x→bet 72% 55% 4% bux→y 92% 44% 33% x→bux 57% 59% 9% omxbb→y 75% 36% 17% x→omxbb 75% 74% 48% omxr→y 23% 41% 2% x→omxr 72% 42% 42% omxt→y 71% 25% 15% x→omxt 74% 85% 64% omxv→y 64% 43% 20% x→omxv 65% 45% 37% px→y 86% 82% 43% x→px 65% 25% 6% pxglob→y 85% 83% 45% x→pxglob 66% 27% 6% sax→y 22% 12% 0% x→sax 55% 56% 5% wig→y 87% 64% 25% x→wig 35% 53% 15% wig20→y 93% 53% 15% x→wig20 18% 54% 22% sofix→y 31% 76% 15% x→sofix 93% 60% 38% sbi20→y 42% 42% 7% x→sbi20 82% 48% 23% taking into account number of lags (table a1) we notice that the biggest number of cases, when h0 is rejected for all 10 lags, is obtained for relations: wig20→y (11 times for 13 considered cases), wig→y (11) atx→y (10), x→sofix (12) and x→sbi20 (10). the biggest number of cases, when h0 is 2 see charemza, deadman (1997), osińska (2008). dorota witkowska, krzysztof kompa, aleksandra matuszewska-janica dynamic econometric models 12 (2012) 19–32 24 not rejected for any considered lag, is observed for relations: sofix→y (6), omxr→y (6), sax→y (5), x→wig20 (8) and x→wig (7). for weekly data the greatest percentage of rejections is obtained for following relations: pxglob→y (83%), px→y (82%), x→omxt (85%), x→omxbb (74%). such results denotes that weekly changes of pxglob and px cause weekly changes of other analysed indexes most often. omxt and omxbb are the most sensitive to changes of other indexes. the smallest number of h0 rejection is observed for following relations: sax→y (12%), omxt→y (25%), x→px (25%) and x→pxglob (27%). the weekly changes of index sax cause (in granger sense) the changes of other indexes most rarely. the same result we obtain for daily data but, in contradistinction to daily changes, px and pxglob are the less sensitive to weekly changes of other indexes appear (table a2). with change from daily to weekly data, number of relations when h0 is rejected for all lags (from 1 to 10) is decreasing, and number of cases when h0 is not rejected for any lag is increasing. it can be interpreted that by broadening of the time interval for returns calculations, the number of causal relations (in granger sense) is reduced. for monthly data the greatest percentage of rejections is obtained for the following relations: atx→y (59%), pxglob→y (45%), px→y (43%), x→omxt (64%), x→omxbb (48%). while the smallest number of h0 rejections is observed for: sax→y (0%), omxr→y (5%), sbi20→y (7%), x→atx (2%) and x→bet (4%) – table a3. in comparison to results, obtained for weekly data, number of causal (in granger sense) relations is decreasing. as it is visible in tables a1 – a3, we obtain similar results for wig and wig20 since both indexes cause changes of other investigated indexes while they do no influence wig20 and wig for daily data. however there are two exceptions for: − weekly data since wig20→sax but ¬wig→sax, and bux→wig20 but ¬bux→wig, − monthly data since wig→sbi20 but ¬wig20→sbi20, and omxr→wig20 but ¬omxr→wig. therefore it does not matter if warsaw stock exchange is represented by wig (performance, all share index) or wig20 (price, blue-chip index). one could also notice that (granger) causal short-run relation between wig and wig20 for weekly and monthly returns is bilateral (wig↔wig20) while for daily observation only changes of wig20 cause changes of wig (wig20→wig). the last statement could be explained by high capitalization of companies represented by wig203. 3 capitalization of wig20 is 69.5% of whole market capitalization represented by wig (www.gpw.pl, september 23, 2011). analysis of linkages between central and eastern european capital markets dynamic econometric models 12 (2012) 19–33 25 the next step of investigation is cointegration analysis provided by johansen method4. as it was mentioned, cointegration analysis is applied in order to check if effective international portfolio (risk) diversification between two capital markets from cee region is possible. number of cointegrating vectors are presented in tables 4–6. table 4. number of cointegrating vectors for pairs of indexes – daily data bet 1* bux 0 1 omxbb 0 1 0 omxr 1 1* 0 0 omxt 0 1 0 1* 0 omxv 0 1 0 1 0 1 px 0 0 0 0 1* 0 0 pxglob 0 0 0 0 1* 0 0 0 sax 0 0 0 0 0 0 0 0 0 wig 0 0 1* 0 0 0 0 0 0 0 wig20 0 1 0 0 0 0 0 0 0 0 2 sofix 0 1 0 0 0 0 0 0 0 0 0 0 sbi20 0 0 0 0 0 0 0 0 0 0 1 1 0 x \ y atx bet bux omxbb omxr omxt omxv px pxglob sax wig wig20 sofix note: hypotheses are verified at the significance level α=0.05; * represents statistical significance at the 0.1. for daily data all indexes but one, i.e. sax, are cointegrated with other indexes from cee capital markets. the greatest number of long-run relations is observed for bet (8 cases from 13 analyzed). thus we can conclude that bulgarian stock exchange could be the most sensitive market for international shocks (in the region) so it creates the less number of diversified portfolios. two cointegrating vectors are observed in relations wig – wig20, therefore we suppose that indexes from wse have stronger relationship among themselves then with other indexes. for weekly data we observe smaller number of long-run relations than for daily data. indexes omxr and bet build the biggest number of cointegrating relations, 4 and 3, respectively. indexes bux, omxbb, sofix and sbi20 are not cointegrated with other analyzed indexes. we observe that wig and wig20 are cointegrated only between themselves with 2 cointegrating vectors only in shorter period. it seems to be two reasons of this phenomenon. firstly, johansen tests results are sensitive on investigation period (see gilmore et al., 2008 and pascual, 2003 among others). secondly, wse could be not influenced by changes that appear on other cee capital markets. 4 usually johansen tests statistics λtrace and λmax yield the same results but in some cases they are different. in such situation it is accepted λmax test indication (the λmax test is considered superior to the λtrace test, see kennedy, 2003, p. 355). dorota witkowska, krzysztof kompa, aleksandra matuszewska-janica dynamic econometric models 12 (2012) 19–32 26 table 5. number of cointegrating vectors for pairs of indexes – weekly data bet 0 bux 0 0 omxbb 0 0 0 omxr 1 0 0 0 omxt 0 1 0 0 0 omxv 0 1 0 0 0 0 px 0 0 0 0 1* 0 0 pxglob 0 0 0 0 1* 0 0 0 sax 0 1* 0 0 1* 0 0 0 0 wig 0 0 0 0 0 0 0 0 0 0 wig20 0 0 0 0 0 0 0 0 0 0 2 a sofix 0 0 0 0 0 0 0 0 0 0 0 0 sbi20 0 0 0 0 0 0 0 0 0 0 0 0 0 x \ y atx bet bux omxbb omxr omxt omxv px pxglob sax wig wig20 sofix note: hypotheses are verified at the significance level α=0.05; * represents statistical significance at the 0.1. a – 2 cointegrating vectors are obtained for sample pw2, for sample pw1 h0 that cointegration does not exist is not rejected. table 6. number of cointegrating vectors for pairs of indexes – monthly data bet 0 bux 0 0 omxbb 0 0 0 omxr 0 0 1* 1 omxt 0 1* 0 0 0 omxv 0 0 0 0 0 0 px 0 1* 0 0 1* 1 0 pxglob 0 1* 0 0 1* 1 0 0 sax 0 0 0 0 0 0 0 0 0 wig 0 0 0 0 0 0 0 0 0 0 wig20 1 0 0 0 0 1* 0 0 0 0 2 sofix 0 0 0 0 0 0 0 0 0 0 0 0 sbi20 0 0 0 0 0 0 0 0 0 0 0 0 0 x \ y atx bet bux omxbb omxr omxt omxv px pxglob sax wig wig20 sofix note: hypotheses are verified at the significance level α=0.05; * represents statistical significance at the 0.1. for monthly data we observe smaller number of long-run relations than for daily data however number of linkages is bigger than for weekly data. indexes omxr, omxt and bet build the biggest number of cointegrating relations – 3 each. indexes omxv, sax, sofix and sbi20 are not cointegrated with other investigated indexes. wig and wig20 are cointegrated with 2 vectors. in opposite to wig, only wig20 is cointegrated with other foreign indexes (i.e. atx and omxt). analysis of linkages between central and eastern european capital markets dynamic econometric models 12 (2012) 19–33 27 in literature it is remarked that johansen tests power does not increase when the higher frequency data are used but with the time span of the data (see hakkio, rush, 1991; diamandis 2009). hence we can suppose that indication of the johansen test is more trustworthy for weekly or monthly data than for daily ones. so taking into consideration two capital markets from cee region, international portfolio risk diversification can be achieved. but we also have to take into account sensitiveness on the changes that appear at the world biggest markets since these shocks are quickly transmitted into global market. in the last step of the analysis, vecm models for relations wig or wig20 with other foreign index are estimated. the obtained results are presented in table 7. table 7. selected results of the estimation of the vecm models (for wig and wig20 indexes) pair of indexes variable order in the model equation ecm parameter wig i bux daily wig, bux first for wig second for bux –0.0036 * 0.0031 wig i sbi20 daily sbi20, wig first for sbi20 second for wig –0.0021 *** 0.0045 wig20 i bet daily bet, wig20 first for bet second for wig20 –0.0010 *** 0.0003 wig20 i sbi20 daily sbi20, wig20 first for sbi20 second for wig20 –0.0014 *** 0.0010 wig i sbi20 weekly sbi20, wig first for sbi20 second for wig –0.0106 *** 0.0021 wig20 i sbi20 weekly sbi20, wig20 first for sbi20 second for wig20 –0.0070 *** 0.0052 wig20 i atx monthly wig20, atx first for wig20 second for atx – 0.0070** 0.0253 wig20 i omxt monthly wig20, omxt first for wig20 second for omxt –0.1301*** –0.0542 note: ecm parameters significant *** – at the level 0.01, ** – at the level 0.05, * – at the level 0.1. error correction mechanism is significant for all presented cases. for daily observations, index wig in relation to bux has the highest speed of adjustment (circa 0.3% of the discrepancy in these two indexes from the previous day is eliminated in present day). while for weekly data the highest speed of adjustment has index sbi20 with relation to wig, and for monthly data the highest speed of adjustment has index omxt with relation to wig20. we can observe that restoring the equilibrium is quicker (from period to period) for monthly data. conclusions integration of financial markets has important implications since highly integrated markets are not isolated from international shocks. it could be also dorota witkowska, krzysztof kompa, aleksandra matuszewska-janica dynamic econometric models 12 (2012) 19–32 28 the reason that the effective portfolio risk diversification between integrated markets cannot be achieved5. for investigated time series many short-run granger causal relationships are found out. it is also noticed that with broadening of intervals (for which returns are computed) many relations disappear, thus the data frequency does matter6. it is also worth mentioning that for the 10 years period of observation only a few long-run relationships are diagnosed, similarly to the results obtained by other researchers (e.g. gilmore et al, 2008; pascual, 2003). the results show that changes at the warsaw stock exchange cause (in granger sense) changes at other capital markets from cee region. while wse indexes are less sensitive to the changes that appear at other investigated markets. thus it could be considered as a premise that wse is the most developed capital market in cee region7 since situation at wse influences other markets in the region while it is not sensitive on shocks that appear in other cee markets. references bailey, w., choi, j.j. (2005), international market linkages, journal of economics and business, 55, 399–404. bailey, w., stulz, r. (1989), the pricing of stock index in a general equilibrium model, journal of financial and quantitative analysis, 24, 1–12. baur, d.g. (2004), what is co-movement? eur working paper no. 20759 http://ssrn.com/ abstract=570585 (17.10.2011). booth, g. g.; martikainen, t. tse y. (1997), price and volatility spillovers in scandinavian stock markets, journal of banking and finance, 21, 811–823. caporale, g.m., spagnolo, n. (2010), stock market integration between three ceecs, economics and finance working paper series, working paper no. 10–09, http://bura.brunel.ac.uk/bitstream/2438/5056/1/1009%5b1%5d.pdf (17.10.2011). charemza, w., deadman, d.f. (1997), new directions in econometric practice, pwe, warsaw [polish edition]. chen, g., firth, m., rui, o. (2002), stock market linkages: evidence from latin america, journal of banking and finance, 26(6), 1113–1141. choudhry, t. (1996), interdependence of stock markets: evidence from europe during the 1920s and the 1930s, applied financial economics, 6, 243–249. choudhry, t., (1997), stochastic trends in stock prices: evidence from latin american markets, journal of macroeconomics, 19, 285–304. claessens, s., forbes, k.j. (ed.) (2001), international financial contagion: how it spreads and how it can be stopped, kluwer academic publishing, boston. 5 see e.g. hassan, naka (1996), phylaktis, ravazzolo (2005), syriopoulos (2007). numerous studies concerning long-run relationships among stock exchanges are quoted in the last mentioned paper. 6 for daily data h0 is rejected for every lag (from 1 to 10) in 74 cases in investigated 182 relations, for weekly data we obtained only 39 such cases while taking into consideration monthly data only 15 such cases are observed. 7 another explanation of mentioned observation is clamming that matured and great capital markets transmit changes to emerging markets. analysis of linkages between central and eastern european capital markets dynamic econometric models 12 (2012) 19–33 29 diamandis, p.f. (2009), international stock market linkages: evidence from latin america, global finance journal, 20, 13–30. dubinskas, p., stunguriene, s. (2010), alterations in the financial markets of the baltic countries and russia in the period of economic downturn, technological and economic development of economy, 16(3), 502–515. elder, j., kennedy, p.e. (2001), testing for unit roots: what should students be taught?, journal of economic education, 31(2), 137–146. engle, r.f. sumsel r. (1993), common volatility in international equity markets, journal of business and economic statistics, 11, 167–176. eun, c., shim, s. (1989), international transmission of stock market movements, journal financial and quantitative analysis, 24, 241–256. fischer, k.p., palasvirta, a.p. (1990), high road to a global marketplace: the international transmission of stock market fluctuations, the financial review, eastern finance association, 25(3), 371–94. forbes, k.j., rigobon, r. (2002), no contagion, only interdependence: measuring stock market comovements, journal of finance, american finance association, 57(5), 2223–2261. ghosh, a., saidi, r., johnson, k.h. (1999), who moves the asia-pacific stock markets: u.s. or japan? empirical evidence based on the theory of cointegration, financial review, 34, 159–170. gilmore c.g., lucey b.m., mcmanus g.m. (2008), the dynamics of central european equity market comovements, the quarterly review of economics and finance, 48, 605–622. hakkio, g.s., rush, m. (1991), cointegration: how short is the long run?, journal of international money and finance, 10, 571–581. hamao, y., masulis, r. w., ng, v. (1990), correlations in price changes and volatility across international stock markets, review of financial studies, 3, 281–307. hanousek j., kočenda e. (2009), intraday price discovery in emerging european stock markets, gerge-ei working paper series 382, http://www.cerge-ei.cz/pdf/wp/wp382.pdf (17.10.2011). hanousek j., kočenda e., kutan a.m. (2008), the reaction of asset prices to macroeconomic announcements in new eu markets: evidence from intraday data, gerge-ei working paper series 349, http://www.cerge-ei.cz/pdf/wp/wp349.pdf (17.10.2011). hassan, m.k., naka a. (1996) short-run and long-run dynamic linkages among international stock markets, international review of economics and finance, 5(4), 387–405. huang, b.n., yang c.w., hu j.w.s. (2000), causality and cointegration of stock markets among the united states, japan and the south china growth triangle, international review of financial analysis, 9(3), 281–297. janakiraman, s., lamba, a.s. (1998), an empirical examination of linkages between pacificbasin stock markets, journal of international financial, markets, institutions and money, 8, 155–173. karolyi, g.a., stulz, r.m, (1996), why do markets move together? an investigation of u.s.japan stock return comovements, journal of finance, american finance association, 51(3), 951–86. kasa k. (1992), common stochastic trends in international stock markets, journal of monetary economics, 29, 95–124. kearney c., lucey b.m. (2004), international equity market integration: theory, evidence and implications, international review of financial analysis, 13, 571–583. kearney, c. (2000), the determination and international transmission of stock market volatility, global finance journal, 11, 1–22. kennedy, p. (2003), a guide to econometrics, blackwell publishing. kim, s.w. rogers, j.h. (1995), international stock price spillovers and market liberalization: evidence from korea, japan, and the united states, journal of empirical finance, 2(2), 117–133. dorota witkowska, krzysztof kompa, aleksandra matuszewska-janica dynamic econometric models 12 (2012) 19–32 30 koedijk, k.c.g., campbell, r.a.j., kofman, p. (2002), increased correlation in bear markets. financial analysts journal, 58(1), 87–94. kompa, k. (2010), capital markets in transitional countries. comparison of stock exchanges in europe, in witkowska, d., nermend, k. (ed.), regional analysis: globalization, integration, transformation, wydawnictwo uniwersytetu szczecińskiego, szczecin, 125–152. koutmos, g. (1996), modelling the dynamic interdependence of major european stock markets, journal of business finance and accounting, 23, 975–988. koutmos, g., booth, g.g. (1995), asymmetric volatility transmission in international stock markets, journal of international money and finance, 14, 747–762. kuçukcolak, n. (2008), co-integration of the turkish equity market with greek and other european union equity markets, international research journal of finance and economics, 13, 58–73. kwan, a.c.c., sim, a.h.b., cotsomitis, j.a. (1995), the causal relationships between equity indices on world exchanges, applied economics, 27, 33–37. li, h., majerowska, e. (2008), testing stock market linkages for poland and hungary: a multivariate garch approach, research in international business and finance, 22, 247–266. lin, w.-l., engle, r. f., ito, t. (1994), do bulls and bears move across borders? international transmission of stock returns and volatility, review of financial studies, 7, 507–538. longin, f. m., solnik, b. (1995), is the correlation in international equity returns constant: 1960–1990, journal of international money and finance, 14, 3–26. longin, f., solnik, b., (2001), extreme correlation of international equity markets, journal of finance, 56(2), 649–676. macdonald, r. (2001), transformation of external shocks and capital market integration, in schroder, m. (ed.), the new capital markets in central and eastern europe, springer verlag, 210–245. maddala, g.s., kim, i.-m. (1998), unit roots, cointegration, and structural change, cambridge university press, cambridge. masih, r., masih, a.m.m. (2001), long and short term dynamic causal transmission amongst international stock markets, journal of international money and finance, 20, 563–587. matuszewska-janica, a. (2010), short-run relationship between selected central european stock exchanges: causality and exogeneity analysis, uniwersytet szczeciński, zeszyty naukowe nr 612(28), tom rynek kapitałowy, 519–532. matuszewska-janica, a. (2011), long-run relationships between selected central european indexes, international advanced in economic research, 17, 157–168. meric, i., meric, g. (1989), potential gains from international portfolio diversification and intertemporal stability and seasonality in international stock market relationships, journal of banking and finance, 13(4/5), 627–640. narayan, p., smyth, r., nandha m. (2004), interdependence and dynamic linkages between the emerging stock markets of south asia, accounting and finance, 44(3), 419–439. ng, a. (2000), volatility spillover effects from japan and the us to the pacific-basin, journal of international money and finance, 19, 207–233. osińska, m. (2008), econometric analysis of causal relationships, wydawnictwo naukowe uniwersytetu mikołaja kopernika, toruń. panton, d.b., lessing, p.v. joy, m.o. (1976), co-movement of international equity markets: a taxonomic approach, journal of financial and quantitative analysis, 11, 415–32. pascual, a.g. (2003), assessing european stock market integration, economic letters, 78, 197– 203. phylaktis k., ravazzolo f. (2005), stock market linkages in emerging markets: implications for international portfolio diversification, international financial markets, institutions and money, 15, 91–106. richards, a.j. (1995), comovements in national stock market returns: evidence of predictability, but not cointegration, journal of monetary economics, 36(3), 631–654. analysis of linkages between central and eastern european capital markets dynamic econometric models 12 (2012) 19–33 31 rigobon, r., (1999), on the measurement of the international propagation of shocks, nber working papers 7354, http://www.nber.org/papers/w7354.pdf (17.10.2011). roca, e.d. (1999), short-term and long-term price linkages between the equity markets of australia and its major trading partners, applied financial economics, 9, 501 –511. serletis, a., king, m. (1997), common stochastic trends and convergence of european union stock markets, the manchester school, 65(1), 44–57. serwa, d., bohl, m.,t. (2005), financial contagion vulnerability and resistance: a comparison of european stock markets, economic systems, 29(3), 344–362. shachmurove, y., witkowska, d. (2001), dynamic interrelation among major world stock market: a neural network analysis, international journal of business, 6(1), 1–22. sharkasi, a., ruskin, h., crane, m. (2004), interdependence between emerging and major markets", in antoch, j. (ed.), compstat 2004 – proceedings in computational statistics 16th symposium prague, czech republic, physica-verlag, springer, 1783–1790. siklos, p.l., ng, p. (2001), integration among asia-pacific and international stock markets: common stochastic trends and regime shifts, pacific economic review, 6, 89–110. syllignakis, m.n., kouretas g.p. (2010), dynamic correlation analysis of financial contagion: evidence from the central and eastern european markets, international review of economic and finance, 20, 717–732. syriopoulos t. (2007), dynamic linkages between emerging european and developed stock markets: has the emu any impact?, international review of financial analysis, 16, 41– 60. voronkova s. (2004), equity market integration in central european emerging markets: a cointegration analysis with shifting regimes, international review of financial analysis, 13, 633–647. watson, j. (1980), the stationarity of inter-country correlation coefficients: a note, journal of business finance and accounting, 7, 297–299. witkowska, d. (1999), neural networks application to analysis of daily stock returns at the largest stock markets, in szczepaniak (ed.), computational intelligence and applications, physica-verlag heidelberg new york, 351–364. yang, j., kolari, j.w., sutanto, p.w. (2004), on the stability of long-run relationships between emerging and us stock markets, journal of multinational financial management, 14(3), 233–248. analiza relacji pomiędzy rynkami kapitałowymi europy środkowej i wschodniej z a r y s t r e ś c i. celem analizy jest ocena związków krótkookresowych (w zakresie przyczynowości) i długookresowych (kointegracja) pomiędzy rynkami kapitałowymi europy środkowej i wschodniej, a w szczególności pomiędzy giełdą w warszawie i pozostałymi rynkami. analizie poddano dzienne, tygodniowe i miesięczne stopy zwrotu indeksów notowanych na tych rynkach. badania obejmują okres od stycznia 2000 do listopada 2010. s ł o w a k l u c z o w e: giełdy europy środkowej i wschodniej, rynki rozwijające się, kointegracja, przyczynowość w sensie grangera, krótkookresowe i długookresowe relacje pomiędzy rynkami. dorota witkowska, krzysztof kompa, aleksandra matuszewska-janica dynamic econometric models 12 (2012) 19–32 32 appendix table a1. lags for h0 rejection in granger causality test: daily data (pd1, pd2) x \ y atx bet bux omxbb omxr omxt omxv px pxglob sax wig wig20 sofix sbi20 atx x 1-10 1-10 1-10 1-10 1-10 1-10 1-10 1-10 7 1-10 1-10 bet 1-10 x 1-7 1-3 2 1-10 1-10 4;5;7 1-10 1-9 bux 1-10 1-10 x 1-10 1;2; 4-10 1-10 1-10 1-10 1-10 1;4-10 4-10 5-10 1-10 1-10 omxbb 8-10 3-10 4-10 x 1-10 1;2;4-10 1-10 4-10 4-10 2-10 1;4-10 1 1-10 1-10 omxr 8-10 5-10 7-10 x 8-10 8-10 1-10 6 omxt 4-10 5-10 4-10 1-10 1-10 x 1-10 4-10 4-10 2-10 1-10 1-10 omxv 2;7-10 3-10 2-10 1;6-10 1-10 1-10 x 7-10 7-10 2-6;8-10 1-10 1-10 px 1-10 2-10 1-7 1-10 1-10 1-10 1-10 x 1-3 4-10 2-10 2;5-10 1-10 1-10 pxglob 1-10 2-10 1-7 1-10 1-10 1-10 1-10 1 x 4-10 2-10 2;3; 5-10 1-10 1-10 sax 3 3-10 3-7 5-10 1;3 x 1-4 1 3 wig 1-10 1-10 1-10 1-10 1-10 1-10 1-10 1-10 1-10 4;9;10 x 1-10 1-10 wig20 1-10 1-10 1-10 1-10 1-3 1-10 1-10 1-10 1-10 3-10 1-10 x 1-10 1-10 sofix 1-10 6-10 6-10 6-10 6-10 1-10 x 1-10 sbi20 3-10 3-10 7 9 4-10 2;5;6 10 4-10 4-10 1-10 x note: bolded are cases when the h0 is rejected for all considered lags, shaded when no h0 is rejected. table a2. lags for h0 rejection in granger causality test: weekly data (pw1, pw2) x \ y atx bet bux omxbb omxr omxt omxv px pxglob sax wig wig20 sofix sbi20 atx x 1-5; 8-10 2;3;8;9 1-10 1-10 9 8 8;9 2-7;10 3-10 3-10 1-10 7-10 bet 2-10 x 2;3;6-8 2-10 6;7 2-10 4-9 2-10 2-10 3-10 2-4;6-8 2-10 1-10 8;9 bux 1-8; 10 x 1-10 4-7 1-10 5 6 2-10 1 4-10 5;7-10 omxbb 3 1 2-8 x 1-10 1-4; 8-10 1 1-4;10 2-5;8 2-7 1;3 1-3 omxr 1-3; 9;10 2-10 1-9 x 4-10 1 1-5;7; 9;10 1-5;7; 9;10 7;8 1;2; 6;8 omxt 2-4 8-10 1;3-10 x 1;2;8;10 2-5; 7-10 2-7;10 omxv 3;6;7; 9;10 9;10 2-10 2-7 1;2;4; 5;7 1-10 x 1-3 2-4; 8;9 2;3;8;9 1-4 1-3 px 1;4-10 1-10 1-10 1-10 4-8 1-10 4-10 x 1-10 1-10 1-10 1-10 3-10 pxglob 1;4-10 1-10 1-10 1-10 4-8 1-10 2;4-10 x 1-10 1-10 1-10 1-10 3-10 sax 1-10 x 3-5; 7;9;10 wig 1;4-6 1-8 1-10 1-10 4-7 1-10 1-9 x 2-10 1-10 1;2; 4-10 wig20 4-8 1-10 1 1-10 4-7 1-10 1-5;6;8 2-4 2-10 x 1;2; 4-10 1;2;4 sofix 1-5; 7-9 1-10 1-10 2-10 1-10 1-10 1-4 1-5;7 1-10 1-8 1-7 x 1-7 sbi20 5-10 6-10 3-10 4-10 1-10 1-10 3-4 3-6; 8;10 x note: bolded are cases when the h0 is rejected for all considered lags, shaded when no h0 is rejected. analysis of linkages between central and eastern european capital markets dynamic econometric models 12 (2012) 19–33 33 table a3. lags for h0 rejection in granger causality test: monthly data (pm1, pm2) x \ y atx bet bux omxbb omxr omxt omxv px pxglob sax wig wig20 sofix sbi20 atx x 1-10 1-10 1-10 1-10 1;4-10 5-10 2;5-10 1-10 1-4;9;10 bet x 1-5;8 2-5 1-6 2;4;8-10 1-3 1-6 bux x 1-10 1-6;8 1-10 1-5 3 3;4 1-4;6 1;3;4 omxbb 3 x 1-4;7-9 1-10 1 6-8 omxr 1 x 3 1 omxt 3;4;6 3;4 1-10 3;7-10 x omxv 1;2 2 1-4;9 x 2 1;3;5 1-7 1;6 1;3;4; 6;7 px 1-10 1-4 1-10 1-10 x 3;6-9 3-9 1-6 2-5 pxglob 1-10 1-5 1-10 1-10 x 3;6-9 3-9 1-7 2-5 sax x wig 1-3;7 1;3 1-10 1-4;7-9 x 1-4 1-3;4 1;2 wig20 1;2 1 1-4; 7;9;10 1-3 1-4;10 x 1;2 sofix 1-8 1;2;5-7 1;8;9 x 1;9;10 sbi20 2 2 2 1-6 x note: bolded are cases when the h0 is rejected for all considered lags, shaded when no h0 is rejected. introduction 1. data description 2. results conclusions references appendix dynamic econometric models © 2012 nicolaus copernicus university press. all rights reserved. http://www.dem.umk.pl/dem dynamic econometric models vol. 12 (2012) 53−71 submitted october 29, 2011 issn accepted july 1, 2012 1234-3862 maciej kostrzewski* bayesian pricing of the optimal-replication strategy for european option in the jd(m)j model† a b s t r a c t. in incomplete markets replication strategies may not exist and pricing of derivatives is not an easy task. this paper presents an application of bertsimas, kogan and lo’s algorithm of determining an optimal-replication strategy. in the merton model the likelihood function is a product of a mixture of infinite number of components. in the paper this number is assumed to be equal to a fixed value m+1. to determine the optimal strategy, we should estimate unknown parameters. to this end we resort to bayesian estimation techniques. the presented methodology is exemplified by an empirical research. k e y w o r d s: incomplete markets, bayesian inference, jump-diffusion process, pricing of derivatives. j e l classification: c11, c15, c58, c61, c63, g17 introduction the black-scholes model assumes a continuous path of underlying instrument values. pricing european options under the black-scholes model is an easy task (black, scholes, 1973). unfortunately, if we additionally include a component responsible for jumps, we obtain a model of a risky instrument in an incomplete market. then, the pricing of options is more of a challenge. in general, replication strategies do not exist (lamberton, lapeyre, 2000; shreve, 2004). bertsimas, kogan and lo (2001) propose an algorithm of determining an optimal-replication strategy. the optimality is understood in a meansquared sense. apart from the major drawbacks of the strategy – it is self * correspondence to: 1. maciej kostrzewski, department of econometrics and operational research, cracow university of economics, ul. rakowicka 27, 31-510 krakow, poland or 1. maciej kostrzewski, faculty of applied mathematics, agh university of science and technology, al. mickiewicza 30, 30-059 krakow, poland, e-mail: maciej.kostrzewski@uek.krakow.pl † the research was supported by the polish ministry of science and higher education. research project 2010-2012; no. n n111 429139. maciej kostrzewski dynamic econometric models 12 (2012) 53–71 54 financing but it is not an admissible strategy, and its calculation may be timeconsuming – it plays a significant role in hedging and pricing derivatives. to determine the optimal-replication strategy, the unknown parameters of the model need to be estimated. in this paper we resort to the bayesian estimation techniques. accounting for the parameter uncertainty inherent to the estimated parameters, for which the bayesian methodology is widely appreciated, is relevant not only to the estimation itself, but extends also to the pricing of optimal strategy, providing the researcher with a full (posterior) distribution of the strategy cost (instead of a single value). 1. the jump-diffusion model with m-jumps let ( )p,,ℑω denote a probability space. we consider a standard wiener process ( ) 0≥= ttww , a poisson process ( ) 0≥= ttnn with intensity 0>λ , and a family of independent random variables { },...2,1: == jqq j . the variables jq ’s have gaussian distributions: ( )2,~ qqj nq σµ . it is also assumed that σ –algebras generated by w, n and q are independent. in the merton model (merton, 1976) the price of a risky instrument is governed by a jump-diffusion process ( ) 0≥= ttpp which is the solution of the equation: ( ) .1 ttqtttt dnpedwpdtpdp −++= σµ the first two elements on the right-hand side define a pure diffusion process. the last element corresponds to jumps. ( ) 0≥ttp is an adapted and rightcontinuous process. it can be shown that: . 2 1 exp 1 2 0         ++      −= ∑ = j n j tt qwtpp t σσµ the price between consecutive jumps is governed by a geometric brownian motion. the process p has a finite number of jumps on each interval [ ]t,0 . the logarithm of the price is the solution of the equation: . 2 1 ln 2 ttt qdndwdtpd ++      −= σσµ hence, for a given time step 0>∆ we obtain: ( ) ( ) ( ) . 2 1 lnln 1 2 j n nj tttt qwwpp t t ∑ ∆+ += ∆+∆+ +−+∆      −+= σσµ it follows that the probability density function of logarithmic rates of return is given by (hanson, westman, 2002): bayesian pricing of the optimal-replication strategy for the european option… dynamic econometric models 12 (2012) 53–71 55 ( )( ) ( ) ( ) ( )( )kkxxp qqk k k tp tp 222 2 1 ! 0ln ,;exp σσµσµφλ λ +∆+∆−∆−∑= ∆ ∞ = ∆+ , (1) where ( )2,; sm⋅φ denotes the density of a normal distribution with mean m and variance 2s . therefore, the likelihood function is given by the product of an infinite mixture of normal densities, which highly complicates the statistical inference for the model. in order to define a jump-diffusion model with m jumps, let us consider a finite approximation of series (1): ( )( )       +∆+∆      − ∆ ∆−∑ ∞ = kkx k qq k k 222 0 , 2 1 ; ! exp σσµσµφ λ λ , ( )( ) ,, 2 1 ; ! exp 222 0       +∆+∆      − ∆ ∆−≈ ∑ = kkx k qq km k σσµσµφ λ λ (2) where { }...,2,1,0∈m is a fixed constant. in the black-scholes framework, the process of logarithmic returns of risky instrument is governed by an arithmetic brownian motion which is a pure diffusion process. under 0=m the above approximation collapses to the density of this arithmetic brownian motion with time step ∆ (kloeden, platen, 1992). in the general case the integral of sum (2) may not equal one. therefore, to obtain a probability density function, let us normalize the approximation given by (2): ( )       +∆+∆      −= ∑ = kkxwxp qqik m k i 222 0 , 2 1 ; σσµσµφθ , (3) with , ! )( ! )( 1 0 − =       ∆∆ = ∑ m j jk k jk w λλ mk ...,,0= , being the normalizing weights1. in the paper the logarithmic rates of return ( ),, 21 xxx = are assumed to follow the distribution defined by (3), and the resulting model is termed the jumpdiffusion model with m jumps, or jd(m)j, in short. the construction of the process restricts the number of jumps over any time interval ∆ to m, with the magnitude of each jump model with normal distribution ( )2, qqn σµ . finally, let us note that the jump-diffusion specification under study is some approximation to the merton model. therefore, and on a more statistical note, estimators of jd(m)j model parameters could be treated as approximations of the merton model parameters. 1 see frühwirth-schnatter (2006) for a thorough exposition on mixture modeling. maciej kostrzewski dynamic econometric models 12 (2012) 53–71 56 2. bayesian inference for the jd(m)j model in the jd(m)j model there are five unknown parameters ( ) θ∈22 ,,,, qq σµλσµ , where ( ) ( ) ( ) 5,0,0,0 rrr ⊂∞××∞×∞×=θ . the likelihood function is given by: ( ) ., 2 1 ; 222 01       +∆+∆      −= ∑∏ == kkxwxp qqik m k n i σσµσµφθ (4) if we observe a path of some jd(m)j process we do not know whether the observations or which of them have resulted from jumps. moreover, we are not able to (directly) identify an impact of the pure diffusion process and the jumps. in other words, we do not know which component of sum (3) is “responsible” for each observation. to solve this problem we introduce latent variables ( )nzzz ...,,1= such that { }mzi ...,,1,0∈ and ( ) ji wjzp == θ , where { }ni ...,,1∈ and { }mj ...,,1,0∈ . by means of iz ’s, we can assess the contribution made by the jumps (as compared with the pure diffusion compound) to explain each of n observations. increments of the poisson process n are independent, variables nzz ,...,1 are also independent. consequently, ( )       +∆+∆      −= iqiqiii zzxzxp 222 , 2 1 ;, σσµσµφθ and ( ) ., 2 1 ;, 222 1       +∆+∆      −= ∏ = iqiqi n i zzxzxp σσµσµφθ it is advisable to consider the following reparametrization of the model parameters: ∆= λl , 21σσ =h , 2 1 q qh σ= , under which vector of all the n+15 unknown quantities is given by: ( ) ( ).,,,,,...,,, 1 qqn hlhzzz µµθ σ= the bayesian model is defined by the joint distribution of the observables x, the hidden variables z and the parameters θ : ( ) ( ) ( ).,,,, θθθ zpzxpzxp = under a common assumption of the mutual prior independence among the covariates of θ , the joint prior density is formed: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ),, 1 θµµθθθ σ i n i qq zphpplphpppzpzp = ∏== bayesian pricing of the optimal-replication strategy for the european option… dynamic econometric models 12 (2012) 53–71 57 where: ( ) ( ) ( ) ,,,0for !! , 1 0 mk jk wwjzp jm j k kji =         ∆∆ === − = ∑ λλ θ ( ) ( ),,; 2µµµφµ smp = ( ) ( ) ( ),2/exp2/2 σνσσ σ ahhhp −∝ − ( ) ( ) ( ),2/exp2/2 lllp l −∝ −ν ( ) ( ),,; 2qqqq smp µφµ = ( ) ( ) ( ).2/exp2/2 qqq bhhhp q −∝ −ν such a choice of the prior structure (normal distributions for µ and qµ , the gamma distributions       2 , 2 a gamma σ ν and       2 , 2 b gamma q ν for σh and qh , respectively, and the 2 lν χ distribution for l ) densities guarantees that the posterior density is a bounded function even though the likelihood function is unbounded (lin, huang, 2002). the prior structure is determined under 1== ba , ∆= 10lν , 01.0=µm , 1 2 =µs , 5=σν , 01.0=qm , 1 2 =qs , 5=qν . posterior characteristics of the unknown quantities are calculated via the markov chain monte carlo (mcmc) methods (gamerman, lopes, 2006), combining the gibbs sampler, the independence and the sequential metropolishastings algorithms, as well as the acceptance-rejection sampling. for more details on the technicalities we refer to kostrzewski (2011). 3. the optimal-replication strategy pricing and hedging derivatives are among investors’ fundamental problems. investors employ replication strategies to hedge derivatives. unfortunately, in the case of incomplete markets such a strategy may not exist. some idea is to create a self-financing strategy, the value of which is “close” (at maturity) to the one of the derivative’s payoff function. we apply the results of bertsimas, kogan and lo (2001) to define and calculate the optimal-replication strategy for portfolios comprising a risky asset and riskless bonds. the approach involves buying, selling, borrowing and lending the portfolio constituents. let tp and tb denote price of the risky instrument and value of the riskless investment at tt ≤≤0 respectively. the payoff of an european option at maturity t is denoted by ( )tpf . finally, let tθ be the amount of stocks in the portfolio at time t. then tttt bpv +=θ is the value of the portfolio at t. bertsimas, kogan and lo (2001) consider a mean-squared-error criterion to define the optimal-replication maciej kostrzewski dynamic econometric models 12 (2012) 53–71 58 strategy *tθ , under which * tv is the value of the optimal portfolio. it follows that * tv and * tθ minimize: ( )[ ]( )02 vpfve tt − over { }tv θ,0 . moreover, { } ( )[ ]( )02 ,0 min vpfve tt v t −=∗ θ ε constitutes the minimum replication error, that is an error of fitting the strategy into the payoff f at t. if the replication strategy exists, then 0=∗ε and ( )tt pfv = * . the error ∗ε is construed as a relative measure of the market incompleteness, with its relativity justified by ∗ε corresponding only to a given derivative and a given model. to evaluate the optimal-replication strategy bertsimas, kogan and lo (2001) make some additional assumptions: 1. there are no taxes and transaction costs. 2. purchasing, selling, borrowing and shortsale are possible without any restrictions. 3. the borrowing and lending interest rate r is constant and equal zero. 4. p is a markov process. 5. trading takes place at known and fixed times { }ni ttt ,...,0∈ , where ttt n =<= 00 . to simplify the notation let iti ≡ . the aim is to calculate strategy ( )ii pvi ,,∗θ , the initial value ∗0v of the optimal portfolio and the error ∗ε . let ix be a logarithmic rate of return such that ( ).explnexp 11 ii i i ii xpp p pp =              = ++ bellman’s principle of optimality (bertsekas, 1995) yields the following theorem: theorem 1 (bertsimas, kogan, lo, 2001) if ( ) ( ) ( )( )( )iinn nki pvk iii pvpfvepvj kk ,min, 2 1 ,, −= −≤≤ θ , then: ( ) ( )( ) ( ) ( ) ( )( ),,,min, ,, 111 ,, 2 iiiiipviiii nnnnn pvpvjepvj pfvpvj ii +++= −= θ for 1,...,0 −= ni . bayesian pricing of the optimal-replication strategy for the european option… dynamic econometric models 12 (2012) 53–71 59 theorem 1 suggests that the strategy is set recursively. the problem of optimal replication is solved via stochastic dynamic programming. the main results are formulated in the theorem below. theorem 2 (bertsimas, kogan, lo, 2001) under the above assumptions: 1. there are functions: ( )ii pa , ( )ii pb and ( )ipc , such that: ( ) ( ) ( )[ ] ( )iiiiiiiiii pcpbvpapvj +−− 2, , .,...,0 ni = 2. ( ) ( ) ( )iiiiiii pqvpppvi −=∗ ,,θ , where functions ia , ib , ic , ip and iq are evaluated recursively. starting with 1)( =nn pa , )()( nnn pfpb = , 0)( =nn pc , the calculations for 0,...,1−= ni proceed as follows: ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ,2111 11111 iiiii iiiiiii ppppae ppppbpae ii pp −⋅ −⋅⋅ +++ +++++= ( ) ( ) ( )[ ] ( ) ( )[ ] ,2111 111 iiiii iiiii ppppae ppppae ii pq −⋅ −⋅ +++ +++= ( ) ( ) ( ) ( )( ) ],1[ 2111 iiiiiiiii ppppqpaepa −⋅−⋅= +++ ( ) ( ) ( ) ( ) ( ) ( )( )⋅−⋅−⋅⋅= +++++ iiiiiiiipaii pppppbpaepb ii 11111 1 [ ( ) ( )( ) ],1 1 iiiii ppppq −⋅−⋅ + ( ) ( ) ( ) ( ) ( )( ) +−⋅−⋅= +++++ ][ 2 11111 iiiiiiiiiii ppppppbpaepc ( ) ( ) ( ) .][][ 211 iiiiiii pbpappce ⋅−+ ++ 3. ( ) 0>ii pa , ( ) 0≥ii pc for 0,...,1−= ni . 4. under the optimal-replication strategy ( )ii pvi ,,∗θ we obtain: ( ) ( ) ( )[ ] ( ),, 00 2 00000000 pcpbvpapvj +−= ( )000 pbv =∗ and ( ).00 pc=∗ε properties (bertsimas, kogan, lo, 2001) a) the error of replication ( )00 pc=∗ε is the same for put and call options. b) if prices tp follow a geometric brownian motion and ∞→n , then the cost ∗0v of the optimal-replication strategy converges to the black-scholes price. c) ( )00 pb meets the put-call parity. d) ( )ii pvi ,,∗θ is the self-financing strategy which does not guarantee 0≥∗iv . e) the value of the optimal-replication strategy could be lower or higher at maturity t than the value of the payoff function. maciej kostrzewski dynamic econometric models 12 (2012) 53–71 60 the optimal-replication strategy could be less or more attractive than other strategies, e.g. the delta-hedging strategy. it is because the optimal-replication strategy is optimal only in the mean-squared-error sense. in general, calculation of expectations defined in theorem 2 may not be straightforward. in the case of the jd(m)j models numerical techniques should be employed to approximate their values. to calculate the cost of the optimal-replication strategy *0v and the relative measure of market incompleteness ∗ε the conditional expectations defined in theorem 2 need to be evaluated. obviously, these are given by relevant integrals, as for instance: ( ) ( )[ ]=−⋅ +++ iiiii ppppae 111 ( )( ) ( )( ) ( ) =⋅−⋅∫= +∞∞− dxpxppxpxpa iiiii expexp1 ( )( ) ( )( ) ( ) ,2 1 exp 2exp2exp 0 dy y pympympaw m k ikkikkiik −+⋅+∞ ∞− = +∫= ∑ σσπ where ( ) ( )[ ] 1!0! − ∆ = ∆ ∑= j m jkk jk w λλ , ( ) km qk µσµ +∆−= 221 and kqk 222 σσσ +∆= . analytical calculations of such formulae are difficult or positively impossible, which is why numerical approximations, such as the piecewise cubic hermite interpolation and gauss-hermit quadrature, are utilized. all numerical calculations are carried out in r using the pracma and glmmml packages. 4. empirical studies in this section we present the results of bayesian estimation, model comparison and pricing of the optimal-replication strategy. the calculations are performed for two stock market indices wig20 and s&p100. the wig20 is a stock market index comprising 20 biggest and most liquid companies on the warsaw stock exchange (wse)2. the considered time series x consists of 946 daily logarithmic rates of return on the wig20 index closing quotations from june 5, 2007 to march 11, 20113. the s&p100 index includes 100 leading us stocks recorded by standard & poor’s. the considered data x contains 1,077 daily log-returns on the index over a period from april 2, 2007 to july 8, 20114. daily closing quotations of the wig20 and s&p100 indices are presented in figure 1, whereas figure 2 plots the logarithmic rates of return. 2 www.gpw.pl. 3 the data were downloaded from www.gpwinfostrefa.pl. 4 the data were downloaded from http://finance.yahoo.com/. bayesian pricing of the optimal-replication strategy for the european option… dynamic econometric models 12 (2012) 53–71 61 wig20 s&p100 figure 1. daily closing quotations of the wig20 and s&p100 indices the horizontal lines present bands of plus/minus two or three standard deviations (dashed and dotted lines, respectively). wig20 s&p100 figure 2. daily logarithmic rates of return on the wig20 and s&p100 closing quotations as evidenced in figure 2 the outlying log-returns on the s&p100 index are more prominent than the ones featured by the wig20 series, which may hint at the jump component playing a more crucial role in modeling the former. 4.1. wig20 it is assumed that time interval between consecutive observations equals =∆ 1/252. for the wig20 series we restrict the analysis to two model specifications: jd(0)j (i.e. a pure diffusion process) and jd(1)j (i.e. the one allowing for a single jump over a given time interval ∆). maciej kostrzewski dynamic econometric models 12 (2012) 53–71 62 4.1.1. general results table 1 presents posterior means and standard deviations (in parentheses) of the parameters. the results are based on 600,000 and 1,000,000 draws of posterior distributions, preceded by 10,000 and 300,000 burn-in passes for m=0 and m=1, respectively. the results of the mcmc sampler are robust to the choice of the starting points. convergence of the chains is confirmed by the cumsum statistics (yu, mykland, 1998), as well as the ergodic means and standard deviations plots. noteworthy, posterior characteristics of the pure diffusion parameters, i.e. µ and 2σ , are close to their jd(1)j counterparts, which may hint at there being no need for jumps to be accounted for. the conclusion is also supported by the close to zero posterior mean of the jump intensity λ, accompanied with relatively large posterior dispersion of the parameters table 1. posterior means and standard deviations (in parentheses) of the parameters for the wig20 index parameters jd(0)j jd(1)j λ – 0.0557 (0.3053) µ -0.0364 (0.1556) -0.0346 (0.1545) µq – 0.0085 (0.9770) σ2 0.0917 (0.0042) 0.0919 (0.0044) σ2q – 0.3520 (1.3100) we now focus on the choice of the appropriate value of m. the model with the highest posterior probability is referred to as the best one. the best model points the value of m. we have to compare: ( ) ( ) ( ) ( ) ( ) ( ) ( )jjdxpjjdpjjdxpjjdp jjdxpjjdp xjjdp )1()1()0()0( )0()0( )0( + = and ( ) ( ).)0(1)1( xjjdpxjjdp −= the newton-raftery estimators (newton, raftery, 1994; raftery, newton, satagopan, krivitsky, 2007) are employed to assess the posterior probabilities ( )jjdxp )0( and ( )jjdxp )1( . these estimators are consistent, but their asymptotic variances do not exist. in practice, the values of the estimator may not stable. a longer monte carlo chain (of 1,000,000 draws) was generated to increase the credibility of the estimator. bayesian pricing of the optimal-replication strategy for the european option… dynamic econometric models 12 (2012) 53–71 63 under equal prior probabilities of each model, i.e. ( ) ( )jjdpjjdp )1()0( = , we obtain ( ) ( ).|)1(|)0( xjjdpxjjdp ≈ however, invoking occam’s razor that promotes parsimony (and thereby models with lower number of parameters) we set ( ) 22)0( −∝jjdp and ( ) 52)1( −∝jjdp , which results in ( ) 9.0|)0( ≈xjjdp . the jd(0)j model is more likely a posteriori than the jd(1)j specification. in other words, jumps are non-essential in modeling dynamics of daily (closing) quotations of the wig20 index5. 4.1.2. calculating the optimal-replication strategy cost under market completeness of the black-scholes model replication strategies do exist. continuous trading opportunity is one of the model’s underlying assumptions. however, in practice this assumption is quite unrealistic. if we limit trading opportunities to discrete times we get an incomplete model (bertsimas, kogan, lo, 2001). in the jd(0)j framework and under the assumption of the fixed time δ between consecutive trading times, the replication strategy may not exist. further, we calculate the costs and errors of the optimal strategies for some european options. let us consider two european call options. the date of pricing the optimal strategy is march 14, 2011, and the maturity date t is march 18, 2011. strike prices are equal 2700=k and 2800=k . the closing quotation value of the wig20 index on march 14, 2011 equals 2757.76. the first option is in the money and the second one is out of the money. the riskless interest rate is arbitrarily set at 0362.0=r (r equals an arithmetic mean of wibid on and wibor on on march 14, 2011). the theory of optimal-replication strategy was originally presented under the restriction of 0=r , but, fortunately, it could be generalized so as to incorporate any constant riskless rate 0>r . the pillar of the extension is normalization of all prices with the price of a zero-coupon bond (bertsimas, kogan, lo, 2001). figures 3 and 4 display posterior distributions of the optimal-replication strategy cost *0v and the relative measure of the market incompleteness ∗ε for each strike price. the histograms are calculated on the basis of 1,000 states of markov chains. the maturity t is specified as 252/4 , which may appear a short period of time, but is long enough to judge the convergence of the optimal-replication strategy to the replication strategy (the strategy exists in black-scholes framework). for the time being let us assume that the unknown parameters equal the assessed posterior means, i.e. 03644597.0−=µ and 09171522.02 =σ . let l denote the number of times the portfolio changes over the duration of the 5 the barndorff-nielsen and shephard’s nonparametric test also rejects jumps in the considered time series (barndorff-nielsen, shephard, 2006). maciej kostrzewski dynamic econometric models 12 (2012) 53–71 64 v0 * for k=2700 75 76 77 78 79 80 81 0. 0 0. 1 0. 2 0. 3 0. 4 ε* for k=2700 figure 3. histograms of the posterior distributions of *0v and ∗ε for 2700=k v0 * for k=2800 23 24 25 26 27 28 0. 0 0. 1 0. 2 0. 3 0. 4 ε* for k=2800 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 0. 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 figure 4. histograms of the posterior distributions of *0v and ∗ε for 2800=k options (a strategy is a sequence of portfolios). tables 2 and 3 present the cost of the optimal-replication strategy *0v and the relative measure of the market incompleteness ∗ε for each strike price k. the prices of options calculated under the black-scholes assumptions are presented in the last row of table 2. recall that the market completeness of the black-scholes model warrants a zero replication error, i.e. 0=∗ε . we note that as the number l of times the portfolio changes over the option duration increases the optimal-replication strategy cost converges to the blackscholes price. the relation is accompanied by a systematic decrease in the replication error (see table 3), indicating that the market is “nearing” completeness. the prices of the options on march 14, 2011 equaled 52 and 5, for strike prices k=2700 and k=2800, respectively. posterior histograms and expected values of *0v suggest that hedging of the options by the optimal-replication is 13 14 15 16 17 18 0.0 0.2 0.4 0.6 0.8 bayesian pricing of the optimal-replication strategy for the european option… dynamic econometric models 12 (2012) 53–71 65 table 2. values of *0v calculated under 03644597.0−=µ and 09171522.0 2 =σ for increasing values of l, along with the black-scholes (bs) prices l * 0v k=2700 k=2800 1 77.7705 26.1447 4 77.7768 24.7248 10 77.7477 25.0945 30 77.7488 25.0621 bs 77.7427 25.0372 table 3. values of ∗ε calculated under 03644597.0−=µ and 09171522.02 =σ for increasing values of l l ∗ε k=2700 k=2800 1 27.8907 28.6173 4 15.1117 16.6562 10 9.8949 10.5713 30 5.8611 6.2656 100 3.2616 3.4959 200 2.3230 2.4952 500 1.4778 1.5844 bs 0 0 strategy expensive in comparison with the prices of the options. note that the above results depend on estimation of the model’s parameters and the choice of the observation set. if the estimation is based on a shorter series, avoiding the period of time with more volatile changes of the index, the estimation and pricing results are affected. we additionally consider a dataset from may 5, 2010 to march 11, 2011. then the values of the relative measure of market incompleteness ∗ε are smaller than in the case of the full sample, and so is the posterior mean of the volatility parameter σ , with its value declining from 0.03 in the case of the full sample model to 0.02 for the trimmed series. figure 5 presents the costs of the optimal-replication strategy *0v for a strike price k=2700 and two sets of observations. however, the cost of the new strategy is still high (or the price of the option is low). 4.2. s&p100 let us consider the s&p100 index and three model specifications: jd(0)j, jd(1)j and jd(10)j. it is assumed that the time interval between consecutive observations equals 252/1=∆ . maciej kostrzewski dynamic econometric models 12 (2012) 53–71 66 v0 * (05-jun-2007 to 11-mar-2011) 75 76 77 78 79 80 81 0.0 0.1 0.2 0.3 0.4 v0 * (05-may-2010 to 11-mar-2011) 65 66 67 68 69 0.0 0.1 0.2 0.3 0.4 0.5 figure 5. costs of the optimal-replication strategy *0v for the strike price k=2700 and two sets of observations: 05-jun-2007 to 11-mar-2011 and 05-may-2010 to 11-mar-2011 4.2.1. general results table 4 contains results of bayesian estimation posterior means and standard deviations (in parentheses). the outcomes are based on 1,000,000 draws of the markov chain and 25,000 burn-in passes. the results of the mcmc sampler are robust to the choice of the starting points. convergence of the chains is confirmed by the cumsum statistics (yu, mykland, 1998), as well as the ergodic means and standard deviations plots. posterior means of the pure diffusion parameters µ and 2σ calculated in the jd(1)j and jd(10)j models – though almost identical across the two specifications – differ quite substantially from their counterparts in the jd(0)j model (i.e. the one that precludes any jumps). particularly, note that e( 2σ |y, jd(0)j) = 0.0677 as opposed to e( 2σ |y, jd(m)j) = 0.047 for m=1 and m=10. the difference is justified by the jump component “absorbing” some of the log-returns’ volatility, whereas exclusion of jumps in the jd(0)j specification is compensated with a higher value of the volatility parameter’s posterior mean. noteworthy, the posterior results for the jd(m)j specifications featuring m>0 are very close, which may be indicative of there being no empirical need for allowing for more than a single jump per ∆. particularly, posterior means of the jump intensity parameter λ in both the jd(1)j and the jd(10)j model consistently imply that on average there are 4 jumps per year. bayesian pricing of the optimal-replication strategy for the european option… dynamic econometric models 12 (2012) 53–71 67 table 4. posterior means and standard deviations (in parentheses) of the jd(m)j models’ parameters for the s&p100 index parameters jd(0)j jd(1)j jd(10)j λ – 4.4234 (1.3363) 4.3507 (1.3030) µ 0.0145 (0.1256) 0.0426 (0.1089) 0.0430 (0.1085) µq – -0.0090 (0.1850) -0.0087 (0.1845) σ2 0.0677 (0.0029) 0.0470 (0.0028) 0.0470 (0.0028) σ2q – 0.5440 (1.1183) 0.5451 (1.3625) turning to the formal pair-wise model comparison, we calculate decimal logarithms of bayes factors (bernardo, smith, 2002): ( ) ( ) ,17 |jjd(0) |jjd(1) log)(log 100,110 ≈      = xp xp b ( ) ( ) ( ) .7.1 |jjd(10) |jjd(1) loglog 1010,110 ≈      = xp xp b it appears that the jd(1)j specification beats the competition, being as much as ca. 17 orders of magnitude more likely a posteriori than the simplest model structure (jd(0)j)6. although only marginally, the former is also favored against the other jump-diffusion specification, i.e. jd(10)j, which seems to be penalized for its excessively large number m=10 of jumps allowed per ∆. admittedly, the result fits in well with the overall pursuit for parsimony. 4.2.2. calculating the optimal-replication strategy cost we confine our further considerations to the jd(1)j model. it is known that markets are incomplete when sources of randomness outnumber the underlying traded risky instruments (björk, 2004). in the jd(1)j specification there are three sources of randomness – the wiener process w, the poisson process n, and random variables q . in our setting we consider a market with only one risky underlying instrument (a stock market index) accompanied by as much as three sources of randomness, so the market is incomplete. therefore, we resort to the optimal-replication strategies for selected options. let us consider two european call options. a date of pricing of the optimal strategy is july 11, 2011, and the maturity date t of the options is july 15, 2011. strike prices equal 590=k and 610=k . the closing quotation of the 6 the barndorff-nielsen and shephard’s nonparametric test reject the pure diffusion at significance level 0.05 (p-value equals 0.011). maciej kostrzewski dynamic econometric models 12 (2012) 53–71 68 s&p100 index on july 11, 2011 equals 588.15. both of the options are out of the money. the riskless interest rate is set at 0075.0=r and it equals the fed funds discount rate at the considered option duration. table 5 presents posterior means and standard deviations (in parentheses) of the optimal-replication strategy (initial) cost *0v and the relative measure of the market incompleteness ∗ε for each strike price. on the day of the pricing, according to our knowledge, there were no transactions of selling the considered options. table 5. posterior means (and standard deviations) of *0v and ∗ε calculated for the s&p100 index as an underlying instrument quantity k=590 k=610 * 0v 12.12 (2.8927) 6.619 (3.0739) *ε 35.68 (11.6726) 35.47 (12.8357) table 6 contains quantiles of the posterior distributions of *0v and ∗ε . in general, the call option with a lower strike price is more expensive than the option with a higher strike price. note that the cost of the optimal-replication strategy *0v is higher for the more attractive option. table 6. quantiles of the posterior distributions of *0v and ∗ε calculated for the s&p100 index as an underlying instrument orders of the quantiles k=590 k=610 * 0v *ε *0v *ε 5% 8.4071 20.2538 2.2598 16.2821 25% 9.9695 27.0364 4.6371 26.7550 50% 11.4782 32.7958 6.2447 33.4317 75% 13.5252 42.2325 8.1295 43.2222 95% 17.8427 57.2116 12.6826 58.7495 figures 6 and 7 display histograms of the posterior distributions of the optimal-replication strategy cost *0v and the relative measure of the market incompleteness ∗ε . these histograms are based on (only) 150 (randomly chosen) states of the markov chains. the reason behind such a small sample is timeconsuming calculations of the optimal-replication strategy for each parameter vector. these calculations took about twenty hours on a standard pc. the application of parallel calculations reduced that time to seven hours. a fairly large dispersion of the posterior distributions of *0v and ∗ε may stem from a relatively large parameter uncertainty (as evidenced by the posterior standard deviations). bayesian pricing of the optimal-replication strategy for the european option… dynamic econometric models 12 (2012) 53–71 69 v0 * for k=590 5 10 15 20 25 0 .0 0 0 .0 5 0 .1 0 0 .1 5 ∗ε for k=590 10 20 30 40 50 60 70 80 0 .0 0 0 0 .0 0 5 0 .0 1 0 0 .0 1 5 0 .0 2 0 0 .0 2 5 0 .0 3 0 figure 6. histograms of the posterior distributions of *0v and ∗ε for 590=k v0 * for k=610 0 5 10 15 0 .0 0 0 .0 5 0 .1 0 0 .1 5 ∗ε for k=610 0 20 40 60 80 0 .0 0 0 0 .0 0 5 0 .0 1 0 0 .0 1 5 0 .0 2 0 0 .0 2 5 0 .0 3 0 figure 7. histograms of the posterior distributions of *0v and ∗ε for 610=k 5. conclusions this paper concerns the issue of option hedging in incomplete market models using stochastic dynamic programming and bayesian statistics. familiar models of option pricing are complete. unfortunately, the assumptions these structures usually rest upon are quite unrealistic. for instance, the black-scholes model is hinged upon continuous trading and continuous paths of a risky underlying instrument. relaxing these assumptions leads to incomplete market models, such as the jd(m)j structures considered in the present study. it is shown that incorporation of jumps in modeling financial time series may improve the model fit (as compared with a pure diffusion process). unformaciej kostrzewski dynamic econometric models 12 (2012) 53–71 70 tunately, the market incompleteness in the models featuring jumps (jd(m)j with m>0) renders the task of pricing and hedging derivatives more demanding. in general, as the replication strategy does not exist, the investor needs to resort to some optimal strategy. in the study we succeeded in employing the optimalreplication strategy algorithm, derived by bertsimas, kogan and lo (2001), in the jd(m)j framework. contrary to what seems a common practice in the financial mathematics works, where the model’s parameters are set arbitrarily, we estimate the parameters using bayesian methodology, taking advantage of its accounting for the parameters uncertainty. moreover, the results are further employed to infer upon the degree of market incompleteness as well as to price the optimalreplication strategy. specifically, posterior densities (rather than point values solely) of both the optimal strategy costs along and its relative error are calculated (using the mcmc techniques), providing us with some insight into their uncertainty. acknowledgements useful comments and remarks by two anonymous referees are highly appreciated. the author would also like to thank łukasz kwiatkowski for language verification of the manuscript. references barndorff-nielsen, o.e., shephard, n. (2006), econometrics of testing for jumps in financial economics using bipower variation, journal of financial econometrics, 4, 1, 1–30. bernardo, j. m., smith, a. f. m. (2002), bayesian theory, wiley series in probability and statistics. bertsekas, d. (1995), dynamic programming and optimal control, vol. i, athena scientific, belmont, ma. bertsimas, d., kogan l., lo, a.w. (2001), hedging derivative securities and incomplete markets: an ε-arbitrage approach, operations research, 49, 3, 372–397. björk, t. (2004), arbitrage theory in continuous time, oxford university press. black, f., scholes, m. (1973), the pricing of options and corporate liabilities, journal of political economy, 81, 637–654. frühwirth-schnatter, s. (2006), finite mixture and markov switching models, springer science + business media, lcc. gamerman, d., lopes, h.f. (2006), markov chain monte carlo. stochastic simulation for bayesian inference, chapman & hall/crc. hanson, f.b., westman, j.j. (2002), stochastic analysis of jump-diffusions for financial logreturn processes, stochastic theory and control, 280/2002, 169–183. kloeden, p.e., platen, e. (1992), numerical solution of stochastic differential equations, springer-verlag, berlin-heidelberg-new york. kostrzewski, m. (2011), bayesian inference for the jump-diffusion model with m jumps, working paper: http://home.agh.edu.pl/ kostrzew/bayesianinferencejdmj.pdf. lamberton, d., lapeyre, b. (2000), introduction to stochastic calculus applied to finance, chapman & hall/crc. bayesian pricing of the optimal-replication strategy for the european option… dynamic econometric models 12 (2012) 53–71 71 lin, s-j., huang, m-t. (2002), estimating jump-diffusion models using the mcmc simulation, national tsing hua university department of economics nthu working paper series, working paper 0215e, october 2002. merton, r.c. (1976), option pricing when underlying stock return rates are discontinuous, journal of financial economics, 3, 141–183. newton, m.a., raftery, a.e. (1994), approximate bayesian inference by the weighted likelihood bootstrap (with discussion), journal of the royal statistical society b, 56 (1), 3–48. raftery, a. e., newton, m. a., satagopan, j. m., krivitsky, p. n. (2007), estimating the integrated likelihood via posterior simulation using the harmonic mean identity, bayesian statistics, 8, 1–45. schweizer, m. (1992), variance-optimal hedging in discrete time, mathematics of operations research, 20, 1–31. shreve, s.e. (2004), stochastic calculus for finance ii. continuous-time models, springer science+business media, inc. yu, b., mykland, p. (1998), looking at markov samplers through cumsum path plots: a simple diagnostic idea, statistics and computing, 8, 275–286. bayesowska wycena kosztu optymalnej strategii replikującej europejską opcję w modelu jd(m)j z a r y s t r e ś c i. wycena opcji w modelu niezupełnym jest nietrywialnym zagadnieniem. przykładem modelu niezupełnego jest wprowadzony przez mertona model dyfuzji ze skokami. gęstość logarytmu procesu dyfuzji ze skokami jest nieskończoną mieszanką rozkładów normalnych. w badaniu przyjęto, że liczba mieszanek jest skończona. otrzymany w ten sposób model nazwano modelem jd(m)j. w praktyce parametry modelu są nieznane i wymagają estymacji. w badaniu zastosowano wnioskowanie bayesowskie. jd(m)j jest modelem niezupełnym dla którego, w ogólnym przypadku, nie można wskazać strategii replikujących instrumenty pochodne. w badaniu zaprezentowano algorytm wyznaczający optymalną w sensie średniokwadratowym strategię replikującą europejski instrument pochodny. do zilustrowania omówionej teorii wykorzystano indeksy wig20 i s&p100. przedstawiona metodologia jest użyteczna dla inwestorów, którzy chcą uwzględnić w wycenie instrumentów pochodnych oraz analizach szacowania ryzyka ,,niepewność” estymacji parametrów modelu. s ł o w a k l u c z o w e: rynki niezupełne, wnioskowanie bayesowskie, procesy dyfuzji ze skokami, wycena instrumentów pochodnych. introduction 1. the jump-diffusion model with m-jumps 2. bayesian inference for the jd(m)j model 3. the optimal-replication strategy 4. empirical studies 5. conclusions references dynamic econometric models © 2012 nicolaus copernicus university press. all rights reserved. http://www.dem.umk.pl/dem dynamic econometric models vol. 12 (2012) 35−52 submitted april 30, 2012 issn accepted october 8, 2012 1234-3862 katarzyna bień-barkowska* “does it take volume to move the eur/pln fx rates?” evidence from quantile regressions a b s t r a c t this study investigates the impact of trading volume on selected quantiles of the eur/pln return distribution. empirical results obtained with the quantile regression approach confirm that an increase in the turnover is associated with a significant increase in the dispersion of the corresponding return distribution. we divided the trading volume into its expected (anticipated) and unexpected (unanticipated) component and found that the unexpected volume shocks have a significantly larger impact on the dispersion of the return distribution. we also observed that the volume-return relationship is nonlinear; the dependence is stronger with more extreme quantiles. moreover, after accounting for a conditional volatility measure as a controlling explanatory factor for the quantile dynamics, the impact of the expected volume declines yet remains significant especially for the most extreme quantiles. k e y w o r d s: volume-return relationship, market microstructure, fx trading, quantile regression. j e l classification: c22, g15 introduction research on price – volume relationship has a long history in the literature of both theoretical and empirical finance. positive contemporaneous correlation between trading volume and price volatility is already a well-documented observation with early studies on the topic traced back to the seventies. in this decade, the positive relationship between the selected measures of price variability and trading volume has been demonstrated for stock markets in various publications (crouch, 1970; epps and epps, 1976; morgan, 1976; westerfield, 1977, among others). karpoff (1987) presents a vast survey of the early litera * correspondence to: 1. katarzyna bień-barkowska, institute of econometrics, warsaw school of economics, ul. madalińskiego 6/8, 02-513 warsaw, poland or 2. katarzyna bieńbarkowska, financial system department, national bank of poland, ul. świętokrzyska 11/21, 00-919, warsaw, poland, e-mail: katarzyna.bien@sgh.waw.pl katarzyna bień-barkowska dynamic econometric models 12 (2012) 35–52 36 ture summarizing the results of 19 empirical studies conducted throughout the seventies and eighties focusing on the volume-return relationship evidenced by daily and intraday data from the stock, bond and commodity future markets. further evidence for this positive relationship can be found in numerous studies from the nineties and more recently (lamoureoux and lastrapes, 1990; gallant et al., 1992; jones et al., 1994; bohl and henke, 2003; luckley, 2005; doman, 2008; doman, 2011; and others). bessembinder and seguin (1993) suggest splitting the trading volume into its anticipated and unanticipated component. given the well documented fact, that the volume is highly autocorrelated, it is also forecastable, hence the authors differentiate its expected and unexpected part and evidence that the unexpected volume shock has between two and thirteen times greater effect on the volatility of stock prices. different informative meanings of the expected and the unexpected volume have also been found in other studies (brown-hruska and kuserk, 1995; gurgul et al., 2005; huang et al., 2006). there are at least three strands of the literature on market microstructure that could explain the relationship between trading volume and return variability. the first one is known as the sequential information arrival hypothesis (siah) (c.f. copeland, 1976; jennings et al., 1981). according to this theory, all traders cannot simultaneously absorb the arrival of new market information. therefore, a revision of their expectations occurs sequentially and the process in which new information is impounded into the price can spread out over time. only after all traders are able to react and trade, the final equilibrium price is set; this explains the lead-lag relationship between volatility and trading volume. accordingly, intensive trading and a high trading volume can help to identify periods where prices continue to adjust to informational shocks. the second explanation of the positive contemporaneous correlation between return and volume arises from the idea of theoretical information models. in such models traders are allowed to trade different sizes; better-informed traders initiate larger transactions and their activity has an adverse selection effect on the price (c.f., easley, o’hara, 1987). in another model, the informational content of trading intensity has been outlined. a long duration between consecutive trades indicates that there was no new information, whereas a short duration increases the probability that better-informed traders have increased the overall trading intensity. hence, market makers change quoted price by increasing the bid-ask spread as a weapon against an adverse selection risk reflected by an increased number of trades (easley and o’hara, 1992). other information models that predict a positive relationship between trading volume and price volatility are developed in the studies of admati and pfleiderer (1988), blume et al. (1994), easley, kiefer and o’hara (1997), and malinova and park (2011). the third explanation for the significant positive volume-return relationship arises from the mixture of distributions hypothesis (mdh) and has more of a statistical than an economic background (clark, 1973; epps and epps, 1976; “does it take volume to move the eur/pln fx rates?”… dynamic econometric models 12 (2012) 35–52 37 tauchen and pitts, 1983). according to the mdh, a bivariate distribution of volume and price change variables is conditional upon an information variable such that both variables react simultaneously to the arrival of news and are driven by this unobservable factor. the aim of this paper is to shed light on the intraday relationship between return and volatility in the eur/pln currency pair on the interbank spot market. we will use trade data from the reuters dealing 3000 spot matching system, a very popular brokerage trading platform that can automatically match all incoming buy and sell orders once their prices agree. inspired by the study of chuang et al. (2009), we describe the returnvolume relationship with the help of quantile regressions (qr). such an approach allows us to generalize the results of many empirical studies concerning the relationship between trading volume and the measure of price variability. the majority of empirical studies demonstrate that it is common to introduce volume as an additional explanatory factor into the garch specification for the conditional variance of the return distribution (lamoureux and lastrapes, 1990; bohl and henke, 2003; gurgul et al., 2005; majand and yung, 2006; and others). such a modeling strategy measures only the impact of the trading volume on the second central moment of the conditional return distribution. the qr approach is much different in that it is semiparametric and allows for an analysis of the impact of some explanatory factors on the selected quantiles of the return distribution without making any assumptions about their parametric form (i.e., gaussian, student’s t, generalized gamma) or about the parametric specification of its conditional mean and variance. models that are members of the garch family make an usual implicit assumption about a type of a parametric location-scale distribution for financial returns where the first two moments (i.e., the conditional expectation and the conditional variance) are described in a dynamic fashion. the qr approach does not impose such parametric assumptions but instead concentrates on the quantiles. accordingly, the qr approach has an obvious upper hand over the standard garch approach: the impact of the explanatory variables can be different for different quantiles. an impact of trading volume can be different for the τ-quantile than it is for the (1-τ)-quantile (where τ denotes a corresponding probability level). hence, we are able to infer whether the arrival of new information has a different impact on the probability of an extreme fx rate increase versus an extreme fx rate decrease. such a situation could be explained by the possibly of a time-varying skewness and/or kurtosis of the return distribution that could also depend on a latent information arrival variable. within this study we decompose trading volume into its expected (forcastable) and unexpected (unpredictable) parts and compare their impact on the selected quantiles of the eur/pln return distribution. moreover, we check to see if the trading volume captures extra information behind the quantile dynamics katarzyna bień-barkowska dynamic econometric models 12 (2012) 35–52 38 when confronted with the standard garch volatility forecasts as an intuitive and natural explanatory factor. 1. the econometric approach 1.1. trading volume decomposition in order to investigate the volume – return relationship one must typically distinguish between the so-called expected (anticipated) and unexpected (unanticipated) trading volume (andersen, 1996; bjonnes et al., 2003). bearing in mind that the volume variable is highly autocorrelated, the expected volume is the result of more persistent fluctuations in liquidity needs whereas unexpected volume should be unpredictable by traders and should approximate a new information arrival. differentiation between what is expected and unexpected volume is typically performed using arima models (c.f., bessembinder and seguin, 1993; hartmann, 1999; bjonnes et al., 2003). however, such models usually require logarithmic transformation of the volume variable in order to avoid its nonnegativity and/or in order to diminish heteroskedasticity, especially for high frequency data. such a transformation may distort the potential relationship between the variables and so we propose a different procedure here. in order to preserve an original time series (unchanged due to a logarithmic transformation), we apply the autoregressive conditional duration (acd) models of engle and russell (1998). the acd were initially used to describe a highly autocorrelated time series of durations (time spells) between selected events (i.e., transactions or price changes). more recently these models were used to describe other financial variables including transaction volume in the studies of manganelli (2005) and doman (2008, 2011) or the bid-ask spread in nolte (2008). the acd models are well designed for serially correlated variables with a strictly positive domain. here we have used the acd (1,1) model with the burr distribution for the error term proposed by grammig and maurer (2000). the model for the trading volume variable tvol can be written as follows: ,= tttvol εψ (1) where )|(= 1−ψ ttt fvole , 1−tf denotes an information set up at the time point 1−t (containing all past realizations of tvol ), tε denotes an error term, and { } ...~ diitε ),( 2σκburr such that 1)( =te ε . the conditional expectation of the dependent variable tvol is described as follows: ,= 12110 −− +ψ+ψ ttt volβββ (2) “does it take volume to move the eur/pln fx rates?”… dynamic econometric models 12 (2012) 35–52 39 this model can be estimated using the ml method. the log likelihood function has the following form: ( ,)(1ln1 1 ln1)(lnln=)( 2 2 1=    ⋅⋅+⋅      +− ⋅−+⋅−θ − ∑ κκξσ σ κξκκ tt tt n t vol vollogl (3) where       −γ      +γ       +γ⋅ ψ       + κσκ σ σ ξ κ 111 1 1 1 )( = 2 2 1 1 2 tt , κσ <<0 2 and )(⋅γ denotes the gamma function. accordingly, the expected volume tvolexp, is defined as an estimate of the conditional expectation tψ̂ (i.e., it is conditional upon all past observations of the volume variable) whereas an unexpected volume is defined as the residual tttun volvol ψ= ˆexp, . 1.2. quantile regressions for the eur/pln returns taking the trading volume as an explanatory variable in the qr setup we are able to check its impact on the dispersion of the return distribution in a very explicit manner. in the qr setting we can “jointly” capture the impact that the trading volume exerts on the general shape (i.e., skewness, kurtosis or variance). as mentioned previously, most popular financial econometric models typically neglect the possible effect of explanatory variables on the skewness or kurtosis of the distribution. furthermore, the popular garch models rule out a potential asymmetric impact on the trading volume on the tail probability of a large price upswing versus a large price falls. in order to check the impact of the trading volume on the selected quantiles of the return distribution we used simple linear qr models where each of them corresponds to a selected conditional quantile )|( trtq xτ : , ),(=)|( 1exp,,3exp,,2exp,,1 1,1,0 − − +++ ++ tuntunt ttr volvolvol srq t τττ τττ γγγ υαατ νx (4) where tr denotes the logarithmic rate of return on the eur/pln exchange rate between the moments t and 1−t ,τ denotes a corresponding probability level and ),( υτνs depicts the intraday seasonality factor given as the fast fourier form (fff): katarzyna bień-barkowska dynamic econometric models 12 (2012) 35–52 40 [ ]][2cos][2sin=),( ,2,12 2 1= ,0 υπνυπνυνυ ττττ lls ll l ⋅+⋅+⋅ −∑ν , (5) where υ denotes a time-of-day variable standardised on the interval ].1,0[ [ ,,),4cos(),4sin(),2cos(),2sin(,,,1 exp,exp,1 tunttt volvolτr πτπτπτπτ−=x ]ttunvol 1exp, − is a vector of selected explanatory variables. the fff diurnality component that we apply has been advocated by andersen and bollerslev (1996). the diurnality function can therefore be depicted as a number of sine and cosine functions and should smoothly depict the systematic intraday seasonality pattern in the dispersion of the return distribution. this standard methodology assumes a two-step procedure: the intraday returns are first deseasonalized (divided by the obtained seasonality pattern) and then garch models are estimated for the filtered returns. we allow for an additive seasonality pattern for each of the estimated quantile regressions, which allow us to capture systematic intraday regularities in the unrestricted shape of the conditional distribution. as previously mentioned, this is done in a semiparametric setup and is much more general such that we can capture different diurnality patterns for different quantile levels. in each of qr regressions given by equation (4) we have also introduced a lagged tr in order to account for possible autocorrelation. the parameter corresponding to the autoregressive term can be different for different quantiles, as evidenced by baur et al. (2011). the suggested model for the conditional quantile of the return distribution can be criticized as being too parsimonious. given that it contains only the past return, intraday seasonality component and volume variables as the major driving forces1, its ability to account for the volatility clustering effect may be rather limited. thus, in the second model that we propose we have included a return volatility forecast as an additional driving factor for the quantile dynamics. chuang et al. (2009) proposed the addition of a lagged value of 2tr as a natural proxy for the volatility. an alternative to this is to account for possible persistency in the conditional quantiles with the application of the conditional autoregressive value-at-risk (caviar) models of engle and manganelli (2004). within this framework quantile dynamics are captured in the form of autoregressive specifications with an absolute value for the past return as newinformation variable. however, the application of an autoregressive specification rules out the applicability of standard linear programming algorithms to estimate qr. accordingly, the estimation process based on the genetic algorithm or the nelder-mead simplex algorithm and quasi-newton method (with the necessity of computing loops for recursive quantile functions) is quite timeconsuming. taking into account that we plan to estimate several quantile re 1 the lag structure for the volume variables has been selected on the grounds of their statistical significance. “does it take volume to move the eur/pln fx rates?”… dynamic econometric models 12 (2012) 35–52 41 gression models for several probability levels, this would likely be inefficient. therefore, we decided to approximate the volatility variable with the conditional standard deviation estimate 1ˆ −tσ , obtained from the garch(1,1) with a student’s t distribution for an error term2 model. as such, a volatility measure3 uses whole information from the history of the return process at 1−t . it is also intended to describe the persistency of the quantile dynamics in a more adequate manner than 21−tr : . ),(ˆ=)|( 1exp,,3exp,,2exp,,1 1,21,1,0 − −− +++ +++ tuntunt tttr volvolvol srq t τττ ττττ γγγ τνσααατ x (6) for a given probability τ , qr estimates can be obtained as a minimum of the following objective function of asymmetrically weighted absolute deviations: ( )∑ = < −−= n t t ttxr rt tt 1 }{ minargˆ ττ τ γx1γ γγ , (7) where [ ]tννννν 3214321010 ,,,,,,,,, γγγαατ =γ denotes a parameter vector and [ ,,),4cos(),4sin(),2cos(),2sin(,,ˆ,,1 exp,exp,1 tuntttt volvolτr πτπτπτπτσ−=x ]ttunvol 1exp, − is a vector of the corresponding explanatory variables. nondifferentiable objective function (7) can be minimized using the linear programming methods described in koenker (2005, p. 170-202). the limiting covariance matrix of ( )ττ γγ −ˆn takes the form of the huber sandwich (huber, 1967): ( ) ( )11)1(,0ˆ −−−→− ttt hjhnt ττττ γγ , (8) where 2 the volatility estimate 1ˆ −tσ refers to deseasonalized returns: trtt srr ,= , thus it accounts for the volatility left on top of its cyclical behavior. in our empirical application an intraday seasonality factor trs , has been estimated with the kernel regression of absolute returns on a time-ofday variable. we use a quartic kernel with bandwidth computed as 2.78sn -1/5, where s is the standard deviation of the data. for details regarding the estimation procedure please refer to (bauwens and veredas, 2004). an alternative would be to estimate trs , by means of cubic splines as suggested by giot (2005) or by means of the fast fourier form as suggested by anderson and bollerslev (1997). 3 introduction of the volatility estimate into the qr model for financial returns has been also adapted in taylor (1999). parsimonious garch(1,1) specification succeeded to depict the volatility clustering in a satisfactory way. katarzyna bień-barkowska dynamic econometric models 12 (2012) 35–52 42 t t t t tt tj xx∑ = −= 1 1 and ( ))(lim 1 1 τqfth t t t t t t t t xx∑ = − ∞→ = (koenker, 2005, p. 74). the term ( ))(τqft denotes the conditional density of tr evaluated at the τth conditional quantile, )(τq . in order to estimate the matrix th , the term ( ))(τqft must be evaluated. it is typically replaced by its consistent estimate ( ))(ˆ τqft obtained with the help of nonparametric methods. koenker (2005, p. 80) shows a method to estimate the density function evaluated at a given quantile )(τq with the help of the sparsity estimation methods proposed by hendricks and koenker (1991), (i.e., as a difference quotient): ( ) )ˆˆ2)(ˆ tt hh t ttt hqf −+ −= τττ γγ(x , (9) where th denotes a bandwidth ( 0lim = ∞→ t t h ); selecting the proper bandwidth is discussed by koenker (2005, p. 139-140). another possibility for the covariance matrix estimation, known as a powell sandwich, would be to estimate th via kernel estimation: t ttt t tttt hrknhh xxγx )()(ˆ 1 −= ∑− , (10) where )(⋅k denotes a proper kernel function (e.g., powell kernel) (c.f., koenker, 2005, p. 80). 2. empirical results 2.1. data an empirical study of the volume-return relationship has been performed with trade data from the reuters dealing 3000 spot matching system. this is a liquid electronic brokerage system that operates as an order-driven market. it can be estimated that trading with the reuters dealing 3000 spot matching system accounts for about 60 % of all interbank spot transactions in the polish zloty market in 20084. the data utilized is comprised of transactions conducted between januaryjune 2008 with respect to the eur/pln currency pair. the eur/pln exchange rate is quoted as a quantity of zlotys per one euro. during the period of study the zloty followed an appreciating trend with respect to the euro. each transaction is marked with the date, the exact time, the rate and the quantity (in mil 4 trading of the polish zloty takes place on offshore markets (mainly between london banks) as well as locally in poland. the datasets used in this analysis take into account both of these trading venues. “does it take volume to move the eur/pln fx rates?”… dynamic econometric models 12 (2012) 35–52 43 lions) of eur. trading on the interbank market is heavily concentrated on business days between the hours of 8:00 and 18:00 central european time (cet). in order to limit the undesired impact of particularly thin trading periods we have excluded observations registered on weekends and on business days between the hours of 18:00 and 8:00 cet. we have also excluded days with exceptionally low liquidity due to national holidays. for data aggregated in a 15minute frequency, we define the following variables: (1) tvol is the trading volume (turnover) between the moments t and 1−t , expressed in m. eur, and (2) tr is the logarithmic rate of return on the eur/pln exchange rate defined as 4 1 10))()((= ⋅− − m t m tt plnplnr where m tp denotes the mid price. the data frequency is chosen as a compromise between the need for observing the intraday instantaneous fluctuation of selected market characteristics and the necessity of avoiding distorted results due to the effects of slow trading periods. because trading volume demonstrates strong intraday seasonality we have divided the volume variable by the corresponding seasonality component: tt svolvol = . as suggested by bauwens and veredas (2004), the intraday seasonality factor ts has been estimated using the kernel regression of tvol on a time-of-day variable5. estimation of the burr-acd models6 has been performed on a diurnally adjusted series7. with the obtained parameter estimates we defined the expected and the unexpected volume variables. as can be observed in figure 1, the expected trading volume reflects forcastable fluctuations in the trading turnover whereas the unexpected volume reflects unanticipated volume shocks. 2.2. modeling the volume-return relationship with the qr in order to obtain an intuitive picture of the relationship between distinct explanatory variables and eur/pln return distribution, in figure 2 we depict preliminary univariate quantile regressions. in the left upper corner of figure 2 we present a scatter plot of ( ttun rvol ,exp, ), as well as the conditional quantile estimates, tuntunr volvolq t exp,,1,0exp, ˆˆ=)|(ˆ ττ λλτ + , for some selected probability levels: { }99.0;95.0;9.0;75.0;5.0;25.0;1.0;05.0;01.0∈τ . the most striking observation is a strong positive relationship between the unexpected trading volume 5 we use a quartic kernel with bandwidth computed as 2.78sn -1/5, where s is the standard deviation of the data. 6 estimation has been performed in gauss 8.0 with the application of library maxlik. 7 the trading volume is stationary. we checked for the presence of deterministic as well as stochastic trends and rejected the null of unit root with the augmented dickey-fuller test (pvalue=0.0000). katarzyna bień-barkowska dynamic econometric models 12 (2012) 35–52 44 0 1000 2000 3000 4000 5000 0 5 10 15 observation number e xp ec te d vo lu m e 0 1000 2000 3000 4000 5000 0 5 10 15 observation number u ne xp ec te d vo lu m e figure 1. expected (left panel) and unexpected (right panel) volume and the dispersion of the return distribution. moreover, we can also observe that the obtained slope parameter estimates vary across quantiles; such an observation also been found by chuang et al. (2009). accordingly, the largest impact on the unexpected volume can be observed with the most extreme quantiles corresponding to the tails of the return distribution. if we turn our attention to the expected volume variable, however, the results seem to be different. although this variable is also positively linked to the dispersion of the eur/pln return distribution, the scale of the effect is much smaller. thus, the anticipated volume seems to have less of an impact on the probability of large price movements. in terms of the impact of the lagged return variable, the following tendency is found: after large (positive or negative) returns, the tail probability of observing further large (either positive or negative) movements increases. in order to confirm this effect we applied a nonparametric quantile estimation technique8. we can see that the dispersion of the return distribution rises in of the wake of large price movements (upswings or drops). to account for this effect it is reasonable to allow for a forecasted volatility estimate as a factor that is positively related to the dispersion of the distribution. we estimated the qr models given by equations (4) (model i) and (6) (model ii) using the “quantreg” library (version 4.79) written by roger koenker under the r9. the further inference has also been carried out with the help of these programming codes. qr regressions have been estimated for a dense grid of probability levels ( 99.0,...,02.0,01.0=τ ); thus, for each model we have 8 we applied a piecewise cubic polynomial with three knots (koenker, 2005). 9 the library can be downloaded from the cran website: http://cran.r-project.org. “does it take volume to move the eur/pln fx rates?”… dynamic econometric models 12 (2012) 35–52 45 0 5 10 15 -3 0 -2 0 -1 0 0 10 20 30 unexpected volume e u r /p l n re tu rn (b as is p oi nt s 1 2 3 4 5 6 -3 0 -2 0 -1 0 0 10 20 30 expected volume e u r /p l n re tu rn (b as is p oi nt s -30 -20 -10 0 10 20 30 -3 0 -2 0 -1 0 0 10 20 30 return, lag1 e u r /p l n re tu rn (b as is p oi nt s 4 6 8 10 -3 0 -2 0 -1 0 0 10 20 30 volatility e u r /p l n re tu rn (b as is p oi nt s figure 2. volume return relationship. estimates of quantile regressions for probabilities: { }99.0;95.0;9.0;75.0;5.0;25.0;1.0;05.0;01.0∈τ allowed for 99 different vectors of parameter estimates that have been plotted in figure 3 (model 1) and figure 4 (model 2). the asymptotic standard errors for each of these specifications have been derived with the help of sparsity estimation methods (see equation (4) and (5))10. these equations have been used for the evaluation of the 90% confidence interval for each of the obtained parameter estimates. as can be anticipated from figure 3, the seasonality factor of the return distribution is different across the quantiles (in the figures 3 and 4 the symbols s1, s2, s3, s4, s5 are defined as τs =1 , )2sin(2 πτ=s , )2cos(3 πτ=s , 10 we have also experimented with the powel kernel and the bootstrap, but it did not influence the further inference. katarzyna bień-barkowska dynamic econometric models 12 (2012) 35–52 46 )4sin(4 πτ=s and )4cos(5 πτ=s ). as the diurnality component ),( υτνs seems illegible in figure 3, we decided to plot the joint diurnality pattern for all of the selected probability levels in figure 5. as can be seen from the obtained surface, the intraday seasonality pattern is most pronounced for the most extreme quantiles of the return distribution. for the middle quantiles (i.e., quantiles surrounding the median, τ=0.5), humps in the surface are rather negligible and the surface is rather flat. the diurnality pattern for the quantiles corresponding to the lower tail of the distribution demonstrate that the probability of observing large drops in the eur/pln rate systematically rises early in the morning (the quantile is “shifted to the left”). this effect is rather symmetric because in the upper quantiles the value of the seasonality function is relatively higher throughout the early morning period (the quantile is “shifted to the right”). what is also striking is the probability of large upswings in the eur/pln exchange rate (polish zloty depreciation) in the late afternoon as the upper tail probability is systematically higher late in the afternoon (just before 18.00 cet). concerning the impact of the volume variables, figure 3 demonstrates that it is the unexpected component of the trading volume that is the most responsible for dispersion of the return distribution. the impact of the unexpected volume is about four times larger than that of the expected volume for the 0.99 quantile and about three times larger for the 0.01 quantile. therefore, the unanticipated volume brings more information with regards to large upswings of the eur/pln exchange rate (i.e., the polish zloty depreciation). generally, the impact of the unexpected volume is indisputable as the variable is statistically significant for the probability levels { }51.0,...,02.0,01.0∈τ and { }99.0...,,6.0,59.0∈τ (at a 5% significance level). the effect is also different for different quantiles with the most striking impact placed on the probabilities of the most extreme price movements. the expected volume was derived as a predictable tendency in the level of a trading volume. at time t this variable uses information about the turnover at time t-1. although it is defined for the moment t, it is simply an anticipation of the trading volume given the history of its observations. thus, it can partially capture a potential lead-lag relationships between returns and volumes. if the expected volume is high, the dispersion of the return distribution rises. the impact that this variable exerts on the quantiles of the distribution is significantly different from zero for the probabilities { }19.0...,,02.0,01.0∈τ and { }99.0...,,65.0,64.0∈τ . however, the scale of this effect is not as strong as the effect evidenced in the case of unpredictable volume shocks. on top of the expected volume, significant impact on selected quantiles have also lagged unexpected volume shocks (recorded at t-1). however, the impact of this variable is significant only for { }07.0...,,04.0,03.0∈τ . “does it take volume to move the eur/pln fx rates?”… dynamic econometric models 12 (2012) 35–52 47 0.0 0.4 0.8 -6 -2 2 (intercept) 0.0 0.4 0.8 -2 2 6 s1 0.0 0.4 0.8 -1 0 1 2 3 s2 0.0 0.4 0.8 -1 .0 0. 0 1. 0 s3 0.0 0.4 0.8 -2 0 1 2 s4 0.0 0.4 0.8 -1 .0 0. 0 1. 0 s5 0.0 0.4 0.8 -3 -1 0 1 expected_vol 0.0 0.4 0.8 -4 0 2 4 6 unexpected_ 0.0 0.4 0.8 -0 .5 0. 5 1. 0 unexpected_ 0.0 0.4 0.8 -0 .1 5 -0 .0 5 return_lag1 figure 3. parameter estimates for quantile regressions (model i). shaded areas depict the 90% confidence interval figure 4 shows that if we account for garch-type volatility forecast, 1ˆ −tσ , as an additional driving force of quantile dynamics, the obtained parameter estimates change. the impact of the unexpected volume remains the same as without the volatile ty variable; however, the role of the expected volume declines in a noteworthy fashion (i.e., it is about two times smaller than in the model i) and it remains significantly different from zero for quantile levels { }02.0,01.0∈τ , { }87.0...,,84.0,83.0∈τ and { }98.0...,,94.0,93.0∈τ . findings like this are rather easy to justify. the information contained in the past realizations of the trading volume is, to a large extent, impounded in the fx prices that are set in the market until time t-1. therefore, if these trading volumes are to a large extent reflected by the volatility forecasts 1ˆ −tσ (which condition on the inforkatarzyna bień-barkowska dynamic econometric models 12 (2012) 35–52 48 mation at t-1), the amount of information that can be attributed only to the historical volume will significantly decline. 0.0 0.4 0.8 -4 0 2 4 (intercept) 0.0 0.4 0.8 -2 0 2 4 s1 0.0 0.4 0.8 -1 0 1 2 3 s2 0.0 0.4 0.8 -0 .5 0. 5 1. 0 s3 0.0 0.4 0.8 -1 0 1 2 s4 0.0 0.4 0.8 -1 .0 0. 0 1. 0 s5 0.0 0.4 0.8 -1 .5 -0 .5 0. 5 expected_vol 0.0 0.4 0.8 -4 0 2 4 6 unexpected_ 0.0 0.4 0.8 -1 .0 0. 0 1. 0 unexpected_ 0.0 0.4 0.8 -0 .1 5 -0 .0 5 return_lag1 0.0 0.4 0.8 -8 -4 0 4 volatility figure 4. parameter estimates for quantile regressions (model ii). the shaded areas depict the 90% confidence interval the obtained parameter estimates may also suggest that the return distribution is skewed. this can be observed from the different parameter estimates corresponding to the volatility forecasts that were obtained for lower and upper quantiles (in a symmetric case they should be equal). conclusions the results suggest that trading volume has a significant impact on the variability of the eur/pln rate fluctuations. we also show that the unexpected (unanticipated) component of this variable has a significantly stronger impact than the expected (predictable) component. the scale of this impact varies “does it take volume to move the eur/pln fx rates?”… dynamic econometric models 12 (2012) 35–52 49 across quantiles and is most pronounced in the tails of the return distribution (i.e., for the most extreme price movements). our study contributes to the scarce literature on the volume-return relationship in fx markets. figure 5. the diurnality pattern for quantiles of the return distribution studies on this topic have for the most part been conducted for the lower frequencies including only daily or monthly data. as the fx market is extremely liquid and transparent in comparison to capital markets, the reaction to new information arrival should also be extremely prompt, which justifies the application of high frequency data. moreover, as outlined by cheung et al. (2009), applying the qr approach enables one to study the impact of trading on the general shape of the return distribution. thus, this approach is complementary to the methods based solely on the conditional variance. references admati, a., pfleiderer, p. (1988), a theory of intraday patterns: volume and price variability, review of financial studies, 1, 3–40. andersen, t. (1996), return volatility and trading volume: an information flow interpretation of stochastic volatility, journal of finance, 51, 169–204. andersen, t., bollerslev, t. (1996), intraday periodicity and volatility persistence in financial markets, journal of empirical finance, 4, 115–158. baur, d.g., dimpfl t., jung, r.c. (2011), stock return autocorrelations revisited: a quantile regression approach, journal of empirical finance, 19, 251–265. katarzyna bień-barkowska dynamic econometric models 12 (2012) 35–52 50 bauwens, l., veredas, d. (2004), the stochastic conditional duration model: a latent variable model for the analysis of financial durations, journal of econometrics, 119, 381–412. bessembinder, h., seguin, p.j. (1994), price volatility, trading volume and market depth: evidence from futures markets, journal of financial and quantitative analysis, 28, 21–39. bień, k. (2006), advanced acd models – presentation and the example of application, statistical review, 53 (1), 90–108. blume, l., easley, d., o’hara (1994), market statistics and technical analysis: the role of volume, journal of finance, 49, 153–181. bjonnes, g.h., rime, d., solheim, h.o. (2003), volume and volatility in the fx market: does it matter who you are?, cesifo working paper no. 783. bohl, m., t., henke, h. (2003), trading volume and stock market activity: the polish case, international review of financial analysis, 12, 513–525. brown-hruska, s., kuserk, g. (1995), volatility, volume and the notion of balance in the s&p500 cash and futures markets, journal of futures markets, 15(6), 677–689. chuang, c-c., kuan c-m., lin, h-y. (2009), causality in quantiles and dynamic stock returnvolume relationships, journal of banking and finance, 33, 1351–1360. copeland, t.e. (1976), a model of asset trading under assumption of sequential information arrival”, journal of finance, 31, 1149–1168. crouch, r.l. (1970), the volume of transactions and price changes on the new york stock exchange, financial analysts journal, 26, 104–109. clark, p. (1973), a subordinated stochastic process model with finite variance for speculative prices, econometrica, 41, 135–155. darrat, a.f., rahman s., zhong m. (2007), intraday trading volume and return volatility of the djia stocks: a note, journal of banking and finance, 31(9), 2711–2729. deo, m., srinivasan k., devanadhen, k. (2008), the empirical relationship between stock returns, trading volume and volatility: evidence from select asia-pacific stock market, european journal of economics, finance and administrative sciences, 12, 58–68. diebold, f.x., gunther, t.a., tay, a.s. (1998), evaluating density forecasts with applications to financial risk management, international economic review, 39, 863–883. doman, m. (2008), information impact on stock price dynamics, dynamic econometric models, 8, 13–20. doman, m. (2011), mikrostruktura giełd papierów wartościowych, wydawnictwo uniwersytetu ekonomicznego w poznaniu, poznań. engle, r.f., manganelli, s. (2004), caviar: conditional autoregressive value at risk by regression quantiles, journal of business & economic statistics, 22(4), 367–381. engle, r.f., russell, j.r. (1998), autoregressive conditional duration; a new approach for irregularly spaced transaction data, econometrica, 66, 1127–1162. easley, d., kiefer, n., o’hara m. (1997), one day in the life of a very common stock, review of financial studies, 10, 805–835. easley, d., o’hara m. (1987), price, trade size and information in securities markets, journal of financial economics, 19, 69–90. easley, d., o’hara m. (1992), time and the process of security price adjustment, journal of finance, 47, 576–605. epps, t.w., epps, m.l., (1976), the stochastic dependence of security price changes an transaction volumes: implications for the mixture of distribution hypothesis, econometrica, 44, 305–321. “does it take volume to move the eur/pln fx rates?”… dynamic econometric models 12 (2012) 35–52 51 fleming, j., kirby, c., ostdiek, b. (2006), stochastic volatility, trading, volume, and the flow of information, journal of business, 79(3), p. 1551–1590. gallant, a.r., rossi, p.e., tauchen, g. (1992), stock prices and volume, the review of financial studies, 5, 199–242. giot, p. (2000), time transformations, intraday data and volatility models, journal of computational finance, 4(2), 31–62. giot, p. (2005), market risk models for intraday data, european journal of finance, 11, 309– 324. grammig, j., maurer, k. (2000), non-monotonic hazard functions and the autoregressive conditional duration model, econometrics journal, 3, 16–38. gurgul, h., majdosz, p., mestel, r. (2005), joint dynamics of prices and trading volume on the polish stock market, managing global transitions, 3(2), 139–156. hartmann, p. (1999), trading volumes and transaction costs in the foreign exchange market. evidence from daily dollar-yen spot data, journal of banking and finance, 23, 801– 824. hendricks, w., koenker, r. (1991), hierarchical spline models for conditional quantiles and the demand for electricity, journal of the american statistical association, 87, 58–68. huber, p. (1967), behavior of maximum likelihood estimates under nonstandard conditions, in proceedings of the 5th berkeley symposium on mathematical statistics and probability, berkeley: university of california press. huang, c-m., lin, t-y., yu, c-h., hoe, s-y. (2006), volatility-volume relationships among types of traders considering investment limitation to foreign investors, review of pacific basin financial markets and policies, 9(4), p. 575–596. jennings, r.h., starks, l.t., fellingham, j.c. (1981), an equilibrium model of asset trading with sequential information arrival, journal of finance, 36, 143–161. jones, c.m., kaul, g., lipson, m.l. (1994), transactions, volume, and volatility, the review of financial studies, 7(4), 631–651. karpoff, j.m. (1987), what drives the volume-volatility relationship on euronext paris?, the journal of financial and quantitative analysis, 22(1), 109–126. koenker, r. (2005), quantile regressions, cambridge university press, new york. luckley, b.m. (2005), does volume provide information? evidence from the irish stock market, applied financial economic letters, 1, 105–109. lamoureux, c., lastrapes, w. (1990), heteroskedasticity in stock return data: volume versus garch effects, journal of finance, 45, 220–229. louhichi, w. (2011), the relation between price changes and trading volume: a survey, international review of financial analysis, 200–206. malinova, k., park, a. (2011), trading volume in dealer markets, journal of financial and quantitative analysis, 45, 1447–1489. majand, m., young, k. (2006), a garch examination of the relationship between volume and price variability, journal of futures markets, 11(5), 613–621. manganelli, s. (2005), duration, volume and volatility impact of trades, journal of financial markets, 8, 377–399. morgan, i. g. (1976), stock prices and heteroskedasticity, journal of business, 49, 496–508. nolte, i. (2008), modeling a multivariate trading process, journal of financial econometrics, 6, 143–170. tauchen, g., pitts, m. (1983), the price variability-volume relationship on speculative markets, econometrica, 51, 485–505. katarzyna bień-barkowska dynamic econometric models 12 (2012) 35–52 52 taylor, j.w. (1999), a quantile regression approach to estimating the distribution of multiperiod returns, journal of derivatives, 7, 64–78. westerfield, r. (1977), the distribution of common stock price changes: an application of trans? actions time and subordinated stochastic models, journal of financial and quantitative analysis, 12, 743–765. „czy wolumen transakcji wpływa na zmiany kursu eur/pln?” wnioski płynące z zastosowania regresji kwantylowych z a r y s t r e ś c i w artykule dokonano badania wpływu wolumenu transakcyjnego na wartość wybranych kwantyli rozkładu stóp zwrotu z kursu eur/pln. wyniki empiryczne otrzymane na podstawie regresji kwantylowych potwierdziły, że wzrost obrotów ma statystycznie istotny wpływ na dyspersję rozkładu stóp zwrotu. w badaniu dokonano podziału wolumenu transakcyjnego na dwie części: tzw. wolumen oczekiwany przez uczestników rynku i tzw. wolumen nieoczekiwany przez uczestników rynku oraz wykazano, że to wolumen nieoczekiwany ma dużo większy wpływ na dyspersję badanego rozkładu. zaobserwowano również, że relacja pomiędzy wolumenem a stopą zwrotu ma charakter nieliniowy, tzn. jest najsilniejsza dla najbardziej ekstremalnych kwantyli. wykazano, że w konsekwencji uwzględnienia miary warunkowej zmienności (jako dodatkowego czynnika wyjaśniającego dynamikę kwantyli stóp zwrotu) wpływ oczekiwanej wartości wolumenu transakcyjnego ulega zmniejszeniu, ale wciąż pozostaje istotny statystycznie, szczególnie dla najbardziej ekstremalnych kwantyli. s ł o w a k l u c z o w e: relacja wolumen-stopa zwrotu, mikrostruktura rynku, obrót na rynku walutowym, regresja kwantylowa acknowledgements the author wants to thank the thomson reuters for providing the data from the reuters dealing 3000 spot matching system and the two anonymous referees for their valuable comments. the views and opinions presented in the paper are those of the author and do not necessary reflect those of the national bank of poland. introduction 1. the econometric approach 2. empirical results conclusions references dynamic econometric models © 2016 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2016.008 vol. 16 (2016) 145−164 submitted november 8, 2016 issn (online) 2450-7067 accepted december 21, 2016 issn (print) 1234-3862 magdalena osińska, tadeusz kufel, marcin błażejowski, paweł kufel * modelling and forecasting business cycle in cee countries using a threshold approach  a b s t r a c t. we propose to apply a time-series-based nonlinear mechanism in the threshold autoregression (tar) form in order to examine business cycles in central and eastern european economies and compare them to the entire eu business cycle. the threshold variables, such as consumer price index, short and long interest rates, unemployment rate and an exchange rate vs. the u.s. dollar, have been considered. the purpose of the paper is to model and to predict business cycles in central and east european (cee) economies (the eu member states) and compare them to business cycles of the entire eu28 area and eurozone eu19. we found that the exogenous mechanism played an important role in diagnosing the phases of business cycles in cee economies, which is in line with the entire eu economic area. the results of business cycle forecasting using bootstrap technique are quite promising, while bootstrap confidence intervals are used for diagnosis. k e y w o r d s: business cycle, central and eastern economies, threshold models, forecasting, bootstrap j e l classification: c24, c53, e32. * correspondence to: magdalena osińska, nicolaus copernicus university, faculty of economic sciences and management, 11a gagarina street, 87-100 toruń, poland, e-mail: emo@umk.pl; tadeusz kufel , nicolaus copernicus university, faculty of economic sciences and management, 11a gagarina street, 87-100 toruń, poland, e-mail: tadeusz.kufel@umk.pl; marcin błażejowski, wsb university in toruń, e-mail: marcin.blazejowski@wsb.torun.pl; paweł kufel, wsb university in toruń, e-mail: pawel.kufel@wsb.torun.pl.  this work was financed from national center of science – grant no 2012/07/b/hs4/02927. magdalena osińska, tadeusz kufel, marcin błażejowski, paweł kufel dynamic econometric models 16 (2016) 145–164 146 introduction the purpose of the paper is to model and predict a business cycle in central and east european (cee) economies (the eu member states) and to compare it to the business cycle of the entire eu28 area and eurozone eu19. our analysis is based on the theory of economic convergence, introduced by barro, sala-i-martin (1992); however, newest empirical facts resulting from the economic crisis of the recent years have been taken into consideration, too. for at least two last decades, it has been assumed that central and eastern european economies have undergone the process of catching up with the most developed western european economies. since 2004 (as well as before), huge structural funds have been spent to speed up the entire process of economic unification of the eu area. common currency – the euro – has become one of the symbols of the unification. at present, 19 of 28 economies use this currency, eliminating one of the internal risk factors but exposing themselves to the external ones. however, the economic crisis of 2007–2009 has broken the process of economic convergence and reveals many differences among the countries. it is worth noting that, after 2004, central and east european (cee) countries usually experienced a sigma-type convergence and a conditional beta-type convergence (see kluth, 2016). in recent literature, two important issues can be found. the first is related to the problem of economic convergence and divergence (decoupling). the hypothesis of decoupling between business cycles in the developed and emerging countries after 2009 has become the subject of a widespread academic debate. in their paper, claassen, kabundi, loots (2013) stated that decoupling between advanced and emerging economies took place, but in recent years the process of re-coupling has started. kawa (2013) demonstrated that regardless the fact that many cee countries introduced the defense mechanisms against the shocks to their regulation systems in the 90s, they remain vulnerable to the external debt, budget deficit and foreign trade imbalance. stańczyk, wyrobek (2013) have analyzed the issue of business cycle synchronization between the usa and emerging economies in 1995– 2009. the authors concluded that no evidence was found that emerging economies as a whole and in subgroups had their business cycles synchronized. however, cycles in many emerging economies were more synchronized with the us economy, particularly at the time of the global economic crisis. they also stated that unexpected and unusual phenomena like global crises, disrupted the relationships among the economies observed in a “normal” state of development. modelling and forecasting business cycle in cee countries… dynamic econometric models 16 (2016) 145–164 147 the second issue – more recent and yet more open to questions – is focused on the problem of the so-called “middle-income trap”. our choice of the central and eastern economies has been motivated by the fact that they are threatened by what has become known as the middle-income trap. this means that compared to the richest economies, their per-capita income stays low and there is little chance to overcome this difficulty. spence (2011) refers to the middle-income countries as to those in the 5,000–10,000 usd range of per capita income. he indicates that in developing countries “the industries that drove the growth in the early period start to become globally uncompetitive due to rising wages”. among the cee countries, only bulgaria and romania entered this range in 2015. the countries analyzed in-depth in the paper have their gdp per capita above 10,000 usd, but some of them, particularly poland, hungary and slovakia, are not very far from this limit. the concept of middle-income trap is still a subject of the economic debate. recently, it has been widely discussed in im, rossenblatt (2015). if the middle-income trap is what the developing european economies are trying to avoid, they should speed up the catching up process by undertaking intensive reforms. that is why the business cycle forecasts must be taken into account. our research is a part of a bigger project and it has been preceded by earlier reports. osińska, kufel, błażejowski, kufel (2016a) have examined the business cycles synchronization within the eu economies in comparison to the u.s. and japan, using spectral analysis methodology. both the quarterly and monthly data has been analyzed, supporting the same results. they found that most european countries, including the cee countries, had their economic cycles synchronized with the entire eu. only hungary represented the opposite case. as for the u.s.a. and japanese business cycles, the two were more synchronized together compared with the eu. in the paper by osińska, kufel, błażejowski, kufel (2015), a business cycle clock methodology for the same economies has been applied. in the paper by osińska, kufel, błażejowski, kufel (2016b) the threshold autoregression models (tar) has been applied to reveal the most likely threshold mechanisms, which underlie the business cycles in the eu economies. the concept that debt/gdp ratio could be the indicator of the changes between the business cycle phases, has been rejected. in the reported research, the quarterly data of 1995–2014 were analyzed. the gross domestic product (gdp) growth rate was traditionally used as the business cycle measure. the following research questions have been formed: which threshold variable(s) help to reveal a threshold mechanism within the business cycles observed in central and eastern european economies; what magdalena osińska, tadeusz kufel, marcin błażejowski, paweł kufel dynamic econometric models 16 (2016) 145–164 148 is the scope of differences between the regimes, what is the forecasting ability of the estimated tar models and what similarities/differences between the observed countries can be indicated. the methodology assumes using a stationary tar model as the basis for applying bootstrap technique. the threshold variable prediction was a key factor in the entire procedure. bootstrap confidence intervals have been used for ex ante forecast evaluation. the novelty of the paper lies in defining a simulation-based procedure for forecasting tar models and its application to the business cycle in cee countries. the paper is organized as follows: in section two, the data has been analyzed from both perspectives: the economic convergence process and business cycle analysis; in section three, the research methodology has been described. in section four, the empirical results have been presented. in section five the conclusion is summarised. 1. characteristics of the data one of the most popular perspectives of classification of economies is based on the criterion of initial wealth measured by the gdp per inhabitant. the initial wealth is crucial for understanding the individual process of economic development and the final stage proves the convergence of a given economy along its long-run path. as it has been already mentioned, the process of catching up may be slowed down while the country experiences the middleincome trap. the new eu member states that belong to the central and eastern european group are bulgaria, the czech republic, estonia, croatia, hungary, lithuania, latvia, poland, romania, the slovak republic and slovenia. in the beginning of the analyzed period i.e., in 1995, all these countries had their gdp per inhabitant below 10,000 usd, while in 2015 only bulgaria and romania had the gdp per capita remaining below this limit. in the same year, slovenia’s gdp, with its 20,713.1 usd (in current prices), surpassed portugal and greece, where both countries were included in the group eu15. this means that some of the newest eu member states that acceded the eu in 2004, managed to make a successful progress in the process of economic convergence, measured by the dispersion from the average level. this process was broken by the recession of 2007–2009, when each country was faced with its own economic decisions being more or less in line with the eu economic policy (osińska and kluth, 2011). countries like poland and hungary, with their gdp p.c. equal to 12,494.5 usd and 12,259.1 usd, respectively, may be concerned about the middle-income trap unless the structure of the productive sectors of the economies will change. modelling and forecasting business cycle in cee countries… dynamic econometric models 16 (2016) 145–164 149 slovakia with the gdp p.c. of 15,962.6 usd, seems to be more resistant and similar to the czech republic and estonia, where the gdp p.c. exceeds 17,000 usd. the study uses quarterly data from 1995–2014, taken from the oecd database study. in several cases, some data were not complete, therefore, we decided to limit our investigation to the countries where databases were as broad as possible. thus, the following countries have been examined: the czech republic, estonia, hungary, poland, slovakia and slovenia. the mechanism of the creation of economic cycle in these countries has been the subject of comparison with both the entire eu28 and the entire eurozone ea19. the figure 1 shows the business cycle dynamics for three selected countries and the eu28. eu28 hungary poland slovenia figure 1. business cycles in selected cee countries and in the eu28 the original gdp series (seasonally adjusted) has been transformed for extracting a business cycle by taking logs and filtering by the hodrickprescott filter with the smoothing parameter equal to 1,600 (hodrick, prescott, 1997). in figure 1, the differences among selected countries, considering the amplitude and phase of the cycle, can be noticed. in the threshold autoregression (tar) models, the assumption of threshold variable is of key importance. in our study, we assumed that the set of magdalena osińska, tadeusz kufel, marcin błażejowski, paweł kufel dynamic econometric models 16 (2016) 145–164 150 exogenous thresholds consists of the following: the consumer price index, short interest rate, long interest rate, unemployment rate and exchange rate. the choice of variables has been determined by data availability and their relation to business cycle analysis. it can be mentioned that when compared to the reference variable, namely the gdp growth rate, the cpi and long interest rate are considered as exhibiting the same changes over time, while the short interest rate and exchange rate are leading indicators for the cycle. the reaction of the unemployment rate usually lags behind. these assumptions are empirically verified; however, they may differ among the countries (zarnowitz, 1999). all these series were checked for stationarity using the adf/kpss approach and finally they were taken at both levels (i(1)) and (i(0)). 2. the model and forecasting procedure the threshold autoregression models have been applied to the u.s. business cycle modeling by tong (tong, 1990). recently, their particular assessment of economic growth has been determined in the publications such as by and niebuhr, 2005 who shed the light on the regional perspective of growth in germany and al., 2013, who introduced a dynamic panel threshold model to estimate inflation thresholds for a long-term economic growth, to mention only a few. in our research, tar models have been used as well because that allows considering different threshold variables as playing a possible rule in regime changes. when the lagged endogenous variable is a threshold variable, the model is known as a self-exciting threshold autoregression (setar). this difference allows identifying exogenous or endogenous mechanism of changes between the regimes that correspond to business cycle phases. this interpretation coincides with the endogenous and exogenous growth idea in economics. let t y denotes k-dimensional random vector. the model of the following form: tttt j t j t j t j t chyayby   1 , (1) where tj is a random variable taking values of finite set of natural numbers  p,...,3,2,1 , tjb , tja , tjh are kk – dimensional matrices of the coefficients, t is the k – dimensional white noise, tjc is a constant vector is called a canonical form of the threshold model. it defines a wide class of the models depending on the choice of tj . modelling and forecasting business cycle in cee countries… dynamic econometric models 16 (2016) 145–164 151 when tj is the function of an exogenous variable, say, tx ),,1},{( mixx itt  then we obtain a tar model. the tar( ) model is defined in the following way: t j k i it j i j t hyy j      1 0 (2) conditionally on pjrx jdt ,...,1,  . the dtx  is called a threshold variable. the more convenient form of presenting (2) is as follows:                1110 21 22 1 2 1 2 0 1 11 1 1 1 1 0 ... ... ... 22 11 pdtt p kt p kt pp dttktkt dttktkt t rxforhyy rxrforhyy rxforhyy y pp     (3) in a setar model the threshold variable is lagged endogenous variable namely, t y . it is useful to present the two regimes model with the i(x) function of the form:           01 00 dt dt xwhen xwhen xi (4) and the corresponding tar(2, k1, k2) model       tdtktkt ktktt xiyy yyy       22110 11110 ... ... . (5) if all k10 ,...,β,ββ parameters are zeros then (5) becomes the linear autoregressive model. when the autoregressive model is considered, its stationarity becomes the crucial point. for the linear autoregressive model, the conditions of stationarity are well known and easy to satisfy (see: greene, 1993). in the case of setar or tar, the problem is much more complicated. even stationary models within the regimes do not guarantee the stationarity of the whole system. giordano, niglio and vitale (2012) analyzed this problem, basing it on the papers by petruccelli and woolford (1984) and chan et al. (1985). in the case of two regime setar model (3) when k is greater than 1, the following stationarity conditions must be satisfied (an, huang, 1996; lin, 1999): magdalena osińska, tadeusz kufel, marcin błażejowski, paweł kufel dynamic econometric models 16 (2016) 145–164 152       k i j ij k i j ij a a 1 )( 1 )( .1||max 1||max estimation of tar/setar models can be done by conditional ordinary least squares or maximum likelihood methods. however, the forecasting is more complex. first of all, it is important to determine the predicted values of the threshold variable in the forecasting possibility. for this reason, we started our procedure for the tar models first because they underlay the exogenous threshold mechanism. our proposal consists of the application of a simulation methodology for forecasting both: the threshold variable dt x  as well as the endogenous variable t y , which is the which is the business cycle (as presented in figure 1). the idea of forecasting tar models has been developed by brown and mariano (1984). forecasting procedure proposed and applied in this paper has been presented in gretl environment (threshold_models package). it is based on the already-estimated tar model and it is implemented in one of the two common simulation approaches, i.e., the bootstrap simulation technique (see dvison et al., 1986) or monte carlo method (see rubinstein and kroese, 2011). a simulation-based forecasting procedure applies to both endogenous and threshold variables, but the starting point is to simulate the possible paths of the threshold variable. the whole procedure is carried out in the following steps: 1. estimation of the predictive model for the threshold variable. the sarima(p,d,q)(p,d,q) approach with the specification selection based on the schwarz information criterion (bic) is applied. 2. generating a noise for the simulation procedure:  when in the bootstrap approach, residuals from estimated sarima model are used (empirical distribution),  when in the monte carlo approach, random numbers are drawn from one of the theoretical distributions, i.e., normal or t-student with a mean of zero and a standard deviation equal to the value from the estimated sarima model. 3. addition of phase noise to the values of threshold variable, reestimation of the sarima model and prediction of the threshold variable for h periods ahead. modelling and forecasting business cycle in cee countries… dynamic econometric models 16 (2016) 145–164 153 4. determination of regimes in which the endogenous variable will be in the future (on the basis of the threshold variable forecasts generated in the previous stage). 5. usage of the already-estimated tar model to generate forecasts of the endogenous variable. 6. stages 2–5 are repeated n times. 3. the empirical results the following quarterly data from 1995–2014 are included and have been analyzed (short names are given in brackets): the gdp growth rate (gdp), unemployment rate (uemp), interest rates (long ir and short ir), cpi and first differences of cpi, exchange rates in usd (exr) and its first differences. it was assumed that the gdp growth rate was the endogenous variable and the remaining lagged variables were supposed to be thresholds for regime changes. the regimes correspond to the phases of economic cycle. to eliminate non-stationarity, the original gdp series were de-trended using hp filter where λ=1600. all the original data were seasonally adjusted, transformed into logs and tested for stationarity using adf-gls and kpss tests. the number of regimes was restricted to maximum three for the following reasons: the relatively short time series and reasonable interpretation of the business cycle in case of prosperity, recession and the intermediary states of increasing and decreasing of the gdp. the number of regimes has been chosen based on the quartiles of the threshold variable. in practice, the following model has been considered:             2 3 1 3 1 3 0 21 2 1 2 1 2 0 1 1 1 1 1 1 0 33 22 11 rxforgdpgdp rxrforgdpgdp rxforgdpgdp gdp dttktkt dttktkt dttktkt t       where a set of threshold variables dt x  is the same as it was described at the beginning of this section. as many threshold models were to be estimated we decided to use bayesian information criterion (bic) for the model selection. the results of model selection are presented in tables 1, 2 and 3. magdalena osińska, tadeusz kufel, marcin błażejowski, paweł kufel dynamic econometric models 16 (2016) 145–164 154 table 1: values of schwartz criterion and threshold values for czech republic, estonia and hungary czech rep. estonia hungary bic threshold bic threshold bic threshold exr –522,202 20,314 –558,056 176,364 d_exr –520,460 –0,042 –540,131 –0,032 cpi –522,902 1,757 –307,094 3,057 –544,147 6,906 d_cpi –527,509 –0,018 –292,824 –0,02 0,020 –542,474 –0,01 0,010 longir –372,092 3,502 –395,920 7,469 d_longir –374,295 –0,001 –392,023 –0,001 shortir –522,206 3,843 –364,478 3,968 d_shortir –519,079 –0,001 –344,437 –0,004 unemp –416,655 7,283 –265,276 7,474 –409,457 7,382 d_unemp –408,035 –0,245 –270,077 –0,89 0,383 –398,233 0,040 setar –522,752 –0,005 –371,486 –0,002 –525,239 –0,001 sd_exr –469,528 26,594 –456,666 174,182 sd_d_exr –471,243 –0,012 –450,686 0,015 sd_cpi –469,097 1,757 7,689 –272,050 3,057 –453,485 6,906 sd_d_cpi –472,006 –0,018 –250,539 –0,020 –450,934 –0,012 sd_longir –355,049 4,193 –351,032 6,807 sd_d_longir –346,873 –0,003 –341,508 –0,001 sd_shortir –475,691 2,090 9,762 –314,583 3,955 6,359 sd_d_shortir –477,825 –0,004 –301,225 –0,001 sd_unemp –392,193 6,709 –225,906 10,154 –359,549 7,400 sd_d_unemp –382,344 –0,245 –224,844 –0,140 –350,690 0,040 sd_setar –464,230 –0,023 –309,123 –0,016 –438,251 –0,001 note: bic – value of schwarz criterion, treshold – value of threshold (if one value is given means model with 2 regimes, if 2 values id given means model with 3 regimes), the best value of bic for each country has been bolded, d_ – means first differences, sd_ – means seasonal differences. empirical analysis of business cycles in cee countries has revealed the most likely threshold variables in the countries in terms of interest. these are: the cpi in the case of slovenia and δcpi in the case of the czech republic, a short interest rate in the case of estonia and the exchange rate against the usd for the other four cases, i.e., hungary, poland, the eu28 and the eu19. in the case of slovak republic, the difference in the exchange rate has been indicated. it shows the importance of the exchange rate channel in the risk exposure of such economic bodies as the european union and its member states. the two other countries were more sensitive to the consumer price changes and the last one, namely, estonia, to the monetary policy changes. obviously, the statistical identification of the thresholds may be limited by the availability of data, but the most likely differences for the mechanism change within similar economic system area is interesting. the level of economic development of particular eu countries remains still diversified and determines the expected results. for these reasons, we assumed the same set modelling and forecasting business cycle in cee countries… dynamic econometric models 16 (2016) 145–164 155 of threshold variables that were the subject of testing for the gdp growth rate. tables 4 and 5 present the estimated tar models for selected thresholds. table 2: values of schwartz criterion and threshold values for poland, slovak republic, estonia and slovenia poland slovak rep. slovenia bic threshold bic threshold bic threshold exr –498,012 2,772 –449,867 0,759 –502,767 0,763 0,803 d_exr –493,983 0,006 –457,259 –0,009 –509,749 –0,03 –0,01 cpi –491,671 4,224 –447,620 2,882 –519,869 5,641 d_cpi –492,383 –0,035 –451,712 –0,01 0,014 –500,665 –0,018 longir –357,629 5,750 –306,867 4,640 –291,729 4,647 d_longir –357,032 –0,004 –273,577 –0,001 –294,096 –0,001 shortir –492,587 4,590 20,660 –312,699 3,697 4,728 d_shortir –500,652 –0,013 –294,841 0,000 unemp –491,760 13,031 –444,902 12,354 –378,125 6,025 d_unemp –490,523 –0,410 –439,242 –0,392 –382,968 –0,256 setar –480,468 0,002 –450,458 –0,004 –475,356 –0,003 sd_exr –423,530 3,175 3,824 –400,520 0,762 –422,020 0,800 sd_d_exr –426,900 –0,023 –389,912 –0,032 –416,761 –0,007 sd_cpi –421,851 2,084 13,050 –388,697 3,009 5,655 –416,292 6,562 sd_d_cpi –428,644 –0,036 –389,507 –0,030 –425,120 –0,008 sd_longir –309,168 5,232 –255,755 4,640 –264,005 4,647 sd_d_longir –307,192 –0,004 –253,535 –0,001 0,002 –259,146 –0,004 sd_shortir –436,382 6,883 21,770 –265,775 3,697 sd_d_shortir –425,519 –0,012 –267,981 –0,003 sd_unemp –423,913 10,022 –389,320 12,377 –333,151 6,025 sd_d_unemp –422,224 –0,410 –388,072 –0,099 –325,931 –0,256 sd_setar –412,556 –0,013 –381,351 –0,010 –397,089 0,004 note: bic – value of schwarz criterion, treshold – value of threshold (if one value is given means model with 2 regimes, if 2 values id given means model with 3 regimes), the best value of bic for each country has been bolded, d_ – means first differences, sd_ – means seasonal differences. only in one case, i.e., the eu28, a three-regime model has been selected. in the other cases, two-regime models have been preferred for the data. figure 2 illustrates the “goodness-to-fit” calculator of the empirical tar model. which is very good in all cases. magdalena osińska, tadeusz kufel, marcin błażejowski, paweł kufel dynamic econometric models 16 (2016) 145–164 156 table 3: values of schwartz criterion and threshold values for european union and euro area note: bic – value of schwarz criterion, treshold – value of threshold (if one value is given means model with 2 regimes, if 2 values id given means model with 3 regimes), the best value of bic for each country has been bolded, d_ – means first differences, sd_ – means seasonal differences. table 4: estimated threshold models with lowest bic values for czech rep., estonia, hungary and poland czech rep. d_cpi (–4) estonia shortir (–3) hungary exr(–3) poland exr(–3) r1 const –0,006*** 0,002 –0,002 –0,003 r1 yt–1 1,133*** 0,518*** 0,61*** 0,167 r1 yt–2 –0,533*** 0,898*** r1 yt–3 –0,016 r1 yt–4 –0,539*** r2 const 0,001 –0,005* 0,000 0,001 r2 yt–1 1,313*** 0,951*** 1,312*** 0,841*** r2 yt–2 –0,413*** 0,294* –0,175 r2 yt–3 0,091 –0,476*** –0,34*** r2 yt–4 –0,025 r2 yt–5 –0,177*** note: *** – 1% significance level, ** – 5% significance level, * – 10% significance level, r1 means regime 1, r2 – regime 2, r3 – regime 3. eu28 ea19 bic threshold bic threshold exr –529,737 1,120 1,312 –573,899 1,250 d_exr –525,366 0,011 0,033 –560,899 0,009 cpi –347,452 0,550 –392,309 1,200 2,250 d_cpi –303,169 –0,016 0,014 –374,549 –0,001 longir –325,275 3,840 –572,221 4,445 d_longir –331,965 –0,002 –563,732 –0,001 shortir –564,237 3,550 d_shortir –573,089 –0,001 0,001 unemp d_unemp setar –505,451 –0,001 0,008 –548,812 –0,004 0,008 sd_exr –445,998 1,122 1,312 –497,023 1,250 sd_d_exr –441,013 0,010 –491,274 0,009 sd_cpi –292,039 0,550 –345,132 1,650 sd_d_cpi –274,322 –0,016 –334,099 –0,001 sd_longir –295,131 4,200 –498,525 4,495 sd_d_longir –284,914 –0,002 –494,773 –0,001 sd_shortir –500,722 3,450 sd_d_shortir –494,463 –0,004 0,001 sd_unemp sd_d_unemp sd_setar –429,992 –0,011 –472,383 –0,001 modelling and forecasting business cycle in cee countries… dynamic econometric models 16 (2016) 145–164 157 table 5: estimated threshold models with lowest bic values for slovak rep., slovenia, european union and euro area. slovak rep. d_exr(–1) slovenia cpi (–2) european union exr(–5) euro area exr(–2) r1 const 0,002 0,001 0,000 –0,001 r1 yt–1 0,931*** 1,13*** 1,398*** 0,866*** r1 yt–2 0,181 –0,539*** r1 yt–3 –0,178 r1 yt–4 –0,195** r2 const –0,004 –0,001 0,001 0,000 r2 yt–1 0,525*** 0,587*** 1,663*** 1,45*** r2 yt–2 –0,729*** –0,626*** r2 yt–3 r2 yt–4 r2 yt–5 r2 const –0,004*** r3 yt–1 1,277*** r3 yt–2 –0,883*** r3 yt–3 0,548*** r3 yt–4 –0,342*** note: *** – 1% significance level, ** – 5% significance level, * – 10% significance level, r1 means regime 1, r2 – regime 2, r3 – regime 3. czech republic ue28 figure 2. actual and fitted (based on tar model) values of business cycle in czech republic and eu28 business cycle prediction was the next step of the analysis. according to the procedure described in section 3, we used bootstrap technique for both forecasting the threshold variable and forecasting the business cycle using 1,000 replications. the results of forecasting both the threshold variables indicated in table 1 and the endogenous variable, are shown in figure 3. two cases have been omitted. the first is the case of the slovak republic, where the exchange rate difference serves as a threshold. the bootstrap forecasts for this variable are stable over a certain constant, thus the forecasts cannot be magdalena osińska, tadeusz kufel, marcin błażejowski, paweł kufel dynamic econometric models 16 (2016) 145–164 158 interpreted properly. the second case, the eurozone (eu19), is pretty similar to the eu28. czech rep. – δcpi czech rep. – bootstrap forecast estonia – short ir estonia – bootstrap forecast hungary – huf/usd hungary – bootstrap forecast figure 3. bootstrap forecasts of threshold variables and business cycle note: mean and median are shown. 90% and 95% confidence intervals are shadowed. a vertical line on figures placed on the lhs of the table separates the sample and forecasting period. all the figures have been prepared in gretl package: threshold_models. modelling and forecasting business cycle in cee countries… dynamic econometric models 16 (2016) 145–164 159 poland – pln/usd poland – bootstrap forecast slovenia – cpi slovenia – bootstrap forecast eu28 – eur/usd eu28 – bootstrap forecast figure 3. continued note: mean and median are shown. 90% and 95% confidence intervals are shadowed. a vertical line on figures placed on the lhs of the table separates the sample and forecasting period. all the figures have been prepared in gretl package: threshold_models. it can easily be noted that the results of forecasting a business cycle using a threshold autoregression (tar) model strongly depends on the results of forecasting the threshold variable. a proper selection of the regime is of magdalena osińska, tadeusz kufel, marcin błażejowski, paweł kufel dynamic econometric models 16 (2016) 145–164 160 particular importance. if the threshold tends to penetrate into one regime, the forecast seems to be more precise and the confidence intervals are narrower. in the opposite case, when the threshold is expected to penetrate two or more regimes, the difference between a mean and a median and the confidence interval limits are greater. in the abovementioned cases, the following conclusions can be made: 1. for estonia – a short interest rate, hungary – huf/usd and slovenia – cpi the threshold variables penetrate only one regime in the forecasted period. 2. for poland, the exchange rate of pln/usd penetrates one regime, apart from three values that fall into the second one. 3. for the czech republic where δcpi and eu28 – eur/usd the ratio of values that penetrate two regimes, is almost the same. empirical results of forecasting quarterly gdp growth rates in selected economies on the basis of tar models using a simulation approach, revealed different possible paths. for such economies as the czech republic, estonia, the european union (ue28) (and the eurozone eu19), the forecasts form the characteristic plume or ribbons. for poland, the majority of realizations of simulation-based forecasts hit just one possible path and only in a few iterations different results were obtained. finally, for hungary and slovenia, all 1,000 forecast values generated in the bootstrap procedure were identical. as concerns the forecasted tendency of business cycle in 5 cases, the phase of recovery has been indicated. only in the case of estonia a slowdown has been shown. conclusions in 1995–2014, the cee as well as all the eu economies experienced a business cycle. its amplitude and phase were diversified among the countries but in general, they were similar. in 2007–2009, the economies were exposed to the global financial and economic recession. thereafter, economic development divergence processes started. the recession revealed complicated economic and social situations in many countries. at the time of their accession to the eu, cee countries optimistically developed their economies. they lowered inflation, improved the economic efficiency and developed many economic institutions. slovenia and estonia became the leaders of institutional changes in central european countries. at present, some of the cee countries are facing a different problem, namely, how to avoid the middle-income trap and how to improve their competitiveness. modelling and forecasting business cycle in cee countries… dynamic econometric models 16 (2016) 145–164 161 in this paper, we defined and applied a time series-based nonlinear mechanism in the threshold autoregression (tar) form in order to examine a business cycle in central and eastern european economies compared to the entire eu business cycle. threshold variables, such as consumer price index, short and long interest rates, unemployment rate, exchange rate vs. the u.s. dollar have been considered. the purpose of the paper was to model and predict business cycles in central and east european (cee) economies (the eu member states) and compare them to the business cycle of the entire eu28 area and eurozone eu19. we found that the exogenous mechanism played an important role in diagnosing the phases of business cycle in cee economies, which is in line with the entire eu economic area. the results of business cycles forecasting using bootstrap technique are quite promising, while bootstrap confidence intervals are used for diagnosis. it is indicated that the results of forecasting a business cycle using a threshold autoregression (tar) model strongly depends on the results of forecasting the threshold variable. among the threshold variables, the following were confirmed by the data: the short interest rate (estonia), huf/usd (hungary), cpi (slovenia), pln/usd (poland), δcpi (czech republic) and eur/usd (eu28 and eu19). the proper selection of the regime is of particular importance. if the threshold tends to penetrate into one regime, the forecast seems to be more precise and the confidence intervals are narrower. in the opposite case, when the threshold is expected to enter two or more regimes, the difference between a mean and a median and the confidence interval limits are greater. in 5 cases, business cycle forecasts show a recovery phase. only in the case of estonia a slowdown has been predicted. although many analyses have been undertaken in the last few years on the monetary and fiscal policy instruments corresponding to different phases of the economic cycle, a proper diagnosis is still an open issue. the quality of institutions, state integrity, the position of the economy (core or peripheral), and the middle-income trap are some examples of states that might affect the economic growth pattern in different countries, including the eu member states. references an, h., huang, f. (1996), the geometrical ergodicity of nonlinear autoregressive models, statistica sinica, 6, 943–956, doi: http://dx.doi.org/10.1016/0167-7152(94)00082-j. barro, r. j., sala-i-martin, x. (1992). convergence. journal of political economy, 100(2), 223–251, doi: http://dx.doi.org/10.1086/261816. magdalena osińska, tadeusz kufel, marcin błażejowski, paweł kufel dynamic econometric models 16 (2016) 145–164 162 brown b. w., mariano r. s. (1984), residual – based procedures for prediction and estimation in a nonlinear simultaneous system, econometrica, 52, 321–344. chan, k. s., petruccelli, j. d., tong, h., woolford, s. w. (1985), a multiple-threshold ar(1) model, journal of applied probability, 22(2), 267–279, doi: http://dx.doi.org/10.1017/s0021900200037748. chen, c. w., so, m. k., liu, f.-c. (2011), a review of threshold time series models in finance, statistics and its interface, 4(2), 167–181, doi: http://dx.doi.org/10.4310/sii.2011.v4.n2.a12. chen, r., tsay, r. s. (1991), on the ergodicity of tar(1) processes, the annals of applied probability, 1(4), 613–634, doi: http://dx.doi.org/10.1214/aoap/1177005841. claassen, c., kabundi, a., loots e. (2013), decoupling between emerging and advanced economies. an exploratory analysis, paper presented at the 11th ebes (eurasia business and economics society) conference, ekaterinburg. durlauf, s. n., johnson, p. a., temple, j. r. (2005), growth econometrics in aghion, p. durlauf, s. (ed.), handbook of economic growth, 555–677, elsevier. dvison, a., hinkley, d. v., schechtman, e. (1986), efficient bootstrap simulation, biometrika, 73(3), 555–566, doi: http://dx.doi.org/10.2307/2336519. eyraud, l., weber, a. (2013), the challenge of debt reduction during fiscal consolidation. imf working paper 13/67, international monetary fund. funke, m., niebuhr, a. (2005), threshold effects and regional economic growth–evidence from west germany, economic modelling, 22, 61–80, doi: http://dx.doi.org/10.1016/j.econmod.2004.05.001. gnegne, y., jawadi, f. (2013), boundedness and nonlinearities in public debt dynamics: a tar assessment, economic modelling, 34, 154–160, doi: http://dx.doi.org/10.1257/aer.100.2.573. granger, c. w. j., teräsvirta, t. (1993), modelling non-linear economic relationships, oup catalogue, oxford university press. greene, w. h. (1993), econometric analysis, macmillan, new york, 2 edition. hamilton, j. d. (1994), time series analysis, vol. 2, princeton university press princeton. herndon, t., ash, m., pollin, r. (2014), does high public debt consistently stifle economic growth? a critique of reinhart and rogoff, cambridge journal of economics, 38(2), 257–279, doi: http://dx.doi.org/10.1093/cje/bet075. hodrick, r.j., prescott, e.c. (1997), postwar u.s. business cycles: an empirical investigation, journal of money, credit and banking, 29(1), 1–16. ilzetzki, e. (2011), fiscal policy and debt dynamics in developing countries, policy research working paper wps5666, the world bank group. im, f.g., rosenblatt, d. (2015), middle-income traps: a conceptual and empirical survey, journal of international commerce, economics and policy, 6(3), 155–194, doi: http://dx.doi.org/10.1142/s1793993315500131. kapetanios, g., shin, y. (2006), unit root tests in three-regime setar model, econometrics journal, 9(2), 252–278, doi: http://dx.doi.org/10.1111/j.1368-423x.2006.00184.x. kawa, p. (2011), wpływ zaburzeń na rynkach finansowych na wzrost gospodarczy w krajach na średnim poziomie rozwoju (theiimpact of financial turmoil on economic growth in the countries the average level of development) in wojtyna, a. (ed.) kryzys finansowy i jego skutki dla krajów na średnim poziomie rozwoju. pwe. kluth, k., (2016), statystyczna i ekonometryczna analiza konwergencji gospodarczej i społecznej państw unii europejskiej (statistical and econometric analysis of economic and social convergence eu countries), wyd. umk, toruń, forthcoming. modelling and forecasting business cycle in cee countries… dynamic econometric models 16 (2016) 145–164 163 kourtellos, a., stengos, t., tan, c. m. (2013), the effect of public debt on growth in multiple regimes, journal of macroeconomics, 38, 35–43, doi: http://dx.doi.org/10.1016/j.jmacro.2013.08.023. kremer, s., bick, a., nautz, d. (2013). inflation and growth: new evidence from a dynamic panel threshold analysis, empirical economics, 44(2), 861–878, doi: http://dx.doi.org/10.1007/s00181-012-0553-9. krugman, p. (2012), end this depression now! ww norton & company. ling, s. (1999), on the probabilistic properties of a double threshold arma conditional heteroskedastic model. journal of applied probability, 36(3), 688–705, doi: http://dx.doi.org/10.1239/jap/1032374627. mendieta-muñoz, i. (2016), on the interaction between economic growth and business cycles, macroeconomic dynamics, 1, 1–41, doi: http://dx.doi.org/10.1017/s1365100515000796. misztal, p. (2011), dług publiczny i wzrost gospodarczy w krajach członkowskich unii europejskiej, zeszyty naukowe szkoły głównej gospodarstwa wiejskiego w warszawie. polityki europejskie, finanse i marketing 5(54), 101–114. mota, p. r., fernandes, a. l. c., nicolescu, a.-c. (2012), the recent dynamics of public debt in the european union: a matter of fundamentals or the result of a failed monetary experiment? fep working papers 467, universidade do porto, faculdade de economia do porto. niglio, m., giordano, f., vitale, c. d. (2012), on the stationarity of the threshold autoregressive process: the two regimes case in giommi, a. (ed.) 46th scientific meeting of the italian statistical society. osińska, m., kluth, k. (2011), konwergencja gospodarcza krajów europy środkowowschodniej do poziomu ue w latach 1995–2009 (economic convergence of central and east european countries to the eu level in 1995–2009), zeszyty naukowe wsg w bydgoszczy ekonomia, 3, 141–150. osińska, m., kufel, t., blazejowski, m., kufel, p. (2015), zegar cyklu koniunkturalnego państw ue i usa w latach 1995–2013 w świetle badań synchronizacji, (business cycle clock for the eu and the usa in 1995–2013 in the light of synchronization research), prace naukowe uniwersytetu ekonomicznego we wrocławiu, taksonomia, 25, 138–146, doi: http://dx.doi.org/10.15611/pn.2015.385.15. osińska, m., kufel, t., blazejowski, m., kufel, p. (2016a), business cycle synchronization in eu economies after the recession of the years 2007–2009, argumenta oeceonomica, 2(37), doi: http://dx.doi.org/10.15611/aoe.2016.2.01. osińska, m., kufel, t., blazejowski, m., kufel, p. (2016b), does economic growth really depend on the magnitude of debt? a threshold model approach, university library of munich, mpra paper, 71476. petruccelli, j. d., woolford, s. w. (1984). a threshold ar(1) model, journal of applied probability, 21(2), 270–286, doi: http://dx.doi.org/10.1017/s0021900200024670. rubinstein, r. y., kroese, d. p. (2011), simulation and the monte carlo method, john wiley & sons. saleh, a. s., harvie, c. (2005). the budget deficit and economic performance: a survey, the singapore economic review, 50(02), 211–243, doi: http://dx.doi.org/10.1142/s0217590805001986. schclarek, a. (2005), debt and economic growth in developing and industrial countries. working papers 2005:34, lund university, department of economics. spence, m. (2011), the next convergence. the future of economic growth in a multispeed world, new york: farrar, straus and giroux. http://dx.doi.org/10.1142/s0217590805001986 magdalena osińska, tadeusz kufel, marcin błażejowski, paweł kufel dynamic econometric models 16 (2016) 145–164 164 stańczyk, z., wyrobek, j. (2013), zastosowanie analizy spektralnej do weryfikacji hipotezy o rozłączeniu się gospodarek rozwiniętych i wschodzących (spectral analysis-based verification of the decoupling hypothesis between developed and emerging economies), zeszyty naukowe uniwersytetu ekonomicznego w krakowie, 920, 5–21. tong, h. (1990), non-linear time series: a dynamical system approach, oxford university press, oxford. tong, h. (2007). birth of the threshold time series model, statistica sinica, 17(1), 8–14, doi: http://dx.doi.org/10.1142/9789812836281_0001. tong, h. (2011), threshold models in time series analysis–30 years on, statistics and its interface, 4(2), 107–118, doi: http://dx.doi.org/10.4310/sii.2011.v4.n2.a1. zarnowitz, v. (1999), theory and history behind business cycles: are the 1990s the onset of a golden age? journal of economic perspectives, 13(2), 69–90, doi: http://dx.doi.org/10.1257/jep.13.2.69. modelowanie i prognozowanie cyklu koniunkturalnego w krajach europy środkowej i wschodniej za pomocą podejścia progowego z a r y s t r e ś c i : artykuł przedstawia badanie cykli koniunkturalnych gospodarek państw europy środkowej i wschodniej w porównaniu do gospodarki unii europejskiej przy wykorzystaniu nieliniowego podejścia – modeli progowych (tar). rozważanymi zmiennymi progowymi są: stopa inflacji, krótko i długoterminowa stopa procentowa, stopa bezrobocia oraz kurs walutowy do dolara. celem artykułu jest modelowanie oraz prognozowanie cyklu koniunkturalnego w państwach europy środkowej i wschodniej oraz porównanie ich do cykli dla całej unii europejskiej oraz strefy euro. prognozowanie cyklu koniunkturalnego za pomocą technik bootstrapowych daje obiecujące wyniki, szczególnie gdy wykorzystywane są bootstrapowe przedziały ufności. s ł o w a k l u c z o w e: cykl koniunkturalny, business cycle, central and eastern economies, threshold models, forecasting, bootstrap. microsoft word 03_pajor_a.docx dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 anna pajor cracow university of economics bayesian optimal portfolio selection in the msf-sbekk model† a b s t r a c t. the aim of this paper is to investigate the predictive properties of the msf-scalar bekk(1,1) model in context of portfolio optimization. the msf-sbekk model has been proposed as a feasible tool for analyzing multidimensional financial data (large n), but this research examines forecasting abilities of this model for n = 2, since for bivariate data we can obtain and compare predictive distributions of the portfolio in many other multivariate sv specifications. also, approximate posterior results in the msf-sbekk model (based on preliminary estimates of nuisance matrix parameters) are compared with the exact ones. k e y w o r d s: portfolio analysis, msv models, msf-sbekk model, forecasting. introduction it is well known that in portfolio selection (computing the weights of the assets in the portfolio) correlations among the assets are essential. the weights of the minimum variance portfolio depend on the conditional covariance matrix (see aguilar, west, 2000, pajor, 2009). thus for active portfolio management multivariate time series forecasts should be applied. the aim of the paper is to examine the predictive properties of the msfsbekk model (being the hybrid of the multiplicative stochastic factor and scalar bekk specifications; hence the model is called msf-sbekk; see osiewalski, pajor, 2009) in context of the optimal portfolio selection problem. the multi-period minimum conditional variance portfolio is considered (as in pajor, 2009). in the optimization process we use the predictive distributions of future returns and the predictive conditional covariance matrices obtained from the bayesian msf-sbekk and other multivariate stochastic volatility (msv) models. in order to compare predictive results in the msf-sbekk model with † research supported by a grant from the cracow university of economics. the author would like to thank janusz jaworski for language verification of the manuscript. anna pajor 42 those obtained in other msv specifications, we consider only bivariate portfolios. the bivariate stochastic volatility models are used to describe the daily exchange rate of the euro against the polish zloty and the daily exchange rate of the us dollar against the polish zloty. based on these two currencies we consider the bayesian portfolio selection problem. in the next section we briefly present the msf-sbekk model. section 3 is devoted to the optimal portfolio construction. in section 4 we present and discuss the empirical results. some concluding remarks are presented in the last section. 1. bayesian msf-sbekk model let xj,t denote the price of asset j (or the exchange rate as in our application) at time t for j = 1,2, ..., n and t = 1, 2, ..., t+s. the vector of growth rates yt =(y1,t, y2,t, ..., yn,t), where yj,t = 100 ln (xj,t/xj,t-1), is modelled using the basic var(1) framework: ttt ξryδy  1 , t = 1, 2, ... ,t, t+1, ..., t+s, (1) where { tξ } is a process with time-varying volatility, t denotes the number of the observations used in estimation, and s is the forecast horizon, δ is a n-dimensional vector, r is a nn matrix of parameters. following osiewalski and pajor (2009), for tξ we assume the so-called type i msf-sbekk(1,1) hybrid specification: tttt g εhξ 2/1 , (2) tgtt gg   1lnln , ),(~})','{( 1]1)1[(  nntt iin i0ε  , 1 (3)   111 ')1(   tttt hξξah  . (4) that is, tξ is conditionally normal with mean vector 0 and covariance matrix gtht, where gt is a latent process and ht is a square matrix of order n that has the scalar bekk(1,1) structure . thus, the conditional distribution of yt, given its past and latent variables, is normal with mean 1 tt ryδy and covariance matrix gtht. the model defined by (2)-(4) includes as special cases two simple basic structures. when 0g and  = 0 we have the scalar bekk(1,1) model, while β = 0 and γ = 0 lead to the msf model (see osiewalski, pajor, 2009). note that the model has one latent process which helps in explaining outlying observations, and time-varying conditional correlations as in the scalar bekk(1,1) structure. 1 { )','( tt ε } is a sequence of independent and identically distributed normal random vectors with mean vector zero and covariance matrix in+1. bayesian optimal portfolio selection in the msf-sbekk model 43 in (4) a is a free symmetric positive definite matrix of order n; for a-1 we assume the wishart prior with n degrees of freedom and mean in; β and γ are free scalar parameters, jointly uniformly distributed over the unit simplex. as regards initial conditions for ht, we take h0 = h0in and treat h0 > 0 as an additional parameter, a priori exponentially distributed with mean 1. for the parameters of the latent process we use the same priors as osiewalski, pajor (2009); for  : normal with mean 0 and variance 100, truncated to (-1, 1), for 2g : exponential with mean 200; g0 is equal 1. the n(n+1) elements of )')'((0 rδδ vec are assumed to be a priori independent of remaining parameters, with the n(0, in(n+1)) prior truncated by the restriction that all eigenvalues of r lie inside the unit circle. in this paper we also want to check how the approximation proposed and explained in osiewalski, pajor (2009) influences the predictive distribution of future logarithmic returns and, in consequence, the optimal portfolio composition. therefore we apply this approximation. that is, we use ordinary least squares (ols) for the var(1) parameters and replace a by the empirical covariance matrix of the ols residuals from the var(1) part. the bayesian analysis for the remaining parameters and future return rates is based on the conditional posterior and predictive distributions given the particular values of vector δ0 and matrix a. all distributions are sampled using the gibbs scheme with metropolis-hastings steps, as shown in detail in osiewalski, pajor (2009). 2. portfolio selection problem in the msf-sbekk model we denote by tθ the latent variable vector at time t, by θ the parameter vector, and we assume that: a) ttt εσξ 2/1 , where ),(~}{ nt iin i0ε , b) tς is a function of the latent variables θ for t , and of the past of tξ , i.e. );,( 1 tt    ξθσς , c) the vector tξ , conditional on );,( 1 t   ξθ , is independent of );( t θ . in pajor (2009) it was assumed that );( tt  θσς . now we relax the assumption, allowing tς to depend on the past of tξ as in the msf-sbekk model. the s-period portfolio at time t is defined by a vector wt+s|t = (w1,t+s|t, w2,t+s|t, ..., wn,t+s|t), where wi,t+s|t is the fraction of wealth invested in asset i (1  i  n). the return on the portfolio that places weight wi,t+s|t on asset i at time t is approximately a weighted average of the returns on anna pajor 44 the individual assets. the weight applied to each return is the fraction of the portfolio invested in that asset: tstw n i tstitstitstw rzwr |, 1 |,|,|, ~      , (5) where zi,t+s|t is the rate of return on the asset i from the period t to t+s, i.e.      st tt titsti yz 1 ,|, (i = 1, ..., n). if tst |σ is the matrix of conditional covariances of zt+s|t = (z1,t+s|t, z2,t+s|t, ..., zn,t+s|t), then the conditional variance of return on the portfolio is ),...,,|~( |, sttttstwrvar  θθ = tsttsttsttstv ||| 2 | '   wσw , (6) where t is the -algebra generated by ε and θ for t , i.e. );,( tt    θε . the vector of the rates of return at time t+k (k > 0, k  s) satisfies: . 1 1 0         k j jt jk t k k j j kt ξryrδry (7) based on equation (7) we have: , 1 011 1 0 |            s j jt js i i s k t k s k k j j tst ξryrδrz (8) since 0θθξξ  ),...,,|'( stttjtite  for i  j, the conditional covariance matrix of zt+s|t in the msf-sbekk(1,1) model becomes: ,)'()( 1 0 * 0 |           s j js i i jt js i i tst rσrς (9) where .),...,,|'(* stttjtjtjt e   θθξξς  finally, the conditional variance of return on the portfolio is: ),...,,|~( |, sttttstwrvar  θθ = .)'()(' | 1 0 * 0 | tst s j js i i jt js i i tst           wrσrw it is easy to show that in the msf-sbekk(1,1) model: ,),...,,|'( 1111   ttsttttt ge hθθξξ  ,')1(),...,,|( 11 tttsttttt e hξξaθθhh    and for 2 < k  s: bayesian optimal portfolio selection in the msf-sbekk model 45 )],,...,,|()()1[( ),...,,|'( 11 stttktktkt stttktkt egg e     θθha θθξξ   ).,...,,|()()1( ),...,,|( 11 stttktkt stttkt eg e     θθha θθh   consequently2: ).()1(1 ),...,,|'( 1 1 1 2 1 1                       k j jkttkt k i i j jktkt stttktkt gggg e ha θθξξ the most popular approach assumes that the investor selects the portfolio with minimum variance (see markowitz, 1959, elton, gruber, 1991). here we assume that the conditional variance of the portfolio is minimized and that short sales are allowed (wi,t+s|t < 0 reflects a short selling). then the problem for the investor reduces to solving the quadratic programming problem: tsttsttst tst ||| 'min |   wσw w subject to w1,t+s|t + w2,t+s|t + ... +wn,t+s|t = 1. in this way we obtain so-called the minimum conditional variance portfolio (the portfolio that has the lowest risk of any feasible portfolio): , ' 1 | 1 | |, ισι ις w       tst tst tstmv (10) which has a return: , ' ' 1 | | 1 | |, ισι zσι        tst tsttst tstmvr (11) and the conditional variance at time t: , ' 1 ),...,,|'( 1 | 2 |,|, ισι θθyw     tst tstmvstttsttstmv vvar  (12) where ι is an n1 vector of ones. next we consider a s-period portfolio selection problem where the investor minimizes the conditional variance of the portfolio with a given level of return * |,|, ~ tstptstw rr   . this problem reduces to solving the quadratic programming problem: 2 a very similar result was obtained by piotr de silva in his unpublished master’s dissertation. anna pajor 46 tsttsttst tst ||| 'min |   wσw w subject to         .' ,1' * |,|| | sttptsttst tst rzw ιw when * |,|, ~ tstptstw rr   , the solution for the s-period portfolio is: . )'()')('( ))(''( 2 | 1 || 1 || 1 | | * |, 1 || 1 | 1 || 1 | |,* tsttsttsttsttsttst tsttstptsttsttsttsttsttst tstmvr r p                     zσιzσzισι zισιzσσιzς w (13) it is important to stress that the classic portfolio choice scheme assumes the covariance matrix and expected returns at time t to be known. in our bayesian models the minimum conditional variance portfolio ( tstmv |, w ), and the minimum conditional variance portfolio with a given level of return, ( tstmvrp |, *  w ) are random vectors as measurable functions of zt+s|t, and tst |σ . hence, the predictive distributions of tstmv |, w , and tstmvrp |,*  w (also, of tstmvv |,  , and tstmvrp v |,*  ) are induced by the distribution of zt+s|t, and tst |σ . in practice, to compute the weights of the assets in the portfolio we must use some characteristic of these predictive distributions. as the predictive mean (for tstmv |, w or tstmvrp |,*  w ) may not exist, we consider the predictive medians of tstimv |,, w and tstimvrp |,,*  w , denoted by )',...,( |,,|,1,|, op tstnmv op tstmv op tstmv ww  w and )',...,( |,,|,|, *** op tstnmvr op tstmvr op tstmvr ppp ww  w , and defined respectively by conditions: 5.0}|pr{ |,,|,,   yww op tstimvtstimv and 5.0}|pr{ |,,|,,   yww op tstimvtstimv , 5.0}|pr{ |,,|,, **   yww op tstimvrtstimvr pp and ,5.0}|pr{ |,,|,, **   yww op tstimvrtstimvr pp for i = 1, ..., n-1, and      1 1 |,,|,, 1 n i op tstimv op tstnmv ww ,      1 1 |,,|,, ** 1 n i op tstimvr op tstnmvr pp ww . in multivariate stochastic variance models there is no analytical solution for the optimal portfolio selection problem even for n = 2 assets. to evaluate the quantiles of the predictive distributions of tstmv |, w and tstmvrp |,*  w , and then find the portfolio, we use markov chain monte carlo methods – the gibbs sampler with the metropolis-hastings algorithm. bayesian optimal portfolio selection in the msf-sbekk model 47 3. empirical results as the dataset we use the same daily exchange rates as in pajor (2009). thus, we consider the daily exchange rate of the euro against the polish zloty and the daily exchange rate of the us dollar against the polish zloty from january 2, 2002 to june 29, 2007. the data were downloaded from the website of the national bank of poland. the dataset of the percentage daily logarithmic growth (return) rates, yt, consists of 1388 observations (for each series). as the first growth rates are used as initial conditions, t = 1387 remaining observations on yt are modelled. 3.1. bayesian model comparison in table 1 we rank the models by the increasing value of the decimal logarithm of the bayes factor of var(1)-sjsv against the alternative models. we see that for our dataset the models with three latent processes describe the time-varying conditional covariance matrix much better than the models with one or two latent processes. the var(1)-sjsv model wins our model comparison, being about 8.5 orders of magnitude better than the var(1)-tsveur_usd model. the decimal log of the bayes factor of the var(1)-msf-sbekk model relative to the var(1)-sjsv model is 27.32. the presence of more latent processes improves fit enormously, but seems infeasible for highly dimensional time series. assuming equal prior model probabilities, the var(1)-msfsbekk model is about 20.73 orders of magnitude more probable a posterior than the var(1)-msf model (with the constant conditional correlations), and about 32 orders of magnitude better than the var(1)-sbekk model. note that the var(1)-msf-sbekk model is about 6.6 orders of magnitude better than another hybrid model – the var(1)-msf-dcc model, proposed by osiewalski, pajor (2007). table 1. logs of bayes factors in favour of var(1)-sjsv model model number of latent processes number of parameters log10 (bsjsv,i) rank var(1)-sjsv 3 18 0 1 var(1)-tsveur_usd 3 18 8.51 2 var(1)-tsvusd_eur 3 18 11.10 3 var(1)-jsv 2 15 19.60 4 var(1)-msf-sbekk 1 14 27.32 5 var(1)-msf-sbekk with the approximation 1 14 29.30 6 var(1)-msf-idcc 1 18 32.00 7 var(1)-msf-dcc 1 20 33.88 8 var(1)-msf(sdf) 1 12 48.05 9 var(1)-sbekk 0 12 59.70 10 var(1)-bmsv 2 14 158.51 11 note: the decimal logarithm of the bayes factors were calculated using the newton and raftery method (see newton, raftery, 1994). only the results for the var(1)-msf-sbekk, var(1)-msf and var(1)-sbekk models are new; the remaining ones were obtained by pajor (2009). anna pajor 48 in the bivariate case considered here it is possible to compare exact and approximate bayesian results relate to estimation of the var(1)-msf-sbekk model. thus, in tables 1 we present the decimal logarithm of the bayes factor for both cases. using the approximate bayesian approach proposed by osiewalski, pajor (2009) leads to smaller values of the data density, but it seems that the fit does not significantly change. of course, our model comparison relies on the prior distributions for the parameters of the models, but these prior distributions are not very informative. 3.1. predictive properties of the msf-sbekk models in portfolio selection it is important to investigate the predictive properties of the msf-sbekk model in portfolio selection. in addition, we can examine how the exact and approximate posterior results may differ. thus, in this section we report the results of building the optimal portfolios using the msf-sbekk model. we consider the hypothetical portfolios, which consist of two currencies: the us dollar and euro. we assume that there are no transaction costs and that we may reallocate zloty to long as well as to short positions across the currencies. allocation decisions are made at time t based on the predictive distribution for yt+k and kt σ for k =1, ..., 60. )|( |,1, ytstmvwp  sjsv tsveur_usd msf-sbekk msf-sbekk with app. msf sbekk figure 1. quantiles of the predictive distributions of the minimum conditional variance portfolios (the fractions of wealth invested in the us dollar). the central black lines represent the medians, and the grey lines represent the quantiles of order 0.05, 0.25, 0.75, 0.95, respectively -2 -1 0 1 20 0 707 -0 2 20 0 707 -0 9 20 0 707 -1 6 20 0 707 -2 3 20 0 707 -3 0 20 0 708 -0 6 20 0 708 -1 3 20 0 708 -2 0 20 0 708 -2 7 20 0 709 -0 3 20 0 709 -1 0 20 0 709 -1 7 20 0 709 -2 4 -2 -1 0 1 2 00 707 -0 2 2 00 707 -0 9 2 00 707 -1 6 2 00 707 -2 3 2 00 707 -3 0 2 00 708 -0 6 2 00 708 -1 3 2 00 708 -2 0 2 00 708 -2 7 2 00 709 -0 3 2 00 709 -1 0 2 00 709 -1 7 2 00 709 -2 4 -2 -1 0 1 20 07 -0 702 20 07 -0 709 20 07 -0 716 20 07 -0 723 20 07 -0 730 20 07 -0 806 20 07 -0 813 20 07 -0 820 20 07 -0 827 20 07 -0 903 20 07 -0 910 20 07 -0 917 20 07 -0 924 -2 -1 0 1 20 07 -0 7 -0 2 20 07 -0 7 -0 9 20 07 -0 7 -1 6 20 07 -0 7 -2 3 20 07 -0 7 -3 0 20 07 -0 8 -0 6 20 07 -0 8 -1 3 20 07 -0 8 -2 0 20 07 -0 8 -2 7 20 07 -0 9 -0 3 20 07 -0 9 -1 0 20 07 -0 9 -1 7 20 07 -0 9 -2 4 -2 -1 0 1 20 07 -0 702 20 07 -0 709 20 07 -0 716 20 07 -0 723 20 07 -0 730 20 07 -0 806 20 07 -0 813 20 07 -0 820 20 07 -0 827 20 07 -0 903 20 07 -0 910 20 07 -0 917 20 07 -0 924 -2 -1 0 1 20 07 -0 702 20 07 -0 709 20 07 -0 716 20 07 -0 723 20 07 -0 730 20 07 -0 806 20 07 -0 813 20 07 -0 820 20 07 -0 827 20 07 -0 903 20 07 -0 910 20 07 -0 917 20 07 -0 924 bayesian optimal portfolio selection in the msf-sbekk model 49 in figure 1 we show the quantiles of the predictive distributions of the minimum conditional variance portfolio sttmvw |,1, (the fraction of wealth invested in the us dollar). if the medians of the marginal predictive distributions are treated as point forecasts, in model with time-varying conditional correlation coefficient the optimum weights to invest in the usd/pln are negative, indicating the short sale of the us dollar (the median of the marginal predictive distribution of sttmvw |,1, is equal to about -0.4 in the most probable a posterior model, and about -0.22 in the var(1)-msf-sbekk model). the short position on the us dollar is connected with corresponding long position on the euro. we see that in var(1)-msv models with more than one latent process the predictive distributions are very widely dispersed and fat-tailed, thus leaving us with considerable uncertainty about the future returns of these portfolios. surprisingly, in the var(1)-msv models with one latent process or in the var(1)-sbekk model the minimum conditional variance portfolios are estimated more precisely – the inter-quartile ranges are relatively small. it seams that the var(1)-msf-sbekk and var(1)-sbekk models produce portfolios with lowest risk measured by the conditional variance (see figure 2). note that the predictive distributions of sttmvw |,1, for s = 1, ..., 60 produced by the var(1)-msf-sbekk model are located in areas of high predictive densities obtained in the best model (i.e. var(1)-sjsv). )|( |, ytstmvvp  sjsv tsveur_usd msf-sbekk msf-sbekk with app. msf sbekk figure 2. quantiles of the predictive distributions of the conditional standard deviation of the minimum conditional variance portfolios. the central black lines represent the medians, and the grey lines represent the quantiles of order 0.05, 0.25, 0.75, 0.95, respectively 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 707 -0 2 2 00 707 -0 9 2 00 707 -1 6 2 00 707 -2 3 2 00 707 -3 0 2 00 708 -0 6 2 00 708 -1 3 2 00 708 -2 0 2 00 708 -2 7 2 00 709 -0 3 2 00 709 -1 0 2 00 709 -1 7 2 00 709 -2 4 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 707 -0 2 2 00 707 -0 9 2 00 707 -1 6 2 00 707 -2 3 2 00 707 -3 0 2 00 708 -0 6 2 00 708 -1 3 2 00 708 -2 0 2 00 708 -2 7 2 00 709 -0 3 2 00 709 -1 0 2 00 709 -1 7 2 00 709 -2 4 0 0.5 1 1.5 2 2.5 3 3.5 4 20 0 7 -0 702 20 0 7 -0 709 20 0 7 -0 716 20 0 7 -0 723 20 0 7 -0 730 20 0 7 -0 806 20 0 7 -0 813 20 0 7 -0 820 20 0 7 -0 827 20 0 7 -0 903 20 0 7 -0 910 20 0 7 -0 917 20 0 7 -0 924 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 707 -0 2 2 00 707 -0 9 2 00 707 -1 6 2 00 707 -2 3 2 00 707 -3 0 2 00 708 -0 6 2 00 708 -1 3 2 00 708 -2 0 2 00 708 -2 7 2 00 709 -0 3 2 00 709 -1 0 2 00 709 -1 7 2 00 709 -2 4 0 0.5 1 1.5 2 2.5 3 3.5 4 20 0 70 70 2 20 0 70 70 9 20 0 70 71 6 20 0 70 72 3 20 0 70 73 0 20 0 70 80 6 20 0 70 81 3 20 0 70 82 0 20 0 70 82 7 20 0 70 90 3 20 0 70 91 0 20 0 70 91 7 20 0 70 92 4 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 707 -0 2 2 00 707 -0 9 2 00 707 -1 6 2 00 707 -2 3 2 00 707 -3 0 2 00 708 -0 6 2 00 708 -1 3 2 00 708 -2 0 2 00 708 -2 7 2 00 709 -0 3 2 00 709 -1 0 2 00 709 -1 7 2 00 709 -2 4 anna pajor 50 as in pajor (2009) we can see that the predictive distributions related to the portfolio with bound on return are more diffuse – the inter-quartile ranges are higher (see figure 3 and 4). comparing the minimum conditional variance portfolio and the minimum conditional variance portfolio with the return equal to at least 5%, we can see that the distributions of the forecasted value of sttmvrp w |,* and sttmvrp v |,* are more dispersed and have very thick tails. thus uncertainty connected with the optimal portfolio with return at least 5% on annual base is huge. in all models the quantiles of the conditional standard deviation of the optimal portfolios (see figure 4) indicate increasing volatility with the forecast horizon. )|( |,1,* y tstpmvr wp  sjsv tsveur_usd msf-sbekk msf-sbekk with app. msf sbekk figure 3. quantiles of the predictive distributions of the minimum conditional variance portfolios with the return equal to at least 5% on annual base (the fraction of wealth invested in the us dollar). the central black lines represent the medians, and the grey lines represent the quantiles of order 0.05, 0.25, 0.75, 0.95, respectively finally, as in pajor (2009) we use the medians of tstmvrp w |,1,*  to construct hypothetical portfolios for s = 1, 2, .., 60. let wt = 10000 pln be the initial wealth of the investor at time t (on june 29, 2007). if we assume that there are no transaction costs and the investor uses the median of the predictive distribution of tstmvrp |, *  w (denoted by op tstmvrp w |,1,*  ) to construct optimal portfolio, then the investor’s wealth at time t+s is given by: -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 20 07 -0 702 20 07 -0 709 20 07 -0 716 20 07 -0 723 20 07 -0 730 20 07 -0 806 20 07 -0 813 20 07 -0 820 20 07 -0 827 20 07 -0 903 20 07 -0 910 20 07 -0 917 20 07 -0 924 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 2 0 0 7 -0 70 2 2 0 0 7 -0 70 9 2 0 0 7 -0 71 6 2 0 0 7 -0 72 3 2 0 0 7 -0 73 0 2 0 0 7 -0 80 6 2 0 0 7 -0 81 3 2 0 0 7 -0 82 0 2 0 0 7 -0 82 7 2 0 0 7 -0 90 3 2 0 0 7 -0 91 0 2 0 0 7 -0 91 7 2 0 0 7 -0 92 4 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 2 00 7 -0 7 -0 2 2 00 7 -0 7 -0 9 2 00 7 -0 7 -1 6 2 00 7 -0 7 -2 3 2 00 7 -0 7 -3 0 2 00 7 -0 8 -0 6 2 00 7 -0 8 -1 3 2 00 7 -0 8 -2 0 2 00 7 -0 8 -2 7 2 00 7 -0 9 -0 3 2 00 7 -0 9 -1 0 2 00 7 -0 9 -1 7 2 00 7 -0 9 -2 4 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 2 00 707 -0 2 2 00 707 -0 9 2 00 707 -1 6 2 00 707 -2 3 2 00 707 -3 0 2 00 708 -0 6 2 00 708 -1 3 2 00 708 -2 0 2 00 708 -2 7 2 00 709 -0 3 2 00 709 -1 0 2 00 709 -1 7 2 00 709 -2 4 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 20 07 -0 702 20 07 -0 709 20 07 -0 716 20 07 -0 723 20 07 -0 730 20 07 -0 806 20 07 -0 813 20 07 -0 820 20 07 -0 827 20 07 -0 903 20 07 -0 910 20 07 -0 917 20 07 -0 924 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 20 07 -0 702 20 07 -0 709 20 07 -0 716 20 07 -0 723 20 07 -0 730 20 07 -0 806 20 07 -0 813 20 07 -0 820 20 07 -0 827 20 07 -0 903 20 07 -0 910 20 07 -0 917 20 07 -0 924 bayesian optimal portfolio selection in the msf-sbekk model 51 )]/()/([ ,2,2|,2,,1,1|,1,|, *** tst op tstmvrtst op tstmvrttstmvr xxwxxwww ppp   , s = 1, 2, .., 60. )|( |,* y tstpmvr vp  sjsv tsveur_usd msf-sbekk msf-sbekk with app. msf sbekk figure 4. quantiles of the predictive distributions of the conditional standard deviation of the minimum conditional variance portfolio with the return equal to at least 5% on annual base. the central black lines represent the medians, and the grey lines represent the quantiles of order 0.05, 0.25, 0.75, 0.95, respectively figure 5. wealth of the investor at time t+s for s = 1, ..., 60 (the optimal portfolio is constructed on the medians of tstpmvr w |,*  ) 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 707 -0 2 2 00 707 -0 9 2 00 707 -1 6 2 00 707 -2 3 2 00 707 -3 0 2 00 708 -0 6 2 00 708 -1 3 2 00 708 -2 0 2 00 708 -2 7 2 00 709 -0 3 2 00 709 -1 0 2 00 709 -1 7 2 00 709 -2 4 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 70 70 2 2 00 70 70 9 2 00 70 71 6 2 00 70 72 3 2 00 70 73 0 2 00 70 80 6 2 00 70 81 3 2 00 70 82 0 2 00 70 82 7 2 00 70 90 3 2 00 70 91 0 2 00 70 91 7 2 00 70 92 4 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 7 -0 702 2 00 7 -0 709 2 00 7 -0 716 2 00 7 -0 723 2 00 7 -0 730 2 00 7 -0 806 2 00 7 -0 813 2 00 7 -0 820 2 00 7 -0 827 2 00 7 -0 903 2 00 7 -0 910 2 00 7 -0 917 2 00 7 -0 924 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 707 -0 2 2 00 707 -0 9 2 00 707 -1 6 2 00 707 -2 3 2 00 707 -3 0 2 00 708 -0 6 2 00 708 -1 3 2 00 708 -2 0 2 00 708 -2 7 2 00 709 -0 3 2 00 709 -1 0 2 00 709 -1 7 2 00 709 -2 4 0 0.5 1 1.5 2 2.5 3 3.5 4 20 07 -0 702 20 07 -0 709 20 07 -0 716 20 07 -0 723 20 07 -0 730 20 07 -0 806 20 07 -0 813 20 07 -0 820 20 07 -0 827 20 07 -0 903 20 07 -0 910 20 07 -0 917 20 07 -0 924 0 0.5 1 1.5 2 2.5 3 3.5 4 2 00 7 -0 7 -0 2 2 00 7 -0 7 -0 9 2 00 7 -0 7 -1 6 2 00 7 -0 7 -2 3 2 00 7 -0 7 -3 0 2 00 7 -0 8 -0 6 2 00 7 -0 8 -1 3 2 00 7 -0 8 -2 0 2 00 7 -0 8 -2 7 2 00 7 -0 9 -0 3 2 00 7 -0 9 -1 0 2 00 7 -0 9 -1 7 2 00 7 -0 9 -2 4 9850 10000 10150 10300 10450 20 07 -0 702 20 07 -0 705 20 07 -0 708 20 07 -0 711 20 07 -0 714 20 07 -0 717 20 07 -0 720 20 07 -0 723 20 07 -0 726 20 07 -0 729 20 07 -0 801 20 07 -0 804 20 07 -0 807 20 07 -0 810 20 07 -0 813 20 07 -0 816 20 07 -0 819 20 07 -0 822 20 07 -0 825 20 07 -0 828 20 07 -0 831 20 07 -0 903 20 07 -0 906 20 07 -0 909 20 07 -0 912 20 07 -0 915 20 07 -0 918 20 07 -0 921 20 07 -0 924 jsv sjsv tsv_e_u tsv_u_e msf-sbekk sbekk msf bmsv bank deposit equally-weighted portf olio app. msf-sbekk anna pajor 52 in figure 5, we present the plot of tstmvrp w |,*  for s =1, 2, ..., 60, and compare them with a bank deposit with the interest rate equal to 4.7% on annual base (the quotation of the 3-month warsaw interbank offered rate on june, 29 2007). the best results we obtain in the var(1)-jsv model – at a 2-month horizon the average return of the optimal portfolios is equal to 0.098%, which represents annual return of 24.58%. in the best model (i.e. var(1) – sjsv) the average return of the optimal portfolios is equal to 0.065%, which represents annual return of 16.34%, whereas in the var(1)-msf-sbekk and var(1)sbekk models we have 0.048% and 0.044%, respectively (i.e. 12.02% and 11.05% per annum, respectively). it is important to stress that in the var(1)msf-sbekk model the returns of the hypothetical investments are higher than those of the bank deposit, indicating good forecasting properties of the model. in the var(1)-msf model (with constant conditional correlation) the average return of the portfolio is negative (we obtained -0.006% i.e. -0.16% per annum). thus the sbekk structure is very important in forecasting. in the approximated var(1)-msf-sbekk model the average return is equal to 0.04% (i.e. 9.43% per annum). thus using approximation in the var(1)-msf-sbekk leads to worse predictive results. after two months the return of the optimal portfolio is lower than the interest rate of the bank deposit, but still it is positive. note that the average return of equally-weighted portfolio is equal to -0.047, i.e. -11.80% per annum. conclusions the paper investigates the predictive abilities of the var(1)-msf-sbekk model in portfolio selection. the predictive distributions of the optimal portfolios produced by the var(1)-msf-sbekk model are compared with those obtained in unparsimonious (but more probable a posterior) msv specifications. the predictive distributions of the weights of the optimal portfolios produced by the var(1)-msf-sbekk model are located in areas of high predictive densities obtained in the best msv model (i.e. var(1)-sjsv). unfortunately, in all models the predictive distributions of the optimal portfolio are very spread and have heavy tails. our main finding is that the var(1)-msf-sbekk model is useful (but not very impressive) for building the multi-period optimal minimum conditional variance portfolio. it seems that the approximation proposed by osiewalski, pajor (2009) results in worse predictive properties of the var(1)-msf-bekk model, but for large portfolios this approximation is necessary. bayesian optimal portfolio selection in the msf-sbekk model 53 references aguilar, o., west, m. (2000), bayesian dynamic factor models and portfolio allocation, journal of business and economic statistics, 18, 338–357. elton, j.e., gruber, m.j. (1991), modern portfolio theory and investment analysis, john wiley & sons, inc, new york. markowitz, h.m. (1959), portfolio selection: efficient diversification of investments, new york, john wiely & sons, inc. newton, m.a., raftery, a.e. (1994), approximate bayesian inference by the weighted likelihood bootstrap [with discussion], journal of the royal statistical society b, 56(1), 3–48 osiewalski, j. (2009), new hybrid models of multivariate volatility (a bayesian perspective), przegląd statystyczny (statistical review), 56, z. 1, 15–22. osiewalski, j., pajor, a. (2007), flexibility and parsimony in multivariate financial modelling: a hybrid bivariate dcc–sv model, [in:] financial markets. principles of modeling, forecasting and decision-making (findecon monograph series no.3), [ed.:] w. milo, p. wdowiński, łódź university press, łódź, 11–26. osiewalski, j., pajor, a. (2009), bayesian analysis for hybrid msf–sbekk models of multivariate volatility, central european journal of economic modelling and econometrics, 1(2), 179–202. pajor, a. (2009), bayesian portfolio selection with msv models, przegląd statystyczny (statistical review), 56, z. 1, 40–55. bayesowska optymalizacja portfela w modelu msf-sbekk z a r y s t r e ś c i. celem artykułu jest analiza prognostycznych własności bayesowskiego modelu msf-sbekk w kontekście wyboru optymalnego portfela inwestycyjnego. wykorzystywany w artykule wielowymiarowy proces msf-sbekk posiada elementy struktury skalarnego procesu bekk oraz procesu msf. obecność, w jego definicji, odrębnego czynnika losowego pozwala lepiej opisywać zjawisko grubych ogonów, zaś w strukturze sbekk uzależnia się warunkowe wariancje oraz warunkowe korelacje od przeszłych wartości procesu. proces msf-sbekk posiada zatem nietrywialną strukturę i może być wykorzystany do opisu zależności miedzy stopami zwrotu kilkudziesięciu (a nawet kilkuset) instrumentów finansowych. w artykule dokonane zostało porównanie prognoz uzyskanych w dwuwymiarowym modelu msf-sbekk oraz w innych modelach z klasy msv na przykładzie portfela walutowego, złożonego z kursu dolara amerykańskiego oraz euro. uzyskane wyniki wskazują na dobre własności prognostyczne modelu msfsbekk, choć uproszczenia w sposobie jego estymacji mogą je pogarszać. s ł o w a k l u c z o w e: model msf-sbekk, modele msv, analiza portfelowa, prognoza. microsoft word 04_bien_barkowska_k.docx dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 katarzyna bień-barkowska warsaw school of economics, national bank of poland distribution choice for the asymmetric acd models† a b s t r a c t. in the paper, i generalize the asymmetric autoregressive conditional duration (aacd) model proposed by bauwens and giot (2003) with respect to the generalized gamma and the burr distribution for an error term. i derive the log likelihood functions for the augmented models and show how to check the goodness-of-fit of the distributional assumptions with the application of the probability integral transforms proposed by diebold, gunther and tay (1998). moreover, i present an exemplary empirical application of the asymmetric acd model for the durations between submissions of market or best limit orders on the interbank trading platform for the polish zloty. i test the impact of selected market microstructure factors (i.e. the bid-ask spread, volatility) on the time of order submissions. k e y w o r d s: asymmetric acd model, financial durations, probability integral transforms, market microstructure. introduction econometric models for financial durations (i.e. time spells between selected events of the trading process) have gained an extreme popularity over the last decade. the standard autoregressive conditional duration (acd) model of engle and russell (1998) and its numerous extensions have become a standard tool in modeling irregularly-spaced financial data. the vast survey of different acd specifications has been recently presented in the study of pecurar (2008). the models have been used in order to test selected market microstructure hypotheses (bauwens, giot, 2001; zhang et al., 2001; hautsch, 2004; nolte, 2008, among others), as well as in order to augment the volatility measures (ghysels, jasiak, 1998; giot, 2000; kalimipalli, warga, 2002; grammig, wellner, 2002; giot, 2005). the outline of the acd models has been † this work has been financed from the polish science budget resources in year 2011 as the part of the research project 03/s/0007/11. katarzyna bień-barkowska 56 covered in many econometric textbooks (bauwens, giot, 2001; gouriéroux, jasiak, 2001; tsay, 2002; osińska, 2006). in this paper i focus on the asymmetric acd model (henceforth aacd model) of bauwens and giot (2003). it is a flexible model for the conditional density of financial durations that elapse if one of two possible end states (i.e. events pointed on the micro-scale by appropriate “thinning” the data) occurs. as an exemplary application of the model, the authors suggest a dynamic description of mid-price durations for selected stocks traded on the nyse. they categorize time intervals between subsequent price movements into (1) “a price increase” duration (if price of a stock increases during the duration) and (2) “a price decrease” duration (if a price of a stock decreases during the duration). the aacd model entwines a logarithmic acd (lacd) specification for the conditional expectations of durations, hence the model accounts for the wellknown clustering that characterizes such processes (see bauwens, giot, 2000). on the other hand, the aacd model has a major advantage over the standard acd specification, as it discriminates between end states of the time spells under study. the model allows for a separate description of expected durations ending as a “price increase” and expected durations ending as a “price decrease”. accordingly, it accounts for the different dynamics characterizing each of the two processes. it also allows for an easy inclusion of explanatory factors that – in this particular setup – may exert a totally different impact on the pace according to which one or the other event (end state) occurs. more recently, the aacd model has been used by lo and sapp (2008) to test the impact of different microstructure factors on the choice and timing of market/limit1 order placement in the reuters dealing 2000-2 spot matching system2. the authors account for various explanatory variables that reflect the state of the order book and some other market conditions (i.e. volatility, timeof-day) and check their influence on the expected time to a market order or a limit order arrival on the ask and bid side of the market. the application of the aacd specification allows for the joint modelling of both: (1) the order choice and (2) the time that elapses between subsequent order submissions. both of the aforementioned studies apply the aacd model with the weibull-distributed error term. such a setup allows for a parsimonous and tractable specification. nevertheless, the studies (bauwens, giot, 2001; bauwens et al., 2004) proved the superiority of some more general and flexible distributions (the burr or generalized gamma distribution) over the exponential or weibull ones, as far as the goodness-of-fit of the acd models is concerned. moreover, a simulation study presented in (grammig, mauer, 2000) proved that 1 a market order is an order to immediately buy or sell an asset at the best prevailing bid or ask prices. a limit order is an order to buy or sell an asset at a specified price (or better) and it is executed if it can be matched with an upcoming order on the opposite market side. 2 reuers dealing 2000-2 spot matching system is a fully automated, order-driven, interbank market for currency trading. distribution choice for the asymmetric acd models 57 the distributional misspecification may lead to serious inference problems: loss of efficiency and bias in the estimated parameters of duration expectations. the aim of this paper is to generalize the standard aacd model of bauwens and giot (2003) with respect to its distributional assumptions. i derive the likelihood function for the aacd model with the burr distribution (henceforth the b-aacd model) and with the generalized gamma distribution (henceforth the gg-aacd model). moreover, i show how to derive the probability integral transforms (pit) of diebold, gunther and tay (1998) for the generalized aacd specifications in order to verify their dynamic and distributional goodness-of-fit. the theoretical issues have been supported with the empirical example. as in the study of lo and sapp (2008), i apply the aacd model to durations between the order submissions in the reuters dealing 3000 spot matching system which is a leading interbank trading platform for the eur/pln currency pair. 1. econometric models 1.1. outline of the asymmetric acd model the asymmetric acd model of bauwens and giot (2003) describes the marked point process { , }i ix y , where 1=i i ix t t  is a duration between moments in which certain events occur: it and 1it  , and iy is a qualitative variable indicating a type of an event:  ,iy a b . in the primary study of bauwens and giot (2003), the states a and b correspond, respectively, to “a price increase” and “a price decrease”; whereas in the study of lo and sapp (2008) they refer to “submission of a market order” and “submission of a limit order”. therefore, at the end of each duration one of two events (states, risks) can be observed. this modeling setup can be perceived as if the two potential risks were competing with each other and only one of them: a or b could be realized at it . accordingly, the model belongs to a class of competing risks models. the duration ix is treated as an outcome variable of a function , ,= min( , )i i a i bx x x , where ,i ax and ,i bx are durations that would end up in state a and b , respectively. in this modeling framework, only one duration, i.e. ,i ax (or bix , ), is realized (i.e. the shorter one), which depends on whether an event a or b occurs first. the duration that is not realized is treated as truncated. the conditional bivariate density for a pair ix , iy can be described as: 1 1 1 1 1 ( , | ) = ( | ) ( | ) ( | ) ( | ), a a a b b b i i i i x i i x i i i x i i x i i f x y f h x f s x f h x f s x f      (1) katarzyna bień-barkowska 58 where ( ) ax h  ( ( ) bx h  ) and ( ) ax s  ( ( ) bx s  ) denote the hazard and the survival functions for the aix , ( bix , ) variable, ai and bi are dummy indicators, i.e. = 1ai ( = 1bi ) if state =iy a ( =iy b ) is observed and, analogously, 0= ai ( 0=bi ) if state =iy b ( =iy a ) is observed at the end of duration ix . 1if  denotes a conditioning information set up to time 1t  which contains past realizations of ix and iy . for example, if an event a happens at it , aix , is observed and bix , is truncated. accordingly, the realized duration iai xx , contributes to the conditional density function given by equation (1) via its density function 1( | )ax i if x f  and the unrealized (truncated) duration ibi xx , via its survival function 1( | )bx i is x f  : 1 1 1 1 1 1 ( , | ) = ( | ) ( |, ) ( | ) ( | ) ( | ). a a b a b i i i x i i x i i x i i x i i x i i f x y a f h x f s x f s x f f x f s x f         (2) as proposed by bauwens and giot (2000) or lo and sapp (2008) we parameterize the expectations of the conditional density functions of aix , and bix , with the application of the logarithmic acd model. the conditional duration expectations, , , 1( | )i a i a ie x f   and , , 1( | )i b i b ie x f   , are specified in a dynamic fashion such as both factors (the previously observed states and the past realized durations) can exert an influence on the expected time to an event a or b : , , , 1 1 , , 1 1 1, = ( ln ) ( ln ) , a i a a a a a i i b b a b a i i a i a x i x i                 (3) , , , 1 1 , , 1 1 1, = ( ln ) ( ln ) , b i b a b a b i i b b b b b i i b i b x i x i                 (4) where , ,ln( )i a i a   . the specifications for ,i a and ,i b change with the previously realized state of iy . thus, the expected time to a given event varies with the type of previously observed state and with the length of preceding duration. in the aacd framework, the observed duration ix stems from a mixing process: , 1 , , 1 , , , , , = [ ( | ) ] [ ( | ) ] = [ ] [ ] , a b i i a i i a i i b i i b i a b i a i a i i b i b i x e x f i e x f i i i          (5) distribution choice for the asymmetric acd models 59 where ,i a ( bi, ) is an independent and identically distributed error term with ,( ) = 1i ae  , ( ,( ) 1i be   ). in this setup ,i a and ,i b can both have the generalized gamma or the burr distribution (see appendix 1 for the theoretical outline of these distributions). in order to parameterize the conditional bivariate density for { , }i ix y outlined in equation (1), it is crucial to derive the conditional hazard and the conditional survival functions for ix under given assumptions about the distribution of ai, and ,i b  . if we denote hazard and survival function of an error term ,i a ( bi, ) as ( )ah  ( ( )bh  ) and ( )as  ( ( )bs  ), respectively, under the necessary assumption that ,( ) = 1i ae  ( ,( ) 1i be   ), the conditional hazard and the conditional survival function of , ,i a i a ax   (a similar result holds for , ,i b i b bx   ) can be given as: 1 , , 1 ( | ) , a a i x i i i a i a x h x f h         (6) 1 , ( | ) , a a i x i i i a x s x f s        (7) where , , /i a i a a   and a is the expectation of a burr-distributed (or a generalized gamma-distributed) random variable (see appendix 1). 1.2. aacd model with the burr distribution the burr distribution has two shape parameters  and 2 . lancaster (1990) proves that it can be derived as a gamma mixture of weibull distributions. it contains exponential (if 1  , 2 0  ), weibull (if 2 0  ) and log-logistic (if 2 1  ) distributions as its limiting cases. in contrast to a weibull distribution, the burr distribution allows for a non-monotonic hazard functions, which can extensively improve the goodness-of-fit of the acd models (see grammig, maurer, 2000; bauwens et al., 2004). i apply the formulas for the hazard and survival functions of a burr-distributed random variable (see appendix 1) in equations (6) and (7) in order to derive the hazard and the survival functions ( ) ax h  ( ( ) bx h  ) and ( ) ax s  ( ( ) bx s  ). then, it is straightforward to rewrite the conditional bivariate density outlined in (1) as: katarzyna bień-barkowska 60         2 2 1 1 ,( ) 2 1 ,2 1 1 , 2 ,2 ( , | ) = 1 1 1 . 1 a i a a a a a a a b i b b b b b b b i a i i ab i i i a i i a a i i i b i i b b i i b b i i x f x y f x x x x x                                                  (8) the marginal distribution of ix in the b-aacd model can be derived as:        2 2 ( ) ( ) ( ) 1 1 1 1 1 2 2 , , 1 1 2 2 , , ( | ) = ( , | ) ( , | ) 1 1 1 1 . a b a a b b b a a b b a b b b b i i i i i i i i a i b i i a a i i i b b i i a i i a b i i b f x f f x y a f f x y b f x x x x x x                                                   (9) bauwens and giot derive the conditional (with respect to a current duration and the past filtration 1if  ) transition probabilities between state a and b for the aacd model with the weibull distribution (see bauwens, giot, 2003). in the case of the b-aacd model, such conditional transition probabilities can be given as:         ( ) ( ) 1 1 ( ) 1 1 12 1 2 1 , , 1 1 1 2 2 , , ( , | ) ( | , ) = ( | ) ( ) ( ) . 1 1 a b i i a a a b b b a b a a b b b b b i i i i i i b i i i i a i i a a i b i i b b i a i b i i a a i i i b b i i f x y f f y x f f x f x x x x x x x x                                                   (10) as ( ) 1( | , ) b i i if y x f  depends on ix , ix and iy are not independent. the log-likelihood function of the b-aacd model is obtained as a sum of n logarithms of conditional probabilities given in equation (8) and it can be decomposed into two parts: 1 2 1 2( , ) ( ) ( ),a bl l l      (11) where:     ( 1) , 2 1 1 2 2 ( ) ln ln ln ln 1 1 ln 1 , a a a a i i a a a n a a i a a i i i a i i a l i x x x                                (12) distribution choice for the asymmetric acd models 61     , ( 1) 2 2 1 2 2 ( ) ln ln ln ln 1 1 ln 1 , b b b b i i b b b n b b i b b i i i b i i b l i x x x                                (13) and    2 2, , , , , , , ,, , , , , , , , , , , , , .a a a a a b a a a b a a b b b a b b b a b b b b                 the model can be easily estimated by maximising the joint likelihood given by equation (11). as the two components of the likelihood function ( 1l and 2l ) depend on different parameters, they can also be maximized separately, as suggested by bauwens, giot (2003) in the case of the aacd model with the weibull distribution. 1.3. aacd with the generalized gamma distribution generalized gamma distribution has two shape parameters  and  . as the burr distribution, it allows for different, non-monotonic shapes of the hazard function. it nests a gamma distribution (if 1  ), a weibull distribution (if 1  ) and an exponential distribution (if 1  , 1  ). as the burr distribution, the generalized gamma distribution is often applied to the acd models (see bauwens, giot, 2001; bauwens et al., 2004, among others). substitution of the hazard and the survival of a generalized gamma distribution for ,i a ( ,i b ) into equations (6) and (7) results in the hazard and survival function for ,i ax ( ,i bx ). accordingly, the conditional bivariate density of the pair { , }i ix y is given as:   ( ) 1 1 , , , , 1 , , , ( , | ) = exp( ) 1 ( , ) ( )(1 ( , ) exp( ) 1 ( , ( )(1 ( , ) a i a a a a a a a a a a b i b b b b b b b b b gg i i i i a i i a i i a i a i i ai a a i i a i b i i b i i b i b i ii b b i i b f x y f x x x x x x x x                                                                   , ) ,bb (14) where  denotes the gamma function and i is the incomplete gamma function (see appendix 1 for details). the marginal distribution of ix can be derived as: katarzyna bień-barkowska 62 ( ) ( ) ( ) 1 1 1 , , , , , , ( | ) = ( , | ) ( , | ) exp( ) ( )(1 ( , ) exp( ) ( )(1 ( , ) 1 ( , a a a a a a a a b b b b b b b b gg gg gg i i i i i i i i a i i a i i a i i a a i i a b i i b i i b i i b b i i b i a i f x f f x y a f f x y b f x x x x x x x x x                                                             , ,) 1 ( , ) ,a a b bii a b i i bx        (15) and the conditional transition probabilities between states a and b are: ( ) ( ) 1 1 ( ) 1 1 , , , 1 , , , ( , | ) ( | , ) = ( | ) exp( ) ( )(1 ( , ) exp( ) ( )(1 ( , ) a i a a a a a a a a b i b b b b b b b b gg gg i i i i i i gg i i i a i i a i i a i a a i i a i b i i b i i b i b b i i b a f x y f f y x f f x f x x x x x x                                                            , , , 1 , , , exp( ) ( )(1 ( , ) exp( ) . ( )(1 ( , ) a a a a a a a a b b b b b b b b i i a i i a i i a a i i a b i i b i i b i i b b i i b x x x x x x x x                                              (16) from that it can be seen that ix and iy are not independent. the log-likelihood of the gg-aacd model can be derived as a sum of n logarithms of conditional probabilities given in (14). in a close analogy to the b-aacd model, the log-likelihood can be decomposed into two components, 1 2 1 2( , ) ( ) ( )a bl l l      , where:   1 1 , , 1 , , ( ) ln ln ln( ) ln( ( )) ln(1 ( , ) ln(1 ( , ) , a a a a i a a a a n a a i a a a i a i i a i i i a a i i a a i i a l i x x x x                                        (17)   1 2 , , 1 , , ( ) ln ln ln( ) ln( ( )) ln(1 ( , ) ln(1 ( , ) , b b b b b b b b n b b i b i b b i b i i b i i i b b i i b b i i b l i x x x x                                        (18) and distribution choice for the asymmetric acd models 63    , , , , , , , ,, , , , , , , , , , , , , .a a a a a b a a a b a a b b b a b b b a b b b b                 the estimation may be performed on the joint log-likelihood function or in two steps, separately for 1l and 2l . 1.4. testing the distribution choice with the pit the goodness-of-fit of the acd models can be checked with the probability integral transforms (pit) proposed by diebold, gunther and tay (1998). this testing procedure has been used to check the adequacy of the distribution choice and the quality of the conditional mean specification in numerous studies on financial durations (e.g. bauwens et al., 2004; grammig, mauer, 2000; hautsch, 2004; bień, 2006, among others). in a shortcut, this approach can be presented as following. if  1 1( | ) m i i if x f  denotes a sequence of one-step-ahead density forecasts of the acd model and  1 1( | ) m i i ip x f  is a sequence of conditional densities for the data generating process of financial durations, the acd model will be correctly specified if the following equation holds:    1 11 1( | ) ( | ) m m i i i i i if x f p x f  (19) although the sequence  1 1( | ) m i i ip x f  cannot be observed, diebold, gunther and tay (1998) show that if equation (19) holds true, the sequence of density transforms  iz for durations  ix should be i.i.d. uniformly distributed on (0,1): ( ) , ~ . . . (0,1). ix i i iz f t dt z i i d u   (20) it can be seen from formula (20) that in order to compute the sequence  iz , we need the cumulative distribution function (cdf) for ix under given acd specification. the marginal densities of ix under the b-aacd data generating process (dgp) or under the gg-aacd dgp have been derived in equations (9) and (15), respectively. the sequence of integrated density transforms for the b-aacd model can be calculated as:    2 2 1 1 ˆ ˆ ˆ( ) 2 2ˆ ˆ , , ˆ ˆˆ ˆˆ 1 1 1 ,a a b ba bbi a i i a b i i bz x x                 (21) which can be seen from the proof 1 or the proof 2: katarzyna bień-barkowska 64 proof 1:           2 2 2 2 1 1( ) ˆ 1 ˆ ˆ ˆ ˆ1 2 2ˆ ˆ , , , 1 1 ˆ 1 ˆ ˆ ˆ ˆ1 2 2ˆ ˆ , , , ˆ 1 ˆ ˆ ˆ2 , , ˆ ˆ ˆ ˆˆ ˆ ˆ1 1 ˆ ˆ ˆˆ ˆ ˆ1 1 ˆ ˆ ˆ ˆˆ1 a a a a b ba b b b b b b bb a a a a a b i a i a i i a b i i b i a i b i b i i b a i i a i b a i b a i i a i a dz x x x dx x x x x x                                                                 2 2 ˆ 1 ˆ ˆ ˆ2 , , 1 1 ˆ ˆ2 2ˆ ˆ , , 1 ˆ ˆˆ1 ˆˆ ˆˆ ˆ1 1 ( | ) b b b b b b b ba b i b i i b i b a i i a b i i b b i i x x x x f x f                                       proof 2: if min( , )a bi i ix x x , iz can be computed as: 1 1 1 1 ˆ ˆ ˆˆˆ ( | ) 1 ( | ) 1 ( | ) ( | )i i i i i a i i b i iz cdf x f s x f s x f s x f        analogously, from proof 2, iz for the gg-aacd model can be estimated as:    ˆ ˆ ˆ ˆ( ) , ,ˆ ˆˆ ˆˆ 1 1 ( , ) 1 ( , ) ,a a b bgg i ii a i i a b i i bz x x              (22) application of the pit diagnostic procedures often boils down to checking whether ˆiz is i.i.d. and uniformly distributed. diebold, gunther and tay (1998) and bauwens et al. (2004) emphasize visual inspection of graphs which depict dynamics and distributional properties of ˆiz . 2. empirical example as in the study of lo and sapp (2008), i apply the aacd model to analyze the order submission process in the reuters dealing 3000 spot matching system3 (rdsm). the rdsm system is a fully automated4 order-driven market where the interbank currency trading takes place. on this trading platform currency dealers can submit two major order types, i.e. market orders or limit orders, to buy or sell a given amount of the base currency5. from the viewpoint of the market microstructure, market orders are perceived as liquidity consuming – they are immediately executed against limit orders listed on the 3 the reuters dealing 3000 spot matching system is an updated version of the reuters dealing 2000-2 matching system described by lo and sapp (2008). 4 orders are automatically matched if they arrive to opposite market sides and if their prices agree. 5 in case of the eur/pln currency pair, euro is the base (transaction) currency and zloty is the counter (quote) currency. distribution choice for the asymmetric acd models 65 opposite side of an order book, hence they exhaust liquidity measured as market depth. limit orders are liquidity supplying – they wait for a possible execution in future, hence they replenish depth on the ask or the bid side of a market. i use data on order submissions on the eur/pln market during four days, from 2nd to 5th january 2007. the vast majority of polish zloty trading takes place in the offshore market (between london banks) and in poland. therefore, in order to account for periods when trading is high, i consider orders placed after 8:00 cet and before 18:00 cet6. in the sample there are 10 515 orders, i.e. 4 848 market orders and 5 667 limit orders7. order durations are defined as time intervals between subsequent moments of order submissions. in the first step i deseasonalized durations as suggested in the literature on acd models. i assume a multiplicative intraday seasonality factor is , such as i i ix s x . as suggested by bauwens and veredas (2004), the intraday seasonality factor is has been estimated with the application of the kernel regression of ix on a time-of-day variable8. estimation9 of the aacd models has been conducted on diurnally adjusted durations ix . for the sake of completeness of my study i estimated four specifications of aacd models, the e-aacd model (with the exponential distribution of an error term), the w-aacd model (with the weibull distribution of an error term), the b-aacd model and the gg-aacd model. the specifications of conditional expectations were always the same – as in equations (3)–(4). the bayesian information criterion (bic = 2.6197) favorises the b-aacd model over the gg-aacd (bic = 2.6540), the w-aacd model (bic = 2.7296) and the e-aacd model (bic = 2.9146 ). histograms and autocorrelation functions for the probability integral transforms ˆiz are depicted in figure 1 and 2. as can be seen in figure 1, neither the exponential, nor the weibull distribution are proper for the aacd model. long (but not very long) durations are underrepresented in both specifications and the distribution of the probability integral transforms is far from being uniform. additionally, the weibull distribution demands more observations of a small value than registered in the duration series. both parsimonious distributions are not flexible enough to reflect the shape of the true data generating process. the generalized gamma distribution does not fit the data as 6 similar truncation was performed by lo and sapp (2008). 7 as in (lo, sapp 2008) i accounted for the best limit orders only, i.e. orders that are placed within the best ask and bid prices in the order book. 8 quartic kernel is used, with the bandwidth computed as 2.78sn -1/5, where s is the standard deviation of the data.. for details of the estimation procedure please refer to (bauwens, veredas, 2004). 9 the whole empirical study has been performed in gauss 8.0. katarzyna bień-barkowska 66 well. just as the weibull distribution, it gives too much probability mass to small durations and too little probability mass to a medium-sized durations (from the third to fifth quantile of the ipf distribution). better distribution choice would require less observations in the lower tail of the distribution and more observations of a middle-sized value. although the visual inspection suggests that the b-aacd has won the competition among the models, the choice of the burr distribution is not optimal as well. the standard pearson’s goodness-of-fit statistic10 2̂ for the uniformity of the ( )ˆ biz distribution equals 158.53. because 2*ˆ 30.14  (5% significance level), so the null of uniformity should be rejected. the dynamic properties of the model are not perfect, as the acf function for the pit depicts significant autocorrelation of the first order. nevertheless, once more b-aacd model seems to provide the best fit among selected specifications. the aacd models were parsimoniously parameterized in terms of the conditional mean functions in order to avoid a burdensome estimation, but the literature on the acd models shows that it is very difficult to find a satisfying model as far as its dynamic features are concerned (see bauwens et al., 2004). e-aacd w-aacd gg-aacd b-aacd figure 1. histogram of the probability integral transforms for the aacd models. horizontal lines depict the 99% confidence interval 10     m i i np npn 1 2 2 )( , where m=20 (number of histogram bins), p=1/m. distribution choice for the asymmetric acd models 67 e-aacd w-aacd gg-aacd b-aacd figure 2. the acf function of the probability integral transforms for the aacd models. horizontal lines depict the 99% confidence interval in the last step of this study, i introduce two explanatory variables into the b-aacd specification. i check the impact of the bid-ask spread and the eur/pln volatility – separately – on the expected durations to a market order versus an expected duration to a limit order. there is a large body of theoretical and empirical studies proving that these both factors matter as far as order choice decisions are considered (foucault, 1999; hautsch, 2004; lo, sapp, 2008; among others). the bid-ask (difference between the best ask and bid price at the moment of order submission) and the volatility (the realized volatility estimate during 10 minutes interval before the order submission) were cleared form the intraday seasonality in the same way as the order durations. in table 1, i report the ml estimates and the corresponding p-values (for the robust standard errors) of the b-aacd model. the estimated shape parameter proves that the assumption of the weibull distribution for the error terms is not proper. as 2ˆ 0.9425a  and 2ˆ 1.0777b  , the obtained burr distribution seems to be rather “closer” to a log-logistic distribution (where 2 1)  , than to a weibull one ( 2 0)  . as far as the dynamic properties of the aacd model are concerned, obtained results agree with the results of lo and sapp (2008). for bk  , that is for durations ending with a market order i have ,ˆa a  ,ˆa b and ,ˆa a  ,ˆa b , hence in result of a previously observed market order, the expected duration to another market order shrinks stronger than directly after a limit order. such a result seems to agree with a study of biais et al. (1995) performed for selected stocks traded on the paris bourse. it is katarzyna bień-barkowska 68 predicted there that market orders placed on the same side of a market cluster together as traders: (1) may split large orders into small ones to avoid a huge price impact, (2) mimic each other or (3) react similarly to given events. in this study i do not differentiate between ask and bid side of a market, but i suspect that overall clustering of market orders may be caused by similar behaviour patterns. on the other hand, the complicated dynamic structure of the aacd model makes it extremely difficult to capture all information given by distinct parameters and to interpret them accordingly. as far as expected durations that end with a limit orders are concerned ( k b ), the obtained relations are more unequivocal and easy to interpret. as ,ˆb a  ,ˆb b and ,ˆb a  ,ˆb b , the expected duration to a limit order shrinks more considerably after a market order than after a limit order. market order always “erodes” depth on the ask or bid side of a market which may result in a wider bid-ask spread. therefore it is more profitable to submit a limit order than a market order, large bid-ask spread makes liquidity consumption more costly. the complete dynamics of the model can be captured in a more detailed way with a simulation of the whole process, as proposed by bauwens and giot, (2003). the bid-ask spread has a significant positive impact on the expected time to a market order and it has a negative impact on the expected time to best limit orders. this result agrees with findings of several studies (ranaldo, 2004; verhouven et al., 2003; ellul et al., 2007; lo, sapp, 2008). if the bid-ask spread is large, it becomes more costly for a trader to cross the difference between the best bid and ask prices in order to buy or sell the currency in an immediate way. the bid-ask spread constitutes the cost of such quick transaction. on the other hand, the traders opt for limit orders. as the differences between best bid and ask prices are large, it is much easier for them to compete for the transaction priority by offering a price at least one tick better than the current one (“the tick rule”). an increase in volatility prompts market orders, as it has a negative significant impact on the expected time to a market order submission. a rise in uncertainty about the future movement of the eur/pln rate encourages traders to close their open currency positions or to realize their gains quickly. the positive impact of volatility on the expected time to a limit order can be easily understood with the nature of a limit order. the submission of a limit order is the same as writing of the option – if the fx rate moves in the undesirable direction, it can be executed at an unfavourable price (so called “free option risk” of a limit order). as prices in limit orders are frozen during their lifetime, an increased volatility increases the risk of incurring potential losses. distribution choice for the asymmetric acd models 69 table 1. ml estimates for the b-aacd model ak  (market order) bk  (limit order) parameters estimate p – val estimate p val ak ,  0.0905 0.0404 0.5388 0.0000 bk , 0.3541 0.0000 0.8631 0.0000 k  0.7478 0.0000 0.6912 0.0000 ak ,  0.1638 0.0000 0.0908 0.0000 bk , 0.1647 0.0000 0.2195 0.0000 k  0.9104 0.0000 1.2104 0.0000 2 k 0.5844 0.0000 0.9840 0.0000 bid-ask spread 1.6472 0.0000 -0.6609 0.0000 volatility -0.3537 0.0000 0.2304 0.0000 bic 2.4122 conclusions in the paper i have extended the asymmetric acd model of bauwens, giot (2003) with respect to more general distribution families: the burr and the generalized gamma. as in the study of bauwens and giot (2003), i present the basic properties of the generalized specifications. additionally, we showed an easy way of testing the goodness-of-fit of such a competing risk models with the probability integral transforms. as lo and sapp (2008), i have also presented an exemplary application of the aacd model to the order submission process on the interbank order-driven market for a polish zloty. the obtained results with regards to the impact of the bid-ask spread and the volatility on the order choice confirm the main results from the empirical literature. there are many possible extentions to my study. a most natural one would be a more finance-oriented empirical application. introducing more explanatory factors, reflecting the larger scope of information that can be deduced from the order book or the external market environment would give more insight into the process of market liquidity fluctuation as studied by lo and sapp (2008). second, different functional form for duration expectations or even some more general distributions for the error term would possibly result in a better fit of the model. the pit diagnostic tool points the burr distribution as a best one, although this distribution choice does not seem to be the optimal one. the possible solution would be even more general distribution, such as a generalized f distribution advocated by hautsch (2001) for the acd models. third, the assymetric acd model could be easily extended to more than two competing risks. it could result in a more flexible specification that, in the context of a current empirical study, could also discriminate between orders listed on either bid or ask side of the market. katarzyna bień-barkowska 70 references bauwens, l., giot, p. (2000), the logarithmic acd model: an application to the bid/ask quote process of two nyse stocks, annales d’economie et de statistique, 60, 117–149. bauwens, l., giot, p. (2001), econometric modelling of stock market intraday activity, kluwer academic publishers, boston. bauwens, l., giot, p. (2003), asymmetric acd models: introducing price information in acd models, empirical economics, 28, 709–731. bauwens, l., giot, p., grammig, j., veredas, d. (2004), a comparison of financial duration models via density forecasts, international journal of forecasting 20, 589–609. bauwens, l., veredas, d. (2004), the stochastic conditional duration model: a latent variable model for the analysis of financial durations, journal of econometrics 119, 381–412. biais, b., hillion, p., spatt, c. (1995), an empirical analysis of the limit order book and the order flow in the paris bourse, journal of finance 50, 1655–1689. bień, k. (2006), advanced acd models – presentation and the example of application, statistical review 53 (1), 90–108. diebold, f.x., gunther, t.a., tay, a.s. (1998), evaluating density forecasts with applications to financial risk management, international economic review 39, 863–883. ellul, a., holden c., jain, p., jennings, r. (2007), order dynamics: recent evidence from the nyse, journal of empirical finance, 14, 636–661. engle, r.f., russell, j.r. (1998), autoregressive conditional duration; a new approach for irregularly spaced transaction data, econometrica 66, 1127–1162. foucault, t. (1999), order flow decomposition and trading costs in a dynamic limit order market, journal of financial markets 2, 99–134. ghysels, e., jasiak, j. (1998), garch for irregularly spaced financial data: the acd-garch model, studies in nonlinear dynamics and econometrics 2, 133–149. giot, p. (2000), time transformations, intraday data and volatility models, journal of computational finance, 4(2), 31–62. giot, p. (2005), market risk models for intraday data, european journal of finance, 11, 309–324. gouriéroux, c., jasiak, j. (2001), financial econometrics: problems, models and methods, princeton university press. grammig, j., maurer, k. (2000), non-monotonic hazard functions and the autoregressive conditional duration model, econometrics journal 3, 16–38. grammig, j., wellner, m. (2002), modeling the interdependence of volatility and intertransaction duration process, journal of econometrics, 106, 369–400. hautsch, n. (2001), the generalized f acd model, discussion paper, university of konstanz hautsch, n. (2004), modelling irregularly spaced financial data, springer, berlin. kalimipalli, m., warga, a. (2002), bid-ask spread, volatility and volume in the corporate bond market, journal of fixed income, 3, 31–42. lo, i., sapp, s. (2008), the submission of limit orders or market orders: the role of timing and information in the reuters d2000-2 system, journal of international money and finance, 27, 1056–1073. nolte, i. (2008), modeling a multivariate trading process, journal of financial econometrics, 6, 143–170. osińska, m. (2006), financial econometrics, pwe, warszawa. pacurar. m. (2008), autoregressive conditional duration models in finance: a survey of theoretical and empirical literature, journal of economic surveys, 22 (4), 711–751. ranaldo, a. (2004), order aggresiveness in limit order book markets, journal of financial markets, 7, 53–74. tsay, r. s. (2002), analysis of financial time series, wiley: new york. verhoeven, p., ching, s., ng, h. (2003), determinants of the decision to submit market or limit orders on the asx, pacific-basin finance journal, 12, 1–18. distribution choice for the asymmetric acd models 71 zhang, m. y., russell, j. r., tsay, r. s. (2001), a nonlinear autoregressive conditional duration model with applications to financial transaction data, journal of econometrics 104, 179–207 appendix 1 burr distribution with parameters 0  , 2 0  and scale parameter is set to 1: survival: ( )s   (1+ 2 1 2 ) ,    density: 2 1 2 1 1 ( ) , (1 ) f            hazard: 1 2 ( ) , 1 h          expectation:   1 1 1 2 2(1 ) 2 1 1 1 1                        if 2 .  generalized gamma distribution with parameters 0 , 0 where the scale parameter is set to 1: survival: ( ) 1 ( , )is      , where i is the lower incomplete gamma function: 1 0 ( , ) ( ) t i t e dt           and )( gamma function, density: 1 exp( ) ( ) , ( ) f          hazard: 1 exp( ) ( ) , ( ) ( , )i f               expectation:  1 . ( )         katarzyna bień-barkowska 72 wybór rozkładu składnika losowego w asymetrycznych modelach acd z a r y s t r e ś c i w artykule dokonano uogólnienia asymetrycznego modelu acd, zaproponowanego w pracy (bauwens, giot, 2003) w odniesieniu do nowych rozkładów składnika losowego: rozkładu burra i uogólnionego rozkładu gamma. wyprowadzono funkcję wiarygodności dla rozszerzonych specyfikacji i przedstawiono procedurę testowania jakości dopasowania modeli za pomocą transformat gęstości (diebold i in., 1998). dodatkowo, przedstawiono przykładową aplikację asymetrycznych modeli acd w odniesieniu do odstępów czasu (tzw. czasów trwania) pomiędzy momentami, w których składane są zlecenia z limitem ceny lub zlecenia rynkowe na kierowanym zleceniami międzybankowym kasowym rynku złotego. dokonano weryfikacji wpływu dwóch czynników mikrostruktury rynku (spreadu bid-ask i zmienności) na tempo składania wyróżnionych typów zleceń. s ł o w a k l u c z o w e: asymetryczny model acd, finansowe czasy trwania, transformaty gęstości, mikrostruktura rynku. acknowledgements the author thanks the thomson reuters and the national bank of poland for providing the data from the reuters dealing 3000 spot matching system and professor małgorzata doman for valuable comments and suggestions. the views and opinions presented in the paper are those of the author and do not reflect those of the national bank of poland. dynamic econometric models © 2016 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2016.004 vol. 16 (2016) 65−86 submitted october 25, 2016 issn (online) 2450-7067 accepted december 18, 2016 issn (print) 1234-3862 andrzej geise, mariola piłatowska * asymmetries in the relationship between economic activity and oil prices in the selected eu countries  a b s t r a c t. in this paper the threshold (t-ecm) and linear (ecm) error correction models are estimated to examine the short-run and long-run granger causality in terms of asymmetric and symmetric relationship for seven european union economies (germany, france, denmark, the netherlands, poland, czech republic and the whole eu). the relationship between production, inflation and oil prices are analyzed in the presence of structural break when both, the change in intercept and the change in the slope of the trend function exist. threshold ecms show the asymmetric response of production and inflation to the changes in oil prices in the case of germany, france, poland and the eu. for other economies (netherlands, denmark and czech republic) the reaction was rather symmetric. k e y w o r d s: short-run asymmetry, threshold cointegration, threshold error correction model, granger causality, oil price shock. j e l classification: c32; e23; e32; q43. introduction energy price shocks have important effects not only on economic activity but also on macroeconomic policy of industrialized countries. it is related to the essential meaning of energy in all areas of economic activities. energy * correspondence to: . andrzej geise, nicolaus copernicus university, faculty of economic sciences and management, 13a gagarina street, 87-100 toruń, poland, e-mail: a.geise@umk.pl; mariola piłatowska, nicolaus copernicus university, faculty of economic sciences and management, 13a gagarina street, 87-100 toruń, poland, e-mail: mariola.pilatowska@umk.pl.  this work was financed from faculty of economic sciences and management research grant no. 2185-e. andrzej geise, mariola piłatowska dynamic econometric models 16 (2016) 65–86 66 is pushing all developed and developing countries to ensure the long-term stable energy prices and energy supply. the stable price of energy and independent source of imported energy resources are crucial to the economic development. huge and unexpected increases of energy prices can cause the inflation to rise and reduce real money balances. these changes have negative effects on consumption and economic output. the purpose of this paper is to analyze the shortand long-term dynamics of oil prices, production and inflation in the selected eu countries in the presence of structural break due to the financial crisis. the potential asymmetries over the production are investigated, and then we examine the causality among the oil prices, production and inflation in the context of high and low growth regime. first, the threshold cointegration analysis applied in geise and piłatowska (2014) for 7 european economies is extended to the case of structural break due to the financial crisis not only in an intercept but both intercept and trend coefficient. these results will be compared to those contained in the mentioned article. second, the granger causality among oil prices, production and inflation in the selected eu countries (germany, france, netherlands, denmark, poland, czech republic and the european union as well) is examined based on the linear ecm and threshold ecm models. in geise and piłatowska (2014) the relationships among explanatory variables were analyzed by the threshold error correction model with asymmetric adjustment process. here, the relationships between production, inflation and oil prices are analyzed by both threshold ecm and linear ecm. when the threshold cointegration occurs, the threshold error correction model with asymmetric adjustment process and asymmetric short-run dynamics (t-ecm) is estimated. in other cases the linear ecm models are estimated. the threshold model allows us to identify the relationships among oil prices, production and inflation according to business cycle phases, i.e. the growth phase of the economic business cycle is defined as high growth regime (when the deviations from longrun equilibrium are above the threshold value) and the recession phase is related to low growth regime (when the deviations are below the threshold value). the concept of a research study is presented in scheme 1. the plan of this paper is as follows. section 1 presents short theoretical explanations for symmetric and asymmetric reaction of economic activity to oil price shocks. section 2 reviews the methodology applied in this paper. section 3 shows the results of threshold cointegration analysis and presents asymmetries in the relationship between economicactivity and oil prices… dynamic econometric models 16 (2016) 65–86 67 empirical error correction models. section 4 presents the results of grangercausality analysis. the last section concludes. scheme 1. concept of the study the granger causality among oil prices, production and inflation 1. asymmetries in the transmission of oil price shocks the nature of the relationship between oil prices and economic activity (symmetric or asymmetric) determines the way of oil prices modelling. it helps to understand the magnitude of positive and negative oil price shocks impact on economy and select the model of transmission of oil price shocks among alternative models. the study of transmission of oil price shocks should focus on whether the transmission channel leads to the symmetric or asymmetric response of economic activity in the context of oil importing and oil exporting countries. oil price shocks are transmitted to the economic activity by the direct supply-side channel. increases in the prices of crude oil have a direct impact on the supply of crude oil and supply of high energy-consuming goods. considering oil importing countries in which the production function can be expressed as a functional form of input factors (such as labour, capital and energy), it turns out that the large fluctuations in economic activity are unlikely to observe without a large fluctuation in the variation in energy prices. also, when the smooth production function is assumed, the symmetric response of output to changes in energy input is observed (hamilton, 2008). the direct demand-side channel of oil price shock transmission for oil importing countries indicates that the increase of oil prices leads to changes in consumer’s purchasing power. these changes are related to the income transfer from oil exporting countries to oil importing countries. therefore, eu countries ecm/t-ecm models step ii production (pt) inflation (int) oil prices (bt) ecm (linear error correction model) long-run relationship step i causality test step iii l in e a r c o in te g ra ti o n w it h s y m m e tr ic a d ju s tm e n t two-step enderssiklos approach to analyze threshold cointegration t-ecm (threshold error correction model with asymmetric adjustment) t h re s h ld c o in te g ra ti o n w it h t a r o r m -t a r a d ju s tm e n t granger causality test andrzej geise, mariola piłatowska dynamic econometric models 16 (2016) 65–86 68 increases in oil prices will cause the increases in production (by the demand push) in oil exporting countries, however the oil importing countries will face the production decrease (hamilton, 2011). both direct transmission mechanisms presents the direct and symmetric reaction of economic activity to changes in oil prices. however, some indirect transmission channels amplify the effect of oil price shock and may lead to an asymmetric reaction of economic activity. the sectoral reallocation associated with capital and labour reallocation from energy-intensive sectors to expanding sectors is a result of a positive oil price shock. in the case of oil-importing countries, sectoral reallocation will reinforce the recessionary effect of oil price increasing (davis and haltiwanger, 2001). however hamilton (1988) presents that the lower economic activity and higher unemployment rate are an outcome of the lack of sectoral reallocation. this disturbance is caused by workers who can be confident about the future improvement of their sector and decide not to relocate even if they are offered a job in a different sector. however, both types off reallocation disturbances will lead to an asymmetric reaction of production as a result of a positive or negative oil price shock. another explanation of an asymmetric reaction of economic activity to oil price shocks is connected with precautionary savings. for an oil importing economy the rise of oil price can cause the future decrease in employment and real income, inducing an increase in precautionary savings, which in turn leads to the decrease in production (edelstein and kilian, 2009). for oil importing economies the asymmetric reaction of monetary policy to oil price increases and decreases is still ambiguous. for instance, bernanke et al., (1997) pointed out the asymmetric reaction of the federal reserve to oil price shocks (authors show that the fed responds more aggressively to oil price increases than oil price decreases). however, some other authors argue that the reaction of monetary policy to oil price shocks is insignificant and doesn’t exist – see e.g. hamilton and herrera, (2004). the evidence consistent with the model of asymmetric reaction of economic activity to oil price shocks was provided by herrera et al., (2015). authors used data for 18 oecd countries and pointed out that the sectoral reallocation disturbances and the existence of a precautionary savings, when future employment level was uncertain, implied an asymmetric response of aggregate production. asymmetries in the relationship between economicactivity and oil prices… dynamic econometric models 16 (2016) 65–86 69 2. methods to test the threshold cointegration among production, inflation and brent oil prices in the selected eu countries the enders-granger approach and the enders-siklos approach is employed 1 . this technique is a residual based two-stage estimation and focuses on the residuals from the long-run equilibrium relationship. for this study, the long-run relationship can be expressed as: tttttt dtduinbp   )()( * 4 * 3210 , (1) where tp , tin denote the industrial production index and consumer price index in selected economies respectively, t b stands for brent crude oil prices; all variables are in natural logarithms;  refers to the time break, i.e., the period at which the change in the parameters of the intercept and trend function occurs, )( * tdu is the dummy variable for the break in intercept of production function where 1)( * tdu if t and 0 otherwise, )( * tdt is the variable for a change in the slope of the trend function where tdt t )( *  if t and 0 otherwise, 43210 ,,,,  are parameters to be estimated and t is the disturbance term that may be serially correlated. next, the analysis focuses on the estimation of 1 and 2 coefficients in the following regression:     p i titittttt uii 1 1211 )1(  (2) where tu is a white-noise disturbance, p the number of lags, 1 , 2 and i the coefficients, t in equation (2) is extracted from long-run equation (1). ti term is the heaviside indicator function such that: 1ti if , dt and 0 otherwise (for tar model); or 1ti if ,  dt and 0 otherwise (for m-tar model) where  is the threshold value. the tar model is designed to capture the potential asymmetric movements in residuals t , while the m-tar model is designed to capture the direction of the potential asymmetric movements (fosten et. al, 2012). negative deepness (i.e., 21   ) of the residuals implies that increases of t 1 see geise and piłatowska (2014), piłatowska, włodarczyk and zawada (2014). andrzej geise, mariola piłatowska dynamic econometric models 16 (2016) 65–86 70 tend to persist, whereas decreases tend to revert quickly toward to equilibrium (enders and granger, 1998). the threshold value  can be set as zero or can be estimated according to the chan’s search method 2 . the estimation of the threshold parameter is not obvious. when the threshold parameter is known (equal 0), the simply ols can be used to estimate the slope parameter   , of equation (2). in the case of unknown threshold parameter the conditional ls must be used. to obtain an estimation minimizing the sum of squares or maximizing the log-likelihood, the ols cannot be used, because as the objective function is highly erratic. a solution is obtained through concentration of the objective function. as the slope estimators given a threshold are the ols one. the problem of using highly erratic function can be reduced by concentrating out the minimization problem through   and the corresponding sum of squares )(ssr . the objective function has the form:    ssrminarg  . (3) minimization of (3) is done through a grid search: the values of the variable are sorted, a certain percentage of the first and last values is excluded to ensure a minimal number of observations in each regime, the ssr is estimated for each selected value and the one that minimize the ssr is taken as the estimator (stigler, 2010) before the threshold cointegration analysis is conducted, the tsay linearity test is used to verify the null hypothesis of linearity in threshold variable for different value of delay (d) – see more tsay (1989). testing for threshold cointegration is performed in two steps. first, an φ-test is employed to examine the null hypothesis of no cointegration ( 0: 210  h ) against alternative of cointegration with either tar or mtar threshold adjustment. second, the test f-test is used to verify the null hypothesis of symmetric adjustment ( 210 :  h ). rejection of both hypothesizes indicates the existence of threshold cointegration with an asymmetric adjustment (enders and siklos, 2001). the threshold error correction model (with two regimes) has the form: 2 see chan (1993), geise and piłatowska (2014). asymmetries in the relationship between economicactivity and oil prices… dynamic econometric models 16 (2016) 65–86 71 ttt q i ti q i itiit q i it tt q i ti q i itiit q i itjt binp binpy                   }){1)(( }{)( 321 321 1 1 11 12 1 1 * 1 * 1 * 11 (4) where  tttjt binpy  ,, , 1t represent the error correction terms, t is the threshold variable, which is a continuous and stationary transformation of the data and takes the form of dt  (when tar adjustment mechanism is used) or dt (when m-tar adjustment mechanism is selected),  is the threshold parameter. when the method proposed by enders and siklos (2001) cannot reject the hypothesis of symmetric cointegration the linear error correction model is estimated. the structure of model is determined by the following equation: t q i ti q i itiit q i itjt binpy          321 1 1 11 11 (5) where 1t  is obtained from the estimated long-run equation (1) and t is a white noise disturbance. 3. linear (ecm) and threshold error correction models (t-ecm) of production, inflation and oil prices in the selected eu countries in this section, the threshold cointegration analysis, threshold (t-ecm) and linear error correction models (ecm) for production, inflation and oil prices in germany, france, the netherlands, denmark, poland, czech republic and the european union are discussed. to investigate the presence of asymmetries in the relationship between oil prices and economic activity the monthly data from january 1995 to april 2014 are used. in particular, industrial production index at constant prices of 2010, consumer price index and brent oil prices (each time series expressed in natural logarithms) were used 3 . the results of the adf test revealed that all analyzed series were integrated of first order, i(1) – see geise and piłatowska (2014). therefore, in further analysis the first differences of each series are used. the structural break occurred due to the financial crisis peaked in 2008 may change the nature of the long-run equation for production. therefore we 3 here, the identical sets of variables as in geise and piłatowska (2014) were used. andrzej geise, mariola piłatowska dynamic econometric models 16 (2016) 65–86 72 assume the long-term relationship with both one-time change in the intercept and change in the slope of the trend function (as in perron, 1989). to motivate the use of long-term relationship (with structural break in september 2008), we display the time series for production in analyzed economies in figure 1 with the fitted values of the regression tttt dttdup   211 ~~~~~ , estimated by the ols, where 0 tt dtdu if 09:2008t and 1tdu , tdtt  if 09:2008t . as we can see in figure 1 the production index for all analyzed economies has an upward trend. however when the financial crisis occurs (in 2008–2009) the line slopes downward and the production index is falling. after 2009, production index returns to the pre-crisis path but at a slower pace than previously. figure 1 reveals that the production response to the global financial crisis is higher in developed countries (i.e., germany, france, denmark, european union) and lower in developing countries (i.e., poland and czech republic). a) logs in production in germany b) logs in production in france c) logs in production in denmark d) logs in production in netherlands 4,1 4,2 4,3 4,4 4,5 4,6 4,7 4,8 l o g o f p ro d u c ti o n i n d d e x production (de) fitted 4,4 4,5 4,6 4,7 4,8 l o g o f p ro d u c ti o n i n d e x production (fr) fitted 4,3 4,4 4,5 4,6 4,7 4,8 4,9 lo g o f p ro d u c ti o n i n d e x production (dk) fitted 4,1 4,2 4,3 4,4 4,5 4,6 4,7 l o g o f p ro d u c ti o n i n d e x production (nl) fitted asymmetries in the relationship between economicactivity and oil prices… dynamic econometric models 16 (2016) 65–86 73 e) logs in production in poland f) logs in production in czech rep. g) logs in production in european union figure 1. logs in production and fitted trend line with break for selected eu countries this findings reveal that the structural changes in production caused by the global financial crisis are more apparent in developed economies than in developing economies. the break point (set in september 2008) refers to the time of break, i.e., the period at which the change in the level of series (break in intercept) and lower growth rate (the change in the slope of a trend function) occur. in the case of germany, france, denmark and european union the production response to the financial crisis is sudden and significant. for other economies (the netherlands, poland and czech republic) the change in production index is small but visible – see figure 1. the estimated parameters 3 and 4 for all economies are statistically significant what is an argument for taking into account both a sharp change in the level of the series (break in intercept) and the change in the slope of a trend function – see table 1. 3,4 3,9 4,4 4,9 l o f o f p ro d u c ti o n i n d e x production (pl) fitted 3,6 3,8 4 4,2 4,4 4,6 4,8 5 l o g o f p ro d u c ti o n i n d e x production (cz) fitted 4,3 4,4 4,5 4,6 4,7 4,8 l o g o f p ro d u c ti o n i n d e x production (eu) fitted andrzej geise, mariola piłatowska dynamic econometric models 16 (2016) 65–86 74 table 1. the adf test for cointegration country estimated parameters of long-run equation with structural break adf const. int bt time dut(λ) dtt(λ) levels (c+t) de –9.935 3.077 [7.218] 0.022 [2.166] 0.0017 [14.58] –0.0992 [–10.24] 0.0008 [3.778] –3.573 (0.006) *** fr 0.498 0.911 [3.334] –0.0302 [–4.144] 0.001 [12.13] –0.16 [–21.92] –0.0005 [–3.18] –3.882 (0.002) *** nl –0.262 0.979 [4.452] 0.038 [5.746] 0.001 [12.83] –0.0329 [–4.37] –0.001 [–6.502] –4.673 (<0.01) *** dk –2.735 1.614 [4.425] –0.066 [–6.811] 0.0022 [17.86] –0.220 [–21.36] –0.0009 [–4.425] –4.323 (0.001) *** pl 2.33 0.256 [2.103] 0.044 [2.761] 0.005 [15.95] 0.163 [3.24] –0.001 [–4.892] –2.986 (0.036) ** cz –1.597 1.17 [10.61] 0.072 [7.16] 0.003 [22.13] –0.114 [–10.65] –0.002 [–6.959] –4.051 (0.001) *** eu 0.786 0.792 [4.023] 0.017 [1.947] 0.001 [9.449] –0.043 [–1.833] –0.0006 [–4.353] –3.044 (0.031) ** note: ***, ** and * denote significance at 1%, 5% and 10% level respectively; in parentheses the p-value are given, in brackets the t-statistics are given. before estimating the model, it is necessary to check whether the relationships between the variables are characterized by non-linearity. the tsay method is employed to test for linearity of time series. in order to test for linearity of series, the optimal autoregression lag of model should be determined. for germany and poland the optimal lag p is equal to 1, for france and the eu the lag p is equal to 5. for denmark, netherlands and czech republic the optimal lag p is respectively equal to 3, 1 and 2. table 2 summarizes the result of linearity test. the f(p,d) statistic suggest that the time series for germany, denmark and poland are nonlinear with d=1 (it appears that the threshold break with delay greater than 1 in the case of germany is possible, however the threshold break with delay d=1 as the threshold variable showing the greatest effect). in the case of france and the eu the tsay test rejects the linear hypothesis with delay d=5 with the greatest effect. for netherlands the threshold delay is equal to 11 and for czech republic the tsay test does not reject the null hypothesis of linear relationship – see table 2. when we allow the structural break in intercept and slope of trend function in long-run equation (1), the linear cointegration test reveals the longrun relationship for all analyzed economies – see table 1. however, the engle-granger approach assuming symmetric adjustment toward equilibrium is misspecified if the adjustment is asymmetric. therefore, to test for cointegration with asymmetric adjustment, the threshold cointegration approach proposed by enders and sikols (2001) is used. asymmetries in the relationship between economicactivity and oil prices… dynamic econometric models 16 (2016) 65–86 75 table 2. the tsay (1989) linearity test delay d germany (de) france (fr) denmark (dk) nederland (nl) poland (pl) czech republic (cz) european union (eu) 1 9.660 ** 4.176 ** 4.632 ** 0.634 6.960 ** 1.183 4.620 ** 2 2.009 5.719 ** 0.765 1.126 2.787 2.446 5.011 ** 3 5.394 ** 3.376 ** 0.064 0.682 1.095 0.902 3.240 ** 4 2.598 0.969 1.021 0.568 0.527 0.946 1.788 5 1.754 ** 6.294 ** 1.190 0.309 0.577 1.055 9.086 ** 6 0.417 1.112 4.373 ** 0.277 1.185 0.796 1.138 7 2.741 0.909 1.306 0.552 0.086 1.256 4.458 ** 8 0.546 1.887 2.341 * 1.825 0.314 0.796 4.151 ** 9 2.236 2.232 ** 0.211 0.917 0.049 0.306 2.532 10 1.471 0.961 0.965 2.750 ** 0.020 1.831 0.900 11 0.391 1.035 0.831 3.499 ** 0.895 0.687 0.931 12 7.141 ** 0.334 0.902 1.664 0.503 0.144 0.411 note: ** indicates the rejection of null hypothesis at the 5% significance level. the asymmetric adjustment mechanism (tar or m-tar) is selected using the akaike information criterion (aic). threshold cointegration test finds the evidence for threshold mechanism in the case of germany, france, poland and the european union (both the null hypothesis of no cointegration ( 0: 210  h ) and the null of symmetric adjustment ( 210 :  h ) are rejected). here the m-tar model with zero-value of threshold is used. for three other countries (denmark, netherlands and czech republic) the symmetric cointegration is found (only the null hypothesis of no cointegration is rejected but the null of symmetric adjustment is not rejected) – see table a.1 in appendix. the selection of m-tar model indicates that the direction in which production is moving (its momentum) is more important than whether deviations from long-run equation are above or below the equilibrium (as in tar model). in the case of mentioned economies (germany, france, poland and the eu) the point estimates 1 and 2 have negative signs. it means that adjustment coefficients act to eliminate deviations from long-run relationship. for germany, france and the eu deviations below the threshold adjust faster toward the long-run equilibrium than the deviations above the threshold ( 21   ). we can see that respectively 22.0%, 21.7% and 26.2% of the deviation from equilibrium is corrected in the next period when they are below the threshold, but only 0.8%, 7.3% and 5.9% when they are above the threshold. in the case of poland the deviation above the threshold adjusts faster toward the long-term equilibrium (16.1% of deviation in the next periandrzej geise, mariola piłatowska dynamic econometric models 16 (2016) 65–86 76 od is corrected) than deviations below the threshold (only 4.9% of deviation is corrected). these results showed that the threshold cointegration occurs for developed countries (germany, france and the eu). in the case of poland the threshold cointegration also occurs, however it should be noted, that the value of adjustment coefficient in regime 1 is greater than regime 2 (since 21   ). results for poland are inconsistent with the theory, where the greater value of coefficient 2 provides confirmation of the faster reversion to the long-run equilibrium when the deviations from long-run equation are below the threshold value and the slowdown of economic activity (recession in business cycle) is mitigated by the interaction of fiscal and monetary policy. some other mitigating factors of recession can be distinguished as follows: interaction between foreign trade and domestic economy or dynamic expansion of the service-sector characterized by the greater resilience to temporary downturn in economic activity (orłowska, pangsy-kania, 2003). given the findings of linearity test and threshold cointegration among oil prices, production and inflation, it is possible to estimate the threshold (t-ecm) and linear error correction models (ecm). the results of the threshold ecms for german, france, poland and the european union economies are reported in table 4. in this case, the m-tar-ecm models for germany, france, poland and the european union take the form of equation (5). parameters of an asymmetric adjustment of threshold models are presented in table 3. the results for denmark, netherlands and czech republic are presented in table 4. table 3. adjustment coefficients (γ1 and γ2) of the estimated m-tar-ecm (with =0) models for the selected eu countries regime variables germany (de) france (fr) european union (eu) poland (pl) high growth pt –0.047 [–0.93] 0.052 [0.52] –0.002 [–0.02] –0.14 [–2.65] *** int 0.013 [1.432] 0.045 [2.08] ** 0.005 [0.27] 0.011 [0.99] bt –0.142 [–0.46] 0.788 [1.07] 0.234 [0.33] 0.269 [1.15] low growth pt –0.193 [–4.16] *** –0.106 [–2.46] ** –0.117 [–3.35]*** –0.03 [–0.54] int 0.015 [1.75] * 0.003 [0.32] 0.014 [1.51] 0.016 [1.41] bt 0.143 [0.494] –0.203 [–0.64] 0.333 [0.86] –0.312 [–1.30] note: ***, (**), (*) indicate significance at 1%, (5%) and (10%) level, t-statistics for 1 and 2 in brackets, the ljung-box statistic for all estimated models show the lack of autocorrelation – the full results are available from the authors. the m-tar-ecm model allows for a different speed of adjustment to the long-run equilibrium depending on highor low growth states of the asymmetries in the relationship between economicactivity and oil prices… dynamic econometric models 16 (2016) 65–86 77 economy (i.e. production is above or below the long-run relationship respectively). the point estimates of 1  and 2  in the m-tar-ecm models for t p have a negative sign but the coefficients of low growth regime ( 2  ) are significant in the case of germany, france and the eu. in the case of poland, the coefficient 1  (in high growth regime) is significant – see table 3. the adjustment coefficients 1  and 2  in the model for t p in germany, france and the eu suggest that the speed of adjustment is more rapid for negative than for positive discrepancies (since ).21   in the case of german economy about 19.3% (for france10.6% and for the eu 11.7%) of the deviation from equilibrium is corrected in the next period when production is falling, compared to about 4.7% (france – 5.2% and the eu – 0.2%) when production is rising 4 – see table 3. the greater value of coefficient 2  than 1  provides confirmation that the business cycle is characterized by a shorter low growth phase. it means that the recession phase is mitigated (for example by effective interaction of fiscal and monetary policies – see orłowska, pangsy-kania, 2003, p. 24) because the greater value of 2  , the faster correction of deviations from long-run equilibrium is observed. in the case of poland the results reported in table 3 show that the production deviations above the threshold adjust faster toward the long-run equilibrium than the deviations below the threshold (since 21   ). we can see that only 3.0% of the production deviations from equilibrium are corrected in the next period when they are below the threshold, but 14.0% when they are above the threshold. as can be seen in table 3 the  coefficients in the threshold models for t in and t b in germany, france, poland and the eu have a positive signs. that means the deviations to the long-run equation adjust to the ‘wrong’ direction (have positive signs) in either or both regimes, and additionally parameters 1  and 2  are not always significant. 4 in geise and piłatowska (2014), when the long-run equation (cointegration relationship) allows for a one-time change in the intercept of the trend function, the point estimates of 1 and 2 in models for pt in german’s economy suggest that the speed of adjustment is more rapid for positive than for negative discrepancies (since |1|>|2|). when the long-run specification allows for the change in the intercept and the change in the slope of the trend function the m-tar-ecm model for pt in german’s economy reveals consistent dependencies (|1|<|2|). andrzej geise, mariola piłatowska dynamic econometric models 16 (2016) 65–86 78 as can be seen in table 4 (the results for linear ecm), only in the case of denmark the deviation away from the long-run relationship is corrected not only by movements in production, but also by movements in inflation and oil prices. this can be seen by the negative signs and significance of the error correction coefficient 1 in the ecms for tp , tin and tb . for netherlands and czech republic only in models for t p a negative and statistically significant sign is observed. it means that the long-run equilibrium is restored only by the short-run movements in production. table 4. adjustment coefficients of the estimated ecm models for the selected eu countries variables denmark (dk) netherlands (nl) czech republic (cz) pt –0.272 [–3.63] *** –0.669 [–6.35] *** –0.174 [–2.82] *** int –0.014 [–1.91] * 0.021 [2.88] *** 0.03 [2.98] *** bt –0.467 [–2.13] ** 0.257 [0.88] 0.561 [3.04] *** note: ***, (**), (*) indicate significance at 1%, (5%) and (10%) level, t-statistics for 1 and 2 in brackets, the ljung-box statistic for all estimated models show the lack of autocorrelation – the full results are available from the authors. to summarize, the discussed results from threshold error correction model (m-tar-ecm) confirmed the previous findings with regard to cointegration with asymmetric adjustment for deviations in production in the case of all analyzed economies (germany, france, the eu and poland). this can be seen by statistical significance of either adjustment coefficients which are properly signed. the enders-siklos procedure pointed out that the linear cointegration (symmetric adjustment) occurs for netherlands, denmark and czech republic. also, the results of engle-granger test confirms these findings. 4. granger causality test based on linear and threshold error correction model in the previous section both the threshold error correction models (t-ecm) and linear error correction models (ecm) were estimated. next, the granger causality test is used to investigate the causal relationship among production, inflation and oil prices in the selected eu countries in the context of shortand long-run asymmetries. the f-statistics for granger causality is employed to examine whether all the coefficients of a given first differenced variable are jointly statistically different from zero (short-run causality) and whether the 21 , coefficients of error correction term are significant (long-run causality). moreover, the asymmetries in the relationship between economicactivity and oil prices… dynamic econometric models 16 (2016) 65–86 79 jointly significance of the  coefficients and all the coefficients of a given explanatory variable is tested. the threshold error correction models allows us to distinguish between shortand long-run granger causality in the context of high growth and lower growth state of the economy. table 5 and 6 present the results of the granger-causality test based on the t-ecm and ecm models for oil prices, inflation and production. the results of granger-causality test, based on the threshold ecm models for germany, show that there is short-run unidirectional causal relationship running from oil prices )( t b to inflation )( t in in both economic activity regimes (in the case of france the unidirectional relationship is identified in high momentum period and for the eu economy causal relationship is found in low momentum period). unidirectional causal relationship between oil prices and production exists for the eu in both regimes (for france the unidirectional short-run causality running from oil prices to production is identified in the high momentum period). bidirectional short-run causal relationship between oil prices, production and inflation exists for polish economy in high growth regime – see table 5. the result of granger-causality test for poland should be treated with precaution because the impact of polish economy on the world crude oil prices does not exist. however short-run asymmetric reaction of inflation and production to oil price increases is highly probable. in the case of short-run causality the t-ecm models are able to identify the significance causal relationship from inflation to production for germany in high momentum period (regime 1) – see table 5. the results of granger-causality test, based on linear ecm models show that there is short-run unidirectional causal relationship running from oil prices to production and inflation in the case of denmark (in the case of czech republic the unidirectional relation from tb to tin is identified). for netherlands the short-run causal relationship between explanatory variables is not confirmed – see table 6. in terms of long-run situation, a unidirectional strong causal relationship running from oil prices tb to production tp is found in the case of all analyzed economies. the asymmetric models show that the oil prices affect the production in a high growth regime (when the production is in upward path) in the case of poland, and in low growth regime (when the production is decreasing) in the case of germany, france and the eu. for denmark, netherlands and czech republic the long-term linear relationship from oil prices to production (in the case of denmark also to inflation) exists. this means that oil prices, among explanatory variables, contribute the most to the shortandrzej geise, mariola piłatowska dynamic econometric models 16 (2016) 65–86 80 run adjustment to re-establish the long-run equilibrium in the case of all analyzed economies. table 5. granger causality analysis based on threshold ecm models r e g im e e ff e ct s causes short-term longterm both (short-term and long-term) pt int bt ecmt-1 (pt, ecmt-1) (int, ecmt-1) (bt, ecmt-1) germany (de) i pt – 4.823 ** 1.187 0.876 – 2.538 * 0.968 int 0.167 – 15.12 *** 2.05 1.025 – 9.055 *** bt 0.238 0.374 – 0.21 0.301 0.364 – ii pt – 1.069 1.705 17.31 *** – 5.035 *** 6.603 *** int 1.337 – 2.513 * 3.067 * 1.913 – 2.538 ** bt 2.714 ** 0.52 – 0.244 2.152 * 0.454 – france (fr) i pt – 0.179 5.344 ** 0.266 – 0.923 2.869 * int 0.482 – 11.05 *** 4.313 ** 2.156 – 8.036 *** bt 0.206 1.794 – 1.453 0.575 1.171 – ii pt – 0.823 1.215 6.036 ** – 1.41 1.94 ** int 1.333 – 1.607 0.104 1.384 – 1.433 bt 1.144 1.21 – 0.415 1.039 1.1 – european union (eu) i pt – 0.924 2.966 * 0.982 – 0.472 1.543 int 0.015 – 12.08 *** 0.072 0.059 – 6.56 *** bt 0.539 0.043 – 0.109 0.275 0.091 – ii pt – 0.468 2.533 * 11.23 *** – 3.054 ** 5.489 *** int 1.513 – 0.22 2.27 1.816 – 0.704 bt 1.112 0.769 – 0.737 0.953 0.741 – poland (pl) i pt – 0.263 3.223 ** 7.018 *** – 1.8 4.081 *** int 0.955 – 2.636 * 0.985 1.127 – 2.21 * bt 2.686 ** 2.291 * – 1.34 2.564 ** 2.038 * – ii pt – 1.012 2.216 0.295 – 0.777 1.49 int 1.98 – 0.659 1.996 1.743 – 1.04 bt 0.044 3.119 ** – 1.693 0.625 2.395 * – note: ***, (**), (*) indicate significance at 1%, (5%) and (10%) level. as can be seen, the granger causality test shows the causal relationship from oil prices to production and/or inflation in both, shortand long-run terms. when we take into consideration the results of threshold cointegration under consideration, the granger causality test based on the t-ecm models indicate the existence of causation in terms of high growth (expansion) and asymmetries in the relationship between economicactivity and oil prices… dynamic econometric models 16 (2016) 65–86 81 low growth (recession) regime. the t-ecm models can detect more interesting relations than the linear ecm models. they can identify the relations in terms of changes in a business cycle, showing the asymmetry not only in the case of different regimes but also in the case of different impact on analyzed economies. in this case the threshold models were able to identify the shortrun granger causality in high regime (for poland and france) and/or low regime (in the case of germany and the eu). in the case of denmark and czech republic the causal relationship between oil prices, production and inflation (however for netherlands linear ecms, the short-run granger causality has been not found) exists. table 6. granger causality analysis based on linear ecm models effects causes short-term longterm both (short-term and long-term) pt int bt ecmt-1 (pt, ecmt-1) (int, ecmt-1) (bt, ecmt-1) denmark (dk) pt – 2.9262 * 1.8508 * 13.16 *** – 6.782 *** 4.2919 *** int 2.1071 * – 2.3112 ** 3.6324 * 1.6863 – 2.4941 ** bt 1.5822 0.6198 – 4.5344 ** 1.6961 2.1692 * – netherlands (nl) pt – 0.6688 0.2781 40.28 *** – 20.27 *** 21.857 *** int 0.3858 – 1.197 8.262 *** 4.6314 ** – 2.0303 * bt 0.4852 0.006 – 0.7693 1.3264 0.3855 – czech rep. (cz) pt – 0.0016 1.8884 * 7.957 *** – 4.0741 ** 2.866 ** int 0.3049 – 1.3078 8.856 *** 4.6519 ** – 4.8005 *** bt 1.0928 0.0942 – 9.216 *** 4.6292 ** 4.694 ** – note: ***, (**), (*) indicate significance at 1%, (5%) and (10%) level. conclusions in this paper, we analyzed the results of granger causality test among oil prices, production and inflation in the framework of the threshold (t-ecm) and linear (ecm) error correction models across the set of european union countries which are very diverse with respect to the role of oil in their economy. several important insights emerge from this analysis. first, the threshold cointegration analysis carried out using the residuals obtained from the estimated long-run equation with a structural change in the intercept and a change in the slope of the trend function (structural break occur in september 2008 when the financial crisis peaked) allows to identify andrzej geise, mariola piłatowska dynamic econometric models 16 (2016) 65–86 82 the long-run relationship between production, inflation and oil prices for germany, france, the eu and poland. however, polish economy reverts much faster toward its long-run equilibrium when the deviations are above the threshold value and tends to persist when the deviations are below the threshold. for germany, france and the eu belonging to the developed countries the threshold cointegration has been found in high growth state of the economy. in the case of denmark, netherlands and czech republic the long-run relationship between production, inflation and oil prices is linear. in the case of poland when the residuals were obtained from long-run equation with structural change in the intercept the enders-siklos procedure was not able to identify the long-run relationship between production, inflation and oil prices. adding the change in the slope of the trend function in long-run equation cause the change in the residuals and then the enders-siklos approach was able to find the evidence for existing the long-term relationship between economic activity and prices of crude oil. second, both types of error correction models are capable to detect the significant relationship between oil prices, production and inflation in the analyzed european union economies. the threshold models (t-ecm) outperform the linear ecm models because the former can reveal the asymmetric connections among analyzed variables in terms of short-run and long-run relations, while the latter cannot. when the economy is in the high growth phase of business cycle (deviations form long-run equation are above the threshold – regime 1) the granger causality test finds evidence of short-run relationship running from oil prices to inflation (for germany, france and poland) and production (for france, the eu and poland). in the case, when the economy is in the low growth phase of business cycle (deviations form long-run equation are below the threshold value – regime 2) the short-run causality from oil prices to inflation (for germany and the eu) and production (for the eu) is identified. third, the granger causality test finds the linear short-run causal relationship running from oil prices to production and inflation for denmark and czech republic (with except for netherlands). the long-run relationship running form oil prices to production was confirmed for all analyzed economies (for denmark exists also the relation running form oil prices to inflation). in this study we focused on testing the causal relationship between economic activity and oil prices using the ecms with threshold cointegration and short-run asymmetries. in further studies we will concentrate on investigating the relationship in the framework of multivariate threshold models (e.g. threshold var and markov switching var models). asymmetries in the relationship between economicactivity and oil prices… dynamic econometric models 16 (2016) 65–86 83 references bernanke, b. s., gertler, m., watson, m. (1997), systematic monetary policy and the effects of oil price shocks, brooking papers on economic activity, 1997(1), 91–157, doi: http://dx.doi.org/10.2307/2534702. chan, k. s. (1993), consistency and limiting distribution of the least squares estimator of a threshold autoregressive model, the annals of statistics, 21, 520–533, doi: http://dx.doi.org/10.1214/aos/1176349040. davis, s. j., halitwanger, j. (2001), sectoral job creation and destruction responses to oil price changes, journal of monetary economics, 48, 465–512, doi: http://dx.doi.org/10.1016/s0304-3932(01)00086-1. edelstein, p., kilian, l. (2009) how sensitive are consumer expenditures to retail energy prices?, journal of monetary economics, 56(6), 766–779, doi: http://dx.doi.org/10.1016/j.jmoneco.2009.06.001. enders, w., granger, c. w. j. (1998), unit-root tests and asymmetric adjustment with an example using the term structure of interest rates, journal of business and economic statistics, 16, 304–311, doi: http://dx.doi.org/10.2307/1392506. enders, w., siklos, p. (2001), cointegration and threshold adjustment, journal of business and economic statistics, 19(2), 166–176, doi: http://dx.doi.org/10.1198/073500101316970395. fosten, j., morley, b., taylor, t. (2012), dynamic misspecification in the environmental kuznets curve: evidence from co2 and so2 emissions in the united kingdom, ecological economics, 76, 25–33, doi: http://dx.doi.org/10.1016/j.ecolecon.2012.01.023. geise, a., piłatowska, m. (2014), oil prices, production and inflation in the selected eu countries: threshold cointegration approach, dynamic econometric models, 14, 71–91, doi: http://dx.doi.org/10.12775/dem.2014.004. hamilton, j. d. (1988), a neoclassical model of unemployment and the business cycle, journal of political economy, 96(3), 593–617, doi: http://dx.doi.org/10.1086/261553. hamilton, j. d. (2008), oil and the macroeconomy, in durlauf s., blume l. (eds.) the new palgrave dictionary of economics, palgrave mcmillan ltd, doi: http://dx.doi.org/10.1057/9780230226203.1215. hamilton, j. d. (2011), nonlinearities and the macroeconomic effects of oil prices, macroeconomic dynamics, 15(s3), 364–378, doi: http://dx.doi.org/10.1017/s1365100511000307. hamilton, j. d., herrera, a. m. (2004), oil shocks and aggregate macroeconomic behavior: the role of monetary policy, journal of money, credit and banking, 36(2), 265–286, doi: http://dx.doi.org/10.1353/mcb.2004.0012. herrera, a. m., lagalo, l. g., wada, t. (2015), asymmetries in the response of economic activity to oil price increases and decreases, journal of international money and finance, 50, 108–133, doi: http://dx.doi.org/10.1016/j.jimonfin.2014.09.004. orłowska, r., pangsy-kania, s. (2003), cykle koniunkturalne – teoria, analiza i praktyka (business cycles – theory, analysis and the practice), fundacja rozwoju uniwersytetu gdańskiego, gdańsk. perron, p. (1989), the great crash, the oil price shock and the unit root hypothesis, econometrica, 57(6), 1361–1401, doi: http://dx.doi.org/10.2307/1913712. piłatowska, m., włodarczyk, a., zawada, m. (2014), the environmental kuznets curve in poland – evidence from threshold cointegration analysis, dynamic econometric models, 14, 51–70, doi: http://dx.doi.org/10.12775/dem.2014.003. http://dx.doi.org/10.2307/2534702 http://dx.doi.org/10.1214/aos/1176349040 http://dx.doi.org/10.1016/s0304-3932%2801%2900086-1 http://dx.doi.org/10.1016/j.jmoneco.2009.06.001 http://dx.doi.org/10.2307/1392506 http://dx.doi.org/10.1198/073500101316970395 http://dx.doi.org/10.1016/j.ecolecon.2012.01.023 http://dx.doi.org/10.12775/dem.2014.004 http://dx.doi.org/10.1086/261553 http://dx.doi.org/10.1057/9780230226203.1215 http://dx.doi.org/10.1017/s1365100511000307 http://dx.doi.org/10.1353/mcb.2004.0012 http://dx.doi.org/10.1016/j.jimonfin.2014.09.004 http://dx.doi.org/10.2307/1913712 http://dx.doi.org/10.12775/dem.2014.003 andrzej geise, mariola piłatowska dynamic econometric models 16 (2016) 65–86 84 stigler, m. (2010), threshold cointegration: overview and implementation in r, web-doc, https://cran.r-project.org/web/packages/tsdyn/vignettes/thcointoverview.pdf tsay, r. s. (1989), testing and modeling threshold autoregressive processes, journal of the american statistical association, 84(405), 231–240, doi: http://dx.doi.org/10.1080/01621459.1989.10478760. asymetria zależności między aktywnością gospodarczą oraz cenami ropy naftowej w wybranych państwach unii europejskiej z a r y s t r e ś c i. w pracy zastosowano progowe (t-ecm) oraz liniowe modele korekty błędem (ecm) do badania krótkookresowej i długookresowej przyczynowości w sensie grangera między produkcją, inflacją oraz cenami ropy naftowej w kontekście asymetrycznych reakcji dla siedmiu gospodarek unii europejskiej (tj. niemiec, francji, danii, holandii, polski, czech oraz unii europejskiej jako całości). relacje między aktywnością gospodarczą i cenami ropy naftowej analizowano przy założeniu występowania zmian strukturalnych w równaniach długookresowych, gdzie uwzględniano załamania strukturalne w wyrazie wolnym oraz w nachyleniu funkcji trendu. progowe modele korekty błędem (t-ecm) wskazują na asymetryczną reakcję produkcji oraz inflacji na zmiany cen ropy naftowej w przypadku niemiec, francji, unii europejskiej oraz polski. dla pozostałych gospodarek (tj. danii, holandii oraz czech) reakcja aktywności gospodarczej na szoki naftowej jest symetryczna. s ł o w a k l u c z o w e: krótkookresowa asymetria, kointegracja progowa, progowy model korekty błędem, przyczynowość w sensie grangera, asymetryczny szok naftowy. http://dx.doi.org/10.1080/01621459.1989.10478760 asymmetries in the relationship between economicactivity and oil prices… dynamic econometric models 16 (2016) 65–86 85 appendix table a.1. results of tar and m-tar test for threshold cointegration in the longrun equation with one structural break in production threshold cointegration test =0 0 tar m-tar tar m-tar germany (de)  0 0 0.0152 –0.0131 1 –0.105 [–1.889] * –0.008 [–0.137] –0.101 [–1.806] * –0.078 [–1.629] 2 –0.133 [–2.173] ** –0.22 [–3.887] *** –0.138 [–2.261] ** –0.278 [–3.251] ***  3.681 7.93 ** 3.732 6.144 * f(1–2=0) 0.146 (0.70) 8.451 (0.004) *** 0.244 (0.62) 4.973 (0.03) ** lag 5 5 5 5 aic –1190.5 –1198.6 –1190.6 –1195.2 lb(12) 16.55 (0.17) 16.24 (0.18) 16.48 (0.17) 16.43 (0.17) france (fr)  0 0 0.0232 –0.0084 1 –0.105 [–1.877] * –0.073 [–1.221] –0.062 [–1.061] –0.108 [–2.183] ** 2 –0.201 [–3.199] *** –0.217 [–3.697] *** –0.233 [–3.959] *** –0.256 [–3.276] ***  6.453 ** 7.33 ** 8.165 ** 7.159 ** f(1–2=0) 1.431 (0.233) 3.149 (0.077) * 4.693 (0.031) ** 2.824 (0.094) * lag 2 2 2 2 aic –1297.01 –1298.74 –1297.26 –1298.42 lb(12) 7.31 (0.2) 9.10 (0.11) 5.97 (0.31) 8.50 (0.13) netherlands (nl)  0 0 0.0219 0.0126 1 –0.422 [–4.244] *** –0.364 [–3.722] *** –0.456 [–4.136] *** –0.501 [–4.187] *** 2 –0.378 [–3.859] *** –0.437 [–4.375] *** –0.368 [–4.041] *** –0.359 [–4.113] ***  12.858 *** 12.999 *** 13.084 *** 13.5 *** f(1–2=0) 0.144 (0.7) 0.397 (0.53) 0.55 (0.46) 1.298 (0.26) lag 3 3 3 3 aic –1096.49 –1096.74 –1096.89 –1097.64 lb(12) 12.2 (0.43) 11.71 (0.47) 12.60 (0.4) 12.55 (0.4) denmark (dn)  0 0 –0.0214 –0.0188 1 –0.444 [–4.42] *** –0.511 [–4.812] *** –0.458 [–4.622] *** –0.403 [–4.608] *** 2 –0.461 [–5.09] *** –0.42 [–4.77] *** –0.45 [–4.899] *** –0.547 [–5.017] ***  17.692 *** 17.966 *** 17.682 *** 18.458 ** f(1–2=0) 0.023 (0.88) 0.631 (0.43) 0.005 (0.95) 1.482 (0.22) lag 3 3 3 3 aic –973.8 –974.4 –973.7 –975.2 lb(12) 12.25 (0.43) 12.77 (0.39) 12.17 (0.43) 12.17 (0.43) andrzej geise, mariola piłatowska dynamic econometric models 16 (2016) 65–86 86 table a.1. continued threshold cointegration test =0 0 tar m-tar tar m-tar poland (pl)  0 0 –0.009 0.0124 1 –0.129 [–2.731] *** –0.161 [–3.321] *** –0.135 [–2.856] *** –0.18 [–2.634] *** 2 –0.075 [–1.543] –0.049 [–1.028] –0.069 [–1.413] –0.079 [–2.022] **  4.782 5.927 * 4.947 5.299 f(1–2=0) 0.634 (0.43) 2.838 (0.09) * 0.952 (0.33) 1.675 (0.2) lag 1 1 1 1 aic –1152.07 –1154.29 –1152.39 –1153.12 lb(12) 6.22 (0.9) 6.18 (0.91) 6.23 (0.90) 6.4 (0.89) czech republic (cz)  0 0 –0.014 –0.018 1 –0.208 [–2.91] *** –0.263 [–3.54] *** –0.216 [–3.05] *** –0.231 [–3.94] *** 2 –0.245 [–3.38] *** –0.194 [–2.79] *** –0.237 [–3.24] *** –0.206 [–1.95] ***  8.821 *** 8.975 *** 8.685 ** 8.644 ** f(1–2=0) 0.159 (0.69) 0.517 (0.47) 0.051 (0.82) 0.047 (0.83) lag 2 2 2 2 aic –1040.1 –1040.5 –1040.0 –1039.99 lb(12) 11.89 (0.45) 11.55 (0.48) 11.84 (0.46) 11.76 (0.46) european union (eu)  0 0 –0.0134 0.0068 1 –0.162 [–2.973] *** –0.059 [–1.046] –0.163 [–3.053] *** –0.041 [–0.481] 2 –0.187 [–3.504] *** –0.262 [–4.992] *** –0.188 [–3.436] *** –0.201 [–4.575] ***  10.662 *** 12.979 *** 10.666 *** 10.648 *** f(1–2=0) 0.109 (0.74) 7.309 (0.01) *** 0.115 (0.73) 3.021 (0.08) * lag 0 1 0 1 aic –1264.22 –1264.56 –1264.23 –1260.38 lb(12) 13.62 (0.33) 16.49 (0.17) 13.55 [0.33] 14.63 [0.26] note: ***, ** and * denote significance at 1%, 5% and 10% level respectively; in parentheses the t-statistics are given; in brackets the p-values are given. microsoft word 13_olbryś_j.docx dynamic econometric models vol. 11 – nicolaus copernicus university – toruń – 2011 joanna olbryś bialystok university of technology arch effect in classical market-timing models with lagged market variable: the case of polish market† a b s t r a c t. the main goal of this study is to present the regressions of the garch versions of classical market-timing models of polish equity funds. we examine the models with lagged values of the market factor as an additional variable because of the fisher’s effect1 in the case of the main warsaw stock exchange indexes. the market-timing and selectivity abilities of fund managers are evaluated for the period jan 2003 – june 2011. results on both the hac and the garch estimates are qualitatively similar, and even better in the case of the simpler hac method. for this reason, it is not necessary to estimate the garch versions of market-timing models in the case of polish mutual funds, even despite the strong arch effects that exist in these models. k e y w o r d s: market-timing, non-trading, arch effect, garch model. introduction market-timing is one strategy by which portfolio managers might attempt to obtain returns in excess of those expected of an unmanaged portfolio. one of the benefits of market-timing is the production of a positively skewed distribution of returns. treynor and mazuy (1966) produce a single factor model derived from capm in which a quadratic term is added to reflect the markettiming. the t-m coefficient measures the co-skewness with the benchmark portfolio. henriksson and merton (1981) start from a similar idea, but provide a different interpretation of market-timing ability. adding a term in the capm model that contains a dummy variable based on the difference between market return and the risk-free rate, they permit managers to choose between two levels of market risk: an up-market and a down-market beta (cogneau, hübner, 2009). some other researchers extend market-timing models to multifactor as well as † the author acknowledges the financial support in 2009 – 2011 from the polish ministry of science and higher education within the grant no. n n113 173237. 1 lawrence fisher’s effect (1966) joanna olbryś 186 to conditional versions. in relation to the polish market, in olbryś (2009) the usefulness of the conditional multifactor market-timing models for the investment managers’ performance evaluation on the polish market has been examined. ferson and schadt (1996) use a collection of public information variables. in poland, the suitable variables are: (1) the lagged monthly dividend yield of the wse stock index, (2) the lagged monthly level of the 1m wibor, (3) the lagged monthly measure of the slope of the term structure (olbryś, 2009, p. 522). the evidence on polish market shows that the quality and usefulness of these models is rather low. as for the other multifactor models, in light of the empirical results for the polish market, the influence of the fama and french’s (1993) size (smb) and book-to-market (hml) spread variables, and carhart’s (1997) momentum (wml) factor on the polish equity funds’ market seems to be rather controversial (olbryś, 2010a b; 2012). it is worth stressing that the smb, hml, and wml factors have a diverse explanatory power for the sample of polish funds. another important finding is that the investigated funds are not homogeneous regarding the influence of the size, book-to-market and momentum factors, despite the fact that all of them are polish equity open-end mutual funds (olbryś, 2011a). for this reason, the smb, hml, and wml factors have not been taken into account as explanatory variables in our models. according to the literature, the methods most widely applied in markettiming models estimation are the two proposed by white (1980) or newey-west (1987) (see e.g. ferson, schadt, 1996; bollen, busse, 2001; romacho, cortez, 2006; olbryś, 2010a, b). some previous publications also describe applications of the gls procedure with correction for heteroskedasticity (see e.g. henriksson, merton, 1981; henriksson, 1984) or the fama-macbeth cross-sectional regression approach from 1973 (carhart, 1997). kao et al. (1998) employ an autoregressive conditional heteroskedastic (arch) model, but without testing the arch effects. recent studies in multifactor market-timing models in the case of polish equity funds by olbryś (2010a) present possibilities and examples of applying the seemingly unrelated regression method (sur) which was described by zellner (1962). the author’s recent research provide evidence of pronounced arch effects (engle, 1982) in the market-timing models of polish equity open-end mutual funds. for this reason, the main goal of this study is to present the regression results of the new garch(p, q) models of these funds. we estimate the garch versions of classical market-timing models with lagged values of the market factor as an additional independent variable because of the pronounced fisher’s effect in the case of the main warsaw stock exchange indexes. the market-timing and selectivity abilities of fund managers are evaluated for the period january 2003 – june 2011. in comparison to robust newey-west method results, our findings suggest that the garch(p, q) model is suitable but not necessary for such applications. to the best of author’s knowledge, no such investigation has been undertaken for the polish market. arch effect in classical market-timing models with lagged market variable… 187 the remainder of the paper is organized as follows. section 1 specifies a methodological background and a brief literature review. first, we stress the validity of the non-trading problem and the fisher’s effect in the case of marketindex returns. next, we present classical market-timing models with lagged values of the market factor as additional explanatory variable. we also present a brief theoretical framework concerning the arch(q) and the garch(p,q) models. in the end of section 1, we describe tests for the arch effect in an econometric model. in section 2, we present the data and methodology in the case of polish emerging market and discuss the results obtained. section 3 recalls the main findings and presents the conclusions. 1. methodological background 1.1. non-trading problem and the fisher’s effect it is worth stressing, that the empirical market microstructure literature is an extensive one recently. high-frequency financial data are important in studying a variety of issues related to the trading process and market microstructure (tsay, 2010, p. 231). for some purposes, such aspects of the market microstructure as non-trading or bid-ask spread effects can be safely ignored. however, for other purposes, market microstructure is central (campbell et al., 1997). in 1980 cohen et al. point to various frictions in the trading process that can lead to a distinction between “true” and observed returns. they have focused on the fact that transaction prices differ from what they would otherwise be in a frictionless environment. it has been reported in the literature that some empirical phenomena can be attributed to frictions in the trading process (see e.g. fisher, 1966; scholes and williams, 1977; dimson, 1979), also on the polish capital market (see e.g. doman, doman, 2004; doman, 2010; brzeszczyński et al., 2011; olbryś, 2011b). it is worthwhile to note that two common elements among most of the phenomena are evident, the “interval effect” and the impact of a security’s “thinness”. the evidence that daily market-index returns exhibit a pronounced positive first-order autocorrelation has been called the fisher’s effect since lawrence fisher in 1966 hypothesized its probable cause. fisher suggested it was caused by a non-trading of the component securities. the observed correlation is higher in those indexes that give greater weight to the securities of smaller firms. to detect for the fisher’s effect in the market-index returns, partial autocorrelations functions (pacf) can be calculated. to calculate pacf, first it should be determined (based on the adf test) that the analyzed index series are stationary. in the next step partial autocorrelations functions for individual stationary processes can be calculated and the significance of the first-order daily serial correlation coefficients 1 can be tested, using the quenouille’s test (kufel, 2009). the evaluation of first-order serial correlation is carried out by testing the null hypothesis: joanna olbryś 188 0 1: 0h   (1) if the estimate 1̂ satisfies the inequality 1 1.96 ˆ t   , then we have no reason to reject the null hypothesis (1). the non-trading effect induces potentially serious biases in the moments and co-moments of asset returns such as their means, variances, covariances, betas, and autocorrelation and cross-autocorrelation coefficients (campbell et al., 1997, p. 84). for this reason, busse (1999) proposed lagged values of the market factor as an additional independent variable in the regressions of markettiming models using dimson’s (1979) correction. we emphasize that the polish market is an emerging market and it can be expected that non-trading problem should be more visible than in other, developed markets. 1.2. classical market-timing models with lagged market variable the classical parametric treynor – mazuy market-timing model with lagged values of the market factor as additional explanatory variable can be expressed as: 2 , 1 , 2 , 1 , ,( ) ,p t p p m t p m t p m t p tr r r r            (2) where tfr , is the one-period return on riskless securities, , , ,p t p t f tr r r  is the excess return on portfolio p in the period t, , , ,m t m t f tr r r  is the excess return on portfolio m in the period t, 1, tmr is the lagged excess return on portfolio m in the period t, p measures the selectivity skills of the manager (jensen, 1968), p1 is the systematic risk measure of portfolio p to changes in the market factor returns, 2 p is the systematic risk measure of portfolio p to changes in the lagged market factor returns, p measures the market-timing skills of the manager of portfolio p (the t-m coefficient), and tp, is a residual term, with the following standard capm conditions: ,( ) 0p te   , , , 1( ) 0p t p te     . in a way analogous to (2), the classical parametric henriksson merton model with lagged values of the market factor as additional explanatory variable can be expressed as: , 1 , 2 , 1 , , ,p t p p m t p m t p m t p tr r r y            (3) where tpr , , tmr , , 1, tmr , p , 1p , 2 p , ,p t are as in equation (2), p measures the market-timing skills of the manager of portfolio p (the h-m coefficient), and , ,max{0, }m t m ty r  . arch effect in classical market-timing models with lagged market variable… 189 if the portfolio manager has the ability to forecast security prices, the intercept p in equations (2)–(3) will be positive (jensen, 1968). indeed, it represents the average incremental rate of return on the portfolio per unit time which is due solely to the manager’s ability to forecast future security prices. in this way, ˆ p measures the contribution of security selection to portfolio performance, which corresponds to testing the null hypothesis: 0 : 0ph   (4) i.e., the manager does not have any microforecasting ability. the evaluation of market-timing skills is carried out by testing the null hypothesis: 0 : 0ph   (5) i.e., the manager does not possess any timing ability or does not on his forecast (henriksson 1984). a negative value for the regression estimate ˆp would imply a negative value for market-timing. the size of the estimate ˆp informs us about the manager’s market skills. 1.3. the garch(p, q) model the first model that provides a systematic framework for volatility modeling is the arch model of engle (1982). engle proposed the arch models to capture the serial correlation of volatility (campbell et al., 1997, p. 482). engle suggested the arch model as an alternative to the usual time-series process. more recent studies of financial markets suggest that the phenomenon is quite common (greene, 2002). the basic idea of the arch models is that 1) the innovation t of the regression is serially uncorrelated, but dependent, and 2) the dependence of t can be described by a simple quadratic function of its lagged values (tsay, 2010). the arch(q) regression model is obtained by assuming that the mean of random variable ty , which is drawn from the conditional density function 1( )t tf y y  , is given as tx b , a linear combination of lagged endogenous and exogenous variables included in the information set 1t  , with b a vector of unknown parameters (engle, 1982). formally: 1 2 1 2 0 1 0 ~ ( , ), , ( , , , , ) , 0, 0, 1, , . t t t t t t t q t t t t q i t i i i y n x h y x h h i q                           b b α  (6) joanna olbryś 190 where t is the innovation in a linear regression with 2( )v   , q is the order of the arch(q) process, α is the vector of unknown parameters, th is the variance function. the null hypothesis of white noise disturbances in (6) is: 0 1: 0qh     (7) the garch(p, q) model generalizes the arch(q) model of engle (1982) and is proposed by bollerslev (1986). the garch(p, q) is given by: 1 1 2 1 2 2 0 1 1 0 ~ ( 0 0 1 0 0 1 0 t t t t t t t t t t t q t t t p q p i t i j t j i j i j y n x ,h ), y x , h h( , , , ,h ,h , ,h , , ) h , , ,i , ,q,q , , j , , p, p .                                          b b α β    (8) where t , q , α , th are as in equation (6) and β is a vector of unknown parameters. in the garch(p, q) model, q refers to the number of lags of t and p refers to the number of lags of th to include in the model of the regression variance (adkins, 2010). for 0p  the process reduces to the arch(q) process, and for 0p q  , t is simple white noise. the null hypothesis of white noise disturbances in (8) is: 0 1 1: 0; 0q ph           (9) in the arch(q) process the conditional variance is specified as a linear function of past sample variances only, whereas the garch(p, q) process allows the lagged conditional variances to enter as well (bollerslev 1986). a wide range of garch models have now appeared in the econometric literature (e.g. engle, 2000; fiszeder, 2009). the parameters of garch(p, q) models are almost invariably estimated via maximum likelihood (ml) or quasi-maximum likelihood (qml: see bollerslev, wooldridge, 1992) methods, which bring up the subject of a suitable choice for the conditional distribution of t . several likelihood functions are commonly used in arch (garch) estimation, depending on the distributional assumption of t (tsay, 2010). 1.4. testing for arch effect in an econometric model before estimating the garch(p, q) model it might be useful to test for the arch effect. the simplest approach is to examine the squares of the least arch effect in classical market-timing models with lagged market variable… 191 squares residuals. the autocorrelations of the squares of the residuals provide evidence about arch effects (greene, 2002). two tests are available. the first test is to apply the ljung-box statistics ( )q q (ljung, box, 1978). the null hypothesis is that the first q lags of acf of the squares of the least squares residuals series are zero. in practice, the choice of q may affect the performance of the ( )q q statistic. simulation studies suggest that the choice of ln( )q t , where t is the number of time periods, provides better power performance (tsay, 2010, p. 33). the second test for conditional heteroskedasticity is the lagrange multiplier (lm) test of engle (1982). lee (1991) found that the lm test of white noise disturbances against garch(p, q) disturbances in a linear regression model is equivalent to that against arch(q) disturbances. this implies that under the null hypothesis of white noise, the garch(p, q) effect and the arch(q) effect are locally equivalent alternatives. hence we can proceed by testing the arch(q) effect against the garch(p, q) effect (lee, 1991, pp. 269–270). an lm test of arch(q) against the hypothesis of no arch effects can be carried out by computing 2 2q t r   in the regression of 2 te on a constant and q lagged values. under the null hypothesis of no arch effects, the statistic has a limiting chi-squared distribution with q degrees of freedom. values larger than the critical table value give evidence of the presence of arch (or garch) effects (greene, 2002, p. 244). 2. empirical results 2.1. the fisher’s effect on the warsaw stock exchange the fisher’s effect on the warsaw stock exchange has been detected by olbryś (2011b). the empirical results show a pronounced fisher’s effect in the case of the wig, mwig40 and swig80 series. we observe the most clear effect for the swig80 series. we have no reason to reject the null hypothesis (1) only in the case of the wig20 series. as mentioned above, this evidence is consistent with most of the literature on friction in the trading process because the observed correlation is higher in those indexes that give greater weight to the securities of smaller firms. for the fisher’s effect reason, we can use dimson’s (1979) correction and include lagged values of the market factor (i.e. the main index of wse companies – wig) as an additional independent variable in the regressions of market-timing models of polish equity open-end mutual funds to accommodate infrequent trading (olbryś, 2011a). 2.2. data the period investigated is january 2, 2003 – june 30, 2011 (t=2137 observations). to detect for the arch effect in market-timing models of polish joanna olbryś 192 funds in subsamples of various length, the entire sample has been divided into eight subsamples: p1, p2, p3, p4, p5, p6, p7, p8 (table 1). table 1. subsamples in the period from jan 2, 2003 to june 30, 2011 subsample t p1 jan 2, 2003–june 30, 2011 2137 p2 jan 2, 2004–june 30, 2011 1886 p3 jan 3, 2005–june 30, 2011 1631 p4 jan 2, 2006–june 30, 2011 1380 p5 jan 2, 2007–june 30, 2011 1129 p6 jan 2, 2008–june 30, 2011 880 p7 jan 5, 2009–june 30, 2011 629 p8 jan 4, 2010–june 30, 2011 377 note: t is the number of data points. table 2. equity open-end mutual funds in poland by the end of 2002 equity fund (current name) short name year of creation 1 arka bz wbk fio subfundusz arka akcji arka 1998 2 aviva investors fio subfundusz aviva investors polskich akcji aviva 2002 3 bph fio parasolowy bph subfundusz akcji bph 1999 4 ing parasol fio ing subfundusz akcji ing 1998 5 investor top 25 małych spółek fio investor 25 2002 6 investor akcji dużych spółek fio investor ads 1998 7 investor akcji fio investor 1998 8 legg mason akcji fio legg mason 1999 9 millennium fio subfundusz akcji millennium 2002 10 novo fio subfundusz novo akcji novo 1998 11 pioneer fio subfundusz pioneer akcji polskich pioneer 1995 12 pko akcji – fio pko 1998 13 pzu fio parasolowy subfundusz pzu akcji krakowiak pzu 1999 14 skarbiec fio subfundusz akcji skarbiec – akcji skarbiec 1998 15 unifundusze fio subfundusz unikorona akcje unikorona 1997 note: the source of this table is the polish financial supervision authority http://www.knf.gov.pl (sept 8, 2011). we have examined the performance of 15 selected equity open-end polish mutual funds which were created up to the end of 2002. our dataset includes returns on all the equity funds in existence in poland from 2002 to 2011, therefore our results are free of survivorship bias (table 2). we have studied daily logarithmic excess returns from jan 2003 to june 2011. daily returns on the main index of wse companies are used as the returns on the market portfolio. the daily average of returns on 52-week treasury bills are used as the returns on riskless assets. all calculations were done using gretl 1.9.5. 2.3. arch effect in market-timing models with lagged market variable volatility clustering, which is a common cause of heteroskedasticity, is more likely to be present in financial models built using higher-frequency data, such as daily data (brzeszczyński et al., 2011). to detect for the arch effect in arch effect in classical market-timing models with lagged market variable… 193 market-timing models of polish equity open-end mutual fund portfolios in the period investigated jan 2, 2003 – june 30, 2011, the lm (lagrange multiplier) and the lb (ljung-box) tests have been applied. the empirical results presented in table 3 show strong arch effect in the case of all of the funds. the null hypothesis (7) is rejected in these cases. because we are using daily logarithmic excess returns on fund portfolios, the lm test at the lag 5q has been applied. on the other hand, the lb test at the lag 8)2137ln( q has been used (tsay, 2010). the p -values of all statistics are very close to zero. table 3. the arch effect in market-timing models (2) and (3) of polish equity mutual funds in the entire sample p1 (period from jan 2, 2003 to june 30, 2011) equity fund (short name) t-m model h-m model lm p-value lb p-value lm p-value lb p-value 1 arka 326.8 110-68 157.6 410-30 346.9 710-73 159.5 210-30 2 aviva 257.1 110-53 299.9 410-60 258.3 910-54 306.7 110-61 3 bph 424.6 110-89 434.2 910-89 427.0 410-90 436.5 210-89 4 ing 443.9 110-93 442.4 110-90 445.2 510-94 444.5 510-91 5 investor 25 404.4 310-85 145.1 210-27 390.4 310-82 142.1 810-27 6 investor ads 524.4 410-111 474.7 110-97 531.8 110-112 475.1 110-97 7 investor 460.3 210-97 498.0 110-102 459.4 410-97 497.2 210-102 8 legg mason 402.1 110-84 333.1 310-67 408.6 410-86 334.2 210-67 9 millennium 437.6 210-92 371.2 210-75 439.8 710-93 374.7 410-76 10 novo 622.2 310-132 489.3 110-100 609.5 110-129 485.7 810-100 11 pioneer 423.7 210-89 372.1 110-75 426.6 510-90 374.4 510-76 12 pko 485.9 810-103 379.1 510-77 477.1 710-101 379.0 510-77 13 pzu 402.0 110-84 387.0 110-78 404.9 210-85 391.8 110-79 14 skarbiec 384.4 610-81 427.0 310-87 385.5 410-81 426.7 310-87 15 unikorona 371.4 410-78 519.3 510-107 376.8 210-79 519.7 410-107 note: the table is based on the entire sample p1; t-m (2) is the classical treynor-mazuy model with the lagged excess return on market portfolio m as additional factor; h-m (3) is the classical henriksson-merton model with the lagged excess return on market portfolio m as additional factor; lm is the engle (1982) statistic at the lag equal to five, which should be distributed as chi-squared; lb is the ljung-box (1978) statistic at the lag equal to eight, which should be distributed as chi-squared. tables 4a–4b present further analysis, including more details about empirical results of testing the arch effect in the t-m and the h-m market-timing models. the arch effect has been tested in the case of all funds and in all subsamples. several results in tables 4a–4b are worth special notice. the arch effect disappears as the interval is shortened and only in the case of 5 out of 15 funds (i.e. arka, investor 25, novo, skarbiec and unikorona) it persists in all samples p1–p8. furthermore, if the arch effects are not present in the model, simple ols regression is quite sufficient (brzeszczyński et al., 2011). table 4a. result summary of the arch effect in the t-m and h-m models; subsamples p1–p4 fu nd . n o. p1 p2 p3 p4 t-m h-m t-m h-m t-m h-m t-m h-m lm lb lm lb lm lb lm lb lm lb lm lb lm lb lm lb 1 + + + + + + + + + + + + + + + + 2 + + + + + + + + + + + + + + + + 3 + + + + + + + + + + + + 4 + + + + + + + + + + + + + + + + 5 + + + + + + + + + + + + + + + + 6 + + + + + + + + + + + + + + + + 7 + + + + + + + + + + + + + + + + 8 + + + + + + + + + + + + 9 + + + + + + + + + + + + 10 + + + + + + + + + + + + + + + + 11 + + + + + + + + + + + + 12 + + + + + + + + + + + + + + + + 13 + + + + + + + + + + + + 14 + + + + + + + + + + + + + + + + 15 + + + + + + + + + + + + + + + + table 4b. result summary of the arch effect in the t-m and h-m models; subsamples p5–p8 fu nd . n o. p5 p6 p7 p8 t-m h-m t-m h-m t-m h-m t-m h-m lm lb lm lb lm lb lm lb lm lb lm lb lm lb lm lb 1 + + + + + + + + + + + + + + + + 2 + + + + + + + + + + + + 3 + + + + + + 4 + + + + + + + + + + 5 + + + + + + + + + + + + + + + + 6 + + + + + + + + + + + + 7 + + + + + + + + + + + + + + 8 + + + + + + + + 9 + + + + + + + + 10 + + + + + + + + + + + + + + + + 11 + + + + + + 12 + + + + + + + + + + 13 + + + + + + + + + + + + 14 + + + + + + + + + + + + + + + + 15 + + + + + + + + + + + + + + + + note: table 4a is based on the samples p1– p4 and table 4b is based on the samples p5– p8 (table 1); t-m (2) is the classical treynor-mazuy model with the lagged excess return on market portfolio m as additional factor; h-m (3) is the classical henriksson-merton model with the lagged excess return on market portfolio m as additional factor; lm is the engle (1982) statistic at the lag q, which should be distributed as chi-squared; lb is the ljung-box (1978) statistic at the lag q, which should be distributed as chi-squared; + denotes that statistic value is larger than the critical table value of chi-squared and gives evidence of the presence of arch effect; – denotes that statistic value is smaller than the critical table value of chi-squared. arch effect in classical market-timing models with lagged market variable… 195 2.4. the garch versions of market-timing models of polish equity mutual funds hamilton (2008) stresses that even if the researcher’s primary interest is in estimating the conditional mean, having a correct description of the conditional variance can still be quite important. by incorporating the observed features of the heteroskedasticity into the estimation of the conditional mean, substantially more efficient estimates of the conditional mean can be obtained. the most popular white or newey–west corrections may not fully correct for the influence problems introduced by arch. the testing results from the polish equity table 5a. the hac estimates of the t-m market-timing models of polish equity mutual funds in the entire period from jan 2, 2003 to june 30, 2011 equity fund p̂ 1p̂ 2 p p̂ 2r 1 arka 410-4 ** (110-4) 0.71*** (0.03) 0.14*** (0.03) -1.86** (0.83) 0.66 2 aviva 310-4 ** (110-4) 0.75*** (0.03) 0.15*** (0.02) -1.45** (0.69) 0.73 3 bph 210-5 (110-4) 0.72*** (0.02) 0.12*** (0.02) -0.56 (0.48) 0.76 4 ing -210-6 (110-4) 0.76*** (0.02) 0.13*** (0.02) -0.48 (0.43) 0.74 5 investor 25 210-4 (210-4) 0.39*** (0.02) 0.31*** (0.03) -1.76** (0.85) 0.39 6 investor ads -110-4 (210-4) 0.63*** (0.03) 0.38*** (0.03) -0.55 (1.26) 0.55 7 investor -410-5 (110-4) 0.54*** (0.03) 0.37*** (0.03) -0.75 (0.86) 0.55 8 legg mason 210-4 *(910-5) 0.69*** (0.02) 0.12*** (0.02) -0.68 (0.51) 0.74 9 millennium -410-6 (110-4) 0.69*** (0.02) 0.13*** (0.02) -0.76 (0.57) 0.72 10 novo 810-5 (210-4) 0.48*** (0.03) 0.45*** (0.02) -1.20 (1.14) 0.55 11 pioneer -910-5 (110-4) 0.80*** (0.03) 0.16*** (0.02) -1.13 (0.81) 0.74 12 pko 610-5 (210-4) 0.55*** (0.03) 0.28*** (0.03) -1.66 (1.19) 0.56 13 pzu 310-5 (110-4) 0.71*** (0.02) 0.12*** (0.02) -0.98** (0.49) 0.73 14 skarbiec -410-6 (110-4) 0.46*** (0.03) 0.38*** (0.03) 0.15 (0.73) 0.51 15 unikorona 110-4 (110-4) 0.48*** (0.03) 0.45***(0.03) -0.54 (0.89) 0.55 note: the table is based on the entire sample p1; t-m (2) is the classical treynor-mazuy model with the lagged excess return on market portfolio m as additional factor; the heteroskedastic consistent standard errors are in parentheses next to the coefficient estimates; the values of the adjusted determination coefficient are in the last column; * significant at the 10 per cent level; ** significant at the 5 per cent level; *** significant at the 1 per cent level. mutual funds dataset show pronounced arch effect in market-timing models (tables 3–4). for this reason, the estimation of the market-timing models as the garch(p, q) models is well-founded. although the arch(q) model (6) is simple, it often requires many parameters to adequately describe the volatility process. the modeling procedure of the arch(q) model can also be used to build a garch(p, q) model (8). however, specifying the order of a garch(p, q) model is not easy. only the lower order garch models are used in most applications, i.e. garch(1,1), garch(1,2), garch(2,1), and garch(2,2) models (tsay, 2010). according to the literature, garch(p, q) models are usually compared and selected by the information criterion of akaike (aic) and the information criterion of schwartz (sc). lower values of joanna olbryś 196 the aic and sc indexes indicate the preferred model, that is, the one with the fewest parameters that still provides an adequate fit to the data2. as an example, we present the comparison of the estimation results of market-timing models t-m and h-m of polish equity mutual funds in the entire period from jan 2, 2003 to june 30, 2011. we use the newey-west robust estimates (hac) as well as the robust quasi-maximum likelihood estimates (qml) of the parameters of the suitable garch(p, q) version of the market-timing model. tables 5a–5b provide details on the robust hac estimates of the t-m and h-m market-timing models, respectively. table 5b. the hac estimates of the h-m market-timing models of polish equity mutual funds in the entire period from jan 2, 2003 to june 30, 2011 equity fund p̂ 1p̂ 2 p p̂ 2r 1 arka 710-4 ***(210-4) 0.65*** (0.04) 0.14*** (0.03) -0.14** (0.06) 0.66 2 aviva 610-4 **(210-4) 0.70*** (0.04) 0.15*** (0.02) -0.11** (0.05) 0.73 3 bph 110-4 (110-4) 0.70*** (0.03) 0.12*** (0.02) -0.04 (0.04) 0.76 4 ing 110-4 (110-4) 0.74*** (0.03) 0.13*** (0.02) -0.04 (0.04) 0.74 5 investor 25 410-4 (310-4) 0.33*** (0.03) 0.31*** (0.03) -0.13* (0.06) 0.39 6 investor ads -210-4 (310-4) 0.63*** (0.04) 0.38*** (0.03) -0.01 (0.08) 0.55 7 investor 210-5 (310-4) 0.52***(0.03) 0.37*** (0.03) -0.04 (0.06) 0.55 8 legg mason 310-4 *(110-4) 0.67***(0.03) 0.12*** (0.02) -0.05 (0.04) 0.74 9 millennium 210-4 (210-4) 0.65*** (0.03) 0.13*** (0.02) -0.07 (0.04) 0.72 10 novo 110-4 (310-4) 0.45*** (0.03) 0.45*** (0.03) -0.06 (0.07) 0.55 11 pioneer 910-5 (210-4) 0.76*** (0.04) 0.16*** (0.02) -0.08 (0.06) 0.74 12 pko 310-4 (310-4) 0.50*** (0.04) 0.28*** (0.03) -0.12 (0.07) 0.56 13 pzu 210-4 (210-4) 0.67*** (0.03) 0.12*** (0.02) -0.08* (0.04) 0.73 14 skarbiec -810-5 (310-4) 0.47*** (0.03) 0.38*** (0.03) 0.02 (0.06) 0.51 15 unikorona 110-4 (310-4) 0.47*** (0.03) 0.45*** (0.03) -0.02 (0.06) 0.55 note: the table is based on the entire sample p1; h-m (3) is the classical henriksson-merton model with the lagged excess return on market portfolio m as additional factor; the heteroskedastic consistent standard errors are in parentheses next to the coefficient estimates; the values of the adjusted determination coefficient are in the last column; * significant at the 10 per cent level; ** significant at the 5 per cent level; *** significant at the 1 per cent level. the robust qml estimates of the parameters of the suitable garch(p, q) version of market-timing models are presented in tables 6a–6b, respectively. it is worth stressing that some restrictions for the parameters in the garch(p, q) models (8) can be relaxed. for example, it is not necessary for the 2 parameter in the conditional variance equation in the garch(1,2) model to be nonnegative (fiszeder, 2009). note that in the case of all funds, both for the t-m model and for the h-m model the same variant of the garch(p, q) model has been chosen (tables 6a–6b). 2 when the values of the information criterions aic or sc for different variants of the garch(p, q) models are almost equal, the statistical significance of the parameters in the conditional mean and conditional variance equations of the garch(p, q) model has been analyzed to choose the appropriate model. arch effect in classical market-timing models with lagged market variable… 197 in summary, the results in tables 5a–5b and 6a–6b clearly show that despite the strong arch effect in all models built based on the sample p1, the simpler robust hac method is quite sufficient. therefore, in our opinion, the garch(p, q) model is suitable but not necessary for such applications. table 6a. the garch(p, q) versions of the t-m market-timing models of polish equity mutual funds in the entire period from jan 2, 2003 to june 30, 2011 fu nd . n o. t-m model – conditional mean equation (p, q) conditional variance equation p̂ 1p̂ 2 p p̂ 0̂ 1 2̂ 1̂ 2̂ 1 310-4 (110-4) 0.80 (0.01) 0.04 (0.01) -1.60 (0.52) (1,1) 210-7 (110-7) 0.09 (0.02) 0.90 (0.02) 2 310-4 (710-5) 0.86 (0.01) 0.01 (0.005) -1.95 (0.33) (1,2) 210-7 (610-8) 0.47 (0.12) -0.30 (0.12) 0.85 (0.03) 3 -710-5 (510-5) 0.84 (0.005) 0.005 (0.004) 0.06 (0.18) (1,1) 710-8 (210-8) 0.08 (0.01) 0.91 (0.01) 4 -110-5 (510-5) 0.89 (0.005) 0.003 (0.005) 0.06 (0.21) (1,1) 710-8 (210-8) 0.07 (0.01) 0.91 (0.01) 5 710-5 (210-4) 0.34 (0.02) 0.25 (0.02) -1.06 (1.07) (2,2) 410-6 (110-6) 0.17 (0.04) 0.20 (0.04) -0.16 (0.05) 0.74 (0.05) 6 -210-4 (910-5) 0.94 (0.03) 0.07 (0.02) 0.64 (0.69) (1,2) 410-8 (310-8) 0.28 (0.04) -0.21 (0.04) 0.93 (0.01) 7 -110-4 (710-5) 0.79 (0.01) 0.08 (0.01) -0.28 (0.40) (1,2) 510-8 (310-8) 0.35 (0.04) -0.26 (0.04) 0.91 (0.01) 8 910-5 (610-5) 0.83 (0.007) 0.02 (0.005) -0.009 (0.29) (1,2) 510-8 (210-8) 0.14 (0.03) -0.07 (0.03) 0.93 (0.01) 9 -210-4 (610-5) 0.81 (0.008) 0.009 (0.005) -210-4 (0.31) (2,1) 810-8 (410-8) 0.13 (0.03) 0.33 (0.18) 0.54 (0.18) 10 210-5 (110-4) 0.14 (0.08) 0.64 (0.04) -0.89 (0.79) (1,2) 910-8 (610-8) 0.40 (0.07) -0.30 (0.07) 0.91 (0.02) 11 -210-4 (610-5) 0.89 (0.006) 0.02 (0.005) -0.38 (0.28) (2,1) 910-8 (410-8) 0.13 (0.02) 0.39 (0.10) 0.47 (0.09) 12 -410-5 (910-5) 0.73 (0.02) 0.06 (0.01) -0.34 (0.52) (1,2) 610-8 (310-8) 0.27 (0.04) -0.17 (0.03) 0.91 (0.01) 13 -110-4 (510-5) 0.84 (0.005) 0.01 (0.004) -0.17 (0.18) (1,1) 610-8 (210-8) 0.10 (0.02) 0.90 (0.02) 14 210-5 (110-4) 0.42 (0.05) 0.41 (0.05) 0.35 (0.88) (1,2) 910-7 (410-7) 0.24 (0.04) -0.16 (0.04) 0.91 (0.02) 15 -610-5 (110-4) 0.40 (0.04) 0.53 (0.03) 0.80 (1.08) (1,2) 810-7 (510-7) 0.29 (0.03) -0.20 (0.04) 0.90 (0.03) note: the table is based on the entire sample p1; t-m (2) is the classical treynor-mazuy model with the lagged excess return on market portfolio m as additional factor; the heteroskedastic consistent standard errors are in parentheses below the coefficient estimates; the variance-covariance matrix of the estimated parameters is based on the qml algorithm; the distribution for the innovations is supposed to be normal. joanna olbryś 198 table 6b. the garch(p, q) versions of the h-m market-timing models of polish equity mutual funds in the entire period from jan 2, 2003 to june 30, 2011 fu nd . n o. h-m model – conditional mean equation (p, q) conditional variance equation p̂ 1p̂ 2 p p̂ 0̂ 1 2̂ 1̂ 2̂ 1 510-4 (110-4) 0.75 (0.02) 0.04 (0.01) -0.10 (0.03) (1,1) 210-7 (110-7) 0.09 (0.02) 0.90 (0.02) 2 810-4 (710-5) 0.75 (0.03) 0.02 (0.005) -0.18 (0.03) (1,2) 210-7 (610-8) 0.52 (0.17) -0.37 (0.16) 0.86 (0.03) 3 -610-5 (710-5) 0.83 (0.008) 0.005 (0.004) -410-4 (0.01) (1,1) 710-8 (210-8) 0.08 (0.01) 0.91 (0.01) 4 -410-5 (710-5) 0.89 (0.009) 0.002 (0.005) -0.01 (0.01) (1,1) 710-8 (210-8) 0.08 (0.01) 0.91 (0.01) 5 210-4 (210-4) 0.33 (0.03) 0.25 (0.02) -0.06 (0.06) (2,2) 310-8 (110-8) 0.20 (0.04) -0.20 (0.04) 1.70 (0.06) -0.70 (0.05) 6 -310-4 (110-4) 0.97 (0.03) 0.07 (0.02) 0.05 (0.04) (1,2) 410-8 (310-8) 0.29 (0.04) -0.21 (0.04) 0.93 (0.01) 7 -110-4 (110-4) 0.78 (0.02) 0.08 (0.01) -0.02 (0.02) (1,2) 510-8 (310-8) 0.35 (0.04) -0.25 (0.04) 0.91 (0.01) 8 910-5 (910-5) 0.83 (0.01) 0.02 (0.005) -810-4 (0.02) (1,2) 610-8 (210-8) 0.14 (0.03) -0.07 (0.03) 0.93 (0.01) 9 -710-5 (910-5) 0.81 (0.01) 0.009 (0.005) -0.02 (0.02) (2,1) 810-8 (410-8) 0.13 (0.03) 0.32 (0.18) 0.54 (0.18) 10 -210-4 (110-4) 0.14 (0.06) 0.65 (0.05) 0.02 (0.04) (1,2) 910-8 (510-8) 0.39 (0.07) -0.29 (0.08) 0.91 (0.02) 11 -210-4 (810-5) 0.88 (0.01) 0.02 (0.005) -0.02 (0.02) (2,1) 910-8 (410-8) 0.13 (0.02) 0.39 (0.10) 0.46 (0.09) 12 510-5 (110-4) 0.71 (0.03) 0.06 (0.01) -0.03 (0.03) (1,2) 610-8 (310-8) 0.27 (0.03) -0.18 (0.03) 0.91 (0.01) 13 -210-5 (810-5) 0.83 (0.008) 0.01 (0.004) -0.02 (0.01) (1,1) 610-8 (310-8) 0.10 (0.02) 0.90 (0.02) 14 -110-4 (210-4) 0.44 (0.05) 0.41 (0.05) 0.04 (0.05) (1,2) 910-7 (410-7) 0.24 (0.04) -0.16 (0.04) 0.91 (0.02) 15 -210-4 (210-4) 0.43 (0.04) 0.53 (0.03) 0.07 (0.05) (1,2) 910-7 (510-7) 0.29 (0.03) -0.20 (0.04) 0.90 (0.03) note: the table is based on the entire sample p1; h-m (3) is the classical henriksson-merton model with the lagged excess return on market portfolio m as additional factor; the heteroskedastic consistent standard errors are in parentheses below the coefficient estimates; the variance-covariance matrix of the estimated parameters is based on the qml algorithm; the distribution for the innovations is supposed to be normal. conclusions our research provides evidence of pronounced arch effects in the classical market-timing models of polish open-end mutual funds. we detect for the arch effects in the entire period from jan 2, 2003 to june 30, 2011, as well as for the 7 subperiods. for comparison, we estimate the market-timing models using two methods. results on both the hac and the garch estimates are qualitatively similar, and even better in the case of the simpler hac method. for this reason, it is not necessary to estimate the garch versions of marketarch effect in classical market-timing models with lagged market variable… 199 timing models in the case of polish mutual funds, even despite the strong arch effects that exist in these models. as for the interpretation of the estimated coefficients, our empirical results can be summarized as follows: 1. there is no evidence that equity fund managers are successful in selectivity ( ˆ p ). 2. the levels of systematic risks are significantly positive ( 1̂p ). 3. the regressions including the lagged values of the market factor as an additional explanatory variable are well-founded ( 2 p ). 4. the empirical results show no statistical evidence that polish equity fund managers have outguessed the market in the entire period jan 2, 2003– june 30, 2011 ( ˆp ). probably the point is that mutual fund performance is affected by its operating style and purpose. if the purpose of the fund is to follow the market, its performance will be close to the market and should show no superior performance. therefore, a possible direction for further investigation would be the performance evaluation including the operating style and purpose of the funds as another factor (wermers, 2000). references adkins, l.c. (2010), using gretl for principles of econometrics, 3rd edition, version 1.313. bollen, n.p.b., busse, j.a. (2001), on the timing ability of mutual fund managers, the journal of finance, 56(3), 1075–1094. bollerslev, t. (1986), generalized autoregressive conditional heteroskedasticity, journal of econometrics, 31, 307–327. bollerslev, t., wooldridge, j.m. (1992), quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances, econometric reviews, 11, 143–179. brzeszczyński, j., gajdka, j., schabek, t. (2011), the role of stock size and trading intensity in the magnitude of the „interval effect” in beta estimation: empirical evidence from the polish capital market, emerging markets finance & trade, 47(1), 28–49. busse, j.a. (1999), volatility timing in mutual funds: evidence from daily returns, the review of financial studies, 12, 1009–1041. campbell, j.y., lo a.w., mackinlay, a.o. (1997), the econometric of financial markets, princeton university press, new jersey. carhart, m.m. (1997), on persistence in mutual fund performance, the journal of finance, 52, 57–82. cogneau, p., hübner, g. (2009), the 101 ways to measure portfolio performance, working paper, http://ssrn.com/abstract=1326076 (sept 8, 2011). cohen, k.j., hawawini, g.a., maier, s.f., schwartz, r.a., whitcomb, d.k. (1980), implications of microstructure theory for empirical research on stock price behaviour, the journal of finance, 35, 249–257. dimson, e. (1979), risk measurement when shares are subject to infrequent trading, journal of financial economics, 7, 197–226. doman, m., doman, r. (2004), ekonometryczne modelowanie dynamiki polskiego rynku finansowego, wydawnictwo akademii ekonomicznej w poznaniu, poznań. doman, m. (2010), liquidity and market microstructure noise: evidence from the pekao data, dynamic econometric models, 10, 5–14. joanna olbryś 200 engle, r.f. (1982), autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflations, econometrica, 50, 987–1007. engle, r.f. (ed.) (2000), arch. selected readings, oxford university press. fama, e.f., french, k.r. (1993), common risk factors in the returns on stocks and bonds, journal of financial economics, 33, 3–56. fisher, l. (1966), some new stock market indexes, journal of business, 39, 191–225. fiszeder, p. (2009), modele klasy garch w empirycznych badaniach finansowych (the class of garch models in empirical finance research), wydawnictwo naukowe uniwersytetu mikołaja kopernika, toruń. ferson, w.e., schadt r.w. (1996), measuring fund strategy and performance in changing economic conditions, the journal of finance, 51, 425–461. greene, w.h. (2002), econometric analysis, fifth edition, prentice hall, new jersey. hamilton, j.d. (2008), macroeconomics and arch, working paper 14151, nber working paper series, cambridge. henriksson, r., merton, r. (1981), on market timing and investment performance. ii. statistical procedures for evaluating forecasting skills, journal of business, 54, 513–533. henriksson, r. (1984), market timing and mutual fund performance: an empirical investigation, journal of business, 57, 73–96. jensen, m. (1968), the performance of mutual funds in the period 1945-1964, the journal of finance, 23, 389–416. kao, g.w., cheng, l.t., chan, k.c. (1998), international mutual fund selectivity and market timing during up and down market conditions, the financial review, 33, 127–144. kufel, t. (2009), ekonometria. rozwiązywanie problemów z wykorzystaniem programu gretl, pwn, warszawa. lee, j. h. h. (1991), a lagrange multiplier test for garch models, economics letters, 37, 265–271. ljung, g., box, g.e.p. (1978), on a measure of lack of fit in time series models, biometrika, 66, 67–72. olbryś, j. (2012), wieloczynnikowe hybrydowe modele market-timing polskich funduszy inwestycyjnych (multifactor hybrid market-timing models of polish mutual funds), studia ekonomiczne – zeszyty naukowe, uniwersytet ekonomiczny w katowicach, accepted for publication. olbryś, j. (2011a), book-to-market, size and momentum factors in market-timing models: the case of polish emerging market, research in economics and business: central and eastern europe, tallinn school of economics and business administration of tallinn university of technology, 3(2), accepted for publication. olbryś, j. (2011b), the intertemporal cross price behavior and the “fisher effect” on the warsaw stock exchange, ekonometria 31. theory and applications of quantitative methods. prace naukowe uniwersytetu ekonomicznego we wrocławiu, accepted for publication. olbryś, j. (2010a), orthogonalized factors in market-timing models of polish equity funds, quantitative methods in economics, wuls press, 11(1), 128–138. olbryś, j. (2010b), three-factor market-timing models with fama and french’s spread variables, operations research and decisions, 2/2010, 91–106. olbryś, j. (2009), conditional market-timing models for mutual fund performance evaluation, prace i materiały wydziału zarządzania uniwersytetu gdańskiego, 4/2, 519–532. romacho, j.c., cortez, m.c. (2006), timing and selectivity in portuguese mutual fund performance, research in international business and finance, 20, 348–368. scholes, m., williams, j. (1977), estimating betas from nonsynchronous data, journal of financial economics, 5, 309–327. treynor, j., mazuy, k. (1966), can mutual funds outguess the market?, harvard business review, 44, 131–136. tsay, r.s. (2010), analysis of financial time series, john wiley, new york. arch effect in classical market-timing models with lagged market variable… 201 wermers, r. (2000), mutual fund performance: an empirical decomposition into stock-picking talent, style, transaction costs, and expenses. the journal of finance, 55, 1655–1703. efekt arch w klasycznych modelach market-timing z opóźnioną zmienną rynkową: przypadek rynku polskiego z a r y s t r e ś c i. w artykule przedstawiono badania dokumentujące występowanie efektu arch w klasycznych modelach market-timing z opóźnioną zmienną rynkową w przypadku polskich funduszy akcji, w okresie styczeń 2003-czerwiec 2011. dokonano estymacji wersji garch odpowiednich modeli oraz porównano jakość modeli garch i modeli uzyskanych metodą hac. wyniki wskazują, że modele garch są odpowiednie, ale metoda hac jest wystarczająca, pomimo występowania efektu arch. podano również interpretacje parametrów otrzymanych modeli w badanej grupie funduszy. s ł o w a k l u c z o w e: model market-timing, niesynchroniczne transakcje, efekt arch, model garch. dynamic econometric models © 2016 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2016.001 vol. 16 (2016) 5−20 submitted juny 15, 2016 issn (online) 2450-7067 accepted july 15, 2016 issn (print) 1234-3862 joanna bruzda * quantile forecasting in operational planning and inventory management – an initial empirical verification  a b s t r a c t. in the paper we present our initial results of an empirical verification of different methodologies of quantile forecasting used in operational management to calculate the reorder point or order-up-to level as well as the optimal order quantity according to the newsvendor model. the comparison encompasses 26 procedures including quantile regression, the basic bootstrap method and popular textbook formulas. our results, obtained on the base of 30 time series concerning such diversified phenomena as supermarket sales, passenger transport and water and gas demand, point to the usefulness of regression medians, regression quantiles, bootstrap methods and the procedures available in the sap erp system. k e y w o r d s: linlin loss, quantile forecasting, quantile regression, re-order point, theta method. j e l classification: c21; c44; c53. introduction a quantile forecast of a variable ty is the conditional quantile of ty given the information available till time 1t , i.e., the quantile of order : * correspondence to: joanna bruzda, nicolaus copernicus university, faculty of economic sciences and management, gagarina 13a, 87-100 toruń, poland, e-mail: joanna.bruzda@umk.pl.  this work was financed by the polish national science center with a grant obtained under decision no. dec-2013/09/b/hs4/02716. the paper is a continuation of the research presented in bruzda (2016), based on a different set of forecasting procedures. joanna bruzda dynamic econometric models 16 (2016) 5–20 6 })(;inf{   yfyy (1) of )(f being the conditional distribution of ty given the information set 1  t . the quantity (1) is a solution to the following optimization problem:   1 |,min   t x yxlle , (2) where       yxxy xyxy yxll dla)1( dla ),(   (3) is the double linear loss function, often denoted as linlin (see granger, 1999; gneiting, 2011; bruzda, 2014). the loss (cost) function (3) turns out to be very popular in operational management applications. for example, in the so-called newsvendor problem (the one-period inventory model) the optimal order or production quantity in the next period is given as a quantile forecast of the next period demand d, i.e., assuming d has a continuous distribution, as:            ou u d cc c fq 1 , (4) ( u c and o c are the unit costs of underand overstocking, respectively). in a similar way, setting the next period re-order point (rop) or order-up-to level s for a service level  the following standard textbook formulas are used: ,1)1( d lzldrop   (5a) d tlztlds   )( (5b) ( d denotes the mean demand, l is the lead time, t stands for the inventory cycle period, d denotes the standard deviation of d and z is the -quantile of the standard normal distribution). if the demand is forecasted, quantiles of a predictive distribution (a conditional distribution with estimated parameter values) will be used to compute rop and s, respectively. further in the paper we discuss in some detail the premises of quantile forecasting in operational planning and inventory management, and underquantile forecasting in operational planning and inventory management… dynamic econometric models 16 (2016) 5–20 7 take the task of an empirical verification of different statistical procedures leading to quantile predictions. 1. quantile forecasting in operational planning and inventory management there are numerous procedures of computing quantile forecasts, encompassing parametric, semiparametric and nonparametric methods, approaches utilizing ex ante and ex post forecast errors, and non-simulationand simulation-based (bootstrap and parametric monte carlo) procedures. one can use the many existing methods of forecasting the value at risk in finance (see, e.g., doman and doman, 2009) or computing interval forecasts (and fan charts) of macroeconomic variables (see, e.g., chapter iv in clements, 2005) as well as procedures and formulas tailored to applications in the area of supply chain management (see, e.g., wagner, 2010; ciesielski, 2011, and sap help portal). a description of chosen methods of quantile forecasting with some indications on their potential use in logistics together with a simulation analysis can be found in bruzda (2014). when considering quantile predictions in operational management, one should keep in mind several of its premises. first, to assess the quality of quantile predictions in this area, one will normally rely on different measures than those utilized to examine interval forecasts, in which case the coverage is evaluated, as well as those used to assess forecasts of value at risk, in which case the most important measures are usually the hit ratio and loss functions based exclusively on positive (or, alternatively, negative) forecast errors. table 1 below presents the most important measures of predictive accuracy in the case of unbiased forecasts, which are the solution to the minimization of the mean squared error (mse) loss, accompanied by the corresponding measures which should be used in quantile forecasting. table 1. basic accuracy measures for unbiased and quantile forecasts unbiased forecasts quantile forecasts  teme t tyyihit tpt /)(    termse t 2   teell tt /)1(     note: t y – the realized value of t y ; tpy – the forecasted value of t y ; )(i – the indicator function taking on the value of 1, if the condition in brackets is fulfilled, and 0 otherwise; },0max{ tt ee   (positive forecast errors), },0max{ tt ee   (negative forecast errors), where tptt yye  . joanna bruzda dynamic econometric models 16 (2016) 5–20 8 the measures in the first row of table 1 inform whether the evaluated forecasts can be considered as unbiased or quantile forecasts, respectively, while in the second row there are given mean values of the cost functions for these two sorts of predictions. (in the case of logistic forecasting, the quantile cost function will often be proportional to the real logistic cost – see, for example, the textbook derivation of the formula (4).) in the present paper, the linlin cost function (3) is considered to be the most important measure of forecast accuracy for quantile forecasting in operational management 1 . among other characteristics of quantile predictions in operational planning are the following:  computation of quantile forecasts for a range of quantiles instead of just extreme quantiles,  simultaneous forecasting of a large number of univariate time series and, due to this, the focus on simplified methods  forecasting based on relatively short time series  the need to simultaneously model the conditional mean and conditional variance of time series  the necessity to compute long-term quantile forecasts or, alternatively, quantile forecasts for different sampling rates. the last observation results, among others, from the practice of freezing the master production schedule, the need of long-term optimization of distribution networks, capacity planning or long lead times. 2. empirical verification of quantile forecasting methodologies the dataset used in the study comprises 30 time series from certain publicly available databases. to simplify matters we concentrate on stationary and trend-stationary series (according to the adf tests with lag lengths chosen with the aic) and also assume that an examination of short time series may constitute an interesting addendum to what is already known about the relative performance of different forecasting methodologies. the series are presented in figure 1, whereas below we provide a short description of this dataset 2 : 1 alternative measures of forecasting performance in the inventory management context are those based on the concept of expected shortage. however, as the target here is on quantile forecasting, we concentrate exclusively on the linlin loss, which is the loss minimized with conditional quantiles – see gneiting (2011). 2 for further information see http://www.forecastingprinciples.com/index.php/data and https://cran.r-project.org/web/packages/bayesm/index.html. quantile forecasting in operational planning and inventory management… dynamic econometric models 16 (2016) 5–20 9  x1 – x14 – time series from the m3 forecast competition (category ‘other series’) ‘gas sent’ (6 series) and ‘water demand’ (8 series),  x15 – x23 – weekly data on slice cheese sales in a supermarket chain in the us from the r toolbox ‘bayesm’ (the longest point-of-sale data of length 68 without visible promotional effects were chosen),  x24 – x30 – time series from the t-competition: waste water pipe data (4 weekly time series), weekly train passengers in switzerland (1 series) and 2 monthly series: aircraft capacity utilization in the us and transborders-rail us-mex. all the series have been shortened to the length of the shortest time series of length 68. then observations from 1 to 64 were used to estimate models utilized to compute one-step ahead quantile forecast in the first step. in the next steps, the samples were lengthen recursively by one observation and all models were re-estimated and used to compute further one-step ahead quantile predictions. continuing in this way, 4 forecast were obtained for each time series and each examined quantile 3 . the following methods were compared:  m1 – a procedure refering to the textbook formulas (5a)–(5b), based on a two-step approach, i.e., a quantile forecast is given as etp szy   , where tpy is a forecast of the conditional mean and es is the in-sample standard error (the equation for the conditional mean is estimated by ols),  m2 – as m1 but in the second step the mean absolute deviation (mad) is used, i.e., the quantile forecast is computed as madzytp    2 ,  m3 – as m1 but with a correction of the conditional mean computed nonparametrically, i.e., the forecast is given as eytp  , where e denotes an estimate of the -quantile of residuals, which we compute according to the method implemented in matlab ver. r2015a,  m4 – a basic sap erp approach, i.e., a quantile forecast is given as madzytp    2 but the equation for tpy is estimated by median regression (the least absolute deviation – lad – method),  m5 – median regression combined with the nonparametric estimation of quantiles as in m3, 3 considering a larger number of forecasts (and shorter estimation samples) resulted in a poor behavior of the ml estimates of our models. joanna bruzda dynamic econometric models 16 (2016) 5–20 10  m6 – (partially iterated) weighted ols, in which an auxiliary regression for logarithms of squared ols residuals regressed on the same variables as in the equation for the mean is estimated, which gives weights for an efficient estimation of the equation for the mean, and, in order to forecast the variance, the auxiliary equation is re-estimated on residuals obtained through the efficient estimation 4 ,  m7 – ml estimation applied to the model specified in m6 with starting values being the (appropriately modified) estimates obtained in m6,  m8 – quantile regression (see koenker, 2005). in the procedures m1–m8 the equation for the mean (median, quantile) is an autoregression of order p ( 50  p ) with a polynomial trend of order s ( 20  s ), where the parameters p and s are set automatically with the aicc criterion in its versions for the ols, lad and quantile estimation, respectively.  m9 – the theta method – a modification of the exponential smoothing which was very successful in the m3-forecast competition (see makridakis and hibon, 2000), with parameters estimated here with lad under the initial state set to the first observation in the sample and the starting values: )1,0(~ u and 0drift ; 20 randomizations for  were assumed; the quantile forecast is then computed as in m4,  m10 – the theta model estimated as in m9 combined with smoothing mad as in the sap erp system under the default value of the smoothing constant , i.e.,  – the parameter standing by the absolute value of the last forecast error – is set to 0.3,  m11 – the theta method with smoothing mad and all parameters (, and the drift terms) estimated by quantile regression; the quantile forecasts are computed via the following equation 5 :  ,)ˆ1(ˆˆˆ)ˆ1(ˆˆ 1102110   tttttp maduzyyy    with 4 such an approach produced better forecasts on average (in terms of ratios of the relative linlin loss) than a procedure based on weighted ols without the re-estimation of the auxiliary equation. in both of these procedures, to measure volatility the standard deviation is used, thus the formula for quantile forecasting given in m1 is applied, but this time the standard deviation is forecasted. 5 this specification provided lower values of the relative linlin loss as compared with an equation without the parameter 0 (except for  = 0.25 and both modelling levels and modelling logarithms) quantile forecasting in operational planning and inventory management… dynamic econometric models 16 (2016) 5–20 11 ;0,ˆ,ˆ ,)ˆ1(ˆˆ ,ˆ)ˆ1(ˆˆˆ 111 110 110      madyyyyu madumad yyy ttt ttt ttt   the starting values of the smoothing constants were from the )1,0(u distribution (20 randomizations were assumed), while the starting values of the drift terms were set to 0,  m12–m22 – these are m1–m11, respectively, but applied to logarithms of the data,  m23–m26 – the basic bootstrap method as presented by clements (2005), §4.2.3, based on his formula (4.9) and the percentile method of efron (see the formula (4.4) in clements, 2005), without bias correction; m23 and m24 are performed on levels, while m25 and m26 – on logarithms of the data; in order to better account for estimation errors, in m23 and m25 the residuals from the ols estimation of autoregressive models with trend functions (as chosen in m1) are resampled according to the overlapping block method with block length set to 10, whereas in m24 and m26 a simple resampling is used, i.e., the block length is set to 1; the number of bootstrap replication is set to 250 6 . all computations were performer in matlab ver. 2015a endowed with the newest versions of the optimization and statistics toolboxes, thus they relied on many procedures implemented in these packages, including the nonparametric quantile estimation by the function ‘quantile’. it is worth noticing that among the different forecasting procedures are 4 methods assuming the use of the linlin loss function for estimation purposes, i.e., based on quantile regression (m8, m11, m19, m22). in the case of the linear and linearized models (m8, m19) the estimation relied on linear programming methods (the active-set algorithm was used), while on the other two models nonlinear 6 the procedures m23 and m25 are based on a mixed approach in which an equation for the conditional mean is explicitly specified, whereas the heteroscedasticity of error terms is not modeled. (it is merely accounted for when assessing the variance of estimators in the equation for the mean.) this renders forecasting of the residual variance impossible. we include these procedures here because, somewhat surprisingly, they appear to often provide better forecasts than the other approaches (see table 2), especially when forecasting quantiles near or below the median. this underlines the importance of accounting for estimation errors when computing quantile predictions and leaves space for possible improvements through the use of model-based bootstrap procedures, i.e., an explicit modeling of the conditional volatility and possibly also a better specification of the conditional mean through the inclusion of nonlinearity and/or longer lag lengths. (although it is worth adding that our experimentation with longer lag lengths, in mean terms, did not improve the results presented here.) joanna bruzda dynamic econometric models 16 (2016) 5–20 12 optimization with the function ‘fmincon’ was performed (the interior-point algorithm was applied to the nonlinear objective functions). similar optimization techniques were also utilized in the case of the linear and nonlinear median regression. furthermore, it is also worth noticing that the bootstrap methods m23–m26 (similar to our experiments with the approximation of quantiles of predictive distributions through the formula for interval forecasting in classical regression, which provided slightly less satisfactory results for our dataset than the method m1 and are not presented here) are an attempt to explicitly account for estimation errors associated with equations for the conditional mean in the computation of quantile forecasts. the following quantiles were examined: 0.05; 0.1; 0.25; 0.5; 0.75; 0.9; 0.95. the evaluation of our forecast procedures was based on mean values of the linlin loss for each quantile and each time series. two rankings of the methods were constructed – one for the whole set of quantiles and second for the two largest quantiles (see table 2), showing how often a given procedure produced the best outcomes. besides, we also computed aggregate measures of the form:   , )1(11 1 1        m i t t itp itit y ee tm llrelativem ean    (6) where m is the number of series (m = 30), t is the number of one-step ahead quantile forecasts (t = 4) and  itpy is the tth -quantile forecast for time series i. the measures are also accompanied by the mean values of the hit ratios. in accordance with the way of reporting forecasting results for a larger dataset, the aggregate measures were also presented in their trimmed versions based on datasets without 10% of time series with the smallest and 10% of time series with the largest values of the corresponding mean relative linlin loss or the mean values of hit. all the computations are collated in tables 3–4. the conclusions from our study, based on both the presented results and an analysis of more detailed statistics, are the following. first, the models we use to describe heteroscedasticity suit well the purpose of quantile forecasting for quantiles below the median, as can be seen in table 3, where we find that the ml estimation for variables in levels (the procedure m7) produces the best results in terms of the mean values of the relative ll, while the nonlinear quantile regression (m11 and m22) leads to the lowest values of the trimmed relative ll. it is worth noting that, according to the rankings presented in table 2, the methods based on linear and nonlinear quantile regression (m8, m11, m19, m22) provide the best quantile forecasts in about 27% quantile forecasting in operational planning and inventory management… dynamic econometric models 16 (2016) 5–20 13 of the examined cases and are ranked high, especially in the ranking for all quantiles. interestingly, in the first ranking the nonlinear quantile models have the highest ranks, whereas in the second the linear quantile regression is ranked higher than the methods m11 and m22. in fact, even 16 cases out of the total of 23 when m8 or m19 lead to the best quantile predictions (comp. table 2) concern quantiles 0.75, 0.9 and 0.95. in mean terms, however, linear regression quantiles are not that attractive, since for quantiles 0.9 and 0.95 they are clearly outperformed by median regression with parametric or nonparametric correction of the median, as well as ols with a parametric computation of quantiles. generally, it can be noticed that, for quantiles near the median, the theta method often produces the best outcomes in terms of the relative linlin loss. this seems to support the findings from the m3-forecast competition, where mean values were exclusively forecasted (see makridakis, hibon, 2000). on the other hand, for the largest two quantiles, the lowest values of the mean relative ll are obtained through the lad estimation of simple autoregressive models with trend functions (see table 3). however, depending on the quantile, either parametric (m4, m15) or nonparametric (m5, m16) estimators in the second step lead to the best mean outcomes. nevertheless, the measures collated in the left-hand side of table 3 point to the sap method applied to logarithms of the data (m15) as that producing the best quantile predictions for  = 0.9 and 0.95. interestingly, the rankings presented in table 2 often point to different methods than the mean values of the relative linlin loss as the most valuable solutions to the task of quantile forecasting. for example, they single out the bootstrap methods, especially the procedures m25 and m23. analyzing more detailed statistics shows, however, that the excellent performance of the bootstrap in its different forms (the methods m23–m26) usually takes place for quantiles not very far from the median. this suggests that it may be worth designing a bootstrap procedure which will make it possible to also forecast the variance or, alternatively, one can be interested in performing bootstrapping based on median regression (taking into consideration its excellent performance at forecasting high quantiles). furthermore, it also calls for the inclusion of bootstrap methods in any application of the so-called focus forecasting – forecasting with methods producing the lowest values of chosen loss functions based on ex post forecast errors. among other procedures that certainly should also be consider in focus forecasting are those based on lad estimation (such as m4 or m5) as well as linear and nonlinear quantile regression. joanna bruzda dynamic econometric models 16 (2016) 5–20 14 the most striking finding from our study is probably the performance of the textbook formula m1 in relation to the other methods examined here as far as the largest quantile 0.95 is concerned. it turns out that m1 produces then the best results in terms of the trimmed mean relative ll, while at the same time it is ranked among the worst according to the number of the best outcomes produced for different time series. we treat this finding as an indication that this formula may often constitute a ‘safe choice’, which should not lead to highly suboptimal results, although usually it will be outperformed by other methods used in focus forecasting. analyzing the mean values of hit presented in table 4 it can be noticed that the bootstrap and the smoothing methods perform relatively well in terms of this measure. in particular, the smallest discrepancies between hit and 1 for quantiles 0.25 and 0.5 are for the smoothing procedures. we had to resigned from testing the significance of deviations of the hit ratios for individual time series due to the fact that we operate on short time series and are able to analyze only a small number of forecasts. it can be noticed, however, that the kupiec test performed on the whole set of the analyzed data does not reject its null hypothesis (stating that, for a given method, hit is equal to 1 ) for all the examined quantiles above the median (see table 4). finally, it can also be observed that, although taking logarithms prior to the estimation of our models can improve the forecasting results, it appears that the logarithmic transformation is not always necessary and also not always enough to remove heteroscedasticity from the analyzed time series. final remarks as a summary of our findings from the present examination it is worth underlining that among the most promising methods of quantile forecasting in operational planning and inventory management are procedures based on median and quantile estimation as well as bootstrap techniques. in particular, it seems interesting to join the theta method and the quantile regression methodology since, as it takes place in this study for quantiles 0.05, 0.1, 0.25 and 0.5, such an approach may be competitive to other popular methods of quantile forecasting. 7 7 however, in order to properly assess the performance of this approach in relation to other methods, further studies are required, possibly based on a larger dataset, since in the present study we do not formally test the significance of the differences in predictive abilities of the different approaches. quantile forecasting in operational planning and inventory management… dynamic econometric models 16 (2016) 5–20 15 the above remark stays in accordance with the opinion that regression quantiles should be included in the inventory management toolbox 8 . as was shown in the present study, they can be used both to optimize the value of the smoothing constant  in the mad equation as well as to compute quantile forecasts based on autoregressive specifications with trend functions. some promising areas of further studies are the design of bootstrap procedures based on an explicit modelling of volatility with the aim of a use in inventory planning and an assessment of simulationand non-simulationbased procedures of longer-term quantile forecasting in operational management applications. some of these problems will be discussed further in the present project. references applied forecasting, webpage sponsored by route optimization and heiloo, http://www.appliedforecasting.com/quantiles-forecasts-for-inventory-optimization/ (26.09.2015). bruzda, j. (2014), prognozy kwantylowe w zastosowaniach logistycznych. wprowadzenie do problematyki (quantile forecasts in logistic applications – an introduction), in chaberek m and reszka l. (eds.) zeszyty naukowe uniwersytetu gdańskiego, ekonomika transportu i logistyki (scientific papers of the university of gdańsk, transport and logistics economics), 51, 175–195. bruzda, j. (2016), metody wyznaczania prognoz kwantylowych w logistyce – weryfikacja empiryczna (methods of quantile forecasting in logistic applications – an empirical verification), logistyka (logistics), 5/2016, logistyka – nauka (logistic research), 9–13. ciesielski, m. (2011), zarządzanie łańcuchami dostaw (supply chain management), polskie wydawnictwo ekonomiczne, warszawa. clements, m. p. (2005), evaluating econometric forecasts of economic and financial variables, palgrave macmillan, new york. doman, m., doman, r. (2009), modelowanie zmienności i ryzyka. metody ekonometrii finansowej (risk and volatility modelling. methods of financial econometrics), wolters kluwer business, kraków. forecasting principles, webpage sponsored by the international institute of forecasters, http://www.forecastingprinciples.com/index.php/data (26.09.2015). gneiting, t. (2011), quantiles as optimal point forecasts, international journal of forecasting, 27, 197–207, doi: http://dx.doi.org/10.1016/j.ijforecast.2009.12.015. granger, c. w. j. (1999), outline of forecast theory using generalized cost functions, spanish economic review, 1, 161–173, doi: http://dx.doi.org/10.1007/s101080050007. koenker, r. (2005), quantile regression, cambridge university press, cambridge. 8 see applied forecasting, http://www.appliedforecasting.com/quantiles-forecasts-forinventory-optimization/. http://www.appliedforecasting.com/quantiles-forecasts-for-inventory-optimization/ http://www.forecastingprinciples.com/index.php/data http://ideas.repec.org/a/eee/intfor/v27yi2p197-207.html http://ideas.repec.org/s/eee/intfor.html http://ideas.repec.org/s/eee/intfor.html http://dx.doi.org/10.1016/j.ijforecast.2009.12.015 joanna bruzda dynamic econometric models 16 (2016) 5–20 16 makridakis, s, hibon, m (2000), the m3-competition: results, conclusions and implications, international journal of forecasting, 16, 451–476, doi: http://dx.doi.org/10.1016/s0169-2070(00)00057-1. r archive network, https://cran.r-project.org/web/packages/bayesm/index.html (26.09.2015). sap help portal, http://help.sap.com (26.09.2015). wagner, b. (2010), purchasing and forecasting using sap erp, operations and supply chain management library, http://scn.sap.com/community/uac/operations-and-scmlibrary (26.09.2015). prognozy kwantylowe w planowaniu operacyjnym i zarządzaniu zapasami – wstępna weryfikacja empiryczna z a r y s t r e ś c i. w artykule prezentuje się wyniki wstępnej weryfikacji empirycznej metod prognozowania kwantylowego mających zastosowanie w logistyce do ustalania punktu odnowienia i granicy uzupełniania zapasów czy optymalnej wielkości zamówienia w modelu jednookresowym. porównaniem objęto 26 procedur, a w tym regresję kwantylową, podstawową metodę bootstrapową i popularne formuły podręcznikowe. wyniki otrzymane na bazie analizy 30 szeregów czasowych dotyczących tak różnorodnych zjawisk jak sprzedaż w supermarkecie, przewozy pasażerskie i zużycie gazu i wody wskazują na użyteczność median regresyjnych, kwantyli regresyjnych i procedur dostępnych w sap erp. s ł o w a k l u c z o w e: funkcja straty linlin, prognozy kwantylowe, punkt odnowienia, regresja kwantylowa, wyrównywanie wykładnicze. http://dx.doi.org/10.1016/s0169-2070%2800%2900057-1 https://cran.r-project.org/web/packages/bayesm/index.html http://help.sap.com/ http://scn.sap.com/community/uac/operations-and-scm-library http://scn.sap.com/community/uac/operations-and-scm-library http://scn.sap.com/community/uac/operations-and-scm-library http://scn.sap.com/community/uac/operations-and-scm-library quantile forecasting in operational planning and inventory management… dynamic econometric models 16 (2016) 5–20 17 table 2. rankings of forecasting procedures method percentage (number) of best results method percentage (number) of best results for = 90%, 95% m11 8.10% (17) m8 11.67% (7) m22 8.10% (17) m10 11.67% (7) m21 6.67% (14) m4 8.33% (5) m8 6.19% (13) m19 8.33% (5) m25 6.19% (13) m5 6.67% (4) m10 5.71% (12) m3 5% (3) m23 5.71% (12) m6 5% (3) m24 5.71% (12) m11 5% (3) m4 4.76% (10) m15 5% (3) m19 4.76% (10) m25 5% (3) m5 4.29% (9) m14 3.33% (2) m3 3.81% (8) m17 3.33% (2) m7 3.81% (8) m21 3.33% (2) m9 3.33% (7) m23 3.33% (2) m18 3.33% (7) m1 1.67% (1) m6 2.86% (6) m2 1.67% (1) m15 3.33% (7) m7 1.67% (1) m17 2.86% (6) m9 1.67% (1) m14 2.86% (6) m12 1.67% (1) m20 2.86% (6) m16 1.67% (1) m12 1.90% (4) m20 1.67% (1) m16 1.43% (3) m22 1.67% (1) m1 0.95% (2) m24 1.67% (1) m26 0.95% (2) m13 0% (0) m2 0.48% (1) m18 0% (0) m13 0.48% (1) m26 0% (0) note: the best result means that a method produced the lowest value of the linlin loss (3) for a given series and a given quantile. f ig u re 1 . t h e t im e s e ri e s u se d i n t h e s tu d y 2 0 4 0 6 0 0 5 0 0 0 1 0 0 0 0 x 1 2 0 4 0 6 0 0 5 0 0 0 1 0 0 0 0 x 2 2 0 4 0 6 0 2 0 0 0 4 0 0 0 6 0 0 0 x 3 2 0 4 0 6 0 2 0 0 0 4 0 0 0 6 0 0 0 x 4 2 0 4 0 6 0 2 0 0 0 4 0 0 0 6 0 0 0 x 5 2 0 4 0 6 0 0 5 0 0 0 1 0 0 0 0 x 6 2 0 4 0 6 0 1 5 0 0 2 0 0 0 2 5 0 0 x 7 2 0 4 0 6 0 2 0 0 0 2 5 0 0 3 0 0 0 x 8 2 0 4 0 6 0 2 0 0 0 2 2 0 0 2 4 0 0 x 9 2 0 4 0 6 0 2 0 0 0 2 2 0 0 2 4 0 0 x 1 0 2 0 4 0 6 0 2 0 0 0 2 2 0 0 2 4 0 0 x 1 1 2 0 4 0 6 0 1 5 0 0 2 0 0 0 2 5 0 0 x 1 2 2 0 4 0 6 0 2 0 0 0 2 2 0 0 2 4 0 0 x 1 3 2 0 4 0 6 0 1 5 0 0 2 0 0 0 2 5 0 0 x 1 4 2 0 4 0 6 0 0 5 0 0 0 1 0 0 0 0 x 1 5 2 0 4 0 6 0 2 0 0 0 4 0 0 0 6 0 0 0 x 1 6 2 0 4 0 6 0 012 x 1 0 4 x 1 7 2 0 4 0 6 0 0 5 0 0 0 1 0 0 0 0 x 1 8 2 0 4 0 6 0 0 2 0 0 0 4 0 0 0 x 1 9 2 0 4 0 6 0 0 2 0 0 0 4 0 0 0 x 2 0 2 0 4 0 6 0 0 5 0 0 0 1 0 0 0 0 x 2 1 2 0 4 0 6 0 1 0 0 0 2 0 0 0 3 0 0 0 x 2 2 2 0 4 0 6 0 012 x 1 0 4 x 2 3 2 0 4 0 6 0 0 .51 1 .5 x 1 0 4 x 2 4 2 0 4 0 6 0 0 5 0 0 0 1 0 0 0 0 x 2 5 2 0 4 0 6 0 012 x 1 0 4 x 2 6 2 0 4 0 6 0 024 x 1 0 4 x 2 7 2 0 4 0 6 0 4 4 .55 x 1 0 4 x 2 8 2 0 4 0 6 0 2 0 3 0 4 0 x 2 9 2 0 4 0 6 0 0 5 0 0 1 0 0 0 x 3 0 t a b le 3 . m e a n v a lu e s o f th e r e la ti v e l in l in c o st f u n c ti o n m e th o d s m e a n r e la ti v e l l t ri m m e d m e a n r e la ti v e l l 0 .0 5 0 .1 0 .2 5 0 .5 0 .7 5 0 .9 0 .9 5 0 .0 5 0 .1 0 .2 5 0 .5 0 .7 5 0 .9 0 .9 5 m 1 0 .0 1 6 2 0 .0 2 5 5 0 .0 3 8 6 0 .0 4 4 1 0 .0 3 3 8 0 .0 1 6 8 0 .0 0 9 3 0 .0 1 3 1 0 .0 2 1 6 0 .0 3 3 5 0 .0 3 8 8 0 .0 3 1 2 0 .0 1 5 6 0 .0 0 8 m 2 0 .0 1 5 9 0 .0 2 5 0 .0 3 8 5 0 .0 4 4 1 0 .0 3 3 6 0 .0 1 6 7 0 .0 0 9 3 0 .0 1 2 7 0 .0 2 0 9 0 .0 3 3 3 0 .0 3 8 8 0 .0 3 1 2 0 .0 1 5 6 0 .0 0 8 2 m 3 0 .0 1 6 2 0 .0 2 4 6 0 .0 3 8 5 0 .0 4 4 5 0 .0 3 3 7 0 .0 1 6 9 0 .0 1 0 2 0 .0 1 3 0 .0 2 2 1 0 .0 3 4 0 .0 3 9 2 0 .0 3 0 8 0 .0 1 5 5 0 .0 0 8 7 m 4 0 .0 1 6 7 0 .0 2 4 7 0 .0 3 6 9 0 .0 4 2 8 0 .0 3 2 2 0 .0 1 6 3 0 .0 0 9 2 0 .0 1 4 3 0 .0 2 1 8 0 .0 3 3 2 0 .0 3 9 3 0 .0 3 0 5 0 .0 1 5 2 0 .0 0 8 5 m 5 0 .0 1 6 4 0 .0 2 3 6 0 .0 3 6 2 0 .0 4 2 8 0 .0 3 2 4 0 .0 1 5 8 0 .0 0 9 6 0 .0 1 3 6 0 .0 2 1 0 .0 3 2 6 0 .0 3 9 3 0 .0 3 0 3 0 .0 1 4 7 0 .0 0 8 9 m 6 0 .0 1 8 2 0 .0 4 2 6 0 .0 3 9 0 .0 4 3 8 0 .0 3 3 8 0 .0 1 7 5 0 .0 1 0 1 0 .0 1 3 8 0 .0 2 1 6 0 .0 3 3 7 0 .0 3 8 4 0 .0 3 1 3 0 .0 1 5 7 0 .0 0 8 6 m 7 0 .0 1 2 9 0 .0 2 0 3 0 .0 3 3 2 0 .0 4 2 6 0 .0 3 4 4 0 .0 1 9 2 0 .0 1 1 8 0 .0 1 1 7 0 .0 1 9 2 0 .0 3 0 3 0 .0 3 8 3 0 .0 3 0 9 0 .0 1 6 2 0 .0 0 9 1 m 8 0 .0 1 8 0 .0 2 4 1 0 .0 3 7 8 0 .0 4 2 8 0 .0 3 0 5 0 .0 1 7 6 0 .0 1 1 1 0 .0 1 5 1 0 .0 2 0 6 0 .0 3 2 4 0 .0 3 9 3 0 .0 2 9 4 0 .0 1 6 8 0 .0 0 9 9 m 9 0 .0 1 7 5 0 .0 2 5 7 0 .0 3 4 8 0 .0 3 7 5 0 .0 3 0 2 0 .0 1 6 2 0 .0 0 9 6 0 .0 1 3 8 0 .0 2 2 3 0 .0 3 1 6 0 .0 3 3 4 0 .0 2 8 2 0 .0 1 5 3 0 .0 0 8 8 m 1 0 0 .0 1 7 3 0 .0 2 3 7 0 .0 3 5 4 0 .0 3 7 5 0 .0 3 0 2 0 .0 1 7 2 0 .0 0 9 9 0 .0 1 4 0 .0 2 0 2 0 .0 3 1 3 0 .0 3 3 4 0 .0 2 7 7 0 .0 1 5 7 0 .0 0 8 5 m 1 1 0 .0 1 3 8 0 .0 2 4 1 0 .0 3 4 2 0 .0 3 7 0 .0 3 0 6 0 .0 1 7 3 0 .0 1 0 7 0 .0 1 1 1 0 .0 1 9 2 0 .0 3 0 5 0 .0 3 2 8 0 .0 2 8 2 0 .0 1 6 3 0 .0 0 9 1 m 1 2 0 .0 1 4 4 0 .0 2 2 8 0 .0 3 6 0 .0 4 1 4 0 .0 3 3 2 0 .0 1 7 0 .0 1 0 4 0 .0 1 2 2 0 .0 1 9 7 0 .0 3 2 1 0 .0 3 7 2 0 .0 3 0 6 0 .0 1 5 8 0 .0 0 9 3 m 1 3 0 .0 1 4 2 0 .0 2 2 5 0 .0 3 5 9 0 .0 4 1 4 0 .0 3 3 1 0 .0 1 6 9 0 .0 1 0 2 0 .0 1 2 1 0 .0 1 9 5 0 .0 3 2 1 0 .0 3 7 2 0 .0 3 0 7 0 .0 1 5 8 0 .0 0 9 3 m 1 4 0 .0 1 4 0 .0 2 3 0 .0 3 5 6 0 .0 4 1 8 0 .0 3 2 9 0 .0 1 7 6 0 .0 1 1 2 0 .0 1 2 1 0 .0 2 0 1 0 .0 3 2 1 0 .0 3 8 1 0 .0 2 9 9 0 .0 1 5 9 0 .0 0 9 6 m 1 5 0 .0 1 5 0 .0 2 2 7 0 .0 3 5 0 .0 4 1 1 0 .0 3 1 1 0 .0 1 5 6 0 .0 0 8 9 0 .0 1 2 4 0 .0 1 9 8 0 .0 3 1 9 0 .0 3 7 5 0 .0 2 9 3 0 .0 1 5 3 0 .0 0 8 5 m 1 6 0 .0 1 4 7 0 .0 2 2 2 0 .0 3 5 0 .0 4 1 1 0 .0 3 1 9 0 .0 1 5 6 0 .0 1 0 1 0 .0 1 2 9 0 .0 2 0 .0 3 2 0 .0 3 7 5 0 .0 2 9 9 0 .0 1 5 2 0 .0 0 9 8 m 1 7 0 .0 1 4 7 0 .0 2 2 1 0 .0 3 5 4 0 .0 4 1 5 0 .0 3 2 0 .0 1 7 3 0 .0 1 0 3 0 .0 1 1 9 0 .0 1 8 5 0 .0 3 1 2 0 .0 3 7 9 0 .0 3 0 9 0 .0 1 6 1 0 .0 0 9 3 m 1 8 0 .0 1 3 4 0 .0 2 1 5 0 .0 3 6 3 0 .0 4 3 6 0 .0 3 5 1 0 .0 1 9 3 0 .0 1 1 3 0 .0 1 1 4 0 .0 1 8 7 0 .0 3 1 9 0 .0 3 8 3 0 .0 3 2 3 0 .0 1 6 6 0 .0 0 9 3 m 1 9 0 .0 1 7 5 0 .0 2 4 1 0 .0 3 6 7 0 .0 4 1 1 0 .0 3 0 8 0 .0 1 9 4 0 .0 1 0 9 0 .0 1 4 6 0 .0 2 1 3 0 .0 3 2 5 0 .0 3 7 5 0 .0 2 9 4 0 .0 1 8 6 0 .0 0 9 8 m 2 0 0 .0 1 4 7 0 .0 2 3 0 .0 3 3 5 0 .0 3 7 4 0 .0 3 0 .0 1 6 4 0 .0 0 9 8 0 .0 1 2 0 .0 2 0 2 0 .0 3 0 6 0 .0 3 3 5 0 .0 2 7 5 0 .0 1 5 6 0 .0 0 9 m 2 1 0 .0 1 3 8 0 .0 2 0 9 0 .0 3 4 2 0 .0 3 7 4 0 .0 3 0 3 0 .0 1 7 3 0 .0 0 9 9 0 .0 1 1 7 0 .0 1 8 4 0 .0 3 0 9 0 .0 3 3 5 0 .0 2 7 7 0 .0 1 5 6 0 .0 0 8 7 m 2 2 0 .0 1 4 0 .0 2 2 0 .0 3 3 6 0 .0 3 6 9 0 .0 3 0 9 0 .0 1 7 7 0 .0 1 0 9 0 .0 1 1 9 0 .0 1 7 8 0 .0 3 0 9 0 .0 3 3 0 .0 2 9 0 .0 1 7 1 0 .0 0 9 9 m 2 3 0 .0 1 9 0 .0 2 6 9 0 .0 4 1 6 0 .0 4 7 7 0 .0 3 5 4 0 .0 1 7 5 0 .0 0 9 9 0 .0 1 3 3 0 .0 2 1 7 0 .0 3 4 1 0 .0 3 9 3 0 .0 3 0 4 0 .0 1 5 8 0 .0 0 9 2 m 2 4 0 .0 1 7 9 0 .0 2 5 9 0 .0 4 0 .0 4 5 8 0 .0 3 3 9 0 .0 1 6 5 0 .0 0 9 2 0 .0 1 4 0 .0 2 2 2 0 .0 3 4 1 0 .0 3 9 5 0 .0 3 0 7 0 .0 1 5 9 0 .0 0 8 5 m 2 5 0 .0 1 5 6 0 .0 2 5 1 0 .0 3 9 2 0 .0 4 5 4 0 .0 3 5 6 0 .0 1 9 7 0 .0 1 1 2 0 .0 1 2 6 0 .0 2 0 7 0 .0 3 2 4 0 .0 3 8 5 0 .0 3 1 4 0 .0 1 7 0 .0 0 9 8 m 2 6 0 .0 1 5 1 0 .0 2 3 4 0 .0 3 8 0 .0 4 3 7 0 .0 3 5 0 .0 1 8 3 0 .0 1 0 5 0 .0 1 2 4 0 .0 1 9 9 0 .0 3 3 5 0 .0 3 8 9 0 .0 3 1 9 0 .0 1 7 0 .0 0 9 6 n o te : t h e b e st o u tc o m e s a re i n b o ld . t a b le 4 . m e a n v a lu e s o f th e h it r a ti o s m e th o d s m e a n h it r a ti o s t ri m m e d m e a n h it r a ti o s 0 .0 5 0 .1 0 .2 5 0 .5 0 .7 5 0 .9 0 .9 5 0 .0 5 0 .1 0 .2 5 0 .5 0 .7 5 0 .9 0 .9 5 m 1 0 .9 3 3 3 0 .8 1 6 7 ** 0 .6 9 1 7 0 .3 7 5 ** 0 .2 0 8 3 0 .0 8 3 3 0 .0 3 3 3 0 .9 6 8 8 0 .8 5 4 2 0 .7 0 8 3 0 .3 6 4 6 0 .1 8 7 5 0 .0 5 2 1 0 m 2 0 .9 2 5 0 .8 2 5 0 .6 8 3 3 0 .3 7 5 ** 0 .2 0 8 3 0 .0 8 3 3 0 .0 3 3 3 0 .9 5 8 3 0 .8 6 4 6 0 .6 9 7 9 0 .3 6 4 6 0 .1 8 7 5 0 .0 5 2 1 0 m 3 0 .8 6 6 7 ** 0 .8 ** 0 .6 6 6 7 0 .4 0 .2 0 8 3 0 .0 8 3 3 0 .0 3 3 3 0 .9 0 6 3 0 .8 3 3 3 0 .6 8 7 5 0 .3 8 5 4 0 .1 7 7 1 0 .0 5 2 1 0 m 4 0 .8 9 1 7 0 .8 5 8 3 0 .6 9 1 7 0 .4 0 8 3 0 .2 2 5 0 .0 8 3 3 0 .0 5 0 .9 2 7 1 0 .8 8 5 4 0 .7 1 8 8 0 .3 9 5 8 0 .2 1 8 8 0 .0 5 2 1 0 .0 2 0 8 m 5 0 .8 9 1 7 0 .8 1 6 7 ** 0 .6 2 5 ** 0 .4 0 8 3 0 .2 1 6 7 0 .0 6 6 7 0 .0 3 3 3 0 .9 2 7 1 0 .8 4 3 8 0 .6 3 5 4 0 .3 9 5 8 0 .1 9 7 9 0 .0 3 1 3 0 m 6 0 .9 0 .8 3 3 3 0 .6 7 5 0 .3 6 6 7 ** 0 .2 0 8 3 0 .0 7 5 0 .0 4 1 7 0 .9 2 7 1 0 .8 6 4 6 0 .6 7 7 1 0 .3 5 4 2 0 .1 8 7 5 0 .0 3 1 3 0 .0 1 0 4 m 7 0 .8 9 1 7 0 .8 5 0 .6 3 3 3 ** 0 .3 8 3 3 0 .2 1 6 7 0 .1 0 .0 6 6 7 0 .9 1 6 7 0 .8 8 5 4 0 .6 4 5 8 0 .3 7 5 0 .1 9 7 9 0 .0 5 2 1 0 .0 4 1 7 m 8 0 .8 8 3 3 ** 0 .8 6 6 7 0 .6 3 3 3 ** 0 .4 0 8 3 0 .2 0 .1 0 .0 9 1 7 0 .9 1 6 7 0 .8 9 5 8 0 .6 4 5 8 0 .3 9 5 8 0 .1 6 6 7 0 .0 7 2 9 0 .0 6 2 5 m 9 0 .9 0 8 3 0 .8 5 0 .7 7 5 0 .4 3 3 3 0 .1 7 5 0 .0 5 8 3 0 .0 3 3 3 0 .9 2 7 1 0 .8 8 5 4 0 .8 0 2 1 0 .4 2 7 1 0 .1 5 6 3 0 .0 3 1 3 0 .0 1 0 4 m 1 0 0 .8 9 1 7 0 .8 5 0 .7 2 5 0 .4 3 3 3 0 .1 5 8 3 0 .1 0 8 3 0 .0 5 0 .9 0 6 3 0 .8 5 4 2 0 .7 3 9 6 0 .4 2 7 1 0 .1 3 5 4 0 .0 9 3 8 0 .0 3 1 3 m 1 1 0 .9 0 8 3 0 .8 6 6 7 0 .7 2 5 0 .4 2 5 0 .2 0 8 3 0 .0 9 1 7 0 .0 6 6 7 0 .9 3 7 5 0 .9 0 6 3 0 .7 3 9 6 0 .4 1 6 7 0 .1 9 7 9 0 .0 6 2 5 0 .0 4 1 7 m 1 2 0 .9 2 5 0 .8 1 6 7 ** 0 .7 0 8 3 0 .4 1 6 7 0 .2 0 .0 5 8 3 0 .0 4 1 7 0 .9 5 8 3 0 .8 5 4 2 0 .7 2 9 2 0 .4 0 6 3 0 .1 7 7 1 0 .0 2 0 8 0 .0 1 0 4 m 1 3 0 .9 1 6 7 0 .8 2 5 0 .7 0 8 3 0 .4 1 6 7 0 .2 0 .0 6 6 7 0 .0 4 1 7 0 .9 4 7 9 0 .8 6 4 6 0 .7 2 9 2 0 .4 0 6 3 0 .1 7 7 1 0 .0 3 1 3 0 .0 1 0 4 m 1 4 0 .8 9 1 7 0 .8 0 8 3 ** 0 .6 8 3 3 0 .4 0 8 3 0 .2 0 .1 0 .0 5 0 .9 3 7 5 0 .8 4 3 8 0 .7 0 8 3 0 .3 9 5 8 0 .1 7 7 1 0 .0 5 2 1 0 .0 2 0 8 m 1 5 0 .8 7 5 ** 0 .8 3 3 3 0 .7 0 .4 0 .2 1 6 7 0 .0 7 5 0 .0 2 5 0 .9 0 6 3 0 .8 6 4 6 0 .7 1 8 8 0 .3 9 5 8 0 .2 0 8 3 0 .0 5 2 1 0 m 1 6 0 .9 0 .8 0 8 3 ** 0 .6 5 0 .4 0 .2 0 .0 8 3 3 0 .0 4 1 7 0 .9 2 7 1 0 .8 3 3 3 0 .6 6 6 7 0 .3 9 5 8 0 .1 7 7 1 0 .0 5 2 1 0 .0 1 0 4 m 1 7 0 .8 9 1 7 0 .8 3 3 3 0 .6 5 8 3 0 .4 0 .2 0 .0 5 8 3 0 .0 4 1 7 0 .9 3 7 5 0 .8 7 5 0 .6 7 7 1 0 .3 9 5 8 0 .1 7 7 1 0 .0 2 0 8 0 .0 1 0 4 m 1 8 0 .8 9 1 7 0 .8 2 5 0 .6 2 5 ** 0 .4 0 .2 0 8 3 0 .1 0 .0 6 6 7 0 .9 2 7 1 0 .8 7 5 0 .6 3 5 4 0 .3 9 5 8 0 .1 8 7 5 0 .0 6 2 5 0 .0 4 1 7 m 1 9 0 .8 7 5 ** 0 .8 5 8 3 0 .6 5 0 .4 0 .2 0 .1 4 1 7 0 .0 9 1 7 0 .9 0 6 3 0 .8 8 5 4 0 .6 5 6 3 0 .3 9 5 8 0 .1 7 7 1 0 .1 1 4 6 0 .0 6 2 5 m 2 0 0 .8 8 3 3 ** 0 .8 4 1 7 0 .7 5 0 .4 3 3 3 0 .1 7 5 0 .0 5 8 3 0 .0 2 5 0 .9 2 7 1 0 .8 7 5 0 .7 7 0 8 0 .4 3 7 5 0 .1 5 6 3 0 .0 3 1 3 0 m 2 1 0 .8 8 3 3 ** 0 .8 0 8 3 ** 0 .7 1 6 7 0 .4 3 3 3 0 .1 9 1 7 0 .0 9 1 7 0 .0 4 1 7 0 .8 9 5 8 0 .8 2 2 9 0 .7 3 9 6 0 .4 3 7 5 0 .1 7 7 1 0 .0 7 2 9 0 .0 2 0 8 m 2 2 0 .8 9 1 7 0 .8 5 0 .7 2 5 0 .4 5 0 .2 0 .1 1 6 7 0 .0 5 0 .9 1 6 7 0 .8 8 5 4 0 .7 5 0 .4 5 8 3 0 .1 8 7 5 0 .0 8 3 3 0 .0 3 1 3 m 2 3 0 .9 0 8 3 0 .8 5 0 .6 8 3 3 0 .3 9 1 7 0 .2 1 6 7 0 .0 6 6 7 0 .0 3 3 3 0 .9 3 7 5 0 .8 9 5 8 0 .7 0 8 3 0 .3 7 5 0 .1 8 7 5 0 .0 3 1 3 0 .0 1 0 4 m 2 4 0 .9 0 8 3 0 .8 3 3 3 0 .6 7 5 0 .3 8 3 3 0 .1 9 1 7 0 .0 7 5 0 .0 1 6 7 0 .9 4 7 9 0 .8 7 5 0 .6 9 7 9 0 .3 6 4 6 0 .1 6 6 7 0 .0 4 1 7 0 m 2 5 0 .9 0 8 3 0 .8 5 8 3 0 .6 8 3 3 0 .3 8 3 3 0 .2 0 .1 0 8 3 0 .0 6 6 7 0 .9 4 7 9 0 .9 0 6 3 0 .7 0 8 3 0 .3 6 4 6 0 .1 6 6 7 0 .0 7 2 9 0 .0 3 1 3 m 2 6 0 .9 1 6 7 0 .8 3 3 3 0 .6 8 3 3 0 .4 0 .1 8 3 3 0 .1 0 .0 5 0 .9 5 8 3 0 .8 6 4 6 0 .7 0 8 3 0 .3 8 5 4 0 .1 5 6 3 0 .0 5 2 1 0 .0 2 0 8 n o te : in t h e l e ft -h a n d p a n e l o f t a b le 4 h it r a ti o s si g n if ic a n tl y d e v ia ti n g ( a t th e 1 % l e v e l) f ro m t h e ir o p ti m a l v a lu e s a re d e n o te d b y ‘ * * ’. © 2016 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2016.006 vol. 16 (2016) 87−116 submitted november 30, 2016 issn (online) 2450-7067 accepted december 18, 2016 issn (print) 1234-3862 aneta włodarczyk, iwona otola * analysis of the relationship between market volatility and firms volatility on the polish capital market a b s t r a c t. in this paper we investigate if the strength of firm-market volatility relationship has changed after subprime crisis on the polish capital market. the empirical study concern the selected companies listed on the warsaw stock exchange (wse) from the construction and it sectors in the 2004–2011 period. the volatility measures were computed on the basis of daily low and high prices for companies shares and wig index. for each company arfimax-figarch model with additional exogenous variables, which represented market volatility, was estimated in the stable and the turbulent period. conducted empirical studies have not shown that the negative shocks flowing from the american stock market through investors' behavior channel contributed to the increase in the fraction of firms of the construction and it sectors listed on the wse whose volatility is shaped by market volatility. k e y w o r d s: arfimax-figarch, firm volatility, market volatility, subprime crisis, warsaw stock exchange. j e l classification: g12; d40; c58. introduction volatility of prices of listed companies shares is the subject of interest not only to stock exchange investors but also companies which emit them. high volatility means high probability of change of company market value in the future and this in turn influences its competitiveness on the market. this is * correspondence to: aneta włodarczyk, czestochowa university of technology, faculty of management, 36b armii krajowej street, 42-200 częstochowa, poland, e-mail: aneta.w@interia.pl; iwona otola, czestochowa university of technology, faculty of management, 19b armii krajowej street, 42-200 częstochowa, poland, iwotola@gmail.com. włodarczyk, otola dynamic econometric models 16 (2016) 87–116 88 connected with the fact that high market value of an enterprise in relation to its book value is connected with a simultaneous high market share of a given subject (grabowska, 2013). enterprise competitiveness depends, among others, on the value for the shareholders and clients, financial capability, determining its ability to act and react quickly in the competitive environment and human potential and technologies used in implementing strategic changes (feurer and chaharbaghi, 1994). moreover, the knowledge about linkages between market volatility and firm volatility is of great importance in risk management process and determining the investment portfolio structure, because it helps to understand the nature of individual firm volatility and identify this process determinants. it is not surprising that there exist many articles that were devoted to this issue. we may summarize some important facts about firm volatility on the basis of literature studies. as suggest karolyi (2001) the existence of excessive volatility in stock prices undermines the usefulness of the information about the true value of the company. this clearly does not mean that high volatility is proof of the irrational behavior and inefficient markets and investors. no signals from the market to volatility of stock prices, affirms the conviction that it is the correct measure of risk. interest in the subject matter of volatility, in particular, the possibility of its forecasting is related to the ability to reduce the risk of investment or achieving higher returns from investments. in the literature, there are already several well-documented facts concerning volatility. chen and lai (2013) and other researchers (whaley, 2009; simon, 2003; giot, 2005; peng and ng, 2012) showed a significant feature, which is the asymmetry of volatility. shin and stulz (2000) decomposed shares risk into its market and firmspecific components in order to show that changes in market risk are positively correlated with changes in firm value, but changes in firmspecific risk are negatively correlated with changes in firm value. campbell et al. (2001) used a disaggregated approach to study the volatility of common stocks at the market, industry and firm level. they proposed three variance components, which were estimated monthly using daily data, so their further analysis about volatility components were conducted on the basis of monthly frequency indicators and for this reason they could not consider the impact of volatility clustering effect on evaluation of risk measurement process. phylatkis and xia (2009) investigated the equity market comovement at the sector level and confirmed the sector heterogeneity of the contagion. they showed that investors might find the profitable sectors on the capital markets despite of the prevailing contagion on the market level. chuliá and torró (2011) estimated a conditional capm with multivariate garch-m structure in order to investigate an asymmetric volatility spillover effect between analysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 89 large and small firms in the spanish stock market and they proved the existence of bidirectional relationship between these volatility processes with asymmetric influence of bad and good news for firm volatility. sharma et al. (2011, 2014) showed that firms belonging to different sectors experienced different degrees of association with market volatility, what caused that in some cases firm volatility might be predicted on the basis of market volatility. they proved that commonality in volatility increases with the firm's size. sequeira and lan (2003) found that, the most significant component of aggregate volatility in the international market, with firm-level volatility forming the largest component of total volatility is unsystematic variance of the stock return. they also claim that volatility of the market-level is more important than the volatility of the sector-level in the explanation of the total volatility of stock return, which is also confirmed in earlier research conducted by cavaglia et al. (2000) or griffin and karolyi (1998). that is why both managerial staff in enterprises and investors should be interested in knowing the strength and direction of relationships between shares volatility of particular enterprises and stock market volatility. previous studies on the volatility in the majority relate to market volatility. a small part of the study is devoted to the volatility of the firm-level. volatility of asset prices is also often explained by the mechanism of transmission of crises and the related phenomenon of contagion (baur, 2003; corsetti et al., 2005; claessens and forbes, 2004; le and david, 2014). researchers examining the contagion effect on the capital market level concern only a chosen stock indices, which do not reflect the impact of the volatility of share prices of companies in different sectors. in such an approach the heterogeneity effect of the sectors selected within a given capital market and their various immunity to external shocks and different sensitivity to effects of a financial crisis are omitted (phylatkis and xia, 2009). the impact of the subprime crisis on individual segments of the polish financial market were presented by konopczak et. al. (2010). the analysis shows the local and global conditions that made the interaction of disorders associated with the global financial crisis on the polish financial market was as strong as in developed markets. empirical studies, including the period of the subprime crisis, conducted by otola (2013), concerning the relationship between the wig index and the s&p500 index, had to indicate whether negative shocks arising in the united states have been transferred to polish capital market by stock exchange channel. the high volatility in the capital market, which has been observed in an unstable period is not only the result of the interdependence between the markets but the result of the financial contagion of polish capital market. in this context, we rightly seem to conwłodarczyk, otola dynamic econometric models 16 (2016) 87–116 90 duct the further research on the volatility of the polish capital market on the firm level. taking above into consideration we are interested in examining the direction and strength of relationship between firm-level volatility and marketlevel volatility on the polish capital market in the period of preand post subprime crisis. the aim of this paper is to investigating if the market volatility and firm volatility are related and if the strength of this relationship increase significantly in the turbulent period compared to the stable period. the empirical study concerns the selected companies listed on the warsaw stock exchange from the construction and it sectors in the period from 2 january 2004 to 31 december 2011. in this respect, our contribution to the literature is connected with the calibration of sharma et al. model (2011, 2014) through introducing long memory dynamics and skewed fat-tailed distributions of innovations into the basic model describing the relationship between firm volatility and market volatility. following sadigue and silvapulle (2001) and kang and yoon (2012) we expect that on the polish capital market some investors reacted to new information immediately, whereas others postpone making the investment decision until they confirm the information. it causes that their actions form a non-linear pattern, which characterizes statistical persistence in a time series. we also taking into consideration the possible structural changes that may have affected the long memory property, so we divide the analyzed period into two sub-periods, taking the subprime financial crisis of july 2007 as the structural change point. the above considerations allow us to formulate the following hypothesis: there is no significant increase in the fraction of firms from a given sector of polish capital market whose volatility is strongly and positively related with market volatility in pre-crisis period compared with post-crisis period. the remainder of the paper is organized as follows. section 2 describes the methodology, data properties are presented in section 3. in section 4 we verify the hypothesis about the commonality in volatility in stable and turbulent period. and finally, section 5 concludes the paper. 2. methodology we have modified the model of sharma et al. (2011, 2014), which describes the relationship between firm volatility and stock market volatility by capturing the possible long memory effect in both firm and market volatility process. theoretical frames of the volatility model determines capital asset analysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 91 pricing model (capm) and fractionally integrated generalized autoregressive conditional heteroscedasticity (figarch). in empirical research presented in the paper the authors use the following regression equation, in which firm volatility is conditioned by previous and present market volatility and also by present return on investment into the market portfolio and squared return of market index (sharma et al., 2011): , 2 ,4,31,2,10, ittmitmitmitmiitf rrvvv i    (1) where: vm – stock market volatility, if v – share volatility of i-nth company (i=1, 2, ..,n), rm,t – logarithmic return of stock market index. sharma et al. (2011) incorporated market return and squared market return factors into above equation in order to control for possible spurious dependence between returns and volatility measures. price range estimators are used as volatility estimators that are computed on the basis of full publicly available set of information about financial instrument prices what make them more efficient than volatility estimators calculated as the squared return (fiszeder and perczak, 2013; molnár, 2012; sharma et al., 2014):  daily volatility estimator created by parkinson (1980): ,)]ln()[ln(361.01 2 itit plphv  (2) where: phit – highest price of i-nth financial instrument on t day, plit – lowest price of i-nth financial instrument on t day;  daily volatility estimator proposed by garman and klass (1980): ,)]ln()][ln(12ln2[)]ln()[ln(5.02 22 itititit popcplphv  (3) where: pcit – closing price of i-nth financial instrument on t day, poit – opening price of i-nth financial instrument on t day;  daily volatility estimator defined by rogers and satchell (1991): )].ln())][ln(ln()[ln( )]ln())][ln(ln()[ln(3 itititit itititit pcplpopl pcphpophv   (4) originally, the parkinson, garman-klass, rogers-satchell volatility measures of financial instruments prices are the estimators of unconditionally volatility for the geometric brownian motion determined for time interval [0, t]. therefore, all of analysed here estimators are derived under the assumptions of continuous sampling, no bid-ask spread and constant volatility. moreover, in the case of the parkinson and garman-klass estimators an additional assumption refers to zero drift. it is worth attracting attention on włodarczyk, otola dynamic econometric models 16 (2016) 87–116 92 some properties of range-based volatility estimators, especially in the case when one day is used as a unit of time. it was proved that the garman-klass estimator characterized by the highest efficiency in the case when the assumption of zero drift was met. the efficiency of rangebased volatility estimators for zero drift are as follow: the parkinson – 4.9, the garman-klass – 7.4, the rogers-satchell – 6.0 and they are significantly higher than the efficiency of simply volatility estimator (based on squared daily returns) equals by definition 1 (molnár, 2012). for daily financial time series, mean return is often much smaller than its standard deviation, which is in line with the assumption of zero drift. fiszeder and perczak (2013) derived the expected values of the parkinson and garman-klass estimators for the arithmetic brownian motion with non-zero drift and they proved the unbiasedness of the parkinson and garmann-klass estimators for the process with a zero drift and of the rogers-satchell estimator for any drift. it is worth stressing that the square root of any range-based volatility estimators is a biased estimator of standard deviation of a brownian motion, but this bias is rather small (3– 4%) compared to bias of the square root of the simply volatility estimator (25%). the questionable issue is the use of the range-based volatility estimators for the standardization of the returns. it was shown that returns normalized by means of the garman-klass estimator were approximately normally distributed. another disadvantage concerning the use of range estimators in practice is failure to comply with the assumptions of continuous sampling. prices are observed at discrete points in time (vector of pot, pht, plt, pct for each day) and it causes that the observed high price is below the true high price and observed low price is above the true low price. bid-spread effect works in the opposite direction. these two effects are small for liquid stocks (molnár, 2012). a debatable question is also the adjustment of range-based volatility estimators for the opening jumps, resulting from the situation that most of the assets are not traded continuously for 24h a day (molnár, 2012; fiszeder and perczak, 2013). sharma et al. (2014) limited their empirical studies to modelling the firm volatility shocks ( error term) by using garch(1,1) model with the conditional normal distribution of innovations. in the above equation the authors used full available on a given day information on the market portfolio (opening price, the lowest and the highest price, closing price), which will allow us to eliminate the effect of influencing the research results with one type of data only – closing price. for each analyzed series of share prices volatility of the given enterprises the arfima(p,d,q)-figarch(p,d,q) models were chosen individually in accordance with the following stages: analysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 93  specification of the equation describing the relation between market volatility and firm volatility (1), adjusted for possible autocorrelation dependencies occurring in the company share volatility series;  specification of the conditional variance equation describing the arch effect in the series of residuals from the equation (1) and the selection of the form of innovation density function. the fractionally integrated autoregressive moving average process (arfima) is more flexible econometric tool for modelling the conditional mean process than arma model in the situation when it exhibits long memory properties. similarly, the fractionallly integrated generalized autoregressive conditional heteroskedasticity process, constituting a generalization of the garch model, enables for capturing the persistence in the conditional variance process (bollerslev and mikkelsen, 1996). however, market shocks have a simultaneous impact on the conditional mean and conditional variance. therefore, some recent empirical studies have focused on the analyzing the relationship between the conditional mean and the conditional variance of the process that simultaneously exhibits long memory properties (beine et al., 2002; fiszeder, 2009; kang and yoon, 2012). the arfima(p,d,q)-figarch(p,d,q) model is defined in the following way (arouri et al., 2012): , )](1[)1)(( )()()1)(( 2 2 ttt tt d ttt ttt d h lll h lyll         (5) where: l denotes a lag operator;               00 )()1( )( )1()1( j jjj j d l dj dj l j d l – differential filter of d order, where, )( denotes gamma function; ,...1)( 1 p p lll   q q lll   ...1)( 1 are respectively lags polynomials for the autoregressive part of p order and moving average of q order, whose all roots lie outside the unit circle, i, i are the model parameters;               00 )()1( )( )1()1( s sss s d l ds ds l s d l – differential filter of d order; ,...1)( 1 q q lll   p p lll   ...1)( 1 are respectively włodarczyk, otola dynamic econometric models 16 (2016) 87–116 94 lags polynomials for the arch part of q order and garch part of p order, all the roots of )](1[),( ll   lie outside the unit circle, αi, βi are the model parameters; t  is an innovations series, ).1,0(..~ diit on the basis of d parameter value one can identify the memory type of the process (hosking,1981):  means that the process exhibits negative dependencies between distant observations (anti-persistence, but stationary process),  corresponds to the long-memory stationary process,  means reduction of the arfima(p,d,q) to the stationary arma(p,q) model (short memory process),  indicates a non-stationary process (in particular for the process follows a unit root process – arima(p,1,q) model). for the figarch model the influence of current shocks for volatility forecasts decreases to zero, but at a slower rate than for garch processes. moreover, the autocorrelation function of figarch squared residuals decreases at a hyperbolic rate to zero, which indicates long memory in volatility of series described by this class models. the existence of long memory in volatility process may be recognized on the basis of estimated value of d parameter (fiszeder, 2009):  means that volatility process exhibits long memory property,  means reduction of the figarch(p,d,q) to the garch(p,q) process for which the influence of current volatility on forecasts of conditional variance decays at fast rate (also short memory process in sense of “influence of current volatility for the true conditional variance process”),  means reduction of the figarch(p,d,q) to the igarch(p,q) process for which current volatility has permanent impact on forecasts of conditional variance (short memory in above explained sense). this class of models is estimated by using the quasi-maximum likelihood (qml) estimation method, based on the following log-likelihood function (under the assumption about gaussian distribution of innovations) (fiszeder, 2009; kang and yoon, 2012): ,])[ln( 2 1 )2ln( 2 1 1 2    t t ttgaussian htll  (6) where t denotes the number of observation. it is worth stressing that one of stylized facts of high – frequency time series of financial prices is excess kurtosis and skewness of returns distributions (fiszeder, 2009; laurent, 2013; włodarczyk, 2010). in order to capture the analysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 95 excess kurtosis and fat-tailed effect in the residuals, the student t-distribution may be used as the innovation distribution in arfima-figarch model, and then the log-likelihood function is defined as follows (kang and yoon, 2012; laurent, 2013 ): , 2 1ln)1()ln( 2 1 )]2(ln[ 2 1 2 ln 2 1 ln 1 2                                                  t t t t student h tll      (7) where parameter measures the degree of fat tails of the density function and the lower its values is, the fatter tails of distribution are. the latest specification of arfima-figarch models, which be considered in this work, allows for both the excess skewness and kurtosis of residuals by using the skewed student t-distribution. in this case the log-likelihood function is as follows (kang and yoon, 2012; laurent, 2013 ):   , 2 )( 1ln)1()ln( 2 1 ln 1 2 ln)]2(ln[ 2 1 2 ln 2 1 ln 1 2 2                                                                        t t ists t sentskewedstud tkh k k tll      (8) where: , / if 1 / if 1 t t       ss ss ti   (9) is the indicator function, which is determined on the basis of the mean (s) and standard deviation (σs) of the skewed student t-distribution (kang and yoon, 2012; laurent, 2013 ): , 1 2 2 2 1                        k ks      (10) włodarczyk, otola dynamic econometric models 16 (2016) 87–116 96 .1 1 2 2 22 ss k k         (11) the value of the asymmetry parameter ln(k) determined the type of skewness, that is, if 0)l n ( k , the density is right skewed and if 0)ln( k , the density is left skewed. 3. data description empirical research were conducted for the chosen companies from construction and it sector listed on the main market of the warsaw stock exchange whereas restrictive provisions which these companies had to meet in order to be able to exist on it. to the research 17 companies from the construction sector were chosen in the stable period and during a turbulent three more. the it sector study involved 10 companies during the stable period and 20 in a turbulent period. the authors excluded from the research sample enterprises which in the given sub-period withdrew from the market or their debuts took place in the course of the given sub-period. data were obtained from the notoria service and consist of 2013 daily observations of high, low, open and close prices for analyzed companies from january 2, 2004 to december 30, 2011. the starting point is determining the breakthrough on the basis of which it is possible to distinguish two sub-periods: the stable period which corresponds to low stock market volatility and the turbulent period characterized by high market volatility. first symptoms of growing volatility of main indexes of american market (djia, nasdaq, s&p500) can be observed in july and august of 2007. they are accompanied by bankruptcy of two hedging funds of bear stearns bank connected with the mortgage market in the usa (july 2007), as well as the insolvency of three funds of the french bank bnp paribas investing in the bonds market secured with mortgages subprime (9 august 2007) (otola, 2013; burzała, 2013; dungay et al., 2011). the analyzed period was divided into two sub-samples:  the stable period (02.01.2004–25.07.2007) – 899 observations;  the turbulent period (26.07.2007–31.12.2011) – 1114 observations. due to the prolonging period of instability on the european financial market, caused not only by the occurrence of the subprime crisis, but also the debt crisis in the peripheral countries of the eu zone, the turbulent period in the present analysis was lengthened to the end of the year 2011. selection of sectors was conducted on the following basis. the construction sector is most connected with the country's economy and most susceptianalysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 97 ble to gdp changes. it is also most strongly represented sector on the polish stock market. moreover, since 2004 poland as a member of the european union has been receiving financial resources within european funds to improve the living conditions of inhabitants. in the course of the second programming period (2007–2013) poland received about 242,3 billion pln, the part of which was destined for public infrastructural investments, both in the scope of road construction as well as other public facilities. about 48% of allocated within structural funds resources were used to execute the infrastructure and environment programme. additionally, the decision of uefa in the ii quarter of 2007 to give the right to organize the european football championship – euro 2012 in poland and ukraine was also connected with the necessity to complete numerous construction investments in short time. according to the central statistical office since the year 2005 the construction sector was the fastest growing sector of polish economy. moreover, this sector is the most abundantly represented sector on the warsaw stock exchange. we claim that there is no significant increase in the fraction of firms, belonging to the construction sector, whose volatility is shaped by market volatility in the turbulent period compared to the stable period. thus, one can expect that external shocks transmitted on the stock market in poland in the subprime crisis period and the debt crisis in the euro zone should not be noticeable by the enterprises of this sector. claessens and forbes (2004) emphasised the importance of investor reactions in the contagion process in the liquidity risk context. referring to this theory one can expect that investors having in mind good development perspectives resulting from a substantial financing of this sector with structural funds should not withdraw capital located in the stocks of construction companies. the second select sector, namely it, may be considered as the one based on knowledge, most intellectual and innovative one. the it sector may be divided into three groups due to the form of conducted activity:  production of it equipment (among others comp, elzab, novitus);  software related services (among others asseco poland, comarch, sygnity);  distribution of it solutions (among others arcus, b3system, nnt system). comparing it with the construction sector one should notice that in the analyzed period a lot of enterprises from the it sector put on acquiring capital to conduct activity through emission of shares (see table 1). the resources acquired from the shareholders for development of this sector are used not only on tangible investments but also investments in intellectual capital. moreover, development of new and growth of awareness of needs in the włodarczyk, otola dynamic econometric models 16 (2016) 87–116 98 scope of it also in the sector of small and medium enterprises, the use of structural funds to liquidate disproportions in access to tele-informatic technologies in rural areas are only some of the factors conditioning continuity of operations of this sector companies. the it sector is perceived in the world as one of the most perspective ones, the one which creates new innovative products, such as for example mobile product applications or development of cloud services. customers of products and services of the it sector are primarily enterprises from the public administration sector that to a large extent use for this purpose the resources from the european funds. more important customers of services and products from the it sector are enterprises operating in the banking and tele-communication sectors. such a perception of the it sector by the investors should contribute to the fact that the enterprises listed on the stock market should show resistance to shocks coming from the market. thus, in our opinion commonality in volatility did not increase significantly in the turbulent period compared to the stable period in the it sector on the polish capital market. table 1. descriptive statistics for volatility measures – stable period variable minimum mean maximum standard deviation skewness kurtosis v1_wig 0.0004 0.008 0.157 0.012 5.577 [0.000] 47.622 [0.000] v2_wig 0.0003 0.008 0.195 0.011 7.348 [0.000] 85.857 [0.000] v3_wig 0 0.008 0.269 0.013 9.764 [0.000] 147.92 [0.000] v1_construction 0.013 0.136 1.303 0.130 3.042 [0.000] 14.837 [0.000] v2_construction 0.011 0.131 1.786 0.146 4.248 [0.000] 28.550 [0.000] v3_construction 0.006 0.158 3.506 0.241 5.418 [0.000] 47.534 [0.000] v1_it 0.009 0.116 1.046 0.119 2.759 [0.000] 10.424 [0.000] v2_it 0.010 0.084 1.397 0.081 6.675 [0.000] 78.417 [0.000] v3_it 0.011 0.092 2.179 0.103 10.018 [0.000] 165.52 [0.000] note: all volatility series were scaled by 100, p-value in brackets. on the basis of daily information on four price categories (open, close, high and low) of the wig stock index and shares of particular companies the authors determined volatility series in the stable and turbulent period according to the price range volatility measures (2)–(4). for each sector the authors estimated series of average firm volatility in each of the analyzed subanalysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 99 periods and then on their basis they determined descriptive statistics (tables 1–2). table 2. descriptive statistics for volatility measures – turbulent period variable minimum mean maximum standard deviation skewness kurtosis v1_wig 0.0002 0.015 0.373 0.028 6.125 [0.000] 53.180 [0.000] v2_wig 0.0002 0.013 0.305 0.027 6.340 [0.000] 50.279 [0.000] v3_wig 0 0.013 0.357 0.029 6.651 [0.000] 55.075 [0.000] v1_construction 0.012 0.205 2.481 0.351 2.176 [0.000] 3.837 [0.000] v2_construction 0.010 0.120 3.394 0.195 7.707 [0.000] 103.24 [0.000] v3_construction 0.010 0.100 3.623 0.253 10.149 [0.000] 112.55 [0.000] v1_it 0.061 0.289 1.224 0.170 1.861 [0.000] 4.802 [0.000] v2_it 0.011 0.092 0.571 0.075 2.919 [0.000] 11.721 [0.000] v3_it 0.010 0.104 0.697 0.084 2.626 [0.000] 9.510 [0.000] note: all volatility series were scaled by 100, p-value in brackets. one can observe that the average volatility in the it sector was relatively lower than in the construction sector for all three measures. a similar dependence can be observed for the maximum value of each volatility measures evaluated for the firms in both sectors. it is worth emphasizing that in the stable period market volatility was substantially lower than volatility of firms in both sectors, comparing volatility range determined in accordance with the three analyzed measures. also determined skewness and kurtosis indicate sector differentiation of the volatility process, which may indicate heterogeneity of the construction and it sectors on the warsaw stock exchange. similar conclusions were formulated for the american stock exchange in the paper of sharma, narayan and zheng (2014). the determined average statistics for the series of firm and market volatility took higher values in the turbulent period in comparison with the stable period. also in this period market volatility was substantially lower than the firm volatility in both sectors. due to high kurtosis and skewness of the volatility series determined in accordance with the rogers and satchel formula only volatility calculated according to the parkinson method has been included in the further part of the analysis. shaping of daily volatility for the włodarczyk, otola dynamic econometric models 16 (2016) 87–116 100 figure 1. shaping of daily volatility for the companies from the construction sector in the stable period 1 figure 2. shaping of daily volatility for the companies from the construction sector in the turbulent period 2 1 explanations of used abbreviations for listed companies can be found in table 5. 2 explanations of used abbreviations for listed companies can be found in table 6. analysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 101 figure 3. shaping of daily volatility for the companies from the it sector in the stable period 3 figure 4. shaping of daily volatility for the companies from the it sector in the turbulent period 4 3 explanations of used abbreviations for listed companies can be found in table 7. 4 explanations of used abbreviations for listed companies can be found in table 8. sgn 2008 2010 2 4 sgn ctc 2008 2010 2 4 ctc elz 2008 2010 2.5 7.5 elz acp 2008 2010 2.5 7.5 acp cmr 2008 2010 2 4 cmr mcl 2008 2010 2 4 mcl tlx 2008 2010 2 4 tlx sme 2008 2010 2 4 sme was 2008 2010 2 4 was cdr 2008 2010 2 4 cdr bcm 2008 2010 2 4 bcm atm 2008 2010 2 4 atm cmp 2008 2010 2 4 cmp pcg 2008 2010 5 15 pcg qsm 2008 2010 2 4 qsm u2k 2008 2010 2 4 u2k lsi 2008 2010 2 4 lsi imx 2008 2010 2 4 imx prd 2008 2010 2 4 prd ntt 2008 2010 2 4 ntt włodarczyk, otola dynamic econometric models 16 (2016) 87–116 102 companies from the construction and it sectors in the stable and turbulent periods show figures 1–4. in the stable period the biggest range of daily share price volatility fell for budopol sa company. high daily spread of volatility was also characteristic for such companies as: awbud sa, energopol południe sa, ulma construccion sa. in the turbulent period two companies elkop sa and erbud sa were characterized by very high daily volatility spread. in the stable period the highest spread in daily volatility was observed for the companies: simple sa, wasko sa and calatrava capital sa. in the turbulent period a high daily spread of volatility was observed in the companies: zuk elzab, asseco poland sa, cd project red sa, calatrava capital sa, simple sa, pc guard sa, lsi software sa, ntt system sa. the presented above descriptive statistics for enterprises listed on the warsaw stock exchange from the construction and it sectors let us notice differences both between sectors as well as the stable and turbulent periods. 4. verifying the hypothesis about the commonality in volatility in stable and turbulent period time series volatility v1 of share prices of enterprises from the construction and it sectors in the stable period i.e. 02.01.2004.–25.07.2007 and the turbulent one 26.07.2007–31.12.2011 were subject to modelling in order to verify the occurrence of significant relationships between firm volatility and wig index volatility. due to the characteristic properties of volatility time series, which were the subject of modelling: skewness, high kurtosis, fattailed distribution, autocorrelation dependencies (box-pierce test), occurrence of long memory effect (gph test), occurrence of volatility clustering effect (engle test, mcleod and li test) (see table 3), for each volatility series individual specification of the model in the form (5)–(8) were selected with t-student distribution or skewed t-student distribution (laurent, 2013). additionally, in the conditional mean equation is assumed that the firm volatility is dependent on previous, present and future market volatility and also expected daily change of the wig index and the squared wig index return. the choice of the final form of the firm volatility model has been performed on the basis of :  evaluating significance of the model parameters estimations;  minimizing information criteria of akaike’a (aic) and bayesian schwarz (sc); analysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 103  test results of standardized models residuals which verified the correctness of the model dynamic specification: box-pierce test, engle test, mcleod and li test;  nyblom test for checking the constancy of model parameters over time;  engle and ng set of tests for existing leverage effect and investigating possible misspecification of the conditional variance equation (figarch vs. fieparch), with in particularly covered negative sign bias test and positive sign bias test for verifying the effect of negative and positive shocks on the conditional variance (laurent, 2013). table 3. descriptive statistics and diagnostic tests for volatility series of msw in stable period parameter/test value of statistics mean 0.868 standard deviation 1.410 minimum 0 maximum 14.537 skewness 5.239 [0.000] kurtosis 38.493 [0.000] jarque-bera statistics 59616 [0.000] adf statistics –6.588 d-gph statistics 0.259 [0.000] q box-pierce statistics q(5) 71.1948 [0.000] q(10) 120.783 [0.000] q(20) 145.085 [0.000] q(50) 172.831 [0.000] q mcleod-li statistics q(5) 117.096 [0.000] q(10) 182.624 [0.000] q(20) 204.178 [0.000] q(50) 214.615 [0.000] engle lm statistics arch(1–1) 46.606 [0.000] arch(1–2) 25.877 [0.000] arch(1–5) 23.281 [0.000] arch(1–10) 15.902 [0.000] arch(1–20) 9.590 [0.000] note: p-value in brackets. below we present the detailed results of the estimation of arfimaxfigarch model with t-student innovations for the chosen enterprise from the construction sector – mostostal warszawa sa (msw; see table 4a–4b). włodarczyk, otola dynamic econometric models 16 (2016) 87–116 104 table 4a. estimation results of armax(1,2)-garch(1,1) for msw in stable period parameter parameter estimates cst(m) 0.223 [0.000] vm,t_wig 0.369 [0.374] ar(1) 0.858 [0.000] ma(1) –0.712 [0.000] ma(2) –0.049 [0.044] cst(v) 0.221 [0.025] arch1 0.165 [0.003] garch1 0.757 [0.000] df-student 2.258 [0.000] aic 2.029 sc 2.072 note: p-value in brackets. table 4b. diagnostic residuals tests for msw test estimates of the test statistics q box-pierce statistics for standardized residuals q(5) 8.183 [0.037] q(10) 11.812 [0.107] q(20) 19.714 [0.289] q(50) 54.719 [0.205] q box-pierce statistics for squared standardized residuals q(5) 0.660 [0.883] q(10) 0.916 [0.999] q(20) 2.187 [0.999] q(50) 3.857 [0.999] engle lm statistics arch(1–2) 0.035 [0.965] arch(1–5) 0.128 [0.986] arch(1–10) 0.088 [0.999] jarque-bery for normality 27151 [0.000] joint nyblom test of stability 1.2021 nyblom for cst(m) 0.1501 nyblom for vm,t_wig 0.0541 nyblom for ar(1) 0.0577 nyblom for ma(1) 0.0505 nyblom for ma(2) 0.0803 nyblom for cst(v) 0.1432 nyblom for arch1 0.2187 nyblom for garch1 0.2149 nyblom for df-student 0.0712 sign bias test 0.899 [0.369] negative sign bias test 0.381 [0.703] positive sign bias test 0.275 [0.783] jointly engle–ng test 2.227 [0.527] note: p-value in brackets, p-value (5%) = 0.47 for individual nyblom test. analysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 105 it is worth emphasizing that for each estimated model for the mostostal warszawa company, the parameters indicating the strength and direction of dependencies between enterprise volatility and market volatility was statistically insignificant. the results summarizing estimation of arfimax-figarch class models with included variables describing volatility of the stock market for particular companies from the construction sector in the stable period and turbulent period have been presented in tables 5–6. the analysis of data included in table 5 allows us to formulate the following conclusions. in case of energopol-południe and budopol-wrocław enterprises the relationship between the market volatility and volatility of their share prices was statistically insignificant. for the remaining enterprises the relationship between the wig index volatility and firms volatility was statistically significant and positive. all additional variables describing market volatility (vm,t-1_wig, rm,t, rm,t 2 ) turned out to be statistically insignificant in case of all estimated models. table 5. results on commonality in volatility for construction sector in stable period company model specification estimate of vm,t_wig parameter significance of the influence of others market variables (vm,t-1_wig, rm,t, rm,t2) on firm volatility mostostal export (msx) armax(1,1)figarch(1,0.54,1), t-student distribution 2.069 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model prochem (prm) arfimax(1,0.121,1) garch(1, 1), t-student distribution 0.390 [0.005] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model mostostal zabrze holding (msz) arfimax(1,0.049,1) garch(1, 1), t-student distribution 1.191 [0.013] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model budimex (bdx) arfimax(1,0.021,1) garch(1, 1), skewed t-student distribution 0.090 [0.006] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model elektrobudowa (elb) arfimax(1,0.105,1) garch(1, 1), t-student distribution 0.677 [0.010] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model energoaparatura (enp) arfimax(1,0.086,0) arch(1), t-student distribution 0.539 [0.022] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model energopol południe (epl) arfimax(1,0.185,1) garch(1, 1), t-student distribution 0.075 [0.410] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model włodarczyk, otola dynamic econometric models 16 (2016) 87–116 106 table 5. continued company model specification estimate of vm,t_wig parameter significance of the influence of others market variables (vm,t-1_wig, rm,t, rm,t2) on firm volatility projprzem (pjp) arfimax(1,0.027,0) arch(1), t-student distribution 0.167 [0.003] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model elkop (ekp) armax(1,1) garch(1, 1), t-student distribution 0.023 [0.018] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model ulma construccion polska (ulm) arfimax(0,0.06,1) figarch(1,0.57,1), t-student distribution 0.041 [0.355] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model energomontaż-południe (edp) arfimax(1,0.042,1)garch(1,1), t-student distribution 0.815 [0.045] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model mostostal płock (msp) arx(2)-garch(1,1), t-student distribution 3.642 [0.000] rm,t: 0.081 [0.027] rm,t2: –0.069 [0.001] budopolwrocław (bdl) arfimax(0,0.06,1) figarch(1,0.79,1), t-student distribution 0.024 [0.283] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model awbud (awb) arfimax(0,0.06,1) garch(1,1), t-student distribution 0.921 [0.005] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model pemug (pmg) arx(2)-garch(1,1), t-student distribution 0.748 [0.099] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model instal kraków (ink) armax(1,1)garch(1,1), t-student distribution 2.618 [0.000] rm,t2: –0.076 [0.001] note: p-value in brackets. in the turbulent period for five enterprises: awbud, projprzem, prochem, elkop, ulma construccion the relationship between the wig index volatility and firm volatility was statistically insignificant. for the remaining 15 enterprises statistically significant influence of market volatility on firm volatility was confirmed, while the direction of this dependence was positive. additionally, share price volatility of mostostal warszawa was significantly shaped by market volatility observed on the previous day. in case of the remaining enterprises additional variables did not significantly influence share price volatility. the conducted commonality in volatility analysis for the enterprises listed on the warsaw stock exchange shows that in the stable period 82% of analyzed enterprises were sensitive to the wig index volatility, the appropriate ratio in the turbulent period reached the value of 75%. analysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 107 table 6. results on commonality in volatility for construction sector in turbulent period company model specification estimate of vm,t_wig parameter significance of the influence of others market variables (vm,t-1_wig, rm,t, rm,t2) on firm volatility mostostal export (msx) arfimax(1,0.07,1) garch(1,1), t-student distribution 0.949 [0.007] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model mostostal warszawa (msw) arx(2)-garch(1,1), t-student distribution 0.171 [0.006] vm,t-1_wig: 0.176 [0.040] prochem (prm) armax(1,1)figarch(1,0.68,1), t-student distribution 0.138 [0.483] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model budimex (bdx) arfimax(1,0.103,1) garch(1,1), t-student distribution 0.206 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model mostostal zabrze holding (msz) arfimax(1,0.076,0) garch(1,1), t-student distribution 0.867 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model elektrobudowa (elb) arfimax(1,0.121,1) garch(1,1), t-student distribution 0.122 [0.082] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model ulma construccion polska (ulm) arfimax(1,0.041,1) garch(1,1), t-student distribution 0.227 [0.454] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model energoaparatura (enp) arfimax(1,0.111,0) garch(1,1), t-student distribution 0.508 [0.005] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model polimexmostostal (pxm) armax(1,1) garch(1,1), t-student distribution 1.639 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model mostostal płock (msp) armax(1,1)arch(1), t-student distribution 0.401 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model energopol południe (epl) arfimax(1,0.045,1) garch(1,1), t-student distribution 0.521 [0.012] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model instal kraków (ink) arfimax(0,0.12,1)figarch(1,0.75,1), t-student distribution 0.541 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model projprzem (pjp) armax(1,1) garch(1,1), t-student distribution 0.065 [0.562] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model elkop (ekp) arfimax(1,0.999,0)arch(1), t-student distribution 0.054 [0.189] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model włodarczyk, otola dynamic econometric models 16 (2016) 87–116 108 table 6. continued company model specification estimate of vm,t_wig parameter significance of the influence of others market variables (vm,t-1_wig, rm,t, rm,t2) on firm volatility pbg (pbg) arfimax(1,0.236,1) garch(1,1), t-student distribution 1.041 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model herkules (hrs) arfimax(1,0.063,1) garch(1,1), t-student distribution 1.016 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model erbud (erb) arfimax(1,0.04,1) garch(1,1), t-student distribution 0.269 [0.020] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model energomontażpołudnie (edp) arfimax(1,0.14,1) garch(1,1), t-student distribution 0.457 [0.009] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model budopolwrocław (bdl) arfimax(1,0.087,1) garch(1,1), t-student distribution 0.972 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model awbud (awb) armax(1,1) garch(1,1), t-student distribution –0.018 [0.735] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model note: p-value in brackets. estimation of volatility models for the it sector enterprises in both analyzed periods was conducted in the same manner as in case of the construction sector enterprises (tables 7–8). the estimated models indicate a statistically insignificant relationship between market volatility and share price volatility of it sector companies only in case of macrologic and talex. moreover, included in the modelling process the additional variables describing market volatility, did not have a statistically significant impact on shares volatility of the analyzed enterprises. in the turbulent period for the seven analyzed enterprises from the it sector: comarch, talex, simple, wasko, pc guard, unima 2000, infovide-matrix the relationship between the wig index volatility and share price volatility was statistically insignificant at 10% significance level. in addition, for three enterprises: comarch, procad and ntt system market volatility from the previous day significantly affected share price volatility of these enterprises. on the basis of empirical studies results for it sector one may draw the conclusion that arfimax-figarch specification has been chosen more frequently for the post-crisis period compared to pre-crisis period. therefore, long memory property is not an inherent feature of firm volatility process and the structural change connected with the subprime crisis may influence analysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 109 to the assessment of the long memory property in the volatility 5 . the conducted commonality in volatility analysis for the it sector enterprises listed table 7. results on commonality in volatility for it sector in stable period company model specification estimate of vm,t_wig parameter significance of the influence of others market variables (vm,t-1_wig, rm,t, rm,t2) on firm volatility calatrava capital (ctc) arfimax(1,0.104,1)garch(1,1), t-student distribution 1.186 [0.002] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model zuk elzab (elz) arfimax(1,0.044,1)garch(1,1), t-student distribution 0.362 [0.030] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model asseco poland (acp) arfimax(1,0.108,1) garch(1,1), t-student distribution 0.536 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model comarch (cmr) armax(1,1)garch(1,1), t-student distribution 0.349 [0.003] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model macrologic (mcl) armax(1,1)garch(1,1), t-student distribution 0.392 [0.133] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model talex (tlx) armax(1,1)garch(1,1) with tstudent distribution 0.362 [0.155] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model simple (sme) armax(1,1)garch(1,1), t-student distribution 1.414 [0.061] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model cd project red (cdr) armax(1,1)garch(1,1), t-student distribution 0.949 [0.001] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model sygnity (sgn) armax(1,1)garch(1,1), t-student distribution 0.349 [0.086] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model wasko (was) armax(1,1) garch(1,1), t-student distribution 0.719 [0.056] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model note: p-value in brackets. 5 it is worth underlining that for both the construction and it sectors there were no reasons for changing the basic specification of firm volatility models to exponential garch model, which enables capturing the asymmetric effect of negative and positive shocks for conditional variance (results of sign bias tests). but for deepening the studies over the linkage between market volatility and firm volatility on the polish capital market we will use the asymmetric multivariate garch structure in near future. włodarczyk, otola dynamic econometric models 16 (2016) 87–116 110 on the warsaw stock exchange shows that in the stable period 80% of analyzed enterprises were sensitive to the wig index volatility, while in the turbulent period the fraction of these enterprises dropped to 65%. table 8. results on commonality in volatility for it sector in turbulent period company model specification estimate of vm,t_wig parameter significance of the influence of others market variables (vm,t-1_wig, rm,t, rm,t2) on firm volatility sygnity (sgn) armax(1,1)garch(1,1), t-student distribution 0.633 [0.005] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model calatrava capital (ctc) arfimax(1,0.049,1)garch(1,1), t-student distribution 1.401 [0.013] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model zuk elzab (elz) armax(1,1)figarch(1,0.39,1), t-student distribution 1.494 [0.036] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model asseco poland (acp) arfimax(1,0.64,1)garch(1,1),skewed t-student distribution 1.381 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model comarch (cmr) arfimax(1,0.07,1)figarch(1,0.31,1), t-student distribution 0.085 [0.410] rm,t2: 0.011 [0.005] macrologic (mcl) arfimax(1,0.08,1)figarch(1,0.7,1), t-student distribution 0.020 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model talex (tlx) arfimax(2,0.044,0)garch(1,1), t-student distribution –0.008 [0.942] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model simple (sme) armax(1,1)garch(1,1), t-student distribution 0.365 [0.203] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model wasko (was) armax(1,1)garch(1,1), skewed t-student distribution 0.073 [0.326] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model cd project red (cdr) armax(1,1)garch(1,1), t-student distribution 2.180 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model betacom (bcm) armax(1,1)garch(1,1), t-student distribution 0.706 [0.000] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model atm (atm) armax(1,1)garch(1,1), t-student distribution 0.554 [0.049] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model comp (cmp) armax(1,1)garch(1,1), t-student distribution 0.188 [0.080] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model analysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 111 table 8. continued company model specification estimate of vm,t_wig parameter significance of the influence of others market variables (vm,t-1_wig, rm,t, rm,t2) on firm volatility pc guard (pcg) arfimax(1,0.002,1)garch(1,1), t-student distribution 0.003 [0.731] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model, problem with parameters’ stability qumaksekom (qsm) armax(1,1)garch(1,1), t-student distribution 0.271 [0.021] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model unima 2000 systemyteleinformatyczne (u2k) armax(1,1)garch(1,1), t-student distribution –0.009 [0.661] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model lsi software (lsi) armax(1,1)garch(1,1), t-student distribution 0.807 [0.016] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model infovidematrix (imx) armax(1,1)garch(1,1), t-student distribution 0.043 [0.568] all additional market variables have statistically insignificant impact on the firm volatility – elimination from the model procad (prd) armax(1,1)figarch(1,0.29,1), t-student distribution 0.896 [0.004] vm,t-1_wig : 0.644 [0.005] ntt system (ntt) armax(1,1)garch(1,1), t-student distribution 0.733 [0.000] vm,t-1_wig : 0.539 [0.000] note: p-value in brackets. in order to verify the hypothesis assuming that within a given sector the percentage of enterprises in the stable and turbulent period, for which a significant positive dependence between their share price volatility and stock market volatility is the same, we used a statistical test (krysicki et al. 2012): .: : 211 210 pph pph   the test statistics under the null hypothesis has a standardized normal distribution: ),1,0(arcsin2arcsin2 21 21 2 2 1 1 n nn nn n k n k u              (12) where: p1 – fraction of the firms in the given sector for which the commonality in volatility effect was confirmed in stable period, p2 – fraction of the włodarczyk, otola dynamic econometric models 16 (2016) 87–116 112 firms in the given sector for which the commonality in volatility effect was confirmed in turbulent period, and – sample proportions corresponding to p1 and p2. table 9. statistics for similarity of commonality in volatility effect in stable and turbulent period u statistic (1.95 – 5% critical value) construction sector it sector 0.546 0.875 at 5% significance level there is no reasons for rejecting the hypothesis that commonality in volatility didn’t differ significantly in the turbulent period compared to the stable period in construction sector as well as in it sector on the polish capital market (table 9). the conducted research show that the fractions of enterprises, in the stable and turbulent period, for which the occurrence of significant relationships between their share price volatility and the wig index volatility was proved, did not significantly differ one from the other. this may confirm that analyzed sectors were characterized by resistance to external shocks absorbed by polish stock market through the investor behaviour channel. conclusions the analysis of relationships between market volatility and volatility of enterprises from the construction and it sectors has shown that there is not an increase in the fractions of firms, for which their share volatility in a significant and positive way was connected with stock market volatility in the period corresponding to the subprime crisis and the debt crisis in the euro zone. moreover, one can observe a different relation of these sectors to external shocks coming to polish market, namely in the it sector the fraction of enterprises sensible to the wig index volatility decreased, while in the construction sector it increased. however, the changes were not statistically significant. the financial situation of the enterprises from these sectors was conditioned not only by unforeseen factors connected with reaction of investors to the information about crisis, but first of all economic factors and skills of managers in the scope of adjusting enterprise strategies to the changeable conditions of external environment, making the use of the opportunities and threats filter in building and maintaining enterprise competitive position. on the basis of conducted empirical studies (otola, 2013) concerning financial condition of enterprises from the construction and it sectors we can draw the following conclusions. majority of the analyzed enterprises analysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 113 from the construction sector did not possess financial liquidity which is the determinant of the ability to pay back their liabilities. moreover, in case of over 65% of the examined enterprises their rates of debt exceeded 70%. in addition, majority of the discussed enterprises achieved low sales profitability already at the level of gross profit from sales or operational profit. there were such enterprises among all the examined ones which were characterized by high negative sales profitability. the situation was different in the it sector. this sector enterprises were characterized by average debt level and maintained financial liquidity on the proper level (in accordance with the assumed norms of indexes), and a part of them possessed surplus of turnover assets with reference to the current liabilities, which indicates the excess of liquidity. sales profitability indexes in the stable period were on a higher level than in the turbulent period, in which they can be considered as low and unsatisfactory for investors, nonetheless they were better than the ones from the construction sector. references arouri, m., hammoudeh, s., lahiani, a., nguyen, d. (2012), long memory and structural breaks in modeling the return and volatility dynamics of precious metals, quarterly review of economics and finance, 52, 207–218, doi: http://dx.doi.org/10.1016/j.qref.2012.04.004. baur, d. (2003), testing for contagion – mean and volatility contagion, journal of multinational financial management, 13, 405–422, doi: http://dx.doi.org/10.1016/s1042-444x(03)00018-5. beine, m., laurent, s., lecourt, ch. (2002), accounting for conditional leptokurtosis and closing days effects in figarch models of daily exchange rates, applied financial economics, 12, 589–600, doi: http://dx.doi.org/10.1080/09603100010014041. bollerslev, t., mikkelsen, h.o. (1996), modeling and pricing long memory in stock market volatility, journal of econometrics, 73, 157–160, doi: http://dx.doi.org/10.1016/0304-4076(95)01736-4. burzała, m. (2013), determination of the time of contagion in capital markets based on the switching model, dynamic econometric models, 13, 69–85, doi: http://dx.doi.org/10.12775/dem.2013.004. campbell, j.y., lettau, m., malkiel, b., xu, y. (2001), have individual stocks become more volatile? an empirical exploration of idiosyncratic risk, journal of finance, 56, 1– 43, doi: http://dx.doi.org/10.1111/0022-1082.00318. cavaglia, s., brightman, c., aked, m. (2000), the increasing importance of industry factors, financial analysts journal, 56, 41–54, doi: http://dx.doi.org/10.2469/faj.v56.n5.2389. chen, y., lai, k.k. (2013), examination on the relationship between vhsi, hsi and future realized volatility with kalman filter, eurasian business review, 3, 200–216, doi: http://dx.doi.org/10.14208/ebr.2013.03.02.005. http://dx.doi.org/10.1016/j.qref.2012.04.004 http://dx.doi.org/10.1016/s1042-444x(03)00018-5 http://dx.doi.org/10.1080/09603100010014041 http://dx.doi.org/10.1016/0304-4076(95)01736-4 http://dx.doi.org/10.12775/dem.2013.004 http://dx.doi.org/10.2469/faj.v56.n5.2389 włodarczyk, otola dynamic econometric models 16 (2016) 87–116 114 chuliá, h., torró, h. (2011), firm size and volatility analysis in the spanish stock market, the european journal of finance, 17, 695–715, doi: http://dx.doi.org/10.1080/1351847x.2011.554286. claessens, s., forbes, k. (2004), international financial contagion: the theory, evidence and policy implication, https://pdfs.semanticscholar.org/0fda/b705fde6484961e3679cecd41e449721a132.pdf (2.12.2014). corsetti, g., pericoli, m., sbracia, m. (2005), some contagion, some interdependence, more pitfalls in tests of financial contagion, journal of international money and finance, 24, 1177–1199, doi: http://dx.doi.org/10.1016/j.jimonfin.2005.08.012. dungey, m., fry, r., gonzalez-hermosillo, b., martin, v.l., tang, ch. (2011), contagion and the transmission of financial crisis, in kolb r.w. (ed.), financial contagion: the viral threat to the wealth of nations, john wiley & sons, inc., hoboken, new jersey. feurer, r., chaharbaghi, k. (1994), defining competitiveness: a holistic approach, management decision, 32, 49–58, doi: http://dx.doi.org/10.1108/00251749410054819. fiszeder, p. (2009), modele klasy garch w empirycznych badaniach finansowych (the class of garch models in empirical finance research), wydawnictwo umk, toruń. fiszeder, p., perczak, g. (2013), a new look at variance estimation based on low, high and closing prices taking into account the drift, statistica neerlandica, 67, 456–481, doi: http://dx.doi.org/10.1111/stan.12017. giot, p. (2005), relationships between implied volatility indexes and stock index returns, the journal of portfolio management, 31, 92–100, doi: http://dx.doi.org/10.3905/jpm.2005.500363. grabowska, m. (2013), wartość rynkowa oznaką pozycji i przewagi konkurencyjnej przedsiębiorstw, (the market value as an indication of the position and competitive advantage of enterprises), zeszyty naukowe uniwersytetu szczecińskiego. finanse, rynki finansowe, ubezpieczenia (scientific journal of the university of szczecin, finance, financial markets, insurance), 786 (64/1), 147–155. griffin, j.m., karolyi, g.a. (1998), another look at the role of the industry structure of markets for international diversification strategies, journal of financial economics, 50, 351–373, doi: http://dx.doi.org/10.1016/s0304-405x(98)00041-5. hosking, j.r.m. (1981), fractional differencing, biometrika, 68, 165–176, doi: https://doi.org/10.1093/biomet/68.1.165. kang, s. h., yoon, s. m. (2012), dual long memory properties with skewed and fat-tail distribution, international journal of business and information, 7, 225–249. karolyi, a. (2001), why stock return volatility really matters, strategic investor relations, institutional investor journals series, http://bryongaskin.net/education/mba%20track/current/mba611/assignments/proje ct/whyvolatilitymatters.pdf (15.10.2014). konopczak, m., sieradzki, r., wiernicki, m. (2010), kryzys na światowych rynkach finansowych–wpływ na rynek finansowy w polsce oraz implikacje dla sektora realnego (global financial markets crisis – impact on the polish fnancial market and implications for the real sector of the economy), bank i kredyt (bank and credit), 41, 45–70. krysicki, w., bartos, j., dyczka, w., królikowska, k., wasilewski, m. (2012), rachunek prawdopodobieństwa i statystyka matematyczna w zadaniach, (probability and mathematical statistics in exercises), part 2, pwn, warsaw. http://dx.doi.org/10.1080/1351847x.2011.554286 http://dx.doi.org/10.1016/j.jimonfin.2005.08.012 http://dx.doi.org/10.1108/00251749410054819 http://dx.doi.org/10.1016/s0304-405x(98)00041-5 https://doi.org/10.1093/biomet/68.1.165 analysis of the relationship between market volatility and firms volatility… dynamic econometric models 16 (2016) 87–116 115 laurent, s. (2013), estimating and forecasting arch models using g@rch 7, timberlake consultants ltd, london. le, c., david, d. (2014), asset price volatility and financial contagion analysis using the ms-var framework, eurasian economic review, 4,133–162, doi: http://dx.doi.org/10.1007/s40822-014-0009-y. molnár, p.(2012), properties of range-based volatility estimators, international review of financial analysis, 23, 20–29, doi: http://dx.doi.org/10.1016/j.irfa.2011.06.012. otola, i. (2013), procesy zarządzania przedsiębiorstwami a konkurencyjność w warunkach zarażonego rynku (the management processes of enterprises and competiveness under condition of market contagion), wydawnictwo politechniki częstochowskiej, częstochowa. peng, y., ng, w. (2012), analysing financial contagion and asymmetric market dependence with volatility indices via copulas, annals of finance, 8, 49–74, doi: http://dx.doi.org/10.1007/s10436-011-0181-y. phylatkis, k., xia, l. (2009), equity market comovement and contagion: a sectoral perspective, financial management, 38, 381–409, doi: http://dx.doi.org/10.1111/j.1755-053x.2009.01040.x. sadigue, s., silvapulle, p. (2011), long-term memory in stock market returns: international evidence, international journal of finance and economics, 6, 59–67, doi: http://dx.doi.org/10.1002/ijfe.143. sequeira, j.m., lan, d. (2003), does world-level volatility matter for the average firm in a global equity market?, journal of multinational financial management, 13, 341– –357, doi: http://dx.doi.org/10.1016/s1042-444x(03)00015-x. sharma, s.s., narayan, p.k., zheng, x. (2011), an analysis of firm and market volatility, an financial econometrics series, swp 2011/02, deakin university, australia, https://core.ac.uk/download/pdf/6266325.pdf (12.06.2012). sharma, s.s., narayan, p.k., zheng, x. (2014), an analysis of firm and market volatility, economic systems, 38, 205–220, doi: http://dx.doi.org/10.1016/j.ecosys.2013.12.003. shin, h.h., stulz, r.m. (2000), firm value, risk and growth opportunities, nber working paper 7808, http://www.nber.org/papers/w7808.pdf (2.03.2015). simon, d.p. (2003), the nasdaq volatility index during and after the bubble, the journal of derivatives,11, 9–24, doi: http://dx.doi.org/10.3905/jod.2003.319213. whaley, r.e. (2009), understanding the vix, journal of portfolio management, 35, 98–105, doi: http://dx.doi.org/10.3905/jpm.2009.35.3.098. włodarczyk, a. (2010), testowanie efektu długiej pamięci w zmienności cen metali szlachetnych (testing for long memory in volatility of precious metal prices), roczniki naukowe. seria b, nauki ekonomiczne i informatyka (scientific annals. series b, economic science and informatics), 1–2, 151–170. analiza relacji między zmiennością rynku a zmiennością firm na polskim rynku kapitałowym z a r y s t r e ś c i. w artykule analizowano kształtowanie się zależności między zmiennością rynku a zmiennością cen akcji w spółkach z wybranych sektorów giełdy papierów wartościowych w warszawie przed i po kryzysie subprime. badania empiryczne dotyczą wybranych przedsiębiorstw z sektorów budownictwo i it w latach 2004–2011. miary zmienności zostały obliczone na podstawie najniższego i najwyższego kursu dziennego spółek należących http://dx.doi.org/10.1016/j.irfa.2011.06.012 http://dx.doi.org/10.1016/s1042-444x(03)00015-x http://dx.doi.org/10.1016/j.ecosys.2013.12.003 http://dx.doi.org/10.3905/jpm.2009.35.3.098 włodarczyk, otola dynamic econometric models 16 (2016) 87–116 116 do indeksu wig. dla każdej firmy został oszacowany, zarówno dla okresu stabilnego, jak i niestabilnego, model arfimax-figarch z dodatkowymi zmiennymi egzogenicznymi, odzwierciedlającymi zmienność rynku. przeprowadzone badania empiryczne nie wykazały, że negatywne szoki płynące z rynku amerykańskiego przez kanał zachowań inwestorów przyczyniły się do wzrostu frakcji firm z sektorów budownictwa i it notowanych na gpw, których zmienność jest kształtowana przez zmienność rynku. s ł o w a k l u c z o w e: arfimax-figarch, kryzys subprime, zmienność firm, zmienność rynku, giełda papierów wartościowych w warszawie. © 2017 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2017.003 vol. 17 (2017) 41−57 submitted october 2, 2017 issn (online) 2450-7067 accepted december 17, 2017 issn (print) 1234-3862 paweł kaczmarczyk * microeconometric analysis of telecommunication services market with the use of sarima models a b s t r a c t. the paper presents the results of testing the effectiveness of the multi sectional model in the short-term forecasting of hourly demand for telephone services. the model was based on the integration of the linear regression model with dichotomous independent variables and the sarima model. the regression was used as a filter of modelled variability of the demand. the sarima was applied to model residual variability. the research shows that the proposed integration provides a greater possibility of approximation and prediction in comparison to the non-supported linear regression model. the results of the study provide support for operational planning of telecommunications operator. k e y w o r d s: decision support system; dichotomous regression; sarima model, forecasting. j e l classification: c53; l86; l96. introduction the level of competition in the telecommunication market is getting higher. the number of operators is increasing and the division of telecommunication markets is increasingly greater. it originated from the execution of measures, which were assumed in the lisbon strategy (lisbon european council, 2000) and its current continuation and extension – europe 2020 strategy (begg, 2010; european commission 2010). according to these strategies the telecommunication market should be primarily liberalised (the * correspondence to: paweł kaczmarczyk, the state university of applied sciences in płock, faculty of economic sciences and information technology, 28 gałczyńskiego street, 09-400 płock, poland, e-mail: p.kaczmarczyk@pwszplock.pl. https://www.diki.pl/slownik-angielskiego?q=primarily paweł kaczmarczyk dynamic econometric models 17 (2017) 41–57 42 abolition of restrictions, monopolies and discriminations) and harmonised (common regulations to create fair activity conditions for all telecommunications operators). the premises of the conceptions involve the initiative of the information society building. in connection with increasingly higher level of competition in the telecommunications market, the problem of effective modelling and forecasting of demand for electronic communication services has gained significantly greater importance. sales forecasts of electronic connection services play a particularly important role in the management of a telecommunications operator. these forecasts support planning policy of the operator, because they are the basis for operational planning. within the framework of operational planning of the telecommunications enterprise, decisions relating to price calculation and network management are made that are connected with achievement of operational (short and medium-term) objectives. in terms of literature, this level of planning is defined as the key decision-making field of managers, because the operational planning can strengthen the effectiveness of growth of the enterprise value. due to the fact that operational planning also involves the means to achieve operational objectives, these means may be considered as analytical tools. therefore the use of effective analytical techniques that improve operational management is a source of increase of enterprise value. in order to rationalise the operational planning and finally to strengthen the market position, managerial staff of telecoms enterprises are interested in effective prediction systems (ps) application (dittman, 2004) that belong to one of the decision support systems (dss) subclasses. 1. the purpose and thesis of the research the literature study leads to statements that ps can function alone, or as a part of a broader (multifunction) dss. the effect of the ps work is the prospective information about the external (micro and macro) environment of an enterprise as well as internal characteristics of an enterprise. ps is built the following components: prognostic database, statistical data preprocessing methods, statistical data analysis methods, forecasting methods, computer programs and forecast monitoring system. within the framework of the literature on electronic communications, the contents, which refer to modelling and forecasting of demand for telecommunications services, are not popular. furthermore, it can be noticed that there is a lack of such contents as effectiveness descriptions of the telecommicroeconometric analysis of telecommunication services market… dynamic econometric models 17 (2017) 41–57 43 munications market data mining techniques and methods (including econometric techniques) applied by operators. it is caused by the existence of the previously mentioned significant competition between operators. in practice the transfer of used knowledge does not exist in the field of telecommunications data mining, because operators protect their experience. they treat the knowledge acquired, by using data mining methods, as a part of their competitive advantage (muraszkiewicz, 2000) the study of the predictive potential, which is implemented in available commercial decision support systems (e.g. prophiks, kobat-sair, kobat-sad), encourages one to conduct research into other approaches to forecasting of the demand for telephone services, and to assess their implementation techniques. commercial software to conduct data mining calculation is not always effective in solving tasks, which are important for telecommunications operators. it is ineffective particularly in solving problems where there are more complex data structures and temporal dependences, i.e. sequences of events (muraszkiewicz, 2000). the results of the conducted research, which has been described in this article, relate to one of the ps components, i.e. the internal characteristics forecasting techniques, namely sales expectations techniques. the purpose of this study is to verify the effectiveness of the constructed model in shortterm forecasting of the demand for telephone services. the linear regression model, including dichotomous (binary) explanatory variables, was integrated with the sarima model. this integration was based on the assumption that the regression model was supported by the sarima model. thus the regression model is used as a filter of modelled variability of demand for telephone operator services (response variable). in turns the sarima model is used to reflect the remaining volatility of the demand for electronic communications services, i.e. received after the filtration of the origin variability of the modelled demand. the author formulated the thesis, with regard to approximation and prediction, supported linear regression model enables better results in comparison to non-supported linear regression model. effectiveness comparison of the above mentioned two techniques (integrated and non-integrated model) was verified by means such obtained values as: fit coefficients, autocorrelation coefficients, partial autocorrelation coefficients, and the average errors of expired forecasts ex-post. the calculation study was carried out on the basis of data provided by one of the telecommunications network operators. the range of empirical material consisted of hourly counted seconds of outgoing calls within the framework of: given subscriber group, particular day (e.g. working or nonhttps://www.diki.pl/slownik-angielskiego?q=sales+expectations paweł kaczmarczyk dynamic econometric models 17 (2017) 41–57 44 working), and specific category of connection (7 categories of connections were taken into account). 2. the theoretical conception of the prepared model the configuration of the forecasting model of demand for electronic connection services depends on the forecasting horizon. if the purpose of constructing the model is long-term prediction, apart from obvious quantitative changes, qualitative changes should also be considered. quantitative changes are based on changes in the value of the response variable according to the detected regularity, e.g. the regression function. on the other hand, qualitative changes are transformations of the essential features of the phenomenon, such as the transformation of the existing regularity, which is expressed by the change of parameters or function type of the model (nadolny, 2011). if the purpose of the modelling is short-term and medium-term prediction, the qualitative changes mentioned above do not occur or occur in trace dimension. therefore, it is not necessary to include them in the prognostic process. when a model is created for short and medium forecast horizon, the following factors should be considered: the type of day (typical working day, saturday, sunday, high days, and holidays) hour of the day, category of connection, the type of subscribers, promotions (kaczmarczyk, 2016). the author applied the approach consisting of several segments. the approach is based on the fusion of the results obtained with the use of two different models, i.e. the linear multiple regression model and the sarima (p,d,q)(p,d,q)s model. the first one is used to isolate the linear relationship between the dependent variable and independent variables, and the second one is used to model the residual values of the first model. this is shown in figure 1. in the first segment, the linear (multiple) regression model is estimated. the regression model enables one to obtain typical demand values for telecommunications services that are generated by the specified subscriber group on the particular hours of the given type of day, within the particular category of connection. in the second segment, the residual values are calculated (i.e. cleaning time series of the response variable). the first and the second segments can only execute their tasks in the proposed sequence. the third segment serves to forecast the demand by using the regression model, and the fourth segment serves to forecast the residual values of the regression model. in the fourth segment the sarima model is used to model and forecast a lower variability (after elimination of the multiple relationships microeconometric analysis of telecommunication services market… dynamic econometric models 17 (2017) 41–57 45 included in regression model). the third and fourth segments can work collaterally. the results received by using the forecasting tools were integrated in the fifth segment, i.e. the forecast values obtained with the use of the regression model are corrected by the prognostic residual values. the econometric analysis of high frequency data was researched by kufel (2010). methods of elimination of deterministic components are a very important issue because they have a great impact on the accuracy of forecasts (box et al., 1994; makridakis and wheelwright 1989; makridakis et al., 1998). figure 1. the integration of regression and sarima model 3. the research results in the conducted empirical analyses, the demand for telephone services (response variable) was considered as the hourly call time measurements (sec.) of outgoing connections of the telecommunications operator network. as the classification factors of the demand were assumed: hour within 24 hours, type of 24 hours, connection category, and the kind of subscribers group. within the framework of every mentioned classification factor were defined particular levels. therefore the following 35 variables were defined: bus – business subscribers, ind – individual subscribers, mn – mobile networks, lc – local calls to the same network, lco – local calls to other networks, tc – trunk calls, ic – international calls, oc – other connections, 2. residuals calculation zt = yt – ŷt t = 1, ..., n 1. regression model estimation titi m i t xaay ξ+σ+= 1= 0 t = 1, ..., n 3. prediction of the dependent variable value )( * hnii m i hn xaay + 1= 0+ σ+= 5. results aggregation and forecast accuracy assessment 4. sarima model estimation and prediction of the residual value ( * htz + ) sqdpqdparima ),,)(,,( https://www.diki.pl/slownik-angielskiego?q=collaterally https://www.diki.pl/slownik-angielskiego?q=collaterally https://www.diki.pl/slownik-angielskiego?q=measurement paweł kaczmarczyk dynamic econometric models 17 (2017) 41–57 46 w – working 24 hours, sat – saturday, sun – sunday, and 24 variables to describe particular hours during the day: from 12am – 00:00:00–01:00:00 to 11pm – 23:00:00–00:00:00. hourly averages of the demand for telephone services in 24-hours cycles, within the selected working 24 hours (wednesdays) and generally nonworking 24 hours (sundays) during a year and generated by business or individual subscribers, are presented in figure 2. figure 2. the average time (sec.) of outgoing calls generated by business or individual customers in hours of working and non-working 24 hours the courses of the demand for telecommunications services are different due to the category of connection, subscriber group and type of 24 hours. the analytical sections have different location of the demand extreme. the structure (categorised histogram) of demand values (hourly counted seconds of outgoing calls) generated by business customers, within working and non-working 24 hours, is shown in figure 3. figure 3 was drawn on the basis of the same statistical material that was used to present daily course of demand (generated by business customers) in figure 2. visual analysis leads to the remark that the variables distributions within the framework of business subscribers are different, and the variable lco has the highest observaworking 24 hours (business customers) sundays (business customers) working 24 hours (individual customers) sundays (individual customers) microeconometric analysis of telecommunication services market… dynamic econometric models 17 (2017) 41–57 47 tions in the both types of days. the lowest values were observed in the case of the variable ic and oc. a relatively high value may be observed also within lc variable and may be noticed within working 24 hours as well as within sundays. working 24 hours sundays figure 3. the structure of observations (hourly counted sec.) of outgoing calls generated by business customers during working and non-working 24 hours in the next figure (figure 4), categorised histogram of hourly combined seconds of outgoing calls generated by individual subscribers also during paweł kaczmarczyk dynamic econometric models 17 (2017) 41–57 48 working and non-working 24 hours is presented. it can be noticed in both types of day that the highest observations there are within the framework of local connection to other network and these observations are even higher than in the case business subscribers within this category and during the same type of day. the demand value within remaining categories of connection during both type of 24 hours are significantly lower. working 24 hours sundays figure 4. the structure of observations (hourly counted sec.) of outgoing calls generated by individual customers during working and non-working 24 hours microeconometric analysis of telecommunication services market… dynamic econometric models 17 (2017) 41–57 49 the intervals of hourly counted seconds of outgoing calls are left-open and right-closed. therefore, the intervals (–50000, 0] or (–5000, 0] include the observations that equal to 0. this assumption enables isolation of the observations that equal to 0 and consequently it allows for better overview of the structure (categorised histogram) of the demand for telecommunication services generated by particular subscribers group, within defined type of day. in this analysis, the volumes of demand that equal to 0 constitute significant group of observations and these observations can be isolated in separated interval. the regression model has included 35 dichotomous independent variables, which were specified at the beginning of this study section. independent variables take value 0 or 1. the dependent variable was set by hourly measurements of seconds of outgoings calls of the operator network. multiple regression parameters were estimated from data for the period from 1st january to 20th february of the selected year (14688 cases). the data, which was used to construct the tested models, was very complex. the constructed regression model was based on data relating to six categories of connections and two subscriber groups (12 separated analytical sections of demand). due to the fact that the period from 1th january to 20th february consist of 1224 hours, the modelling of full variability of demand for telecommunication connections services needed to involve 14688 cases. therefore, the number of cases reflects joint analysis of demand in terms of various analytical sections at the same time. due to the fact that ridge regression was estimated, it was necessary to optimise parameter λ. the optimal value of the parameter λ was set at 0.0570. at this value the ridge regression model was the best fitted to the modelled data (r square = 0.4748; std. error of the estimate = 59524.1568; calculated f statistic = 378.4818 on 35 and 14,652 df, p < 0.05). in the obtained regression model only one of the structural parameters, i.e. parameter standing by the variable 09pm, was statistical insignificant – see table a1 in appendix. then the object of the research was autocorrelation function and the partial autocorrelation function of the regression model residuals. the residuals of the regression model are characterised by transparent repetitions in 24-hour cycles. the result of the autocorrelation analysis is presented in figure 5. the regression model was constructed for almost full variability (many levels of applied classification factors) of demand for telecommunications connections services. the estimated model possibilities are too low to model such complex variability. this was the reason that autocorrelation was in the error term. paweł kaczmarczyk dynamic econometric models 17 (2017) 41–57 50 forecasting procedure (by the use of regression and constructed integrated model) was conducted with the use of the period from 21st to 28th of february (2304 forecasts). the predictive activity was conducted in all the assumed analytical section. therefore the demand course was forecasted in section of: 24 hours, subscribers groups and connections categories. the autocorrelation function the partial autocorrelation function figure 5. the autocorrelation function and the partial autocorrelation function of the regression model residuals the demand forecasts accuracy was verified by using the mean absolute error (mae) and the root mean square error (rmse) according to the following formulas: ∑ 1 *1 t nt tt yy nt mae     (1) ∑ 1 2* )( 1 t nt tt yy nt rmse     (2) where t – forecast horizon, and n – a number of observations, which were used to estimated model. the research results, of the forecasting effectiveness of the regression model by the use of the above errors (in sec.), were 43148.92 and 57409.18 for mae and rmse respectively. the analysis of the regression residuals shows that they are characterised by seasonality in daily courses and in particular analytical section. in connection with the seasonality, the thesis can be formulated that the use of the sarima model to reflect residuals variability allows for improving results in terms of approximation and forecasting of the analysed demand. moreover, the analysis of the calculated cook’s distances and the obtained standardised residuals indicate that there are unusual observations, i.e. influence microeconometric analysis of telecommunication services market… dynamic econometric models 17 (2017) 41–57 51 observations or outliers (see figure a1 in appendix). due to the risk of obliteration of real patterns occurring in the studied phenomenon, the unusual observation were not eliminated and not replaced by their estimates (dittmann et al., 2011). general overview of residuals course for business and then individual subscribers is presented in figure 6. figure 6. the regression residuals course the data used to estimate the regression model included patterns from each analytical section of the demand, so the regression residuals course is different in particular intervals of observations. the highest residuals values can be observed in the interval of the local connection to other network generated by individual subscribers (the interval of observations 8089–8832 in january and the interval of observations 12289–12768 in february). the high values of residuals were also noticed in the interval of the local connection to other networks generated by business subscriber, but these observations were obviously lower than in the case of the individual subscribers. these results came from the highest values of the demand for telephone services within the framework of this category of connection in both groups of customers. the analysis of the regression residuals confirms the remarks on categorised histogram and interaction plot (figure 1–3). several sarima models were tested. the maximum likelihood estimation (mle) was applied. two approaches i.e. mle according to melard (1984), also known as exact likelihood, and mle according to mcleod and sales (1983) were used. the maximum likelihood estimation according to salad and macleod (1984) was also used. the criterion for assessing the model fit were squared errors of the sarima model. the initial sum of squared errors (iss), final sum of squared errors (fss) and mean of squared errors (ms) were taken into account. the goodness of fit was also assessed due to the percentage relation of these errors (rss = fss/iss). in the all business subscribers individual subscribers paweł kaczmarczyk dynamic econometric models 17 (2017) 41–57 52 experiments the estimation process was stopped when the convergence criterion (required accuracy) was reached. thus, it was assumed that the changes in the sarima parameters over consecutive iterations should be less than the value of the convergence criterion. the optimal model was sarima (1,0,3)(1,0,4)24. the goodness of the best sarima model fit is presented in table 1. table 1. summary of the sarima (1,0,3)(1,0,4)24 coefficient value iss 49.7153 fss 4.5122 rss 9.0761 ms 0.0003 note: the value of iss, fss, and ms was presented in trillions and rss in per cent. convergence criterion was set at 0.0001. the estimation process reached convergence criterion after cumulatively 46 iterations. the parameters was obtained by the use of mle according to mcleod and sales (9 iterations) and then mle according of melard (37 iterations). all of the model parameters were statistically significant. the values and standard errors of the parameters were juxtaposed in table 2. table 2. parameters and their errors of sarima (1,0,3)(1,0,4)24 coeff. non-seasonal parameters seasonal parameters p(1) q(1) q(2) q(3) p(1) q(1) q(2) q(3) q(4) parameter 0.75 –0.19 –0.19 –0.10 0.97 0.64 0.25 –0.03 –0.11 ase 0.01 0.01 0.01 0.01 0.00 0.01 0.01 0.01 0.01 a t(14684) 82.99 –15.06 –17.11 –11.18 374.08 71.43 24.40 –2.67 –9.88 llc 0.73 –0.21 –0.21 –0.12 0.96 0.62 0.23 –0.05 –0.13 ulc 0.77 –0.16 –0.17 –0.08 0.97 0.66 0.27 –0.01 –0.08 note: non-seasonal parameters and also seasonal parameters are statistically significant at significance level p = 0,05. explanation of the abbreviations: ase – asymptotic standard error, a t(14684) – asymptotic t(14684), llc – lower limit of confidence interval (95%), ulc – upper limit of confidence interval (95%). the course of sarima (1,0,3)(1,0,4)24 residuals (for both subscribers groups) shows that their values are much lower than in the case of the regression model and the variability reflecting of demand for telephone services was improved (figure 7). then the estimated value of the regression residuals (that were obtained by using the sarima model) were applied to correction estimated value of the demand (determined values of regression). the goodness of the fit to data for the final model, which was verified by means of r square, was 0.9480. therefore the level of fit was clearly higher in comparison to the regression, which was not supported by the sarmia model. microeconometric analysis of telecommunication services market… dynamic econometric models 17 (2017) 41–57 53 figure 7. the residuals of sarima (1,0,3)(1,0,4)24 the analysis of obtained values of q box and ljung coefficients and also partial correlation coefficients (figure 8) indicate that they are much lower than the values of these coefficients, which were calculated in the analysis of regression model residuals (figure 5). however they can be considered as not fully satisfactory. it is noticeable that there are still repetitions in 24 cycles (but smaller than previously). the results provide rationales to further research to reduce correlation in error term. the reduction of the correlation in terms of error could be probably achieved by reduction of such high numbers of the analytical section included in the regression model. figure 8. the autocorrelation function and the partial autocorrelation function of the residuals of the supported regression then the forecasting effectiveness of the supported regression model was verified and compared with the previous regression model. average forecasting errors for the forecast period (the same as in the case of nonbusiness subscribers individual subscribers the autocorrelation function the partial autocorrelation function paweł kaczmarczyk dynamic econometric models 17 (2017) 41–57 54 supported regression), accounted according to (1) and (2) formulas, amount to 8488.65 and 15758.65 respectively. these results can be objectively considered as much better than the result obtained by using the previous regression model, i.e. the forecasting effectiveness (in the mean of forecast accuracy) was significantly higher. conclusions in the light of the obtained research, the thesis can be confirmed that the supported regression model enables higher efficiency of approximation and prediction of demand for telecommunications services in comparison with the non-supported regression model. the results encourage further research in the explored field. the fit and forecasts accuracy could probably be higher by the volatility reflecting within the framework of lower number of the analytical sections, for example by the volatility modelling only within the business group, or even only within the business group and working 24-hours. the separation of particular types of day is especially important, because cycles of repetitions of daily demand during the same day in different categories of connection are similar in terms of the phases of the cycles. it is also interesting to try to improve the fit of the constructed model by creating demand patterns in particular analytical sections, i.e. by using averages for the particular hours within the sections. this could contribute to a better fit of the regression model and consequently better fit of the overall model. in this approach one could also use separate information for separate models. further research could concern another realisation of the fourth segment. thus future work may relate to the use of other modelling and forecasting methods for seasonality, i.e. the residuals of the regression model. references begg, i. (2010), europe 2020 and employment. intereconomics, 45, 146–151. box, g. e. p., jenkins, g. m., reinsel, g. c. (1994), time series analysis. forecasting and control, prentice hall, englewood clifs. dittmann, p. (2004), prognozowanie w przedsiębiorstwie. metody i ich zastosowanie (forecasting in enterprise. methods and their application), oficyna ekonomiczna, kraków. dittmann, p., szabela-pasierbińska e., dittmann i., szpulak a. (2011), prognozowanie w zarządzaniu sprzedażą i finansami przedsiębiorstwa (forecasting in sales management and finance of enterprise), wolters kluwer polska, warszawa. europe 2020. a strategy for smart, sustainable and inclusive growth. communication from the commission, european commission, brussels 3.3.2010, https://www.diki.pl/slownik-angielskiego?q=respectively microeconometric analysis of telecommunication services market… dynamic econometric models 17 (2017) 41–57 55 http://www.buildup.eu/sites/default/files/content/com2010_2020en01.pdf (22.07.2017). kaczmarczyk, p. (2016), integrated model of demand for telephone services in terms of microeconometrics, folia oeconomica stetinensia, (16)2, 72–83, doi: https://doi.org/10.1515/foli-2016-0026box. kufel, t. (2010), ekonometryczna analiza cykliczności procesów gospodarczych o wysokiej częstotliwości obserwowania (econometric analysis of the cyclicality of economic processes at high frequency of observations), wydawnictwo naukowe umk, toruń. makridakis, s., wheelwright, s. c. (1989), forecasting methods for management, j. wiley, new york. makridakis, s., wheelwright, s. c., hyndman, r. j. (1998), forecasting methods and applications, j. wiley, new york. mcleod, a. i., sales, p. r. h. (1983), an algorithm for approximate likelihood calculation of arma and seasonal arma models, applied statistics, 32, 211–223. melard, g. (1984), a fast algorithm for the exact likelihood of autoregressive-moving average models. applied statistics, 33, 104–114, doi: http://dx.doi.org/10.2307/2347672. muraszkiewicz, m. (2000), eksploracja danych dla telekomunikacji (data mining for telecommunication), http://www.ploug.org.pl/showhtml.php?file=konf_00/materialy_00, (2.07.2017). nadolny, m. (2011), podstawowe modele dyfuzji dóbr telekomunikacyjnych (basic models of telecommunications goods diffusion) in łyka j. (ed.), wybrane modele matematyczne w ekonomii. globalizacja i rozwój (selected mathematical models in economics. globalisation and growth). wydawnictwo uniwersytetu ekonomicznego, wrocław. presidency conclusions. lisbon european council of 23rd and 24th march 2000, http://www.europarl.europa.eu/summits/lis1_en.htm (24.06.2017). wiśniewski, j.w. (2009), mikroekonometria (microeconometrics), wydawnictwo naukowe umk, toruń. mikroekonometryczna analiza rynku telekomunikacyjnego z wykorzystaniem modeli sarima z a r y s t r e ś c i. w artykule przedstawiono wyniki testów efektywności wieloprzekrojowego modelu w krótkookresowym prognozowaniu cogodzinnego zapotrzebowania na usługi telefoniczne. model został oparty na integracji zero-jedynkowego modelu regresji liniowej i modelu sarima. model regresji spełnia rolę filtra modelowanej zmienności popytu na usługi telefoniczne. model sarima służy do modelowania pozostałej zmienności. badania wykazały, że proponowana integracja zapewnia wyższe możliwości aproksymacyjne i predykcyjne w porównaniu z niezintegrowanym modelem regresji liniowej. wyniki badań stanowią dla operatora wsparcie procesu planowania operacyjnego. s ł o w a k l u c z o w e: system wspomagania decyzji, regresja zero-jedynkowa, model sarima, prognozowanie. http://yadda.icm.edu.pl/yadda/element/bwmeta1.element.ekon-element-59348fa7-e43f-3ca4-90f1-cee018b3fb04 http://dx.doi.org/10.2307/2347672 paweł kaczmarczyk dynamic econometric models 17 (2017) 41–57 56 appendix table a1. the results of the multiple regression estimation variable standardised parameters and errors non-standardised parameters and errors β std. error of β b std. error of b t(14652) 39380.4034* 5205.2949 7.5655 bus –0.0354 0.01798 –5802.9220* 2949.3400 –1.9675 ind 0.0354 0.01798 5802.9224* 2949.3400 1.9675 mn –0.0975 0.01134 –21455.2156* 2495.9717 –8.5959 lc 0.3342 0.01134 73567.5588* 2495.9717 29.4745 lco 0.1548 0.01134 34071.6527* 2495.9717 13.6507 tc –0.0336 0.01134 –7389.2377* 2495.9717 –2.9605 ic –0.1819 0.01134 –40032.2150* 2495.9717 –16.0387 oc –0.1761 0.01134 –38762.5432* 2495.9717 –15.5300 w –0.0962 0.00758 –39485.9177* 3110.7645 –12.6933 sat –0.1020 0.00758 –41857.1040* 3110.7645 –13.4556 sun –0.1032 0.00758 –42371.4863* 3110.7645 –13.6209 12am –0.1036 0.00758 –42524.3610* 3110.7645 –13.6701 01am –0.1038 0.00758 –42627.5665* 3110.7645 –13.7032 02am –0.1029 0.00758 –42233.6372* 3110.7645 –13.5766 03am –0.0936 0.00758 –38416.3277* 3110.7645 –12.3495 04am –0.0525 0.00758 –21547.7074* 3110.7645 –6.9268 05am 0.0348 0.00758 14268.2139* 3110.7645 4.5867 06am 0.1016 0.00758 41724.4466* 3110.7645 13.4129 07am 0.1204 0.00758 49438.9906* 3110.7645 15.8929 08am 0.1124 0.00758 46126.3489* 3110.7645 14.8280 09am 0.1043 0.00758 42806.7784* 3110.7645 13.7609 10am 0.1027 0.00758 42157.8503* 3110.7645 13.5522 11am 0.0852 0.00758 34987.4733* 3110.7645 11.2472 12pm 0.0476 0.00758 19547.3911* 3110.7645 6.2838 01pm 0.0299 0.00758 12294.3716* 3110.7645 3.9522 02pm 0.0297 0.00758 12210.4834* 3110.7645 3.9252 03pm 0.0486 0.00758 19952.9375* 3110.7645 6.4142 04pm 0.0619 0.00758 25430.5249* 3110.7645 8.1750 05pm 0.0327 0.00758 13441.3084* 3110.7645 4.3209 06pm –0.0023 0.00758 –939.0656* 3110.7645 –0.3019 07pm –0.0608 0.00758 –24963.2014* 3110.7645 –8.0248 08pm –0.0912 0.00758 –37420.7441* 3110.7645 –12.0294 09pm 0.0838 0.01712 14808.3710 3027.1566 4.8918 10pm –0.0416 0.01325 –9923.5595* 3158.3647 –3.1420 11pm –0.0644 0.01443 –13852.4642* 3105.0410 –4.4613 note: * denote significance at 5% level. microeconometric analysis of telecommunication services market… dynamic econometric models 17 (2017) 41–57 57 figure a1. normal probability plot of the regression residuals © 2017 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2017.009 vol. 17 (2017) 147−159 submitted november 28, 2017 issn (online) 2450-7067 accepted december 28, 2017 issn (print) 1234-3862 dorota witkowska , krzysztof kompa * how the change of governing party influences the efficiency of financial market in poland a b s t r a c t. financial market seems to be sensitive to political changes, especially when the change of governing party is connected with essential changes of the economic development concepts. such situation took place in poland in 2015, as a result of the presidential and parliamentary elections. the aim of our research is to investigate the changes occurred on the market, represented by some stable growth open mutual funds, and stock indexes: wig and tbsp. analysis is provided applying single index and capm models, classical investment performance measures, and statistical interference. k e y w o r d s: stable growth open mutual funds, investment efficiency, sharpe model, capm, sharpe, treynor and jensen ratios. j e l classification: g11; c12. introduction financial market seems to be sensitive to political changes, especially when the change of governing party is connected with essential changes of the concepts concerning economic development. such situation took place in poland at the end of 2015, as a result of the presidential and parliamentary elections that were won by the law and justice party (pis) which was in an opposition to the government during two last terms. now pis is the largest * correspondence to: krzysztof kompa, warsaw university of life sciences, department of econometrics and statistics, 166 nowoursynowska street, 02-787 warsaw, poland, e-mail: krzysztof_kompa@sggw.pl; dorota witkowska, university of lodz, faculty of management, department of finance and strategic management, 22/26 matejki street, 90-237 łódź, poland, e-mail: dorota.witkowska@uni.lodz.pl. https://orcid.org/0000-0001-9538-9589 http://orcid.org/0000-0002-2810-6654 dorota witkowska, krzysztof kompa dynamic econometric models 17 (2017) 147–159 148 party in the polish parliament having majority in both chambers of parliament. the governing party introduced program called “good change” consisting in populist movements such as decreasing of the retirement age or the 500+ familly programme, etc. which burden the economy and may affect the financial market. therefore, here a question arises how the change of ruling party and their economic program influence the situation of polish financial market. the answer is not easy especially that both sides i.e. governing and opposite parties presented completely different arguments which had rather political than economic character. pis was emphasizing social benefits of proposed programs while the opposition was highlighting the economic consequences and threats for the budget. some economists even forecasted that financial market in poland may collapse since investors do not trust markets with high political risk which comes from social programs together with controversial economic proposals such as increasing taxes from the banking sector or supermarkets. therefore, the aim of our research is to find out how the change on the political scene affected the performance of equity and bond markets, together with stable growth mutual funds (fio), applying single index and capital assets pricing models together with classical investment performance measures and statistical interference. 1. data and methodology our investigation is carried out using daily logarithmic rates of returns from selected financial instruments:  warsaw stock exchange index – wig, representing equity market,  poland’s official treasury bonds index – tbsp.index, representing bond market,  participation units of stable growth open mutual funds (fio): credit agricole stabilnego wzrostu (denoted as ca), kbc fundusz stabilny (as kbc), nationale-nederlanden stabilnego wzrostu (as nn), pioneer stabilnego wzrostu (as pio) and pzu stabilnego wzrostu mazurek (as pzu). the analysis is provided for the period from 10.10.2013 to 9.12.2016 (the whole period including 826 observations). this time span is divided into five pairs of sub-periods according to the selected events which we take into consideration:  the presidential elections: the first round of election – 10.05.2015, the second round of election – 24.05.2015, how the change of governing party influences the efficiency of financial market… dynamic econometric models 17 (2017) 147–159 149  the parliamentary election – 25.10.2015,  the new government appointment – 16.11.2015,  the entry into force 500+ familly program – 1.04.2016. the subperiods are defined assuming that the last observation comes from 9.12.2017, and sharing date is a day when the distinguished event took place. it was our concern to obtain subsamples with similar number of observations (detailed information about sample sizes are in table 2). investigation of returns and risk generated by the investment portfolios constructed by selected funds is conducted in several steps, beginning from the analysis of the basic parameters and applying statistical interference 1 (assuming the significance level 0.05). denoting by: e(r) – expected returns, d 2 (r) – variance of returns, rwig, rtbsp, rfio – returns from wig, tbsp and fio respectively, ,  – parameters of sharpe model or capm, rbefore, rafter, before, after – returns from the portfolio and beta coefficients before and after the considered event, respectively, we verify the null hypotheses concerning: 1. rates of return levels, i.e.: e(rfio) = 0; e(rwig) = 0; e(rtbsp) = 0, 2. parameters of sharpe and capm models, i.e.:  = 0;  = 0, 3. comparisons of parameters values in two considered sub-periods i.e.: e(rbefore) = e(rafter), d 2 (rbefore) = d 2 (rafter), before = after, before = after. we apply the classical tests for verification hypothesis of returns equity:  using the cochran-cox test statistics: (1)  and using the following statistics: (2) where for the k-th period, – average logarithmic rate of return from the selected instrument, – variance of return, – number of observations, b – benchmark: b=0, b=r1 or b=r2. the comparison of returns in both period is provided using statistics (1) and (2). in the letter case benchmark b is defined as an average value of returns obtained in the second considered period (k=1, 2) and the test is provided as two-way test. comparison of variances is provided using the test with fisher statistics: (3) 1 all formulas (1)-(5) are discussed in (witkowska 2016, p. 29-55). dorota witkowska, krzysztof kompa dynamic econometric models 17 (2017) 147–159 150 where, – maximal and minimal variance obtained for the both compared samples. the shape of the probability distribution of logarithmic rates of return is examined on the basis of parametric tests verifying hypothesis that symmetry and kurtosis equal to zero, applying the following statistics: (4) (5) where, at, kt – the third and the fourth central statistical moments of logarithmic rates of returns. the next step of our research is estimation of sharpe and capital assets pricing models on the basis of daily logarithmic rates of return 2 : (6) (7) where for the t-th period, rit – rate of return from participation units of the i-th stable growth open mutual fund; rrt – rate of return from the market index (wig); rrt – rate of return from the risk-free instrument (tbsp); i, i – model parameters, it – random component; t – number of observation (t=1, 2, …, t). parameters of both models are estimated using ols method. analysis of parameter significance in the models is provided using the test statistics 3 : and (8) where, – parameter estimates, – standard estimation errors from the models (6) and (7). comparison of parameters obtained in both comparable periods is made applying the test statistics: and (9a) and (9b) 2 models are widely discussed in literature see for instance: (zamojska 2012, p. 57-60), (witkowska 2016, p. 41-48) 3 these measures are discussed in (witkowska 2016, p. 49-50). how the change of governing party influences the efficiency of financial market… dynamic econometric models 17 (2017) 147–159 151 the performance of mutual funds is provided using classical measures i.e. sharpe, treynor and jensen ratios 4 . the first two measures estimate the obtained risk premium evaluated for the unit of risk and the decision about the efficiency is made by comparing the values of the ratios obtained by the mutual fund and the market index. jensen ratio is the parameter estimate from capm, and investment is efficient if is positive. the comparison of the funds’ efficiency is provided applying measures mentioned above which are evaluated for all considered mutual funds in all analyzed time spans, assuming that wig represents market index, and tbsp is the risk-free instrument. to recognize if the sharpe ratios are equal, we apply jobson-korkie test (jobson, korkie, 1981) with memmel correction (memmel, 2003), using test statistics – see (blitz, van vliet, 2007), (kurach, papla, 2014): (10) where for the k-th period of analysis (k=1,2), wsi – sharpe ratio, ρ12 – correlation coefficient, (i=1, 2). 2. changes of the equity and bond markets in the first step of our analysis we investigate daily rates of return of indexes wig and tbsp. tables 1–3 contain basic characteristics of logarithmic rates of return. bold letters denote rejection of null hypotheses. table 1. basic characteristics of daily logarithmic rates of return from the both benchmarks evaluated for the whole period of analysis basic parameters wig tbsp basic parameters wig tbsp min –5.8250% –0.6366% range 8.8302% 1.2498% max 3.0052% 0.6133% standard deviation 0.9022% 0.1752% arithmetic mean –0.0026% 0.0148% coefficient of variability 343.12 11.80 median 0.0000% 0.0091% interquartile deviation 0.9530% 0.1958% quartile i –0.4495% –0.0774% asymmetry –0.7278 –0.2519 quartile iii 0.5036% 0.1184% kurtosis 4.2876 1.0511 note: bold letters denote rejection of null hypothesis. it is visible (table 1) that expected rate of returns from treasury bonds is significantly positive while average returns from equity market is negative 4 the description of the efficiency ratios, their application and discussion can be found in many publications. good examples could be (borkowski 2014), (perez 2012), (zamojska 2012) and witkowska (2016). dorota witkowska, krzysztof kompa dynamic econometric models 17 (2017) 147–159 152 although the null hypothesis cannot be rejected. tbsp can be treated as riskfree instrument because its variability is very low. time series of daily rates of return from wig and tbsp are asymmetric with positive kurtosis thus they are not normally distributed. however according to the huge number of observation we assume that probability distribution is asymptotical normal. table 2. test statistics verifying the hypothesis about expected returns from both indexes in the considered periods h0: e(rwig) = 0 and h0: e(rtbsp) = 0 number of observations rates of returns from number of observations rates of returns from wig tbsp tbsp tbsp whole period for the presidential elections: 10.10.2013–9.12.2016 whole period for the parliamentary elections: 8.09.2014–9.12.2016 826 –0.0597 1.7347 590 –0.3203 0.9267 period before the first round of presidential elections: 8.10.2013–10.05.2015 period after the first round of presidential elections: 11.05.2015–9.12.2016 409 0.6326 3.0922 417 –0.6079 0.4903 period before the second round of presidential elections: 8.10.2013–22.05.2015 period after the second round of presidential elections: 25.05.2015–9.12.2016 419 0.5652 2.9436 407 –0.5767 0.5054 period before the parliamentary elections: 8.09.2014–23.10.2015 period after the parliamentary elections: 26.10.2015–9.12.2016 295 –0.4297 1.3199 295 –0.0541 –0.0376 period before the new government appointment: 20.10.2014–13.11.2015 period after the new government appointment: 16.11.2015–9.12.2016 280 –0.5547 0.9040 280 0.2423 –0.0931 period before entry into force 500+ familly program: 23.07.2015–31.03.2016 period after entry into force 500+familly program: 1.04.2016–9.12.2016 181 –0.3644 1.0793 181 0.2689 –0.5718 note: bold letters denote rejection of null hypothesis. analyzing returns from both markets in distinguished 12 periods of consideration, one may notice (table 2) that only treasury bonds generated significantly positive rates of return in the periods before both rounds of presidential election and in the whole analyzed period. it is visible that change on the politic scene caused decline in the both markets. however, the performance of the equity market shows slight improvement in the periods after new government appointment and when 500+ family program entered into force. such situation may be a result of investors emotions and expectations which revealed earlier and anticipated both events. the obtained results are also validated by the tests which are used for comparison of returns in considered sub-periods (table 3). higher returns are observed only for tbsp before both rounds of presidential election. equity market was characterized by the significant increase of risk after the government appointment, presidential and parliamentary elections however how the change of governing party influences the efficiency of financial market… dynamic econometric models 17 (2017) 147–159 153 the risk significantly decreased after the 500+ family program went into effect. table 3. test statistics verifying the hypothesis about expected returns and risk of equity and bond markets in considered periods e(rbefore) = e(rafter), d 2 (rbefore) = d 2 (rafter) index test statistics evaluated due to formulas for returns: for risk for returns: for risk (2) (2) (1) (3) (2) (2) (1) (3) presidential elections, the 1st round presidential elections, the 2nd round wig 1.3649 1.1186 0.8652 1.5179 1.2906 1.0112 0.7960 1.5823 tbsp 2.5649 2.5592 1.8116 1.0241 2.4287 2.3494 1.6886 1.0381 parliamentary elections government appointment wig –0.3663 –0.3125 –0.0574 1.3068 –0.8394 –0.7142 –0.1330 1.3810 tbsp 1.3560 1.4143 0.2362 1.0040 0.9925 1.0440 0.1758 1.1066 introduction of 500+ program wig –0.5945 –0.6945 –0.1231 1.3649 tbsp 1.5723 1.8236 0.3247 1.3452 note: positive values of test statistics denote that returns are bigger before the considered event than after. italic letters denote that risk was smaller after the event than before, bold – rejection of null hypothesis. 3. mutual funds market the mutual fund market is represented by five selected stability growth mutual funds. all these funds started their functioning in poland in years 1999– –2003, the “oldest” is fio pzu, and the “youngest” fio credit agricole. analysis is provided for the sub-periods constructed around both rounds of the presidential election because in other sub-periods no essential changes was observed. table 4. values of test statistics (2) verifying the hypothesis about expected returns from mutual funds in considered periods h0: e(rfio) = 0 period ca kbc nn pio pzu whole sample 0.8114 0.4773 0.2734 –0.3305 –0.1463 before the presidential elections 1st round 2.2363 1.4043 1.1328 0.4378 0.4042 after the presidential elections 1st round –0.3498 –0.5631 –0.4387 –1.0512 –0.6572 before the presidential elections 2nd round 2.0612 1.2635 1.0595 0.3959 0.3531 after the presidential elections 2nd round –0.2664 –0.4694 –0.4092 –1.0385 –0.6432 note: bold letters denote rejection of null hypothesis. analyzing expected returns form participation units of mutual funds, it is visible that before both rounds of the presidential election mutual funds generated positive returns while after – the negative ones dorota witkowska, krzysztof kompa dynamic econometric models 17 (2017) 147–159 154 (table 4). however, the null hypothesis is rejected only for fio credit agricole (significantly positive returns are observed before the election). tabele 5. values of statistics (1)–(3) verifying the hypothesis about expected returns and risk in considered periods e(rbefore) = e(rafter), d 2 (rbefore) = d 2 (rafter) period parameter formulas no. ca kbc nn pio pzu presidential elections, the 1st round returns (2) 2.6162 1.8575 1.6271 1.5188 1.0643 (2) 2.2442 2.2198 1.4018 1.4663 1.0493 (1) 1.7034 1.4246 1.0620 1.0549 0.7472 risk (3) 1.3856 1.4007 1.3736 1.0939 1.0489 presidential elections, the 2nd round returns (2) 2.3801 1.6672 1.5480 1.5040 1.0250 (2) 1.9594 1.9103 1.2780 1.3892 0.9671 (1) 1.5127 1.2561 0.9856 1.0205 0.7034 risk (3) 1.4333 1.3515 1.4252 1.1386 1.0912 note: bold letters denote rejection of null hypothesis. table 6. parameter estimates and determination coefficients of sharpe models before after before after the whole period presidential elections, the 1st round presidential elections, the 2nd round beta alfa beta alfa beta alfa beta alfa beta alfa fio credit agricole 0.2445 0.0002 0.2405 0.0000 0.2441 0.0002 0.2409 0.0000 0.2425 0.0001 r2 0.6230 r2 0.6604 r2 0.6176 r2 0.6641 r2 0.6446 fio pzu 0.4285 0.0000 0.3404 0.0000 0.4274 0.0000 0.3410 0.0000 0.3752 0.0000 r2 0.8493 r2 0.7756 r2 0.8444 r2 0.7792 r2 0.8010 fio pioneer 0.3892 0.0000 0.3347 –0.0001 0.3887 0.0000 0.3349 –0.0001 0.3563 0.0000 r2 0.8843 r2 0.9078 r2 0.8821 r2 0.9098 r2 0.8917 fio nationale-nederlanden 0.3323 0.0001 0.3240 0.0000 0.3324 0.0001 0.3240 0.0000 0.3274 0.0001 r2 0.8672 r2 0.9107 r2 0.8651 r2 0.9125 r2 0.8924 fio kbc 0.4163 0.0002 0.2730 0.0000 0.4159 0.0001 0.2732 0.0000 0.3300 0.0001 r2 0.7786 r2 0.7118 r2 0.7748 r2 0.7153 r2 0.7181 note: bold letters denote statistically significant. the better performance of analyzed funds before the election is also proved by the results presented in table 5. as one can see, better performance before the election was visible for fio credit agricole and fio kbc. fio credit agricole and fio nationale-nederlanden were characterized by significantly smaller risk before election however fio kbc generated rehow the change of governing party influences the efficiency of financial market… dynamic econometric models 17 (2017) 147–159 155 turns with smaller volatility after the election. null hypotheses are not rejected for fio pioneer, fio pzu and fio nationale-nederlanden. beta parameters in the single index models and capm are significantly positive however the values of  parameter estimates are rather small (tables 6–7). that is connected with the fact that portfolios of stable growth mutual funds contain a great share of bonds. at the end of january 2017, credit agricole stable growth fund’s portfolio contains only 25% of equity, fio pzu – 30.3%, fio pioneer – 29.3%, fio nationale-nederlanden – 35.3% and fio kbc –38.2%. the structure of the investment funds’ portfolios is also visible when beta parameter estimates are analyzed since the biggest value is observed for fio pzu and kbc while the smallest for fio credit agricole. table 7. parameter estimates and determination coefficients of capm before after before after the whole period presidential elections, the 1st round presidential elections, the 2nd round beta alfa beta alfa beta alfa beta alfa beta alfa fio credit agricole 0.2171 0.0000 0.2072 0.0000 0.2168 0.0000 0.2074 0.0000 0.2112 0.0000 r2 0.8023 r2 0.7528 r2 0.7998 r2 0.7543 r2 0.7727 fio pzu 0.4054 –0.0002 0.3103 –0.0001 0.4042 –0.0002 0.3109 –0.0001 0.3485 –0.0001 r2 0.9109 r2 0.8216 r2 0.9079 r2 0.8232 r2 0.8514 fio pioneer 0.3754 –0.0002 0.3121 –0.0001 0.3752 –0.0002 0.3121 –0.0001 0.3375 –0.0001 r2 0.9083 r2 0.9258 r2 0.9069 r2 0.9262 r2 0.9088 fio nationale-nederlanden 0.3166 –0.0001 0.3012 0.0000 0.3169 –0.0001 0.3010 0.0000 0.3073 –0.0001 r2 0.9173 r2 0.9302 r2 0.9161 r2 0.9305 r2 0.9235 fio kbc 0.3885 0.0000 0.2474 0.0000 0.3879 0.0000 0.2475 0.0000 0.3042 0.0000 r2 0.8464 r2 0.7054 r2 0.8445 r2 0.7068 r2 0.7486 note: bold letters denote statistically significant parameters. value and significance of the alpha parameter is important when capital assets pricing models are taken into consideration since this parameter is an efficiency measure i.e. jensen ratio. in estimated models, none of alphas is significantly positive (table 7). the best portfolio management can be noticed for fio credit agricole and kbc since alphas equaled zero. for the rest of funds jensen ratios were significantly negative at least in the periods before election and for the whole period, it means that the mutual fund mandorota witkowska, krzysztof kompa dynamic econometric models 17 (2017) 147–159 156 agers did not earned enough return given the amount of risk they were taking. in the next step, we compare betas from the models estimated before and after both rounds of the presidential election (table 8). since a positive value of the test statistics means that before the election the parameter was bigger than after the election, it is visible that the risk, measured by beta, significantly lowered after the election for all mutual funds but fio credit agricole. taking into account the quality of management, we notice that it was significantly improved after the election by fio pzu and fio nationale-nederlanden, however one must realize that jensen alphas remained negative. table 8. value of statistics comparing betas estimated in both periods h0: before = after funds period sharpe beta capm beta capm alpha t1 t2 t1 t2 t1 t2 credit agricole i round 0.4255 0.4706 1.8703 1.7048 0.8368 0.6278 ii round 0.3411 0.3750 1.7730 1.5932 0.4965 0.3601 pzu i round 9.8989 9.7889 15.1606 13.3466 –2.4622 –1.7842 ii round 9.6437 9.5620 14.8252 12.9878 –2.2242 –1.5727 pioneer i round 7.7857 10.2830 10.7358 14.5523 –1.5012 –1.6749 ii round 7.7518 10.2474 10.7422 14.3758 –1.1776 –1.2720 nn i round 1.2969 1.6600 3.2745 3.7870 –2.3184 –2.2070 ii round 1.3137 1.6850 3.3972 3.8855 –2.1582 –1.9923 kbc i round 13.0273 16.6628 17.2482 17.9215 0.7775 0.6650 ii round 13.0234 16.6094 17.2693 17.6845 0.4928 0.4073 note: bold letters denote rejection of null hypothesis. the last stage of our investigation consists in evaluation the classical efficiency measures, which are given in tables 9 and 10. treynor ratio uses beta as a measure of risk but some authors apply beta estimated from the single index model – see (domański 2011, p. 64), (perez 2011, p.155), and other – take beta from capm – see (borowski 2014, p. 20), (białek 2009, p. 34). therefore, we use both approaches in our analysis. the main conclusion from our research is that the majority of portfolios were ineffective, except fio credit agricole. inefficiency appeared more often after than before the presidential election. after the election sharpe and treynor ratios show negative risk premium, and they usually decreased in comparison to the first analyzed time span, although jensen alphas for fio pzu and nationale-nederlanden increased in the samples containing observations after the presidential election. how the change of governing party influences the efficiency of financial market… dynamic econometric models 17 (2017) 147–159 157 table 9. values of the efficiency measures evaluated for mutual funds before and after both rounds of presidential election ratio: sharpe treynor (β sharpe) treynor (β capm) jensen alpha fund or index periods before rounds of presidential election 1st round 2nd round 1st round 2nd round 1st round 2nd round 1st round 2nd round ca 0.00392 –0.00015 0.00004 0.00000 0.00004 0.00000 0.00001 0.00001 pzu –0.05050 –0.05011 –0.00044 –0.00043 –0.00047 –0.00046 –0.00018 –0.00017 pio –0.05753 –0.05635 –0.00049 –0.00048 –0.00051 –0.00049 –0.00019 –0.00017 nn –0.03614 –0.03591 –0.00031 –0.00031 –0.00033 –0.00032 –0.00010 –0.00009 kbc –0.00056 –0.00459 –0.00001 –0.00004 –0.00001 –0.00004 0.00000 –0.00001 wig –0.00173 –0.00372 –0.00001 –0.00003 –0.00001 –0.00003 x x periods after rounds of presidential election ca –0.03148 –0.02775 –0.00038 –0.00034 –0.00045 –0.00040 –0.00002 –0.00001 pzu –0.04315 –0.04270 –0.00049 –0.00048 –0.00053 –0.00053 –0.00006 –0.00006 pio –0.06353 –0.06314 –0.00066 –0.00066 –0.00071 –0.00071 –0.00037 –0.00012 nn –0.03399 –0.03284 –0.00035 –0.00034 –0.00038 –0.00037 –0.00004 –0.00001 kbc –0.04069 –0.03642 –0.00048 –0.00043 –0.00053 –0.00048 –0.00019 –0.00005 wig –0.03399 –0.02817 –0.00034 –0.00028 –0.00034 –0.00028 x x note: bold letters denote that sharpe and treynor ratios evaluated for mutual funds are bigger than the ones calculated for wig and jensen ratios are statistically significant. table 10. values of the efficiency measures evaluated for whole period sharpe treynor (β sharpe) treynor (β capm) jensen alpha ca –0.01485 –0.00017 –0.00019 0.00000 pzu –0.04640 –0.00047 –0.00050 –0.00011 pio –0.05976 –0.00057 –0.00060 –0.00014 nn –0.03413 –0.00033 –0.00035 –0.00005 kbc –0.01894 –0.00020 –0.00022 –0.00001 wig –0.01938 –0.00017 –0.00017 x note: bold letters denote that sharpe and treynor ratios evaluated for mutual funds are bigger than the ones calculated for wig and jensen ratios are statistically significant. here the question arises if changes of efficiency measures observed in comparable periods are statistically significant. to verify such hypothesis the test with statistics (10) is used for sharpe ratio, together with the test statistics (1) and (2), applied for average values of sharpe and treynor ratios, evaluated for all analyzed funds. as one may notice in table 11, neither differences of sharpe ratios between periods nor differences between mutual fund and market index are significant. however, if average values of treynors ratios are compared the better investment performance before the election is proved (table 12) since the higher risk premium was obtained. dorota witkowska, krzysztof kompa dynamic econometric models 17 (2017) 147–159 158 table 11. values of test statistics in jobson – korkie test funds or index comparison of sharpe ratios in two sub-periods between fio credit agricole and wig 1st round 2nd round ca –0.03148 –0.02775 1st round of presidential election 0.17623 pzu –0.04315 –0.04270 pio –0.06353 –0.06314 2nd round of presidential election 0.11181 nn –0.03399 –0.03284 kbc –0.04069 –0.03642 in the whole period of analysis 0.20728 wig –0.03399 –0.02817 table 12. values of test statistics for average ratios no. of formula sharpe treynor (β sharpe) treynor (β capm) i round ii round i round ii round i round ii round (2) 1.6922 1.4133 2.7795 2.5495 2.9429 2.8190 (2) 3.9387 2.6980 5.4554 4.0513 5.5685 4.2752 (1) 1.5547 1.2520 2.4765 2.1578 2.6019 2.3534 note: bold letters denote rejection of null hypothesis. conclusion our research show that in the whole period of analysis (covering more than three years), statistically significant and positive returns were generated only by the bond market, represented by index tbsp. the mutual stabile growth fund fio credit agricole also obtained positive rates of return but only in the periods before both rounds of the presidential election. taking into account other considered instruments we cannot reject the hypotheses about zero returns. comparison of the return and risk level let us conclude that: 1. returns from treasury bonds significantly decreased after both rounds of the presidential election, 2. risk of the bond market significantly decreased after entry into force 500+ family program, 3. returns from equity market did not significantly change but rates of return were lower after both rounds of the presidential election, 4. equity market risk increased after both rounds of the presidential election, the parliamentary election and appointment of the government, 5. equity market risk decreased after the program 500+ started. how the change of governing party influences the efficiency of financial market… dynamic econometric models 17 (2017) 147–159 159 references białek, j. (2009), konstrukcja miar efektywności otwartych funduszy emerytalnych (construction of efficiency measures for open pension funds), wydawnictwo uł, łódź. blitz, d., van vliet, p. (2007), the volatility effect, journal of portfolio management, 34(1), 102–113, doi: https://doi.org/10.3905/jpm.2007.698039. borowski, k. (2014), miary efektywności zarządzania na rynkach finansowych (management efficiency measures on financial markets), difin, warszawa. domański, c. (2011) (red.), nieklasyczne metody oceny efektywności i ryzyka. otwarte fundusze emerytalne (non-classic methods of risk and performance evaluation. open pension funds), pwe, warszawa. jobson, j. d., korkie, b. m. (1981), performance hypothesis testing with the sharpe and treynor measures, journal of finance, 36(4), 889–908, doi: https://doi.org/10.1111/j.1540-6261.1981.tb04891.x. kurach, r., papla, d. (2014), inwestycje alternatywne w portfelach otwartych funduszy emerytalnych (alternative investments in open pension funds’ portfolios), optimum. studia ekonomiczne, 1(67), 71–81, doi: https://doi.org/10.15290/ose.2014.01.67.06. memmel, c. (2003), performance hypothesis testing with the sharpe ratio, finance letters, 1(1), 21–23. perez, k. (2012), efektywność funduszy inwestycyjnych (efficiency of mutual funds), difin, warszawa. witkowska, d. (2016), zmiana warunków funkcjonowania a efektywność inwestycyjna otwartych funduszy emerytalnych (the change of functioning conditions vs investment efficiency of open pension funds), wydawnictwo uł, łódź. zamojska, a. (2012), efektywność funduszy inwestycyjnych w polsce. studium teoretycznoempiryczne (efficiency of mutual funds in poland. theoretical-empirical study), c.h. beck, warszawa. jak wpływa zmiana partii rządzącej na efektywność rynku finansowego w polsce z a r y s t r e ś c i. rynek finansowy wydaje się być wrażliwym na zmiany polityczne, zwłaszcza gdy zmiana partii rządzącej wiąże się z zasadniczymi zmianami koncepcji rozwoju gospodarczego. taka sytuacja miała miejsce w polsce w 2015 roku w wyniku wyborów prezydenckich i parlamentarnych. celem naszych badań jest analiza zmian zachodzących na rynku, reprezentowanym przez niektóre stabilne, otwarte fundusze inwestycyjne oraz indeksy giełdowe: wig i tbsp. przeprowadzono analizę, stosując modele jednoczynnikowe i capm, klasyczne miary efektywności inwestycji i wnioskowanie statystyczne. s ł o w a k l u c z o w e: otwarte fundusze inwestycyjne stabilnego wzrostu; efektywność inwestycji; model sharpe; capm; sharpe, treynor i jensen. © 2018 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2018.002 vol. 18 (2018) 35−47 submitted january 23, 2018 issn (online) 2450-7067 accepted september 5, 2018 issn (print) 1234-3862 kamal p. upadhyaya, raja nag and franklin g. mixon jr.  stock market prices and the macroeconomics of emerging economies: the case of india  a b s t r a c t. this paper investigates the relationship between stock prices and selected macroeconomic variables in india. the empirical results suggest that, in the long run, output growth and the exchange rate are positively related to stock prices, while money supply exhibits a negative relationship to stock market prices. in the short run most of the variation in the stock market is captured by its own innovation, although the exchange rate, the price level and the interest rate seem to have some effect on short-run stock prices. k e y w o r d s: stock prices, indian economy, emerging economies, asian economics j e l classification: e00; e44 introduction despite having the seventh largest economy in the world, india is also currently home to one of the world’s fastest growing economies. this phenomenon is relatively recent. however, the indian economy was more or less stagnant until the early 1980s. after enacting some major economic reforms  correspondence to: kamal p. upadhyaya, university of new haven, department of economics, 300 boston post road, west haven, ct 06516, united states, e-mail: kupadhyaya@newhaven.edu; raja nag, new york institute of technology, department of accounting and finance, old westbury campus, old westbury, ny 11568, united states, e-mail: dnag@nyit.edu; franklin g. mixon, jr., columbus state university, center for economic education, 4225 university avenue, columbus, ga 31907, united states, e-mail: mixon_franklin@columbusstate.edu.  the authors thank two anonymous referees for many helpful comments on a prior version. any remaining errors are our own. kamal p. upadhyaya, raja nag and franklin g. mixon, jr. dynamic econometric models 18 (2018) 35–47 36 during the 1980s, the growth rate of india’s economy accelerated, at an average 6.27 percent annual growth from 1980 to 2014. in 2014, india’s economic growth rate stood at 7.24 percent, making that economy the fourteenth fastest growing economy in the world. in 2015, this growth rate increased to 7.36 percent, establishing india as the world’s ninth fastest growing economy. in response to this level of gdp growth, stock prices in india have also been growing steadily. in 2003, total indian stock market capitalization was $279 billion dollars (slightly more than 40% of gdp), increasing to $1.516 trillion (over 70% of gdp) in 2015. as a result, the growing indian stock market has become an attractive investment opportunity not only for domestic investors but also for foreign investors. there are several factors that contribute to, and influence, a company’s stock price. it is in part determined by a firm’s own company fundamentals, such as earning per share, dividend per share, book value and other internal factors that may affect the growth of the firm. in addition to its own internal factors, various external factors, such as domestic as well as foreign macroeconomic variables, also affect the firm’s stock price 1 . according to shleifer and vishny (1997), successful corporate governance systems have significant legal protections for investors. unfortunately, developing countries often lack such legal protection, which has led to concentrated ownership of equity and large investors. thus, the findings from studies such as those cited above with regard to the relationship between macroeconomic variables and stock market prices cannot necessarily be generalized for an emerging economy such as india. as such, the objective of this study is to explore, identify and analyze the effects of various macroeconomic variables on equity prices in the indian stock market, and to determine if these effects differ from those of developed countries such as the u.s., u.k. and japan. 1. literature review there is abundant theoretical literature that links the stock market with different macroeconomic variables (e.g., see fama, 1981 and 1990; chen, roll and ross, 1986). empirical studies on the validity on this issue, however, began only few decades ago. one of the early studies on this topic is by mukherjee and naka (1995), which studied the relationship between macroeconomic variables such as the exchange rate, money supply, inflation, in 1 given the strong link between the financial market and the macroeconomy, andreou, ghysels and kourtellos (2013) use daily financial data to predict quarterly real economic activity. stock market prices and the macroeconomics of emerging economies dynamic econometric models 18 (2018) 35–47 37 dustrial production, long term government bond rates and call money rates and the japanese stock market using a vector error correction model (vecm). their findings suggest that there exists a long-run relationship between these macroeconomic variables and stock prices in japan 2 . maysami and koh (2000) developed a vector error correction model (vecm) to study the relationship between different macroeconomic variables and the stock market in singapore. their findings suggest that only price levels, money supply, short-term and long-term interest rates and exchange rates are cointegrated with singapore’s stock market. they also found that industrial production and trade, which essentially are the measures of real economic activity in singapore, are not cointegrated with its stock market. wongbangpo and sharma (2002) also employ a vecm model to study the effects of different macroeconomic variables such as gross national product, the consumer price index, money supply, interest rate and the exchange rate on the stock prices in five asean countries (indonesia, malaysia, philippines, singapore and thailand). from the estimated model they observed both a short-term and long-term relationship between the various macroeconomic variables and stock prices. they also suggest that there exists a bi-directional causality between these macroeconomic variables and stock prices in every sample country in their study. abugri (2008) investigates the effect of macroeconomic volatility on stock returns in four latin american countries (argentina, brazil, chile and mexico) using a vector autoregressive (var) model. the macroeconomic variables considered in the study include exchange rates, interest rates, industrial production and money supply. in addition to these macroeconomic variables they also included the msci world index and the u.s.three-month treasury yield. based on their empirical findings they argue that the effect of macroeconomic variables on stock returns varies from country to country. 2 many researchers have studied the effects of macroeconomic variables on stock markets in different countries (e.g., see fama, 1981; chen, 1986; geske and roll, 1983; huang and kracaw, 1984; kwon, shin and bacon, 1997; humpe and macmillan, 2008). most of these studies are based on data from developed countries, such as the united states, united kingdom, and japan, where the financial sector is well developed. corporate governance is at a mature state in these developed countries (shleifer and vishny, 1997). although, according to fama’s efficient market hypothesis (emh) continuum, the stock market in the u.s. is semi-strong efficient, it is still the most efficient capital market in the world. this is due largely to its well-evolved and mature corporate governance structure. kamal p. upadhyaya, raja nag and franklin g. mixon, jr. dynamic econometric models 18 (2018) 35–47 38 thus, it is not possible to determine, a priori, the effect of a change of any individual macroeconomic variable on market returns 3 . humpe and macmillan (2009) examine, using johansen’s (1991) multivariate cointegration test, the importance of various macroeconomic variables in explaining long-term movements in the stock markets in both the u.s. and japan. in doing so they include industrial production, the cpi, money supply, the long-term interest rate and stock prices in the statistical model.their findings suggest that in the u.s., stock prices are positively related to industrial production and negatively related to both the cpi and long-term interest rates. the relationship between the stock returns and the u.s. money supply is found to be positive but insignificant. in case of japan, stock prices are positively affected by industrial production but negatively affected by the money supply, while industrial production is negatively influenced by both the cpi and long-run interest rates. these authors conclude that the slump in the japanese economy during the 1990s, and the subsequent liquidity trap experience there, may be responsible for these contrasting results. lastly, al-tamami and rahman (2011) study the factors affecting stock prices in the united arab emirates (uae). in doing so they develop a linear regression model in which stock prices are a function of earnings per share, dividend per share, oil prices, gdp, the cpi, interest rates and the money supply. their analysis of time series data from 17 companies, divided into two groups – banks and non-banks – for the period 1995–2005 suggests that both gdp and the money supply positively affects both stock prices for banks and non-banks, whereas the cpi has a negative effect on both groups of stock prices. the interest rate fails to impact stock prices in either subgroup. 2. methodology and data based on the economics literature discussed above, the following model is developed, sp = f (iip, ms, wpi, r, exch), (1) where the dependent variable, sp, represents india’s stock price index. on the right-hand side are national output, iip, as measured by the index of industrial production, the m1 money supply, ms, the price level, wpi, as 3 an additional finding of interest in this study is that the global variables (i.e., the msci world index and the u.s. treasury yield) are more consistent in explaining stock returns across the countries than are the domestic macroeconomic variables. stock market prices and the macroeconomics of emerging economies dynamic econometric models 18 (2018) 35–47 39 measured by the wholesale price index, the nominal interest rate, r, as measured by the 10-year treasury rate, and the exchange rate, exch 4 . an increase in economic activity not only increases corporate profitability, it may also increase expected future cash flows. this in turn will have a positive effect on stock prices. accordingly, a decrease in economic activity may lower corporate profitability and expected future cash flows, which in turn negatively affects stock prices. therefore, we expect a negative relationship between stock prices (sp) and economic output (iip). the relationship between the money supply and stock prices is not as straightforward. on one hand, an increase in the money supply, ceteris paribus, creates an excess supply of money balances and an excess demand for equity, resulting to an increase in equity prices (dhakal, kandil and sharma, 1993; wongbangpo and sharma, 2002). on the other hand, an increase in the money supply may lead to inflation, which can lower corporate profits by increasing costs. therefore, it is possible that an increase in the money supply will negatively affect stock prices. as such, the relationship between ms and sp is an empirical question. the expected relationship between sp and wpi follows a similar path. in general, the price level is expected to have a negative effect on stock prices, given that inflation raises the costs of production, lowers profit and reduces future cash flows of a firm (fama and schwert, 1977; fama, 1981; chen et al., 1986; defina, 1991). wongbangpo and sharma (2002) argue, however, that inflation can have a positive effect on stock prices given because equities serve as a hedge against inflation given that they represent claims on real assets. the relationship between the interest rate and the stock prices is expected to be negative. there are primarily two reasons for this expectation. first, an increase in the interest rate makes interest earning assets more attractive, leading investors to reallocate their portfolios by substituting equity for other assets. second, an increase in the interest rate can reduce corporate profitability because of the concomitant increase in financing costs. in either case, stock prices (sp) are negatively related to the interest rate (r). next, an increase in the exchange rate (i.e., currency depreciation) makes domestic goods (imported goods) relatively cheaper (more expensive) and, thus, more (less) competitive in the world (domestic) market. under this circumstance, the volume of exports should increase and at the same time importcompeting domestic goods production also should increase. under such 4 ideally gdp would have been more appropriate than the index of industrial production (iip) as a measure of overall economic activity in india. however, a lack of monthly data for gdp resulted in our use of industrial production data as a measure of economic activity. kamal p. upadhyaya, raja nag and franklin g. mixon, jr. dynamic econometric models 18 (2018) 35–47 40 a scenario, corporate profits, as well as future cash flows, should rise. the opposite is true in case of a decrease in the exchange rate (i.e., currency appreciation). therefore, we expect a positive relationship between the exchange rate (exch) and stock prices (sp). monthly data from january 2006 to march 2016 are used to test the hypotheses developed above. for sp (stock prices), the bombay stock exchange index (sensex) is used 5 . sensex is a free-float market-weighted stock market index of established companies which are listed in the bombay stock exchange (bse). it is the s&p bse sensitive index, which is widely considered as the benchmark index of domestic stock markets of india. the sensex has a free-float market capitalization of over $450 billion (us). data on ipp data are obtained from the ministry of statistics and program implementation of the government of india., while data for the money supply (ms), the price index (wpi), the interest rate (r) and the exchange rate (exch) are obtained from various issues of international financial statistics, which is published by the international monetary fund. all of the data series, with the exception of the interest rate, are transformed into natural logs before the empirical estimation. 3. estimation and empirical findings macroeconomic time series data are typically not stationary. the use of non-stationary data series produces spurious results. therefore, it is important to test the stationarity of the each data series before estimating the model. to ensure the stationarity of the data series employed in this study, both an augmented dickey-fuller test (said and dickey, 1984) and phillipsperron test (phillips and perron, 1988; perron, 1988) are conducted. results from these tests are reported in table 1. as indicated there, both the augmented dickey-fuller test and the phillips-perron test statistics do not reject the null hypothesis of “the existence of a unit root” in level form, while they do reject this null in the case of first differences for each of the series. these results suggest that all of the data series are integrated of order one, or are i(1). after establishing the stationarity of the data series, johansen’s cointegration test (johansen, 1988 and 1991; johansen and juselius, 1990) is conducted in order to test for long-run relationships among the variables. the aic criterion is used to identify the optimal lag length. the johansen’s 5 the data for this variable is collected from cnbc money control: (http://www.moneycontrol.com/stocks/hist_index_result.php?indian_indices=4). stock market prices and the macroeconomics of emerging economies dynamic econometric models 18 (2018) 35–47 41 cointegration test results are reported in table 2, and indicate that one may reject the null hypothesis of “no cointegration”. the long-run relationship between stock price indices (sp) and the macroeconomic variables are derived after normalizing the coefficient of sp to one, which occurs as follows 6 : sp = 30.61iip *** −10.07ms *** − 9.40wpi + 0.53r + 12.77exch *** (2) (4.51) (−3.13) (−1.54) (1.32) (3.43) table 1. unit root test results augmented dickey-fuller phillips-perron variable level fd level fd logsp −2.66 −10.07*** −2.92 −10.06*** logiip −2.54 −3.86** −2.47 −21.55*** logms −2.05 −6.75*** −2.99 −13.89*** logwpi −2.47 −6.44*** −1.76 −6.39*** r –2.99 –6.31*** –3.32** –11.92*** logexch −2.59 −7.92*** −2.20 −7.84*** notes: fd = first-difference. *** (**) denotes significance at the 1% ( 5%) critical level. as indicated in (2) above, the coefficient of iip, which is a proxy for overall economic activity, is both positive and statistically significant. this finding is consistent with the notion that an increase (decrease) in economic activity increases (decreases) corporate profits and expected future cash flows, which in turn increases (decreases) stock prices. table 2. johansen’s cointegration test results h0 eigen value trace statistic 5% critical value r = 0 0.314 123.50*** 95.75 r ≤ 1 0.224 80.22*** 69.82 r ≤ 2 0.205 50.97** 47.85 r ≤ 3 0.129 24.53 29.79 r ≤ 4 0.058 8.64 15.49 r ≤ 5 0.015 1.73 3.84 note: *** (**) signifies rejection of hypothesis at 1% (5%) critical level. as discussed in the previous section of this study, the relationship between the money supply and stock prices is not straightforward, given that on the one hand an increase in money supply may increase stock prices by creating an excess supply of money balances, while on the other hand an increase in the money supply may lower stock prices by increasing the price level. in case of india, it appears as though an increase in the money supply 6 the figures in the parentheses are t-values for the corresponding coefficients, where *** indicates significance at the 1% critical level. kamal p. upadhyaya, raja nag and franklin g. mixon, jr. dynamic econometric models 18 (2018) 35–47 42 has a negative effect on stock prices in the long run, presumably because of its effect on the price level. this finding is consistent with wongbangpo and sharma’s (2002) findings for indonesia and the philippines. the coefficient on the price level (wpi) is negative, which suggests that an increase in the price level raises the production costs, lowers corporate profits and reduces expected future cash flows. the coefficient, however, is not statistically significant at the conventional level. likewise, the coefficient attached to the interest rate (r), which carries an unexpected sign, is not statistically significant. wongbangpo and sharma (2002) similarly found a positive long-run effect of long-term interest rates on stock prices for both indonesia and the philippines. lastly, the coefficient of exch, which shows the long-run effect of the exchange rate (exch) on stock prices, is positive and statistically significant. this finding is consistent with that of mukherjee and naka (1995) in the case of japan and indonesia, as well as that for the philippines and malaysia in the study by wongbangpo and sharma (2002). after establishing the long run relationships between the variables in (1) above, we proceed to identify the short run relationship of the various macroeconomic variables in (1) stock prices (sp) in india. although recent studies provide examples of the efficacy of various types of granger causality (e.g., piłatowska, m., włodarczyk, a., zawada, 2014; syczewska, 2014; geise, a. and piłatowska, 2016), the relationships examined in this study are best understood through the granger causality test derived from the vecm estimation 7 . the vecm granger causality test results are reported in table 3. as reported there, when all the macroeconomic variables are taken together the chi-square value for sp is statistically significant. this indicates that national output (iip), the money supply (ms), the price level (wpi), the interest rate (r) and the exchange rate (exch) jointly granger-cause stock prices (sp). accordingly, we can observe that, with exception to price level (wpi), each of the variables in the model is jointly granger-caused by the other variables. 7 vector error correction (vecm) includes lags of the dependent variables, in addition to its own lags. econometircs software used (e-views-7) to estimate this equation also generates the cointegration equation which is reported in (2). our methodology here is consistent with other similar studies (maysami and koh, 2000; wongbangpo and sharma, 2002; abugri, 2008). next, to ensure that the vecm estimation does not suffer from serial correlation, and that it is stable and robust, we conducted a portmantau test (castle and hendry, 2010) for autocorrelation (for up to 10 lags). encouragingly, the test result does not reject the null hypothesis of no autocorrelation. stock market prices and the macroeconomics of emerging economies dynamic econometric models 18 (2018) 35–47 43 table 3. vec granger causality/block exogeneity wald test results excluded variables dependent variable δsp δiip δms δwpi δr δexch all δsp n/a 4.95 10.57 5.29 7.25 25.52*** 58.58*** δiip 0.57 n/a 11.45 18.27** 7.66 9.95 61.67*** δms 3.55 17.71*** n/a 9.48 6.61 5.64 59.65*** δwpi 2.61 4.43 7.54 n/a 4.62 3.54 31.28 δr 2.92 7.14 11.74 11.34 n/a 3.21 54.32** δexch 25.30*** 7.97 6.61 6.78 2.60 n/a 56.53** note: the numbers above are chi-square statistics, where *** (**) signifies the 1% (5%) critical level. the effect of innovation in any variable on stock prices (sp) can be sensitive to the ordering of the variables. most of the literature places stock prices first, which are followed by the output level (i.e., gdp), cpi, money supply and interest rate. given that the exchange rate is “pegged” in most developing countries it is considered as exogenous and typically placed last (see naka and tufte, 1997). the same ordering is followed in this study, with exception to the money supply. the ordering in this case is sp, iip, ms, wpi, r and exch. table 4 reports the variance decomposition, also referred to forecast error variance (fev) decomposition. in addition to the forecast error variance (fev) decomposition of stock prices due to a shock in the macroeconomic variables, table 4 also includes the variance decomposition of other variables due to a shock in other macroeconomic variables, including stock prices. table 4. variance decomposition forecast error variance of stock prices explained by innovations steps ahead sp iip ms wpi r exch 3 89.16 1.40 1.06 1.18 0.35 6.85 6 91.35 1.33 0.75 0.63 1.17 4.77 9 87.33 1.26 0.50 2.11 2.98 5.82 12 85.02 1.15 1.20 3.56 3.28 5.78 forecast error variance of stock prices explained by innovations in sp 3 n/a 1.72 1.05 0.03 6.25 24.36 6 n/a 11.62 1.49 0.24 9.39 35.79 9 n/a 21.40 2.70 1.09 15.81 40.12 12 n/a 22.90 3.41 2.16 21.67 40.72 from the fev it appears that most of the variance in stock prices (sp) in the short horizon is mainly attributable to its own shock (e.g., 89 percent in third period, 91 percent in sixth period, and 85 percent in twelfth period). interestingly, innovations in macroeconomic variables such as the output level (iip) and money supply (ms) do not seem to attribute much in the varikamal p. upadhyaya, raja nag and franklin g. mixon, jr. dynamic econometric models 18 (2018) 35–47 44 ance of stock prices (sp). for example, iip attributes less than two percent and ms attributes around one percent over the time period. the effects of the price level (wpi) and the interest rate (r) on the variance of sp are little more than three percent by the ninth and twelfth time period. the only macroeconomic variable that has a larger effect on the variance of sp is the exchange rate (exch). innovation in this variable attributes five percent or more from period three to period twelve. the effect of innovations on stock prices (sp) and other macroeconomic variables on the variance output level (iip) is also of interest. it appears that a one standard deviation shock in sp contributes to 22.90, 21.67 and 40.72 percent, respectively, in iip (output), the interest rate (r) and the exchange rate (exch), respectively, by within a year (i.e., period 12). the magnitude of the effect of innovation in sp is much smaller in terms of the money supply (ms) and the price level (wpi) compared with other variables. the impulse response functions, which are available from the authors upon request, show the response of stock prices (sp) to a one standard deviation shock to all the macroeconomic variables, and vice versa. in terms of the effect of a shock on sp on itself, the effect initially keeps on rising, it peaks by period 6 and thereafter starts declining, which is compatible with the variance decomposition result. in the case of iip, an innovation on its effect on sp peaks at period two and becomes negative by period 12. likewise, money supply and price level innovations also have some positive effect in the beginning, but eventually a negative effect prevails. interest rate and exchange rate shocks have negative effects from the beginning to period 12. overall, the impulse response functions corroborate the variance decomposition analyses. conclusions in the midst of a growing economy, the stock market in india is also expanding and maturing, thus providing an opportunity for both domestic and foreign investors. india’s stock market, like others, is affected by a number of macroeconomic variables, and while there are several empirical studies investigating the relationship between stock prices and the macroeconomy, particularly for developed economies, such a study of india’s emerging economy is currently missing from the economics literature. as such, this study is to investigate the relationship between stock market prices in india and selected macroeconomic variables, such as overall economic output, the money supply, the price level, the interest rate and the exchange rate. using monthly data from january 2006 to june 2016, both stock market prices and the macroeconomics of emerging economies dynamic econometric models 18 (2018) 35–47 45 short-run and long-run relationships are estimated using mix of approaches, including a vector error correction model (vecm), johansen’s cointegration test, granger causality tests and variance decomposition. the empirical findings suggest that, in the long run, stock market prices in india are positively related to output growth, while money supply growth seems to be negatively related to stock prices. the granger causality test indicates that, as a group, all of the macroeconomic variable in the model granger-cause stock prices in india. likewise, the combination of stock prices and the other variables (as a group) granger-causes each of the macroeconomic variables in model, with the exception of the price level. lastly, variance decomposition and impulse response functions suggest that the stock market dynamically interacts with key macroeconomic variables. more specifically, although most of the variation in the stock market is captured by its own innovation, the exchange rate, the price level and the interest rate seem to have some effect on stock price variation as well. references abugri, b. a. (2008), empirical relationship between macroeconomic volatility and stock returns: evidence from latin american markets, international review of financial analysis, 17, 396–410, doi: https://doi.org/10.1016/j.irfa.2006.09.002. al-tamimi, h. a., alwan, a. a. and rahman, a. a. (2011), factors affecting stock prices in the uae financial markets, journal of transnational management, 16, 3–19, doi: https://doi.org/10.1080/15475778.2011.549441. andreou, e., ghysels, e. and kourtellos, a. (2013), should macroeconomic forecasters usedaily financial data and how? journal of business and economic statistics, 31, 240–251, doi: https://doi.org/10.1080/07350015.2013.767199. castle, j. l. and hendry, d. f. (2010), a low-dimension portmanteau test for nonlinearity, journal of econometrics, 158, 231–245, doi: https://doi.org/10.1016/j.jeconom.2010.01.006. chen, n. f. roll, r. and ross, s. a. (1986), economic forces and the stock market, journal of business, 59, 383–403, doi: https://www.jstor.org/stable/2352710. defina, r. h. (1991), does inflation depress the stock market? federal reserve bank of philadelphia business review, 17, 3–12, doi: https://www.phil.frb.org. dhakal, d., kandil, m. and sharma, s. c. (1993), causality between the money supply and share prices: a var investigation, quarterly journal of business and economics, 32, 52–74, doi: https://www.jstor.org/stable/40473092. fama, e. f. (1981), stock returns, real activity, inflation and money, american economic review, 71, 545–565, doi: https://www.jstor.org/stable/1806180. fama, e. f., and schwert, g. w. (1977), human capital and capital market equilibrium, journal of financial economics, 4, 95–125, doi: https://doi.org/10.1016/0304-405x(77)90038-1. fama, e. f. (1990), stock returns, expected returns, and real activity, journal of finance, 45, 1089–1108, doi: https://www.jstor.org/stable/2328716. https://doi.org/10.1016/j.irfa.2006.09.002 https://doi.org/10.1080/15475778.2011.549441 https://doi.org/10.1080/07350015.2013.767199 https://doi.org/10.1016/j.jeconom.2010.01.006 https://www.jstor.org/stable/2352710?seq=1#page_scan_tab_contents https://www.phil.frb.org/-/media/research-and-data/publications/business-review/1991/brnd91rd.pdf https://www.jstor.org/stable/40473092?seq=1#page_scan_tab_contents https://www.jstor.org/stable/1806180?seq=1#page_scan_tab_contents https://doi.org/10.1016/0304-405x(77)90038-1 https://www.jstor.org/stable/2328716?seq=1#page_scan_tab_contents kamal p. upadhyaya, raja nag and franklin g. mixon, jr. dynamic econometric models 18 (2018) 35–47 46 geise, a. and piłatowska, m. (2016), asymmetries in the relationship between economic activity and oil prices in the selected eu countries, dynamic econometric models 16, 65–86, doi: http://dx.doi.org/10.12775/dem.2016.004. geske, r. and roll, r. (1983), the fiscal and monetary linkage between stock returns and inflation, journal of finance, 38, 1–33, doi: https://doi.org/10.1111/j.1540-6261.1983.tb03623.x. huang, r. d. and kracaw, w. a. (1984), stock market returns and real activity: a note, journal of finance, 39, 267–273, doi: https://www.jstor.org/stable/2327683. humpe, a. and macmillan, p. (2009), can macroeconomic variables explain long–term stock market movements? a comparison of the us and japan, applied financial economics, 19, 111–119, doi: https://doi.org/10.1080/09603100701748956. johansen, s. (1988), statistical analysis of cointegration vectors, journal of economic dynamics and control, 12, 231–254, doi: https://doi.org/10.1016/0165-1889(88)90041-3. johansen, s. (1991), estimation and hypothesis testing of cointegration vectors in gaussian vector autoregressive models, econometrica, 59,1551–1580, doi: https://www.jstor.org/stable/2938278. johansen, s. and juselius, k. (1990), maximum likelihood estimation and inference on cointegration—with applications to the demand for money, oxford bulletin of economics and statistics, 52, 169–210, doi: https://doi.org/10.1111/j.1468-0084.1990.mp52002003.x. kwon, c. s., shin, t. s. and bacon, f. w. (1997), the effect of macroeconomic variables on stock market returns in developing markets, multinational business review, 5, 63–73, doi: https://www.questia.com/library/journal/1p3-16999247. maysami, r. c. and koh, t. s. (2000), a vector error correction model for the singapore stock market, international review of economics and finance, 9, 79–96, doi: https://doi.org/10.1016/s1059-0560(99)00042-8. mukherjee, t. k. and naka, a. (1995), dynamic relations between macroeconomic variables and the japanese stock market: an application of a vector error correction model, journal of financial research, 18, 223–237, doi: https://doi.org/10.1111/j.1475-6803.1995.tb00563.x. naka, a. and tufte, d. (1997), examining impulse response functions in cointegrated systems, applied economics, 29, 1593–1603, doi: https://doi.org/10.1080/00036849700000035. perron, p. (1988), trends and random walks in macroeconomic time series: further evidence from a new approach, journal of economic dynamics and control, 12, 297–332, doi: https://doi.org/10.1016/0165-1889(88)90043-7. phillips, p. c. b. and perron, p. (1988), testing for a unit root in time series regression, biometrika, 75, 335–346, doi: https://www.jstor.org/stable/2336182. piłatowska, m., włodarczyk, a., zawada, m. (2014), the environmental kuznets curve in poland – evidence from threshold cointegration analysis, dynamic econometric models, 14, 51–70, doi: http://dx.doi.org/10.12775/dem.2014.003. said e., and dickey, d. a. (1984), testing for unit roots in autoregressive-moving average models of unknown order, biometrika, 71, 599–607, doi: https://www.jstor.org/stable/2336570. shleifer, a. and vishny, r. w. (1997), a survey of corporate governance, journal of finance, 52, 737–783, doi: https://doi.org/10.1111/j.1540-6261.1997.tb04820.x. http://dx.doi.org/10.12775/dem.2016.004 https://doi.org/10.1111/j.1540-6261.1983.tb03623.x https://www.jstor.org/stable/2327683?seq=1#page_scan_tab_contents https://doi.org/10.1080/09603100701748956 https://doi.org/10.1016/0165-1889(88)90041-3 https://www.jstor.org/stable/2938278?seq=1#page_scan_tab_contents https://doi.org/10.1111/j.1468-0084.1990.mp52002003.x https://www.questia.com/library/journal/1p3-16999247/the-effect-of-macroeconomic-variables-on-stock-market https://doi.org/10.1016/s1059-0560(99)00042-8 https://doi.org/10.1111/j.1475-6803.1995.tb00563.x https://doi.org/10.1080/00036849700000035 https://doi.org/10.1016/0165-1889(88)90043-7 https://www.jstor.org/stable/2336182?seq=1#page_scan_tab_contents https://www.jstor.org/stable/2336570?seq=1%23page_scan_tab_contents https://doi.org/10.1111/j.1540-6261.1997.tb04820.x stock market prices and the macroeconomics of emerging economies dynamic econometric models 18 (2018) 35–47 47 syczewska, e. m. (2014), the eurpln, dax and wig20: the granger causality tests before and during the crisis, dynamic econometric models, 14, 93–104, doi: http://dx.doi.org/10.12775/dem.2014.005. wongbangpo, p. and sharma, s. c. (2002), stock market and macroeconomic fundamental dynamic interaction: asean-5 countries, journal of asian economics, 13, 27–51, doi: https://doi.org/10.1016/s1049-0078(01)00111-7. http://dx.doi.org/10.12775/dem.2014.005 https://doi.org/10.1016/s1049-0078(01)00111-7 © 2017 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2017.007 vol. 17 (2017) 115−127 submitted october 25, 2017 issn (online) 2450-7067 accepted december 28, 2017 issn (print) 1234-3862 ewa majerowska, magdalena gostkowska-drzewicka * determinants of corporate performance: modelling approach a b s t r a c t. this study is to investigate the influence of the selected factors of the capital structure on the corporate performance. an empirical analysis covers a sample of 90 nonfinancial companies traded on the warsaw stock exchange, in the period of 2000–2015. the panel data models for two corporate performance measures such roa and roe were estimated. the company’s capital structure negatively affects its performance. it is in line with pecking order theory and previous studies on capital structure of polish companies. k e y w o r d s: capital structure, efficiency, panel data modelling, roa, roe j e l classification: g32 introduction chakravarty (1986) states that superior financial performance is a way to satisfy investors and could be represented by profitability, growth in sales and market value of the company. these three aspects complement each other (santos, brito, 2012). profitability measures a company’s ability to generate returns. growth reflects firm’s ability to enlarge its size. increasing size, even at the same profitability level, lead to increase company’s absolute profit and cash generation. larger size also can bring economies of * correspondence to: ewa majerowska, faculty of management, university of gdańsk, ul. armii krajowej 101, 81-824 sopot, poland, e-mail: ewa.majerowska@ug.edu.pl; magdalena gostkowska-drzewicka, faculty of management, university of gdańsk, ul. armii krajowej 101, 81-824 sopot, poland, e-mail: mgostkowska@wzr.ug.edu.pl. ewa majerowska, magdalena gostkowska-drzewicka dynamic econometric models 17 (2017) 115–127 116 scale, leading to enhanced future profitability. market value represents the external assessment and expectation of company’s future performance. it is postulated, that company’s value depends on its financing decisions (jensen and meckling, 1976; myers, 1977, 1984). it means that capital structure has a significant impact on corporate performance. it is also argued that profitable companies are less likely to depend on debt than less profitable ones. moreover, it was found that companies with high growth opportunities have a high profitability ratio. also asset structure and size of the firm were found as important factors affecting firm performance level. therefore, one of the most important decisions that must be made by the financial managers is to choose the right combination of debt and equity capital that optimizes a corporate performance defined in terms of return on assets and equity. moreover, it is necessary to identify the major factors influence the relationship between a company’s capital structure and its performance. the purpose of this study is to investigate the influence of the selected factors of the capital structure on the corporate performance of polish companies listed on warsaw stock exchange. obtaining clear and comprehensive answers about the nature of the factors shaping the corporate performance of polish companies is difficult due to the small range of domestic research. previous research on capital structure of polish companies showed that they prefer to finance with internal founds (e.g. hamrol, sieczko 2006; wilimowska, wilimowski 2010; jędrzejczak-gas, 2014). this is in line with pecking order theory, so it is assumed that: h1. a company’s leverage is expected to decrease its performance. h2. growth opportunities affect a company’s performance positively. h3. there is a positive relationship between size and a company’s performance. h4. a company’s tangibility has a negative influence on its performance. an empirical analysis covers a sample of 90 non-financial companies traded on the warsaw stock exchange, in the period of 2000–2015. the panel data models for two corporate performance measures such roa and roe were estimated. the paper is organized in following way: after introduction, the first section discusses the literature review on corporate performance. the second section presents the determinants of corporate performance. the third section presents dataset and methodology of the research. the fourth section shows the empirical models used to investigate the effect of capital structure determinants on corporate performance, and the last one concludes the paper. determinants of corporate performance: modelling approach dynamic econometric models 17 (2017) 115–127 117 1. the literature review on corporate performance corporate performance is an ambiguous term and it is used interchangeably with terms of business or firm performance. it appears in most branches of management and finance and it is of interest to both academic scholars and practicing managers. cameron and whetten (1983) state that the importance of performance in strategic management can be argued along three dimensions, i.e. theoretical, empirical and managerial. theoretically, the concept of business performance is at the center of strategic management and finance. empirically, many research studies employ the construct of performance to examine a variety of strategy content and process issues. the managerial importance of this category is all too evident in many prescriptions offered for performance improvement. venkatraman and ramanujam (1986) offered a scheme of three overlapping concentric circles with the largest one which is representing organizational effectiveness. this broadest domain includes the medium circle representing business performance. this one includes the inner circle representing financial performance. santos and brito (2012) state that business performance is a subset of organizational effectiveness that covers operational and financial outcomes. corporate performance measures could be either financial or operational. financial performance, for example profit maximization, maximizing profit on assets and maximizing shareholders' benefits are at the core of the firm’s effectiveness. operational performance could be measured by growth in sales and growth in market share. it provide a broad definition of performance and focus on the factors that ultimately lead to financial performance (chakravarthy,1986; hoffer and sandberg, 1987). corporate performance could be defined in context of accounting or market measures. the most commonly used corporate performance measures are return on assets (roa), return on equity (roe) and return on investment (roi). they are financial ratios that are assumed to reflect the fulfillment of the economic goals of the company (venkatraman and ramanujam, 1986). these indicators are used by many researchers (e.g. rajan, zingales, 1995; nawaz et al., 2011; addae, nyarko-baasi, hughes, 2013; gupta, 2015, igbinosa, 2015). in the ,,value-based” approach market performance measures are more appropriate than accounting-based ratios (hax and majluf, 1984). such indicators as price per share to the earnings per share (p/e), market value of equity to book value of equity (mbvr), and tobin’s q are often used in many studies (e.g. rajan, zingales, 1995; demsetz, villalonga, 2001; zeitun, tian, 2007; van essen et al., 2015). ewa majerowska, magdalena gostkowska-drzewicka dynamic econometric models 17 (2017) 115–127 118 2. determinants of corporate performance the capital structure theories suggest that leverage level can have a major impact on corporate performance (bandyopadhyay, barua, 2016). some of theoretical predictions on this effect are contradictory. under framework of trade-off theory lewellen and roden (1995) showed that the total debt and the profitability of a company are positively related. hadlock and james (2002) using a sample of 500 non-financial united states firms concluded that companies prefer debt financing because they anticipate higher returns from a higher debt level. fama (1985) argued that bank borrowings could lead to increase in company’s performance, because it avoids the high information costs incurred in public debt offerings through bonds issuance. thus, companies relying more on bank loans are expected to be more profitable. the pecking order theory (donaldson, 1961; ross, 1977; myers and majluf, 1984) proposes a negative relationship, because companies prefer to finance with internal funds rather than debt ones. a negative relationship between capital structure and corporate performance (profitability) was also found by kester (1986), rajan and zingales (1995) wiwattanakantang (1999) and chen, strange (2005). the assumptions of trade-off theory and pecking order theory are often used to explain the factors that determine the capital structure. these factors also could be used to describe the corporate performance. among these factors the most frequently mentioned are: size of the firm, company’s tangibility (asset structure), growth opportunities (e.g. margaritis and psilaki, 2010; banerje and de, 2014; yinusa, et al., 2016). a company’s profit is in line with its sales turnover. this is why it could be said that if a company’s sales increases, there is a probability that its profit will increase. wagner (1995) argues that large companies leads to scale expansion. as a result of such expansion, a company becomes more profitable. further research (harvey et al., 2001) showed that firm size is significantly and nonlinearly related to profitability. it suggests that although bigger companies are likely to experience higher profitability, revenue growth is likely to slow faster in larger firms. tangibility refers to a company’s investments in tangible assets. if these investments are effective, the company’s performance improves. ghosh (2008) found that the greater asset tangibility, the lower the scope for informational asymmetries between insiders and outsiders. it allows for higher leverage with a concomitant positive effect on profitability. this relationship determinants of corporate performance: modelling approach dynamic econometric models 17 (2017) 115–127 119 is u-shaped. it suggests that greater increases in tangibility exerts a positive effect on profits, what is in line with trade-off theory. in turn, in line with pecking order theory there is a negative relationship between tangibility and company’s performance. high profits motivate companies to accumulate surpluses. they are invested in short-term securities. these funds can be spent to fund the investment. in such a situation, the company does not need to have large amounts assets in the form of collateral, as its investments are financed by cash from the sale of its short-term securities. companies in the growth phase have a high performance ratio. they are able to generate profit from investment. so, growth opportunities are expected to be positively related to a company’s performance. literature also indicates other factors of corporate performance. there are age of the firm, risk, debt service capacity, dividend pay-out, development expenditures, degree of operating leverage, ownership structure and ownership type industry affiliation, macro factors (e.g. bandyopadhyay and barua, 2016; margaritis and psillaki, 2010; banerje and de, 2014; zeitun and tian, 2007). 3. dataset and methodology the object of analysis are companies continuously listed on the main market of the warsaw stock exchange in 2000–2015. there were 117 companies, including 6 from the sub-index wig-20, 17 from the sub-index wig-40, 22 from the sub-index wig-80 and 72 not included in any of them. the affiliation of a company to a given sub-index was determined by its composition at the end of december 2015. in this analysis we omitted the fact that the company could change its size during the period of the analysis. of the above companies only entities from the non-financial sector were included into research. therefore, 16 companies were excluded from the sample. in addition, entities that did not submit complete financial statements were also rejected, i.e. 11 companies. finally, 90 companies were qualified for the study, i.e. 77% of the pre-selected entities and the balanced panel was created. in this study we used four variables: growth opportunities, firm’s size and tangibility (asset structure) as determinants of corporate performance. capital structure is measured by total debt ratio to total assets (leverage). growth opportunities are represented by growth of sales (growth). the size is measured by the growth of company's total assets. asset structure is measured by assets tangibility (tang), i.e. the ratio of ewa majerowska, magdalena gostkowska-drzewicka dynamic econometric models 17 (2017) 115–127 120 fixed assets to total assets. as a corporate performance measures we used return on assets (roa) and return on equity (roe). as it was mentioned, in the literature there are more factors that could influence the corporate performance. due to the scope of this research only four factors out of all was taken into account. the selected factors are the factors of the capital structure of companies. so the rest of them was omitted. the two variables, roa and roe are modelled using the panel data approach. the proposed model takes a form: ititit titit tangsize growthleveragey     43 210 (1) where it y is the explained variable, represented by the roa or roe of i’th company in time t, 43210 ,,,,  are the structural parameters and the it  is the error term. the model will be estimated by the pooled ols and using the fixed effects estimator and the random effects estimator. application of three testes, that are the chow f-test of the joint significance of group effects, the breusch-pagan test and the hausman test will allow for selecting the best model. the first of them tests that the pooled ols is adequate, in favor of the fixed effects alternative, the second one tests that the pooled ols is adequate, in favor of the random effects alternative. the null hypothesis of the last test states that the random effects estimator is consistent and more effective than the fixed effect estimator. 4. empirical results 4.1. performance of companies listed on wse in 2000–2015 the fluctuations of roe and roa ratios in companies listed on the warsaw stock exchange in 2000–2015 were similar to each other throughout the period considered (table 1). for this reason, the description focuses only on one of them, i.e. roe. the value of the first quartile, calculated for roe in companies listed on wse, decreased systematically in 2000–2002. in 2002 it reached the lowest level in the whole period, i.e. –13,76%. it means that 25% of companies achieved a value of roe not greater than – 13,76% in 2002. starting from 2003, the ratio was increasing until 2007 when it reached 4,57%, i.e. the highest value in the whole period analysed. in subsequent years, the value of the roe fluctuated. however, only in 2012–2013 it was negative (table 1.). the value of roe has been increasing since 2003 to 2007. determinants of corporate performance: modelling approach dynamic econometric models 17 (2017) 115–127 121 table 1. return on assets (roa) and return on equity (roe) in companies listed on wse in 2000–2015 years roa(%) roe(%) i quartile median iii quartile i quartile median iii quartile 2000 –2,76 2,71 5,67 –1,11 5,17 10,36 2001 –5,82 0,99 4,02 –9,12 2,07 6,09 2002 –4,29 1,76 3,99 –13,76 3,46 7,04 2003 0,61 3,24 6,11 0,97 5,85 12,64 2004 1,83 4,04 7,92 2,64 8,33 17,03 2005 1,03 4,22 9,39 0,54 7,92 16,11 2006 1,38 4,91 10,38 2,22 10,03 19,42 2007 2,69 5,49 10,69 4,75 9,50 17,72 2008 0,72 3,46 6,42 0,92 4,99 12,81 2009 0,60 3,42 6,91 0,80 4,84 10,39 2010 0,93 3,32 5,95 1,25 5,50 10,54 2011 1,05 4,62 6,39 1,61 6,68 10,89 2012 –0,73 2,43 5,14 –1,63 4,08 8,45 2013 0,01 2,71 6,88 –0,17 4,50 10,20 2014 1,33 3,78 6,71 1,25 6,06 11,63 2015 0,12 2,97 6,85 0,16 5,15 11,67 the median, calculated for roe in the entities in 2000–2015, ranged from 2,07% in 2001 to 10,03% in 2006. a similar situation was noted for the third quartile of roe. in 2001 and 2006, it was at the level of 6,09% and 19,42% respectively, which means that 25% of the companies achieved roe not lower than the third quartile. the trends described above are related to the economic situation in poland during the period considered. in the years when the companies achieved the highest profitability level, there was a prosperity in polish economy. in turn, during the economic downturn profitability ratios roe and roa were relatively low (table 1). in the next step the stationarity of the two selected ratios was tested. statistics of the tests for the panel data are given in the table 2. in all cases the null hypothesis that the series is integrated in order of one or higher at the 0.05 level of significance need to be rejected. consequently, it can be assumed that all series are stationary. additionally, the correlations between analysed series were tested. only one statistically significant correlation at the level of 0.05 appeared (see table 3). that was between roa and leverage which means that companies listed on wse prefer to finance with internal founds what is in line with pecking order theory. similar results could be found in most of previous studies for poland (e.g. hamrol, sieczko 2006; wilimowska, wilimowski 2010; jędrzejczak-gas, 2014). ewa majerowska, magdalena gostkowska-drzewicka dynamic econometric models 17 (2017) 115–127 122 table 2. statistics of the stationarity tests of the indicators statistic test roa roe im-pesaran-shin –2.5162 –2.7011 choi meta-tests: inverse chi-square 406.944 458.669 inverse normal test –9.8127 –11.1538 logit test –9.8738 –11.6641 levin-lin-chu pooled adf –3.6030 –35.2900 table 3. correlation coefficients of the indicators, 2000–2015 roa roe leverage growth size tang roa 1 –0.0017 –0.9997 0.0008 0.0039 0.0359 roe 1 –0.0004 –0.0002 –0.0013 0.0355 leverage 1 –0.0010 –0.0041 –0.0360 growth 1 –0.0008 0.0506 size 1 –0.0399 tang 1 4.2. estimated models the proposed model (1) was estimated for both roa and roe ratios. the 90 selected companies in years 2000–2015 constituted the panel data sample. additionally, we propose to estimate the model dividing the sample into four subsamples. the first subsample holds large companies, included in the wig20. there were only 4 of them. to the second one contains mediumsize companies (included in the mwig40). there were 14 of them. the group of 21 companies (included in the swig80 index) constituted the subsample of the small companies. the rest of companies (51) represents the last subsample. the models, for all companies and for subsamples, were estimated using three estimators: the pooled ols, the fixed effects estimator and the random effects estimator. application of such estimators for modelling the corporate performance could be found , for example, in majumdar and chhibber (1999), berger, bonaccorsi di patti e. (2006) or king, santor (2007). then the selection of the most appropriate model was based on the basis of the three tests, mentioned above. results are given below (tables 4 and 5). it can be noticed that in almost all cases (in four out of five cases) the best model for the roa was the one obtained with the fixed effects estimator, and for roe the pooled ols (also in four out of five cases). also the values of the measure of goodness of fit (r-squared), the values of the durbin-watson statistic (dw) and the akaike criterion are presented in the tables. determinants of corporate performance: modelling approach dynamic econometric models 17 (2017) 115–127 123 table 4. estimates of the model of roa and statistics of the tests for companies listed on wse in 2000–2015 variable all companies large companies medium companies small companies rest of companies fixed effects fixed effects fixed effects random effects fixed effects constant 0.1279 0.2334 0.0221 0.1160 0.1139 leverage –0.2379*** –0.4011*** –0.0993** –0.1887*** –0.2379*** growth 0.0000 0.1051* –0.0001 0.0013 0.0000 size –0.0006 0.0456 0.1035*** 0.0425*** –0.0006 tang –0.0159 –0.0788 0.2123*** –0.0745* –0.0140 joint significance test 2.2854#) 3.6343#) 7.3432#) 2.7666#) 2.0205#) breuschpagan test 56.2668#) 0.5627 85.3518#) 13.6845#) 19.8526#) hausman test 14.0096#) na 24.2201#) 8.4723 11.1615#) r squared 0.9995 0.4203 0.4282 na 0.9996 dw 1.4859 1.7440 1.5033 na 1.4954 akaike crit. –214.2246 –104.6516 –444.6726 –554.2151 265.8595 *) **) ***) statistically significant at the level of 0.1, 0.05 and 0.01 respectively; #) the null hypothesis is rejected at 0.05 significance level if the roa ratio is modelled, in all cases the leverage negatively influences its volatility. as it was mentioned such relationship is in line with pecking order theory. it is interesting that for large companies the growth of the company positively affects the roa ratio. it means that they are able to generate profit from investment, what is also in line with pecking order theory. growth is not significant if the medium, small size companies and rest of companies are analysed. for the medium size companies the hypothesis of the negative relationship between asset structure (tang) and their performance is not confirmed. in turn, it is confirmed in the case of small companies. there is a positive relationship between the size of the company (size) and the company’s performance for medium and small size companies. in the case of the rest of companies the size of the company and its tangibility do not affect their performances. estimated models of roe for all companies and small companies did not point out any statistically significant relationships between explanatory and explained variables. for the large, medium-size and the small-size companies the roe was negatively influenced by the leverage and positively by the tang variable. asset structure (tang) refers to a company’s investments in tangible assets. if these investments are effective, the company’s performance improves. in contrary to previous relationships this one is in line with trade-off theory. ewa majerowska, magdalena gostkowska-drzewicka dynamic econometric models 17 (2017) 115–127 124 table 5. estimates of the model of roe and statistics of the tests for companies listed on wse in 2000–2015 variable all companies large companies medium companies small companies rest of companies pooled pooled fixed effects pooled pooled constant –0.9658 0.1237* 0.0699 1.6421 –0.8189 leverage 0.0006 –0.5718*** –0.8700*** –7.0535*** 0.0005 growth –0.0000 0.2200* 0.0001 0.0794 –0.0000 size 0.0003 0.0685 0.1685* 0.8597 –0.0009 tang 4.3894 0.4979*** 1.0623*** 2.0505 6.1764 joint significance test 1.0420 2.3126 6.0486#) 1.3991 1.0197 breuschpagan test 0.0322 0.0449 21.8961#) 1.1299 0.0023 hausman test 0.7468 na 80.6125#) 0.3983 1.0782 r-squared 0.0012 0.2563 0.3272 0.0331 0.0016 dw 1.9694 1.7481 1.6607 1.4721 2.0026 akaike crit. 13339.5 –24.705 231.6411 2616.291 7980.600 *) **) ***) statistically significant at the level of 0.1, 0.05 and 0.01 respectively; #) the null hypothesis is rejected at 0.05 significance level finally, for those models, for which the fixed effects estimator was in use, the test for differing group intercept was applied. for all models the null hypothesis that the groups have the common intercept was rejected at the level of significance of 0.05. it means that there were specific differences among companies in the level of roa and roe. conclusions on the basis of the empirical analysis in the case of roa only the first hypothesis stated in the paper cannot be rejected. the company’s capital structure negatively affects its performance. the rest of the hypotheses need to be rejected, with some exceptions. only for the large companies the second one, that the growth opportunities lead to the increase of the company’s performance, cannot be rejected. also for the medium size companies the third hypothesis cannot be rejected. it means the positive relationship between the size and the performance of the company. in turn, the forth hypothesis, that a company’s asset structure affects its performance negatively need to be rejected. generally, these results are rather consistent with assumptions of pecking order theory than with trade-off theory. estimates of the model of roe for the full panel data and for the subsample of the small companies point the necessity of rejection of all four hypotheses. in the case of the large, medium and the small size companies determinants of corporate performance: modelling approach dynamic econometric models 17 (2017) 115–127 125 separately the first one cannot be rejected, so the negative relationship between the company’s capital structure and its performance is found. the last hypothesis need to be rejected in all cases, even if for the large companies and the medium ones the structural parameters were statistically significant at 0.05 significance level. it means no negative influence of tangibility on performance. the subsample, called the rest of companies, include the companies of different size. the results for this group are close to those for all companies, because they show similar volatilities of variables. both subsample and the full sample include large companies. changes in the size of these companies, expressed in total assets growth, are relatively small, so the variable size is irrelevant. in other words, this growth is too small to influence the value of the explanatory variable. however, we have to remember that the sample size was relatively small, especially the number of large companies taken into account and the short time series. it is due to the short history of the capital market in poland and lack of the data. it would be interesting to extend the analysis and to compare results with outputs from the other markets. one of the ways could be to create the non-balanced panel, so more companies would be included in the sample. references addae, a.a., nyarko-baasi, m., hughes, d. (2013), the effects of capital structure on profitability of listed firms in ghana, european journal of business and management, 5(31), 215–229. bandyopadhyay, a., barua, n.m. (2016), factors determining capital structure and corporate performance in india: studying the business cycle effects, the quarterly review of economics and finance, 61, 160–172, doi: https://dx.doi.org/10.1016/j.qref.2016.01.004. benerje a., de a. (2014), determinants of corporate financial performance relating to capital structure decisions in indian iron and steel industry: an empirical study, paradigm, 18(1), 35–50, doi: https://dx.doi.org/ 0.1177/0971890714540365. berger a.n., bonaccorsi di patti e. (2006), capital structure and firm performance: a new approach to testing agency theory and an application to the banking industry, journal of banking and finance, 30(4), 1065–1102. cameron, k.s., whetten, d.a. (1983), organizational effectiveness: one model or several? in k. s. cameron and d.a. whetten (eds.), organizational effectiveness: a comparison of multiplie methods, new york, academic press, 1–24. chakravarthy, b. s. (1986), measuring strategic performance, strategic management journal, 7, 437–58. chen j., strange, r. (2005), the determinants of capital structure: evidence from chinese listed companies, economic change and restructuring, 38, 11–35, ewa majerowska, magdalena gostkowska-drzewicka dynamic econometric models 17 (2017) 115–127 126 doi: https://dx.doi.org/ 10.1007/s10644-005-4521-7. donaldson, g. (1961), corporate debt capacity: a study of corporate debt policy and the determinants of corporate debt capacity, division of research, boston, harvard graduate school of business administration. demsetz, h., villalonga, b. (2001), ownership structure and corporate performance, journal of corporate finance, 7, 209–233. fama, e. (1985), what’s different about banks?, journal of monetary economics, 15(1), 29–39. gosh, s. (2008), leverage, foreign borrowing and corporate performance: firm-level evidence for india, applied economics letters, 15, 607–616, doi: https://dx.doi.org/ 10.1080/13504850600722047. gupta, p., (2015), an empirical study of relationship between capital structure and profitability of foreign promoters holding companies in india, management edge, 8(1), 80–91. hadlock, c. j., james, c. m. (2002), do banks provide financial slack? journal of finance, 57, 1383–420. hamrol, m., sieczko, j. (2006), czynniki kształtujące strukturę kapitału polskich spółek giełdowych, prace i materiały wydziału zarządzania uniwersytetu gdańskiego, 1, 127–142. harvey, c. r., lins, k. v. and roper, a. h. (2001), the effect of capital structure when expected agency costs are extreme, nber working article no. 8452, cambridge, ma. hax, a.c., majluf, n.s. (1984), strategic management: an integrative perspective, new york, prentice-hall. hoffer, c. w., sandberg, w. r. (1987), improving new venture performance: some guidelines for success, american journal of small business, 12, 11–25. igbinosa, s. (2015), another look at capital structure and corporate performance in emerging markets: the case of nigeria, asian journal of business management, 7(1), 1–12. jensen m.c., meckling w.h. (1976), the theory of the firm: managerial behavior, agency costs and ownership structure, journal of financial economics, 3(4), 305–360. jędrzejczak-gas, j., (2014), influence of the selected factors on the capital structure of enterprises in the construction industry, management, 18(1), 241–254. kester, w. c. (1986), capital and ownership structure: a comparison of united states and japanese manufacturing corporations, financial management, 15(1), 5–16. king m.r., santor e. (2007), family values: ownership structure, performance and capital structure of canadian firms, working paper, bank of canada, 40. lewellen, w. g., roden, d. m. (1995), corporate capital structure decisions : evidence from leveraged buyouts, financial management, 24(2), 76–87. majumdar s.k., chhibber p. (1999), capital structure and performance: evidence from a transition economy on an aspect of corporate governance, public choice, 98, 287–305. margaritis, d., psillaki, p. (2010),capital structure, equity ownership and firm performance, journal of banking & finance, 34, 621– 632, doi: https://dx.doi.org/ 10.1016/j.jbankfin.2009.08.023. myers s.c. (1977), determinants of corporate borrowing, journal of financial economics, 5(2), 147–175. myers s.c. (1984), the capital structure puzzle, journal of finance, vol. 39, no. 3, 575–592. myers, s.c., majluf n.s. (1984), corporate financing and investment decisions when firms have information that investors do not have, journal of financial economics, 13(2), 187–222. determinants of corporate performance: modelling approach dynamic econometric models 17 (2017) 115–127 127 nawaz, a., ali, r., naseem m.a. (2011), relationship between capital structure and firms performance: a case of textile sector in pakistan, global business and management research: an international journal, 3(4), 270–275. rajan, r.g., zingales, l. (1995), what do we know about capital structure? some evidence from international data, the journal of finance, 50(5), 1421–1460. ross, s. (1977), the determination of financial structures: an incentive signalling approach, bell journal of economics, 8, 23–40. santos, j. b., brito, l.a.l. (2012), toward a subjective measurement model for firm performance, brazilian administration review, 9, 95–117. van essen, m., carney, m., gedajlovic, e.r., heugens, p.p.m.a.r. (2015), how does family control influence firm strategy and performance? a meta-analysis of us publicly listed firms, corporate governance: an international review, 23(1), 3–24. venkatraman, n., ramanujam, v. (1986), measurement of business performance in strategy research: a comparison of approaches, academy of management review, 1(4), 801–814. wagner, j. (1995), exports, firm size, and firm dynamics, small business economics,7(1), 29–39. wilimowska, z., wilimowski, m. (2010), wpływ czynników mikroekonomicznych na zarządzanie strukturą kapitałową polskich przedsiębiorstw, in r. knosala (ed.), komputerowo zintegrowane zarządzanie, t. 2, opole: oficyna wydawnicza polskiego towarzystwa zarządzania produkcją. wiwattanakantang, y. (1999), an empirical study on the determinants of the capital structure of thai firms, pacific-basin finance journal, 7(3–4), 371–403. yinusa o.g., somoye r.o.c., alimi o.y., ilo b.m., (2016), firm performance and capital structure choice of firms: evidence from nigeria, journal of knowledge globalization, 9(1), 1–16. zeitun, r. and tian, g. g. (2007), capital structure and corporate performance: evidence from jordan, australasian accounting, business and finance journal, 1(4), 1–24. czynniki kształtujące wyniki finansowe firmy: ujęcie panelowe z a r y s t r e ś c i. celem artykułu jest zbadanie wpływu wybranych czynników struktury kapitału spółek notowanych na gpw na ich rentowność. do badań zakwalifikowano 90 przedsiębiorstw z sektora niefinansowego notowanych na gpw w latach 2000–2015. oszacowano model panelowy dla dwóch miar rentowności, tj. roa i roe. pomiędzy strukturą kapitału (dźwignią finansową) a rentownością występuje związek ujemny, co jest zgodne z teorią hierarchii źródeł finansowania i wynikami dotychczasowych badań w zakresie struktury kapitału polskich przedsiębiorstw. s ł o w a k l u c z o w e: struktura kapitału, efektywność, modelowanie panelowe, rentowność kapitału, rentowność aktywów. © 2017 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2017.011 vol. 17 (2017) 177−189 submitted october 30, 2017 issn (online) 2450-7067 accepted december 28, 2017 issn (print) 1234-3862 rafał warżała * business cycles variability in polish regions in the years 2000–2016 a b s t r a c t. the aim of this article is to study the morphology of regional business cycle in poland. to do this, such parameters like: cycle length, coherence ratio, standard deviation ratio, mean delay, cross-correlation ratio, were calculated. the main conclusion is that regions have different sensitivity to economy "shocks", both positive and negative. the analysis of regional specialization appear varied level of it among individual regions. despite a few exceptions, a correlation between the level of regional specialization and the degree of sensitivity to economic disturbances can be marked. k e y w o r d s: business cycle; region; variability. j e l classification: e32; f21; r11; r12; r30. introduction one of the most important policy conducted in the framework of the european union is – except agricultural policy – the regional policy. since turn of the 60s and 70s, we can observe the process of regions empowerment, perceived as an autonomous economic and social system. this involved promoted by the eu policy tendency to raise the rank of the region in the administrative and economic system. regional policy supported by eu appeared a new problem – uneven dynamic of regional development, result * rafał warżała, university of warmia and masuria in olsztyn, faculty of economics sciences, chair of macroeconomics, 19 prawocheńskiego street, 10 – 719 olsztyn, poland, email: rafal.warzala@uwm.edu.pl. https://orcid.org/0000-0001-8677-977x rafał warżała dynamic econometric models 17 (2017) 177–189 178 ing in a progressive economic divergence of european regions. one of the results of it was, among others, varied resistance to economic crises. the issue of regional business cycles divergence is important by two reasons. first is the perspective of accessing poland to european monetary union. it means that both polish national and regional economies will subject to common european monetary policy, conducted by european central bank. there are studies proving that common monetary policy doesn’t fit all countries. the second reason deals with fiscal policy. in fact this macroeconomic tool of business cycles affecting is still in member states competence, however some of the fiscal policy parameters are regulated at the supranational level (limit of budget deficit or public debt in relation to gdp). moreover, as a consequence of last world economic crisis, it is suggested that fiscal policies conducted by national governments should be more coordinated. in the face of that facts, we can conclude that countries may be deprived of ability to influence their economy through fiscal policies. the purpose of this paper is to investigate the business cycles morphology in poland on a regional perspective. the process of structural homogenization is, on one hand, an important feature of real convergence and on the other hand, one of the determinants of asymmetry shocks. 1. regional business cycles in the light of literature most analysis concentrate on relations between regional and nationwide business cycles. some researchers indicate that national business cycle is a kind of average variability of cyclical changes in the individual regions. however, this aggregated approach to the analysis of business cycles eliminates from the study characteristics of the different regions, and thus eliminates information about specific reactions of regional fluctuations on changing worldwide business activity. carlino and sill (2001, p. 16–17) state, analyzing cyclical changes in real income growth, that there is strong divergence of cycles run between the regional and national cycle. relating to that, there are some indicators presented in the literature examining the convergence rate of individual regions components (crone 2005, p. 13–15). regional business cycles convergence is also critical for proper european monetary union functioning. from the one side, the economic and monetary union may be an important tool to achieve economic and social cohesion. from the other side, economic convergence is also a prerequisite to accomplish an effective emu. in the literature also exist an opinion, that real business cycle convergence stimulates proper functioning of economic and business cycles variability in polish regions in the years 2000–2016 dynamic econometric models 17 (2017) 177–189 179 monetary union and raises net benefits of the union (artis, zhang 1997, p. 26; barrios et al. 2003, p. 31). in general, there are two streams of views on international and interregional results of deepening economic integration in the literature. the first supports the idea that economic integration leads to symmetrical changes, which in turn causes more synchronized business cycles, both on national and regional levels (marelli 2006, p. 38; barrios, lucio 2001, p. 13). the second concept comes from the paul krugman work (1991), who proves that economic integration leads to an increase of regional concentration of industrial production, which in turn will cause sectoral or even regional shocks, enhancing the probability of asymmetric shocks and divergent business cycles (krugman 1993, p. 242–244; krugman 1991, p. 483–498). regions appear individual business cycles characteristics for two reasons (fatas 1997, p. 1):  different production output, resulting from the individual regional specialization, industry-specific shocks, associated with different levels factors of production mobility;  diversified economic policy in the particular regions. the matter of economic shocks asymmetry, coming across the regions is also one of the optimum currency area (oca) criterion. in the literature it is pointed, that business cycles across countries (regions), that belong to an optimum currency area, should be synchronized (mckinnon 2002, p. 343). the aspect of economic shocks asymmetry, after creating the euro-zone have been studied among the euro area member states. the issue of (a)symmetry business cycle fluctuations is crucial in the context of supra-regional (and national) monetary policy conducting. moreover, if fiscal policy is subject to strict controls and harmonization at the supranational level, according to the theory of optimum currency areas, the effects of using such instruments should be predictable and similar to across the common currency area (frenkel, nickel 2002, p. 6). from the theory and oca criteria point of view, also important as the fact of shock, is the way of reacting to any disturbances determined by effectiveness of the union economic policy instruments. if in one country the shock will be positive, while in the other – negative, harmonization of economic policies would be senseless (weimann 2003, p. 4). researches results show, that the effect of deepening international specialization in the eu is ambiguous. this is also confirmed by montoya and de haan’s studies. in their paper from 2007, using the correlation coefficient of the regional cycles with the euro zone benchmark, they found that synchronization has increased on average for the period considered with some rafał warżała dynamic econometric models 17 (2017) 177–189 180 exceptions. but the correlation of the business cycle in some regions with the reference cycle remained low or even decreased (montoya, de haan 2007, p. 15). moreover, as anagnostou, panteladis and tsiapa conclude, the eu business cycle benchmark has greater impact on the more developed regions, whereas, it has less or no effect on the less developed regions. even though the transmitted values of the euro area shocks are very small, the values of shocks are greater for the higher developed regions rather than those to the lower developed regions (anagnstou et al. 2014, p. 24). antonio fatas states, that economic or commercial cooperation between cross border regions can cause high level of business cycle synchronization between them. however, in the same time, regional business fluctuations may appear cyclical desynchronization with national business cycles, to which they belong (fatas 1997, p. 7). to similar results came anagnostou, panteladis and tsiapa, revealing that the national impact–‘‘the border effect’’–is greater on the less developed regions rather than the higher developed ones (anagnostou et al. 2014, p. 24). eliminating transaction costs between regions, as a result of tariff reduction, as well as transport costs and administrative barriers, causes different levels of economic development in the interregional scale. consequently it leads to the geographical industry concentration. reduction of transaction costs causes a divergence between regions in terms of industry structure and expand specialization of individual regions (krugman 1993, p. 245). the degree of regional business cycle synchronization also depends on such factors, as: the scope of historical ties, the level of economic and trade relations and cultural affinity among regions. as a result, some regions may appear higher degree of convergence, even without belonging to one country, and the other, in spite of administrative links will demonstrate rather diversified in this area. in the literature there is also presented studies showing that the common monetary policy, facing with no business cycles convergence, is improper to all members of the common currency area. this is due to the presence in the national or regional scale “asymmetric shocks” (correia, gouveia 2013, p. 91). the conclusions of the literature study are two opposing theoretical concepts. the first of these is the paul krugman specialization concept, who proves that economic integration in the regional dimension leads to a greater degree of development polarization, rather than to its unification. this is the result of externalities taking in the single currency area, economies of scale of dynamic production, in relation to the environment, as well as the development of metropolitan areas. the main conclusion arising from krugman's business cycles variability in polish regions in the years 2000–2016 dynamic econometric models 17 (2017) 177–189 181 model is that the result of single currency area introduction, may be increase the business cycles convergence degree on the state level, while growing up the range of divergence at the regional level (krugman 1991, p. 486–487). the opposite concept, was proposed by frankel and rose. in their paper from 1996, they described results of economic barriers elimination among countries and regions in a single currency area, followed by intensification of trade. the effect of this process in opinion of the authors, is an increase in the synchronization of cyclical fluctuations. another factor contributing to the going up synchronization of business cycle fluctuations is, according to the authors, the implementation of common economic policy on the integrating area. the difference in the approach to effects of the optimum currency area created here lies in the formulation of the idea that positive results in this concept reveals ex-post, i.e. as a result of conduct of the single monetary policy and single currency (frankel, rose 1996, p. 21). similar conclusions resulted from salvador barrios and juan lucio’s paper (2001). they provide evidence on the positive impact of economic integration on regional business cycles correlation, on two neighbouring economies: spain and portugal. 2. methodology of regional business cycles in poland the objectives of the study are the morphological features of regional business cycles in poland, represented by 16 administratively separate local government units, i.e. voivodships. the time range of the analysis embraces a dynamics series of regional industry production in a monthly cross-section for the period from january 2000 to december 2016. the selection of such a range is dictated by the availability of comparable statistical data. the period of 17 years also offers the possibility of separating several complete business cycles, as well as possibility of evaluating differences in their morphological structure. the bases of research in this study are growth cycles (drozdowicz-bieć 2012, p. 15). the first stage of business cycle fluctuations analysis is the elimination of seasonal fluctuations from raw time series. in order to level seasonality, the tramo/seats method was used, which is recommended by eurostat (grudkowska, paśnicka 2007, pp. 8–9). tramo/seats is an arima model based on a seasonal adjustment method developed by victor gomez and agustin maravall. tramo/seats is a two-stage procedure. the first one pre-adjusts and removes the deterministic effects from the series by means of a regression model with arima noises. the second program executes the decomposition rafał warżała dynamic econometric models 17 (2017) 177–189 182 of the time series into components using an arima model (gomez, maravall 2001, p. 9). for the purpose of separating a cyclical factor from the previously deseasonalised empirical data with the use of the tramo/seats method, the christiano-fitzgerald asymmetrical filter was used, which enables the procurement of cycle evaluation at the beginning and at the end of a time series (adamowicz et al. 2008, p. 12). the christiano-fitzgerald random walk filter is a band pass filter and formulate the de-trending and smoothing problem in the frequency domain. in fact that the granularity and finiteness of real life time series do not allow for perfect frequency filtering, the christiano-fitzgerald filter approximate the ideal infinite band pass filter (nilsson, gyomai 2010, p. 7). in the opposite to other filters the christiano-fitzgerald random walk filter uses the whole time series for the calculation of each filtered data point. the advantage of the cf filter is that it is designed to work well on a larger class of time series, converges in the long run to the optimal filter (christianofitzgerald 1999, p. 3). finally, the process of business cycle turning points identification was based on the bry-boschan method (adamowicz et al. 2008, p. 13). the main rules of this algorithm are as follows (mazzi, scocco 2003, p. 17): 1. peaks and troughs must alternate. 2. each phase (from peak to trough or trough to peak) must have a duration of at least six months. 3. a cycle (from peak to peak or from trough to trough) must have a duration of at least 15 months. 4. turning points within six months of the beginning or end of the series are eliminated as are peaks or troughs within 24 months of the beginning or end of the series if any of the points after or before are higher (or lower) than the peak (trough). analysis of the morphological features of business cycles takes measures of variability and dispersion, i.e. the measure of the length of individual phases and cycles, standard deviation, the variability factor, amplitude and intensity factors, as well as analysis of cross correlations. on the basis of the obtained results, an analysis of the morphological features of industrial production was conducted in 16 voivodships of poland. 3. an empirical analysis of convergence in polish regions the first view on regional gdp structure allows to assess the scale of the regional economies diversity. typically agricultural regions, with more than business cycles variability in polish regions in the years 2000–2016 dynamic econometric models 17 (2017) 177–189 183 average agriculture share in gdp, are: kujawy-pomerania, podlasie, warmia-mazury, lublin, lubuskie, lodz, opole, swietokrzyskie and wielkopolska. podlasie, lublin, mazovia and west pomerania are also the provinces with significantly smaller share of industry in gdp. for the contrast, regions with higher services share in gdp, what means remarkable higher than average development, are: lublin, malopolska, mazovia, pomerania and west pomerania. mentioned above regions are located in different parts of poland, and also represent different level of gdp per capita. in the gdp structure lublin dominated by mining and chemical, and in podlasie – the agri-food, wood and machinery. the swietokrzyskie share of agriculture is similar to the national average, but higher than the rest of regions is the share of it in construction. in turn, podkarpacie has the smallest share of agriculture, while the share of industry – the largest of the polish eastern provinces. here major industries are: aviation, electrical engineering, chemical and food products. the above statement may be prompted to explain the varied course of fluctuations investigated regions. analysis of the business cycles morphology will assess the sensitivity of individual polish provinces to volatility both national and supranational economy. the regional gdp structure in polish provinces is presented in table 1. table 1. regional gdp structure in poland in 2000 and 2014 year (in %) agriculture industry services total voivodship\ year 2000 2014 2000 2014 2000 2014 2000–2014 l.p. poland 3.7 4.0 24.2 33.6 72.1 62.4 100.0 1. lower silesia 2.7 2.1 26.2 43.9 71.1 54.0 100.0 2. kujawy-pomerania 4.5 5.8 25.0 34.9 70.5 59.3 100.0 3. lublin 6.9 8.6 18.8 27.3 74.3 64.1 100.0 4. lubuskie 3.9 4.9 23.8 38.1 72.3 57.0 100.0 5. lodz 3.8 5.2 27.0 36.2 69.2 58.6 100.0 6. malopolska 2.3 2.2 23.2 33.4 74.5 64.4 100.0 7. mazovia 3.5 2.6 19.0 22.9 77,5 74.5 100.0 8. opole 4.9 5.7 27.4 38.2 67.7 56.1 100.0 9. podkarpacie 2.9 2.5 26.4 37.0 70.7 60.5 100.0 10. podlasie 6.9 10.9 18.9 27.8 74.2 61.3 100.0 11. pomerania 2.4 3.0 23.5 34.1 74.1 62.9 100.0 12. silesia 1.2 1.0 32.4 43.1 66.4 55.9 100.0 13. swietokrzyskie 4.9 5.9 23.4 36.4 71.7 57.7 100.0 14. warmia-masuria 6.2 8.9 21.5 32.3 72.3 58.8 100.0 15. wielkopolska 6.6 5.6 25.5 36.1 67.9 58.3 100.0 16. west pomerania 4.2 4.1 20.1 28.9 75.7 67.0 100.0 source: statistical yearbook of the regions – poland, http://stat.gov.pl/obszarytematyczne/roczniki-statystyczne/roczniki-statystyczne/rocznik-statystyczny-wojewodztw2015,4,10.html;http://stat.gov.pl/obszary-tematyczne/roczniki-statystyczne/rocznikistatystyczne/rocznik-statystyczny-wojewodztw-2000,4,10.html. http://stat.gov.pl/obszary-tematyczne/roczniki-statystyczne/roczniki-statystyczne/rocznik-statystyczny-wojewodztw-2015,4,10.html http://stat.gov.pl/obszary-tematyczne/roczniki-statystyczne/roczniki-statystyczne/rocznik-statystyczny-wojewodztw-2015,4,10.html http://stat.gov.pl/obszary-tematyczne/roczniki-statystyczne/roczniki-statystyczne/rocznik-statystyczny-wojewodztw-2015,4,10.html http://stat.gov.pl/obszary-tematyczne/roczniki-statystyczne/roczniki-statystyczne/rocznik-statystyczny-wojewodztw-2000,4,10.html http://stat.gov.pl/obszary-tematyczne/roczniki-statystyczne/roczniki-statystyczne/rocznik-statystyczny-wojewodztw-2000,4,10.html rafał warżała dynamic econometric models 17 (2017) 177–189 184 in period of 2000–2014 the regional gdp structure changed. in general the changes were related to industry share growth, while services and in seven regions agriculture share fell. the most remarkable production structure evolution was observed in lower silesia, silesia, malopolska, mazovia, podkarpacie, wielkopolska and west pomerania. in mentioned regions the share of agriculture and services decreased while industry increased. analyzing regional gdp time series in poland, it can be marked serious difference from industry production time series, in morphological fluctuations characteristics 1 . the first is the average coherence level – higher in the case of industrial production. the most coherent gdp time series with the reference series appear in such regions, like: lubuskie (0.66), lodz (0.65), west pomerania (0.52) and silesia (0.57). the more dissimilar to the reference series were malopolska (0.12) and swietokrzyskie (0.18). table 2. bivariate statistics with the poland gdp reference series time series coherence ratio mean delay cross-correlation r0 rmax tmax(1) lower silesia 0.36 –0.64 0.59 0.65 –1 kujawy-pomerania 0.47 0.71 0.53 0.70 1 lodz 0.65 –0.42 0.74 0.77 –1 lublin 0.40 0.54 0.59 0.64 1 lubuskie 0.66 –0.07 0.84 0.84 0 malopolska 0.12 –0.58 0.23 0.27 –1 mazovia 0.54 0.42 0.69 0.72 1 opole 0.39 –0.23 0.53 0.53 0 podkarpacie 0.31 0.56 0.50 0.54 1 podlasie 0.28 0.13 0.51 0.51 0 pomerania 0.53 –0.38 0.68 0.68 0 silesia 0.57 –0.34 0.73 0.73 0 swietokrzyskie 0.18 0.63 0.35 0.39 1 warmia-masuria 0.41 0.16 0.56 0.58 0 wielkopolska 0.48 0.42 0.67 0.71 1 west pomerania 0.52 –0.84 0.64 –0.75 4 (1) the + (–) sign refers to a lead (lag) in quarters with respect to the reference series; r0 – cross correlation value with no shift; rmax – cross-correlation shift maximum value; tmax – shift dimension in quarters. source: own elaborations based on: “monthly reports on the socio-economic situation of lower silesia, kujawy-pomerania, lublin, lubuskie, lodz, malopolska, mazovia, opole, podkarpacie, podlasie, pomerania, silesia, swietokrzyskie, warmia-masuria, wielkopolska and west pomerania voivodship”, local data bank, regional statistical office. 1 the industry production time series morphology was analysed in other work of author, i.e. warżała r. (2016), business cycless in polish regions. an theoretic and empirical study, publisher university of warmia-masuria, olsztyn. business cycles variability in polish regions in the years 2000–2016 dynamic econometric models 17 (2017) 177–189 185 to describe the interdependence level between two time series, the coherence ratio was exploited. coherence is a function of frequency showing what is the correlation coefficient level between two stochastic processes depending on the frequency. some regions appear average leading of business cycles phases with respect to reference series (kujawy-pomerania, lublin, mazovia, podkarpacie, swietokrzyskie, wielkopolska) and some of them are lagged (lower silesia, lodz, malopolska). in the case of six of the polish regions, almost coincident fluctuations with respect to reference series were observed, i.e. lubuskie, opole, podlasie, pomerania, silesia, warmia-masuria. the highest cross-correlation level was in the case of lubuskie (0.84), lodz (0.77), silesia (0.73) and mazovia (0.72). the west pomerania voivodship appeared four months lag of average business cycle and the cross-correlation index relatively high, but negative (–0,75). this region can be perceived as one of the most dissimilar, with respect to the national business fluctuations. table 3. analysis of regional gdp cycles with respect to the reference series phases and cycles average duration reference series p – t p – p t – p t – t poland 4.74 9.35 5.37 9.46 lower silesia 4.81 9.40 5.41 9.60 kujawy-pomerania 5.70 10.30 5.25 10.20 lodz 4.83 9.53 5.21 9.73 lublin 6.14 11.30 6.40 10.40 lubuskie 6.82 11.46 5.53 10.50 malopolska 7.10 14.27 7.36 11.00 mazovia 5.80 11.36 6.20 11.20 opole 7.45 14.20 6.85 12.80 podkarpacie 5.33 9.80 5.50 9.40 podlasie 6.20 11.80 6.10 10.20 pomerania 4.95 9.50 5.15 9.90 silesia 5.45 10.70 6.20 10.90 swietokrzyskie 4.88 12.30 6.70 10.50 warmia-masuria 4.80 9.90 5.80 9.10 wielkopolska 6.40 11.20 5.30 9.90 west pomerania 5.10 9.60 5.40 10.30 explanation: p-p – a business cycle defined by upper turning points, t-t – a business cycle defined by bottom turning points, t-p – the upward phase of the cycle, p-t – the downward phase of the cycle. source: as in table 2. most of the regional business cycle length oscillated around ten quarters. the average longest business cycle were in malopolska, mazovia, opole and swietokrzyskie voivodship. by contrast – the shortest were marked in rafał warżała dynamic econometric models 17 (2017) 177–189 186 lower silesia, lodz, podkarpacie, pomerania, warmia-mazuria and wielkopolska. in most of the regions the upward business cycle phases were longer than the downward one. the inverse relationship revealed kujawypomerania, lubuskie, opole and wielkopolska. regarding this results to the gdp share changes, it can be concluded, that regions with falling share of agriculture and growing share of industry appear on average shorter duration of cycles. moreover, this group of regions is also characterized by longer upward phase of the cycle. this finding goes in line with industry behavior in business cycle, described in literature. apart from length and coherence regional business cycle appear differences in the aspect of volatility and standard deviation. analyzing the average standard deviation of particular regions, we can indicate the voivodships, that are characterized by more balanced business cycle, i.e. kujawypomerania (6.63), silesia (6.85), lublin (7,01), lubuskie (7.06), opole (7.14) and podkarpacie (7.69). the situation is similar to the coefficient of variation index. table 4. the intensity of the poland gdp time series and the individual voivodships in the years 2000–2016 time series standard deviation (in p.p.) coefficient of variation (in%) average amplitude (in %) upward phases downward phases cycles poland 7.73 7.08 2.3 2.4 –0.1 lower silesia 7.36 6.75 2.9 2.6 0.3 kujawy-pomerania 6.63 6.04 2.8 2.7 0.1 lodz 7.24 6.58 2.4 2.7 –0.3 lublin 7.01 6.37 2.8 3.1 –0.3 lubuskie 7.06 6.42 2.7 2.8 –0.1 malopolska 8.04 7.18 2.1 2.5 –0.4 mazovia 10.20 9.08 2.9 2.8 0.1 opole 7.14 6.56 2.9 2.4 0.5 podkarpacie 7.69 7.03 2.0 2.0 0.0 podlasie 8.47 7.74 2.9 2.9 0.0 pomerania 7.59 6.93 2.9 2.7 0.2 silesia 6.85 6.27 2.8 2.7 0.1 swietokrzyskie 7.62 7.04 2.6 2.4 0.2 warmia-masuria 7.85 7.19 2.3 2.5 –0.2 wielkopolska 8.75 7.96 2.7 2.9 –0.2 west pomerania 8.03 7.26 2.7 2.8 –0.1 source: as in table 2. the opposite group were regions with relative high level of standard deviation. these are: mazovia (10.20), wielkopolska (8.75), podlasie (8.47), malopolska (8.04), west pomerania (8.03), warmia-masuria (7.85) and business cycles variability in polish regions in the years 2000–2016 dynamic econometric models 17 (2017) 177–189 187 swietokrzyskie (7.62). this can result from relative higher specialization level. exeption is connected with malopolska, wielkopolska and warmiamasuria voivodships. the difference in volatility is also visible in the average amplitude dimensions. the highest amplitude of upward and downward deviations are characterized by regions with relative high sensitivity to business fluctuations, i.e. regions with upper than average specialization level. such features were observed in lower silesia, kujawy-pomerania, lublin, mazovia, malopolska, opole, podlasie, pomerania, wielkopolska and west pomerania. part of them (lublin, opole, pomerania) appear relative low standard deviation level, what can be interpreted as relative balanced average business cycle fluctuations. the others regional business cycles are characterized by more heterogeneous business fluctuations. conclusions business cycle analysis in regional dimension is crucial because of different regional development, and gdp structure. in the face of deepening regional divergence, it takes conducting adequate individual regional policy, stimulating balanced and sustainable development throughout the country. this is also important in the context of eu funds redistribution. if the process of economic policy decentralization will go on, knowledge about morphology and specificity of regional fluctuations enables to respond appropriately to regional changes in the economic situation. as regards the gdp cyclical fluctuations in polish regions in the 2000– 2016 it can be concluded, that the research issues are not uniform and clear. regions are in different extent susceptible to economy "shocks", both positive and negative. in comparison to industry production fluctuations elaborated in earlier papers, it can be concluded, that gdp fluctuations are more smooth. it is confirmed both by lower standard deviation and coefficient of variation. gdp fluctuations appear different turning points location, as well as the average upward, downward and cycle amplitude. despite a few exceptions it can be observed correlation between the level of regional specialization, and the degree of sensitivity to the economic disturbances. regions, that are less specialized and more diversified production structure, show greater resistance to economic fluctuations. this is confirmed by cycles morphology analysis on a regional basis. to sum up it can be stated, that the eu membership do not seem to have caused any negative effects on regional economies convergence. however, rafał warżała dynamic econometric models 17 (2017) 177–189 188 appropriate economic policies must be designated and implemented whenever economic cohesion risk being affected. references adamowicz, e., dudek, s., pachucki, d., walczyk, k. (2008), synchronization of the business cycle of polish economy with the euro area countries in the context of structure of these economies, publishing irg sgh, warsaw. anagnostou, a., panteladis, i., tsiapa, m. (2015), disentangling different patterns of business cycle synchronicity in the eu regions, empirica, 42(3), 615–641, doi: http://dx.doi.org/10.1007/s10663-014-9268-9. artis, m., zhang, w. (1997), international business cycles and the erm: is there a european business cycle?, international journal of finance and economics, 2(1), 1–16, barrios, s., lucio, j. (2003), economic integration and regional business cycles: evidence from the iberian regions, oxford bulletin of economics and statistics, 65(4), 497–515 carlino, g., sill, k. (2001), regional income fluctuations. common trends and common cycles, review of economics and statistics, 83(3), 446–456. correia, l., gouveia, s. (2013) business cycle synchronization at the regional level: evidence for the portuguese regions, regional and sectoral economic studies, 13–1, 91–108. crone, t.m. (2005), an alternative definition of economic regions in the united states based on similiarities in state business cycles, the review of economics and statistics, 87(4), 617–626. drozdowicz-bieć, m. (2012), cykle i wskaźniki koniunktury (business cycles and indicators), publisher poltext, warsaw. fatas, a. (1997), emu: countries or regions? lessons from the ems experience, european economic review, 41(3–5), 743–751. frankel, j., rose, a. (1996), the endogeneity of the optimum currency area criteria, national bureau of economic research working paper, 5700, 1–33, doi: http://dx.doi.org/10.3386/w5700. frenkel, m., nickel, c. (2002), how symmetric are the shocks adjustment dynamics between the euro area and central and eastern european countries? international monetary fund, imf working paper, 02/222, 1–27. gomez, v., marvall, a. (2001), seasonal adjustment and signal extraction in economic time series, (w:) pena d., tiao g.c., tsay, r.s. (eds.), a course in time series analysis, new york: j. wiley and sons. grudkowska, s., paśnicka, e. (2007), x-12-arima i tramo/seats – empiryczne porównanie metod wyrównania sezonowego w kontekście długości próby (x–12 – arima and tramo/seats – empirical comparison of seasonal alignment methods in the context of the length of the sample), polish national bank, department of public relations, warsaw. krugman, p. (1991), increasing returns and economic geography, journal of political economy, 99(3), 483–499. krugman, p. (1993), lessons of massachusetts for emu, (in:) torres, f., giavazzi, f. (eds.) adjustment and growth in the european monetary union, cepr and cambridge university press, 241–261. business cycles variability in polish regions in the years 2000–2016 dynamic econometric models 17 (2017) 177–189 189 marelli, e. (2007), specialization and convergence of european regions, the european journal of comparative economics, 4(2), 149–178. mazzi, g.l., scocco, m. (2003), business cycles analysis and related software applications, working papers and studies, european communities, luxembourg. mckinnon, r. (2002), optimum currency areas and the european experience, economics of transition, 10(2), 343–364. montoya, l.a., de haan, j. (2007), regional business cycle synchronization in europe?, bruges european economic research papers, college of europe, brugge, natolin. nillson, r., gyomai g. (2010), cycle extraction. a comparison of the phase-average trend method, the hodrick-prescott and christiano-fitzgerald filters, oecd, 23. statistical yearbook of the regions – poland, http://stat.gov.pl/obszary-tematyczne/rocznikistatystyczne/roczniki-statystyczne/rocznik-statystyczny-wojewodztw-2015,4,10.html. www.stat.gov.pl/ local data bank. weimann, m. (2003), oca theory and emu eastern enlargement – an empirical application, deutsche bank research working paper series, 8, 1–33. zmienność cykli koniunkturalnych w polskich regionach w latach 2000–2016 z a r y s t r e ś c i. celem badawczym artykułu jest ocena różnic w budowie morfologicznej regionalnych cykli koniunkturalnych w polsce. wyodrębnione cykle zostały opisane za pomocą takich parametrów, jak: długość cyklu, współczynnik koherencji, odchylenie standardowe, współczynnik korelacji krzyżowej, średnie przesunięcie cyklu. wyniki badań wskazują, że regiony cechują się różną podatnością na „szoki” ekonomiczne, zarówno pozytywne, jak i negatywne. ponadto analiza poziomu specjalizacji regionalnej oraz stopnia wrażliwości na fluktuacje gospodarcze wykazała występowanie pozytywnej korelacji w tym zakresie. s ł o w a k l u c z o w e: cykl koniunkturalny; region; zmienność. http://www.stat.gov.pl/ © 2017 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2017.005 vol. 17 (2017) 81−96 submitted november 6, 2017 issn (online) 2450-7067 accepted december 26, 2017 issn (print) 1234-3862 alicja ganczarek-gamrot, józef stawicki * comparison of certain dynamic estimation methods of value at risk on polish gas market a b s t r a c t. the paper compares the results of the estimation of var made using markov chains as well as linear and non-linear autoregressive models. a comparative analysis was conducted for linear returns of the daily value of the gas base index quoted on the day-ahead market (dam) of the polish power exchange (ppe) in the period commencing on january 2, 2014 and ending on april 13, 2017. the consistency and independence of the exceedances of estimated var were verified applying the kupiec and christoffersen tests. k e y w o r d s: var; markov chain; sarima models; garch models; back testing. j e l classification: c12, c58, g32. introduction accurate risk assessment in markets with dynamic volatility requires that real time positioning be monitored according to the frequency of observations. it is difficult in such a situation to base decisions taken in a short time horizon on the assumption that during the period under review the volatility of quotations is a sequence of independent random variables with the same distribution. in this paper, to estimate the volatility of the gas base index quoted on the day-ahead market (dam) of the polish power exchange (ppe) in the * correspondence to: józef stawicki, nicolaus copernicus university, faculty of economic sciences and management, 11a gagarina street, 87-100 toruń, poland, e-mail: stawicki@umk.pl; alicja ganczarek-gamrot, university of economics in katowice, faculty of informatics and communications, 3 bogucicka street, 40-287 katowice, poland, e-mail: alicja.ganczarek-gamrot@ae.katowice.pl. alicja ganczarek-gamrot, józef stawicki dynamic econometric models 17 (2017) 81–96 82 period from january 2, 2014 to april 13, 2017. value-at-risk was estimated using the following two dynamic approaches: markov chains and autoregressive models. the aim of the study is to evaluate and compare the efficiency of var estimation methods using the kupiec and christoffersen tests for compliance and independence of exceedances. 1. characteristics of gas prices in 2012 on the commodity futures market of the polish power exchange (ppe), commodity futures instruments for gas appeared, and on december 31, 2012 a gas spot market was launched, where since march 2013 continuous quotations of contracts for gas supply have been announced. figure 1.1 presents the time series of the gas_base index quoted from january 2013 (the beginning of the rdn gas operation) until april 2017. the gas_base index value corresponds to the average daily gas price [pln/mwh] from among all transactions concluded on a given day. the index is announced every day of the week including holidays. at the beginning of the introduction of gas contracts, apart from some exceptions, gas prices remained stable. it is only at the end of 2013 that changes in the level of gas prices may be observed, as well as the trend and the seven-day cyclicality. figure 1. the gas_base index [pln/mwh] quoted on the day-ahead market of the polish power exchange between 12 january 2013 and 13 april 2017 for further analysis a time series of daily return rates of the gas_base index was taken for the period from 01 of april 2014 to 13 of april 2017. 40,00 50,00 60,00 70,00 80,00 90,00 100,00 110,00 120,00 130,00 140,00 d a ta 2 0 1 3 -0 4 -2 1 2 0 1 3 -0 6 -0 3 2 0 1 3 -0 7 -1 3 2 0 1 3 -0 8 -2 2 2 0 1 3 -1 0 -0 1 2 0 1 3 -1 1 -1 0 2 0 1 3 -1 2 -2 0 2 0 1 4 -0 1 -3 0 2 0 1 4 -0 4 -0 1 2 0 1 4 -0 5 -1 1 2 0 1 4 -0 6 -2 0 2 0 1 4 -0 7 -3 0 2 0 1 4 -0 9 -0 8 2 0 1 4 -1 0 -1 8 2 0 1 4 -1 1 -2 7 2 0 1 5 -0 1 -0 6 2 0 1 5 -0 2 -1 5 2 0 1 5 -0 3 -2 7 2 0 1 5 -0 5 -0 6 2 0 1 5 -0 6 -1 5 2 0 1 5 -0 7 -2 5 2 0 1 5 -0 9 -0 3 2 0 1 5 -1 0 -1 3 2 0 1 5 -1 1 -2 2 2 0 1 6 -0 1 -0 1 2 0 1 6 -0 2 -1 0 2 0 1 6 -0 3 -2 1 2 0 1 6 -0 4 -3 0 2 0 1 6 -0 6 -0 9 2 0 1 6 -0 7 -1 9 2 0 1 6 -0 8 -2 8 2 0 1 6 -1 0 -0 7 2 0 1 6 -1 1 -1 6 2 0 1 6 -1 2 -2 6 2 0 1 7 -0 2 -0 4 2 0 1 7 -0 3 -1 6 gas base index comparison of certain dynamic estimation methods of value at risk… dynamic econometric models 17 (2017) 81–96 83 figure 2 presents a series of return rates for the gas base index. this series clearly shows periods of very low price volatility, i.e., periods of low risk of gas price changes, as well as periods of increased price volatility. figure 2. time series of return rates of the gas_base index in the period from 02 of january 2014 to 13 of april, 2017 the basic statistical analysis allows at the level of significance of 0.05 to reject the hypothesis that the distribution of returns of gas prices is a normal distribution. the distribution assessment should take into account such characteristics as asymmetry, thick tails and leptokurticity. 2. risk measurement – var the formal definition of var does not take into account the process nature of phenomena and focuses only on random variables: value-at-risk (var) represents such a loss of value that with the probability 1 will not be exceeded during a specified time period (jajuga, 2000):   )( varyyp ttt (1) where: )1,0( – set probability, t – specified duration time of the investment, -0,2 -0,15 -0,1 -0,05 0 0,05 0,1 0,15 0,2 0,25 2 0 1 4 -0 1 -0 2 2 0 1 4 -0 2 -1 8 2 0 1 4 -0 4 -1 3 2 0 1 4 -0 5 -2 3 2 0 1 4 -0 7 -0 2 2 0 1 4 -0 8 -1 1 2 0 1 4 -0 9 -2 0 2 0 1 4 -1 0 -3 0 2 0 1 4 -1 2 -0 9 2 0 1 5 -0 1 -1 8 2 0 1 5 -0 2 -2 7 2 0 1 5 -0 4 -0 8 2 0 1 5 -0 5 -1 8 2 0 1 5 -0 6 -2 7 2 0 1 5 -0 8 -0 6 2 0 1 5 -0 9 -1 5 2 0 1 5 -1 0 -2 5 2 0 1 5 -1 2 -0 4 2 0 1 6 -0 1 -1 3 2 0 1 6 -0 2 -2 2 2 0 1 6 -0 4 -0 2 2 0 1 6 -0 5 -1 2 2 0 1 6 -0 6 -2 1 2 0 1 6 -0 7 -3 1 2 0 1 6 -0 9 -0 9 2 0 1 6 -1 0 -1 9 2 0 1 6 -1 1 -2 8 2 0 1 7 -0 1 -0 7 2 0 1 7 -0 2 -1 6 2 0 1 7 -0 3 -2 8 alicja ganczarek-gamrot, józef stawicki dynamic econometric models 17 (2017) 81–96 84 ty – the present value at the moment t, tty  – random variable, the value at the end of the investment. the classical var valuation methods include the methods of variance – covariance, historical simulation, monte carlo simulation (jajuga, 2000b) the development of financial markets is accompanied by a rapid development of the var measurement theory. at present, in empirical financial studies of time series, which in most cases behave as non-stationary stochastic processes, var estimation uses dynamic methods based on garch models of conditional variance (piontek, 2002; doman, doman. 2009; fiszeder, 2009; trzpiot, 2010; pajor, 2010; ganczarek-gamrot, 2006). in this paper, we will compare the results of var estimation taking into account the methodology of stochastic processes and the theory of markov chains. if ty represents the value at time t, then var estimation is reduced to the estimation of the distribution quantile of returns t ttt t y yy z    . assuming that tz is a stochastic process of returns characterized by the effect of concentration of volatility, the quantile of order  can be estimated as follows (piontek, 2002; doman, doman, 2009): ttt fz    )( 1 (2) where: )( 1   f – quantile of order  of the standardized distribution allowed for in the estimation of conditional variance 2 t , 2 t – conditional variance of the process, t – expected value of the process tz , 3. methods of estimation of value at risk 3.1 markov chains markov chains are a well-known tool used in economics (see: ching, ng 2006; decewicz, 2011; podgórska et al., 2002; stawicki, 2004 and many others). the markov process with a discrete time parameter and a discrete phase space is referred to as markov chain. it is defined by a sequence of stochastic matrixes of the following form:   rrij tp   )((t)p , (3) comparison of certain dynamic estimation methods of value at risk… dynamic econometric models 17 (2017) 81–96 85 i.e., matrixes with positive elements and satisfying additional conditions in the form:   j ijit tp 1)( . (4) by denoting with td the vector of unconditional distribution of random variable ty , i.e.,  rtttt ddd ,,, 21 d , where }pr{ iyd tit  , (5) we determine the probability with which the process at time t reaches the phase state i. the components of the vector td satisfy the following conditions: 0 itit d , (6) and 1  i itt d . (7) the dependence between unconditional distributions of random variables ty and 1ty is expressed by the formula resulting from the theorem on the total probability )(1 ttt pdd   . (8) matrices   rrij tp   )((t)p reflect the mechanism of changes in the distribution of the analysed random variable ty over time. markov chain },{ ntyt  with phase space }...,,2,1{ rs  is called a homogeneous markov chain, if the conditional probabilities )(tpij of transition from phase i to state j within a time unit, i.e., in the time period from )1( t to t , do not depend on the choice of the moment t , that is ijijt ptp  )( . (9) in case of a homogeneous markov chain the dependence (8) and (9) take the following form: pdd  1tt . (10) due to the nature of the data characterising the phenomenon observed, we use microdata or macrodata – these are aggregated data. alicja ganczarek-gamrot, józef stawicki dynamic econometric models 17 (2017) 81–96 86 microdata are understood as observations of an object (or multiple objects) in successive time units as well as registers of the state of the object in a given time unit. observation of a change of state throughout the period t–1 to t allows us to apply the most reliable estimator taking the following form:       t t i t t ij ij tn tn p 2 2 )1( )( ˆ , (11) where:        ot herwise0 st at e in t he wasmoment at t he and st at e in t he was1moment at t heobject when t he1 )( jt it tnij     ot herwise0 st at e in t hemoment at t he object was when t he1 )( it tni this estimator has desirable consistency properties, asymptotic unbiasedness, and has an asymptotic normal distribution of expected value ijij ppe )ˆ( (12) and variance      t t i ijij ij tn pp p 2 )1( )1( )ˆvar( . (13) observation of macrodata, that is of the structure (unconditional decomposition vectors) in subsequent periods requires another apparatus that is not used in this article. the first proposal to apply markov chains to determine var was presented in stawicki's work (2016) while presenting another decision problem. this proposal is not fully satisfactory. the article is intended to compare the results obtained by means of the proposed method and the method is recognized in scientific literature. the idea of estimating var at a given moment using the markov chain model is based on the adequate construction of states. the states for the markov chain model are suitably selected intervals which may contain the return rate. comparison of certain dynamic estimation methods of value at risk… dynamic econometric models 17 (2017) 81–96 87 four states are required for the construction of markov chain. two of them play a special role. the first (marked as 1s ) is the state of threat, taking the form of the following interval: ),(1 vars  and the second – the state which contains the return at the present moment. 3szt  takes the form of the following interval ),[3 yxs  . the other two states complement the entire space of the return. the state 2 s is defined as one taking the form of the interval ),[2 xvars  , and the last state as the interval ),[4  ys value-at-risk is determined in accordance with the accepted rule, according to which the interval 1s is changed empirically and thus the interval 2 s , estimating at each change the matrix of the likelihood of transition to the moment when the likelihood of transition 31p in the matrix p is less than the assumed risk level (this work assumes )05.031 p . the construction of the markov chain described above and the estimation of its parameters, i.e., the elements of the transition matrix, is a model construction closely related to the observed return tz . for this observation, the state 3s is being constructed and an appropriate interval ),(1 vars  is searched. the size of the interval ),[3 yxs  is dictated by the amount of available information and thus by the possibility of estimating the parameter 31p . in this study, the interval )005,0,005,0[  tt zz was accepted for each observation where the standard deviation of the examined return amounted to .0339.0std by taking, for example, an observation of the return 0tz , the state 3s takes the form of the interval )005.0,005.0[3 s . the transition matrix (assuming the parameter )05.031 p takes the form: s1 s2 s3 s4 s1 0.1207 0.2241 0.0172 0.6379 p = s2 0.0622 0.3710 0.1866 0.3802 s3 0.0325 0.3862 0.2805 0.3008 s4 0.0365 0.3744 0.2169 0.3721 the state 1s is presented as the interval  0526.0,1 s thus indicating the value-at-risk = –0.0526. alicja ganczarek-gamrot, józef stawicki dynamic econometric models 17 (2017) 81–96 88 by determining value-at-risk in this way, we obtain a simple way of making var dependent on the value currently observed and taking the form of the function )(zvar . in the case of a white noise process, this function is constant at the set quantile value. for the studied process, the function )(zvar was evaluated in a parabolic form. the question remains, however, by how much the function )(zvar changes if we determine the interval 3s differently, and how this function is related to the type and parameters of the model generating returns. identification of such a function gives one a simple tool for determining var on a current basis. for the purposes of this article, this function is estimated as a quadratic polynomial. 5066.0 0474.0221.0274.7)( 2 2   r zzzvar this function is presented in fig. 3.1. figure 3. the var function based on the return figure 4 presents a selected part of a time series of returns (zt) and the estimated 05.0var for the one-day investment horizon using the theory of markov chains. -0,16 -0,14 -0,12 -0,1 -0,08 -0,06 -0,04 -0,02 0 -0,1 -0,05 0 0,05 0,1 0,15 comparison of certain dynamic estimation methods of value at risk… dynamic econometric models 17 (2017) 81–96 89 figure 4. the results of var estimation for the selected subperiod of 250 observations using markov chains 3.2 autoregressive models in order to compare the results obtained using markov chains, the var was determined applying the classical method by estimating the function approximating the behaviour of a series of returns and the use of the estimated model. the sarima (seasonal auto-regressive integrated moving average) models (p,d,q) (p, d, q) (brockwell, davis, 1996) are used to describe the level of phenomena shaping over time at high frequency of observation, in which autocorrelation and seasonality are used. tsts bqbqzbpbp )()()()( sd s s  , (14) where:    pp 11 1)(,1)( i i iss i i i bpbpbpbp ,    qq 11 1)(,1)( i i iss i i i bqbqbqbq , s – seasonal lag, d – order of series integration, tz – empirical values of series, b – transition operator stt s zzb  ,  – differential operator t s sttt s zbzzz )1(   , t – model residuals. -0,2 -0,15 -0,1 -0,05 0 0,05 0,1 0,15 0,2 0,25 1 1 0 1 9 2 8 3 7 4 6 5 5 6 4 7 3 8 2 9 1 1 0 0 1 0 9 1 1 8 1 2 7 1 3 6 1 4 5 1 5 4 1 6 3 1 7 2 1 8 1 1 9 0 1 9 9 2 0 8 2 1 7 2 2 6 2 3 5 2 4 4 return rate var alicja ganczarek-gamrot, józef stawicki dynamic econometric models 17 (2017) 81–96 90 the residuals t  of a linear autoregressive model do not meet the conditions of white noise and display a significant arch effect, therefore model (14) is complemented by a model allowing for heteroscedasticity of variance: ttt   . (15) for the purposes of this work, out of the numerous class of conditional variance models, we selected a model proposed by glosten, jagannathan and runkle (gjr) in 1993: 2 1 22 1 2 )( jt j jititiit i t s          pq i0 , (16) where: 0 – the value of unconditional variance of the process ( 00 a ), 0pq, and the remaining coefficients are non-negative,         01 00 i i its   , which allows for differences in when impacting variances, past negative values t  . among the models considered for the analysed time series – garch, egarch, aparch, igarch, figarch, fiegarch, fiaparch, gjr (osińska, 2006; fiszeder, 2009; trzpiot, 2010) the best fit to empirical data in the sense of the schwartz criterion (bic) was the gjr model with generalized error distribution (ged). table 1 presents the results of the sarima-gjr model parameter estimation for linear returns for the gas_base index in the time period 02.01.2014–13.04.2017. table 1. the sarima-gjr model parameter estimation parameter parameter estimation standard error t-student statistics p-value p(1) 0.7970 0.0502 15.8639 0.0000 q(1) 0.8905 0.0380 23.4505 0.0000 ps(1) 0.0697 0.0344 2.0229 0.0433 qs(1) 0.9207 0.0163 56.4169 0.0000 0  1.5087 0.7695 1.9610 0.0502 1  0.1760 0.0519 3.3900 0.0007 1  0.5678 0.1334 4.2550 0.0000  0.2310 0.0773 2.9870 0.0029 g.e.d.(df) 1.2288 0.0718 17.1200 0.0000 comparison of certain dynamic estimation methods of value at risk… dynamic econometric models 17 (2017) 81–96 91 the residuals t of the obtained model are characterized by absence of autocorrelation, compliance with ged distribution (figure 5) and absence of the arch effect (p-value = 0.87). acf zt : arima (1,0,1)(1,1,1) reszty ; p. ufności -1,0 -0,5 0,0 0,5 1,0 0 15 +,012 ,0290 14 +,007 ,0290 13 +,045 ,0290 12 -,011 ,0291 11 -,021 ,0291 10 -,024 ,0291 9 -,015 ,0291 8 -,030 ,0291 7 -,001 ,0291 6 +,007 ,0291 5 -,022 ,0291 4 +,018 ,0292 3 +,019 ,0292 2 +,044 ,0292 1 -,039 ,0292 opóźn kor. s.e 0 10,94 ,7571 10,76 ,7051 10,69 ,6367 8,25 ,7652 8,10 ,7047 7,55 ,6723 6,85 ,6531 6,57 ,5842 5,49 ,5999 5,49 ,4824 5,44 ,3645 4,90 ,2982 4,50 ,2122 4,09 ,1293 1,83 ,1762 q p figure 5. evaluation of sarima-gjr model adjustment to empirical series of returns figure 6 presents a selected part of a time series of returns (zt) and the estimated 05.0var for the one-day investment horizon using the theory of stochastic processes (var_sgjr). figure 6. the results of var estimation for a selected subperiod of 250 observations using sgjr -0,25 -0,2 -0,15 -0,1 -0,05 0 0,05 0,1 0,15 0,2 0,25 1 1 0 1 9 2 8 3 7 4 6 5 5 6 4 7 3 8 2 9 1 1 0 0 1 0 9 1 1 8 1 2 7 1 3 6 1 4 5 1 5 4 1 6 3 1 7 2 1 8 1 1 9 0 1 9 9 2 0 8 2 1 7 2 2 6 2 3 5 2 4 4 return rate var alicja ganczarek-gamrot, józef stawicki dynamic econometric models 17 (2017) 81–96 92 4. comparison the results in order to compare the obtained results of the var estimation we used back testing for the hit function   tt tt i  1 )(         )(0 )(1 )(    ttt ttt t varzdla varzdla i , (17) where: t – length of time series, ttz  – the stochastic process ttz  . by means of the following test:  number of var exceedances (proportion of failures test – pof) (kupiec, 1995),  independence of var exceedances (independence test – ind) (christoffersen, 1998). the test for the number of var exceedances (pof) verifies the following hypothesis:   varwh :0 against the alternative hypothesis   varwh :1 where:  – the order of var exceedances var w – the participation of var exceedances in the process of the considered returns. t k wvar    – the participation of var exceedances (k – the number of exceedances), in the series of the considered returns (tthe length of the series). assuming the truth of null hypothesis, the statistics (kupiec, 1995): comparison of certain dynamic estimation methods of value at risk… dynamic econometric models 17 (2017) 81–96 93                                        kkt kkt pof t k t k lr 1 )1( ln2  , (18) has an asymptotic distribution 2  with one degree of freedom. the test for independence of var exceedances (ind) verifies the following hypothesis: :0h var exceedances are independent against the alternative hypothesis :1h var exceedances are dependent to verify the null hypothesis, christoffersen proposed statistics using the markov chain idea:           1 11 00 10 0 1 10 11 00 0 11110101 )1()1( )1( ln2 kkkk kkkk ind wwww ww lr , (19) where: ijk – the number of periods in which jit )( on condition that iit  )(1  ; 10 ii ij ij kk k w   ; var w t k t kk w ˆ1101    , i, j= 0, 1. statistics (3.7) with the assumption of the truth of the null hypothesis has an asymptotic distribution 2  with one degree of freedom. table 2 shows the test results for the estimated var. the number of estimated vars using markov chains is equal to the length of the time series (t=1177). for the var obtained based on the results of the sgjr model, the loss of the first seven values (t=1170) is related to the seasonal variation of a series of return rates. for the analysed time series var0.05 estimation using markov chains gives an almost expected exceedances participation of 0.0535. furthermore, the high value of p = 0.0535 of the kupiec proportion of failures test shows no alicja ganczarek-gamrot, józef stawicki dynamic econometric models 17 (2017) 81–96 94 grounds for rejecting the null hypothesis. for a historical time series of exceedances, there was no single case of day-to-day var exceeding. table 2. results of var0.05 back testing var_m var_sgjr t 1177 1170 k 63 66 w 0.0535 0.0564 k00 1051 1045 k10 63 59 k01 63 59 k11 0 7 w00 0.9434 0.9466 w10 1.0000 0.8939 w01 0.0566 0.0534 w11 0.0000 0.1061 pof lr 0.3014 0.9736 p-value 0.5830 0.3238 ind lr x 2.6466 p-value x 0.1038 var0.05 estimated using the sarima-gjr model is slightly underestimated, the participation of exceedances in the examined series is 0.564, not significantly different from the expected (p-value = 0.3238 in the kupiec proportion of failures test). exceeding the so estimated var can be considered as independent (p-value = 0.1038 in christoffersen test). conclusions the obtained var estimation results are far better than var estimates based on monte carlo simulations without taking into account the dynamics of the observed phenomena and the strong autocorrelation observed during the time series (cf. ganczarek-gamrot, 2015). both methods have a great advantage over the classic approach to value-at-risk estimation. nevertheless, var estimated using markov chains based on the selected empirical series is closer to the correct estimation of loss measured by means of var. references brockwell, p. j., davis, r. a. (1996), introduction to time series and forecasting, springer – verlag, new york, doi: http://dx.doi.org/10.1007/978-1-4757-2526-1. ching, w., ng, m. k. (2006), markov chains models, algorithms and applications, springer science+business media. http://dx.doi.org/10.1007/978-1-4757-2526-1 comparison of certain dynamic estimation methods of value at risk… dynamic econometric models 17 (2017) 81–96 95 christoffersen, p. (1998), evaluating interval forecasts, international economic review, 39, 841–862, doi: http://dx.doi.org/10.2307/2527341. decewicz, a. (2011), probabilistyczne modele badań operacyjnych, oficyna wydawnicza sgh, warszawa. doman, m., doman, r. (2009), modelowanie zmienności i ryzyka, wolter kluwer polska, kraków. fiszeder, p. (2009), modele klasy garch w empirycznych badaniach finansowych, wydawnictwo naukowe umk, toruń. ganczarek, a. (2006), wykorzystanie modeli zmienności wariancji garch w analizie ryzyka na rdn, prace naukowe ae w katowicach: modelowanie preferencji a ryzyko’06 (ed. trzaskalik t.), 357–371. ganczarek-gamrot, a. (2015), porównanie metod estymacji var na polskim rynku gazu, studia ekonomiczne. zeszyty naukowe uniwersytetu ekonomicznego w katowicach, 219, 41–52. glosten, l. r., jagannathan, r., runkle, d. e. (1993), on the relation between expected value and the volatility of the nominal excess return on stocks, journal of finance, 48, 1779–1801, doi: http://dx.doi.org/10.1111/j.1540-6261.1993.tb05128.x. jajuga, k (2000), ryzyko w finansach. ujęcie statystyczne. współczesne problemy badań statystycznych i ekonometrycznych, ae, kraków, 197–208, kupiec, p. (1995), techniques for verifying the accuracy of risk management models, journal of derivatives, 2, 173–184. osińska, m. (2006), ekonometria finansowa, polskie wydawnictwo ekonomiczne, warszawa pajor, a. (2010), wielowymiarowe procesy wariancji stochastycznej w ekonometrii finansowej, ujęcie bajesowskie, wydawnictwo uniwersytetu ekonomicznego w krakowie, kraków. piontek, k. (2002), pomiar ryzyka metodą var a modele ar-garch ze składnikiem losowym o warunkowym rozkładzie z „grubymi ogonami”, rynek kapitałowy. skuteczne inwestowanie, 467–483. podgórska, m., śliwka, p., topolewski, m., wrzosek, m. (2002), łańcuchy markowa w teorii i w zastosowaniach, oficyna wydawnicza sgh, warszawa. schwarz, g. (1978), estimating the dimension of a model, the annals of statistics, 6, 461–464, doi: http://dx.doi.org/10.1214/aos/1176344136. stawicki, j. (2004), wykorzystanie łańcuchów markowa w analizie rynku kapitałowego, wydawnictwo umk, toruń. stawicki, j. (2016), using the first passage times in markov chaine model to support financial decisions on stock exchange, dynamic econometric models, 16, 37–47. trzpiot g., (2010): wielowymiarowe metody statystyczne w analizie ryzyka inwestycyjnego, pwe, warszawa. [www 1] www.polpx.pl. porównanie wybranych dynamicznych metod estymacji var na rynku gazu w polsce z a r y s t r e ś c i: w pracy porównano wyniki estymacji wartości zagrożonej var oszacowanej przy wykorzystaniu łańcuchów markowa oraz modeli autoregresyjnych liniowych i nieliniowych. analizę porównawczą przeprowadzono dla liniowych stóp zwrotu wartości dziennego indeksu gas_base notowanego na rynku dnia następnego (rdn) towarowej http://dx.doi.org/10.2307/2527341 http://dx.doi.org/10.1111/j.1540-6261.1993.tb05128.x http://dx.doi.org/10.1214/aos/1176344136 alicja ganczarek-gamrot, józef stawicki dynamic econometric models 17 (2017) 81–96 96 giełdzie energii (tge) w okresie od 2 stycznia 2014 roku do 13 kwietnia 2017 roku. zgodność i niezależność przekroczeń oszacowanych wartości var zweryfikowano testem kupca oraz christoffersena. s ł o w a k l u c z o w e: var, łańcuch markowa, modele sarima, modele garch, analiza wsteczna. © 2017 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2017.008 vol. 17 (2017) 129−145 submitted november 28, 2017 issn (online) 2450-7067 accepted december 28, 2017 issn (print) 1234-3862 aneta włodarczyk * regime-dependent assessment of the european union aviation allowances price risk  a b s t r a c t. in this article the european union aviation allowances (euaa) price risk, associated with the activity of aircraft operators within the european economic area (eea), has been evaluated across the low and high volatility periods occurring on the carbon permits market. it is found that markov-switching heteroscedasticity models distinguish well between two volatility regimes, as well as three volatility regimes on the euaa futures market, and that the assessments of euaa price risk are clearly different in the regimes. these findings may be explained by the european union emission trading scheme (eu ets) design and the changes in both the eu climate policy rules and global regulations in the scope of co2 emissions by international aviation. k e y w o r d s: european union aviation allowances; eu emission trading scheme; markov-switching model; risk. j e l classification: c40; g32; l93; q53. introduction worrying statistics concerning the over 3% share of co2 emission from international aviation bunkers in the total carbon dioxide emission in the eu countries in 2004 and the forecasts concerning an increase in greenhouse gases emission in international aviation by about 70% in 2020 compared with 2005 led the international community to take interest in the issues of * correspondence to: aneta włodarczyk, czestochowa university of technology, faculty of management, 19b armii krajowej street, 42-200 częstochowa, poland, e-mail: aneta.w@interia.pl.  this work was financed by czestochowa university of technology. aneta włodarczyk dynamic econometric models 17 (2017) 129–145 130 limiting the negative impact of air transport on the natural environment. in 2008 the european commission adopted the directive on including civil aviation into the eu emission trading scheme (eu ets), which refers to the eu’s long-term policy aimed at limiting the greenhouse effect, improvement in the environment quality, increased energy efficiency and growth of renewable energy sources share in the energy consumption structure (directive 2008/101/we). aircraft operators that in the given year carry out aviation operations covered by the attachment 1 to the directive 2003/87/we in the territory of european economic area have been included into the eu ets. the period of civil aviation participation in the eu ets has been divided into the two settlement periods: the first one of them comprising only the year 2012, the aim of which was to adjust the aircraft operators functioning to the functioning scheme of the remaining sectors in the allowances trading system, and the second one (common for all the sectors covered by the eu ets) for the years 2013-2020. the eu ets system operates according to the “cap and trade” principle. a permissible limit of co2 emissions in the given period (“cap”) has been determined for all aircraft operators included into the system, which is gradually decreased over time. in order to asses this limit, data on average annual co2 emission in aviation was used, which covered the reference period of 2004-2006 and came from the european organisation for the safety of air navigation (eurocontrol). in the first settlement period the total amount of european union aviation allowances (euaas) to be allocated for aircraft operators was 97% of the average calculated from historical aviation emissions in the reference period. in the second settlement period the joint annual number of allowances granted to the aviation sector decreased to 95% of the same average emission. within the established limit, in the first settlement period 85% of allowances allocated to cover the annual co2 emission of civil aviation will be granted to aircraft operators free of charge, and 15% of aviation allowances will be sold in the auctioning system. in the period from 1 st january 2013 to 31 st december 2020 in turn, 82% of aviation allowances will be allocated free of charge, 15% will constitute allowances purchased at auctions, and 3% will be moved to a special reserve for new aircraft operators. chin and zhang (2013) presented mathematical formulas and described in detail the method of emission allowances allocation consistent with the directive 2008/101/ec and proposed an alternative method of allowances allocation (the augmented eu ets), which considers energy efficiency of aircraft operators (chin and zhang, 2013). the european commission regulation no 601/2012 imposes on aircraft operators the following duties, which result from their participation in the eu ets sysregime-dependent assessment of the euaa price risk dynamic econometric models 17 (2017) 129–145 131 tem: the necessity to monitor carbon dioxide emission in each year, submitting a verified report on historical carbon dioxide emission to the proper body and its settlement through redemption of a proper amount of emission allowances in the eu redemption registry. aircraft operators may settle their own co2 emissions by means of emission allowances assigned to stationary installations (european union allowances euas) or aviation allowances (euaas). moreover, aircraft operators should possess a plan to monitor annual emissions, approved by the appointed in the given member country competent body. there are sanctions foreseen that can be applied to operators that fail to settle their co2 emissions in the given year, including a prohibition on flights to the european union. aircraft operators also has a possibility to apply for allocation of free co2 emission allowances for the years 2013–2020 from a special reserve, provided that they complied with the obligation to monitor tonne-kilometres throughout 2014. such a system of co2 emission allowances allocation was supposed to be an incentive to invest in environmentally-friendly technologies and modernise the air fleet against an alternative to incur additional costs of purchasing allowances. the emission allowances trading system was considered by the european commission to be the most effective and least expensive instrument to limit the emission of greenhouse gases in aviation in the territory of the eu member states, yet, it encountered criticism. this concerned additional costs generated by the eu ets for the aircraft operators participating in it, which was connected with, among others, purchases of missing euaas, new environmentally-friendly investments, administrative costs. the growing fears of aircraft operators included into the eu ets concerned the loss of their competitive position due to their lower market share, change of entry barriers or lower profits margin (meleo et al., 2016). the most severe objection concerned imposing additional charges on aircraft operators from outside the eu. the charges resulted from participation in the eu ets and were imposed without any prior agreements, which was treated as a breach of the chicago convention on international civil aviation (1944). in response to this the european commission introduced derogating mechanisms: „stop the clock” derogation (decision 2013/337/eu), the exclusions mechanism in the scope of aviation operations covered by the eu ets system (regulation 2014/421). the abovementioned legislative regulations in the scope of obligations concerning reporting the emission from aviation operations within the eu ets in the years 2013–2014, postponing the deadline to settle these emissions for 2013 and 2014, delayed the launch of aviation allowances aneta włodarczyk dynamic econometric models 17 (2017) 129–145 132 trading in the auctioning system. 1 in addition, they influenced the volatility of euaas prices and the volume of euaas derivatives trading, as well as the interest of aircraft operators in the use of this type instruments to manage the co2 emission risk. the inclusion of international aviation into the eu emission trading scheme causes that the management of the euaa price risk is becoming increasingly important for companies covered by the eu ets. the increase in the price allowances volatility in the secondary market and the reduction of free allowances, which are granted to companies, cause the increase in their exposure to price risk. the aim of this article is to evaluate the euaa price risk connected with the activity of aircraft operators within the european economic area, distinguishing between low and high volatility periods occurring on the carbon permits market. volatility and downside risk measures are used to assess this type of risk across different regimes of the euaas' future prices, which are identified by means of markov-switching models. 1. research methodology the co2 emission allowances price risk from the aircraft operators point of view may result in a danger of not achieving by them the expected returns due to a sale of excessive or purchase of insufficient aviation allowances on the secondary market (neutral risk concept). the aviation allowances price risk may be perceived as a danger of sustaining a loss, which can have significant impact on financial results of the aircraft operator (negative risk concept). volatility measures are the tools which are most frequently used to measure risk according to the neutral risk concept, while risk measurement in the negative meaning are conducted according to the downside risk measure (jajuga, 2007). the most frequently determined by practitioners volatility measures include: standard deviation, interquartile range and absolute median deviation. the downside risk measure, which exposes only unfavourable situations for the aircraft operator when the real returns on sale or purchase of aviation allowances was below the average, is semi-standard deviation. the first volatility measures, which have been estimated on the basis of settlement prices of euaas futures contracts, is standard deviation (kuziak, 2011): 1 aviation auctions have been conducted since 1st january 2015. regime-dependent assessment of the euaa price risk dynamic econometric models 17 (2017) 129–145 133      t t t rr t 2 2 )( 1 1  , (1) where: %100*)/ln( 1 ttt ppr – return on sale/purchase of aviation allowance on the secondary market in t period; pt – settlement price of euaa in t period; pt–1 – settlement price of euaa in the period t–1(t = 2,.., t);  – standard deviation; tr – expected value of euaa returns. coefficient of variation is also used in order to assess how much risk accompanies one unit of profit from the investment in the euaa futures market. the lower the value of this measure is, the higher the expected profit on sale/purchase of aviation allowances is at the defined risk level or lower risk at the defined level of profit: %100 r vs   (2) where sv is coefficient of variation. asymmetric risk measure is semi-standard deviation, which can be defined in the following way (booth et al., 2005):    t t td t sv 2 2 1 1 , (3) for       rrr r d t t t t rfor rfor 0 , (4) where sv – semi-standard deviation. this measure reflects an unfavourable for aircraft operators situation, where the returns on sale/purchase of euaas on the secondary market were below their expectations. robust volatility estimators in turn are characterized by the fact that present in the sample outliers have little impact on the estimation results (trzpiot, 2010). this is a very important property for euaas return series, where single outliers occur, which confirms extremely high change of aviation allowances price form period to period. the most frequently applied robust volatility estimator is absolute median deviation, which can be described by means of the following relation (trzpiot, 2010): aneta włodarczyk dynamic econometric models 17 (2017) 129–145 134 }}{{)( i i t t t rmedianrmedianrmad  , (5) where mad(rt) – absolute median deviation of euaa returns. one of the simplest robust volatility estimators is interquartile range (iqr), which ignores up to 25% of lowest and largest returns (trzpiot, 2010): )()( 13 tt rqrqiqr  , (6) where )(1 trq and )(3 trq – respectively the first and third quartile of euaa returns empirical distribution. it is worth stressing that risk has been most often measured under the assumption of the worstscenario in the market, what has been derived from the negative concept of risk and has contributed to the popularization of the downside risk measures. however, this approach may not reflect the change of uncertainty sets with respect to different market environments (e.g. calm and turbulent periods). following liu and chen (2014), in this article markov regime switching models are used to describe the time-varying uncertainty set of the first and second order moments, which are related to two main characteristics of investments in the euaa futures, namely expected profits and risk. as a result, markov-switching dynamic regression models with nregimes (ms(n)-dr(p)) have been used to describe the different dynamism of the euaas returns series, generated by changing risk factors on the market of co2 emission allowances (hamilton, 1990; doornik, 2013): )),(,0(~ ,)(...)()()( 2 t 22110 t tpttptttttt sn rsrsrssr     (7) where: i(st) – the parameter describing the influence of delayed by i-periods euaa returns per the current returns (i=1,2,…,p), t – the error term, )( 2 ts residual variance dependent on the valid at the given moment regime, st – non-observable variable modelled as homogeneous markov chain of n regimes and the transition probabilities matrix   }1,...,1,0{,|   njiji pp . the elements of stochastic matrix p, defining the process probability transition from j regime at t–1 moment to i regime at the t moment, satisfy the markov property (hamilton, 1990): )|( ),...,,|( 1 00221| jsisp isisjsispp tt ttttji     , (8) regime-dependent assessment of the euaa price risk dynamic econometric models 17 (2017) 129–145 135 means that probability of moving to the next regime depends only on the current regime of a system and not on the previous regimes. therefore, the transition probability matrix can be defined in the following way (doornik, 2013):                              111 1 1 0 110 1|11|10|11 1|11|10|11 1|01|00|01       nnnnt nt nt ttt pppns ppps ppps nsss p , (9) at the conditions that guarantee the stochastic structure of this matrix:     1 0 j|i| .,for 0p ,1 n i ji n-10,1,...i,j p (10) stawicki (2004) describing this class of econometric models introduced the concept of a dual stochastic process to emphasize the existence of an unobservable variable that has controlled the changes of regimes in addition to the observable economic variable being the subject of modelling (stawicki, 2004). moreover, on the basis of estimated transition probabilities (elements of stochastic matrix p), the further expected duration of the system in a given regime can be determined (hamitlon, 1990): )( 1 1 | n-1 0, 1,...,i p d ii i    , (11) where: di – average duration of euaas returns in i-th regime. it is also worth stressing that the model (7) considers the heteroscedasticity characteristic for the euaas returns through introduction of markov switching also in the residual variance. the most frequently applied parameter estimation method of markov switching model is the maximum likelihood method (ml), which makes use of fsqp algorithm (feasible sequential quadratic programming) (lawrence and tits, 2001; psaradakis and sola, 1998). a by-product of the markov-switching model parameter estimation is a sequence of smoothed probabilities )|( tt jsp  which make it possible to identify the moment of process switching between the particular volatility regimes. these probabilaneta włodarczyk dynamic econometric models 17 (2017) 129–145 136 ity inferences allow to draw conclusions about the euaa returns process being in a particular regime, although the regime variable st is unobserved. on this basis, the time series of the euaas returns have been divided into observations generated in different volatility regimes, and then risk measures (1)–(6) have been estimated for each sub-sample. additionally, regime classification measure (rcm) has been used to determine the number of regimes in markov switching models (ang and bekaert, 2002):       t t n i tt isp t nnrcm 1 1 0 1 2 ))|(( 1 100)( , (12) where: )|( 1 tt isp – filter regime probabilities series (i= 0,1,…,n–1 and t= 1,2,…,t), 1t – information set available up to time t–1. alternatively, ang and bekaert (2002) suggested to replace the filter regime probabilities by the smoothed probabilities over the entire sample ( )|( tt jsp  ) in (12). for the two-regime model the rcm statistics ranges from 0 to 100 and smaller value of this measure means better regime classification. high values of the rcm may indicate that the markov switching model cannot successfully distinguish between regimes from the behavior of the data. it may point at misspecification of the number of regimes (ang and bekaert, 2002). 2. data and empirical results intercontinental exchange futures europe in london (ice futures europe), adjusting itself to the changing regulations in the scope of the eu climate policy, introduced into trade the euaa futures contracts in february 2012. this product was dedicated primarily to aircraft operators as an instrument of co2 emission risk management, due to a significant increase in their exposition to this type of risk after civil aviation had been included into the eu ets. hedging operations with the use of aviation allowances futures should gain in popularity as the international law provisions are being tightened in the scope of carbon dioxide emission in civil aviation and the principles of the eu ets functioning towards introducing an auctioning system as a basic form of acquiring emission allowances by aircraft operators. an important stage of co2 emission risk management process in the civil aviation sector is the measuring price risk of euaa futures contracts. in the empirical research two types of risk measures have been used for this purpose: volatility risk measure and downside risk measure, which have been estiregime-dependent assessment of the euaa price risk dynamic econometric models 17 (2017) 129–145 137 mated on the basis of weekly returns of euaas futures prices listed on the ice futures in the period from 04.03.2012 to 01.10.2017. 2 the shaping of weekly settlement prices and returns for constructed benchmark series for euaas futures have been presented in fig. 1. figure 1. weekly prices of the euaa futures [eur/tco2] (upper panel) and their returns [%] (lower panel) quoted in the ice futures europe in the period 4.03.2012 – 01.10.2017 table 1. ms( n)-ar(4) models for the euaas returns parameter ms(2)-dr(4) ms(3)-dr(4) regime 0 regime 1 regime 0 regime 1 regime 2 constant 0.696 [0.049] –1.104 [0.350] 0.726 [0.045] –20.529 [0.002] 0.358 [0.654] ar-4 –0.179 [0.019] –0.129 [0.234] –0.212 [0.013] –2.288 [0.001] –0.050 [0.546] sigma 4.113 (0.420) 11.356 (1.090) 3.820 (0.357) 11.089 (3.919) 8.393 (0.808) matrix p r(0, t) r(1, t) r(0, t) r(1, t) r(2, t) r(0, t+1) 0.9349 0.1077 0.9314 0.0000 0.0739 r(1, t+1) 0.0651 0.8923 0.0000 0.2092 0.0470 r(2, t+1) – – 0.0686 0.7908 0.8791 di (in weeks) 22.38 15.43 22.43 1.40 9.46 returns assigned to regime 62.37% (179returns) 37.63% (108returns) 54.70% (157returns) 2.44% (7 returns) 42.86% (123returns) note: p – transition probabilities matrix, standard errors of parameter estimates in parenthesis, p-value in brackets. markov switching heteroscedastisity models (ms(n)-dr(p) for n = 2,3 and p = 1, 2, 3, 4) (7) were estimated to describe the dynamism of the euaas 2 the research uses the december prices of euaa futures quoted on the ice futures europe, presented on the website https://www.quandl.com (access 18.10.2017). aneta włodarczyk dynamic econometric models 17 (2017) 129–145 138 returns series in various volatility regimes. the best results have been presented in table 1. 3 two regimes have been distinguished in the first approach: the low volatility regime (regime 0) that describes the period of calm on the euaas futures market and the high volatility regime (regime 1) characterized by turbulent changes on this market, which are generated mainly by changes to the regulations concerning co2 emission in the aviation sector. the volatility parameter estimated for the regime 1 (11.356) is almost three times higher than the parameter describing the volatility in regime 0 (4.113). the differences between the regimes can be observed also for the parameter describing the expected profit from the euaas futures transactions, which in the low volatility regime is positive and in the high volatility regime negative. moreover, each of the regimes is rather stable, as the estimated transition probabilities of indicating a chance of euaas returns process to remain in the given regime in the next period are relatively high (they amount respectively 0.9349 and 0.8923). therefore, the low volatility regime on the euaas futures market lasts for about 22 weeks, while the high volatility regime lasts on average for over15 weeks. in the second approach three regimes have been distinguished: the low volatility regime (regime 0), the high volatility regime (regime 2) and „spiky” regime (regime 1) for which the standard deviation of the error term took the highest value (11.089) compared to the low and high volatility regimes (respectively 3.82 and 8.393). the average profit from euaas futures transaction is highest in the low volatility regime. the spiky regime is a transitional one, which means that there is a great chance (0.7908) that in the next period it will be replaced by the high volatility regime. only single euaas returns have been assigned to this regime, which indicate extreme changes of euaas futures prices connected with the structural changes in the system of aviation allowances trade. table 2 shows the results of diagnostic tests on standardised residuals from each model, which allow for positive verification of white noise properties for residuals series. the akaike information criterion demonstrates the model of three regimes to be better adjusted to the euaas returns series, 3 basic descriptive statistics and diagnostic tests verifying the presence of structural break occurrence, autocorrelation, volatility clustering, leptokurtosis effects have been determined for both presented in figure 1 variables. the results of conducted diagnostic tests justify the use of markov switching heteroscedastisity models (7) to describe the dynamism of the euaas returns series in various volatility regimes. their results are available upon request from author. regime-dependent assessment of the euaa price risk dynamic econometric models 17 (2017) 129–145 139 while the schwarz criterion reaches the lowest value for models of two regimes. table 2. diagnostic tests for scaled residuals from ms( n)-ar(4) models test ms(2)-dr(4) ms(3)-dr(4) aic sc 6.707 6.809 6.675 6.841 b-p(20) 21.120 [0.390] 18.492 [0.555] arch(1–5) 0.970 [0.437] 1.695 [0.136] j-b 5.614 [0.061] 4.009 [0.135] lr 80.132 [0.000] 99.299 [0.000] rcm 33.547 0.350 note: b-p(k) – box-pierce test of serial correlation up to order k, arch(1-q) – engle’s arch test for heteroscedasticity up to lag q, j-b – jarque-bera test for normality, lr – likelihood ratio test for nonlinearity, rcm – regime classification measure, aic – akaike information criterion, sc – schwarz information criterion, p-value in brackets. it is worth stressing that for both markov-switching heteroscedasticity models, the davies (1987) upperbound for the p-value of the lr test of linearity strongly rejects the linear model (doornik, 2013). the calculated values of the rcm statistics are rather low in the case of each specification of markov switching model, what indicates the correct classification of regimes in the estimated models. figure 2. weekly euaas returns (upper panel), smoothed probabilities for regime 0 (middle panel) and smoothed probabilities for regime 1 (lower panel) in the period 4.03.2012 – 01.10.2017 aneta włodarczyk dynamic econometric models 17 (2017) 129–145 140 figure 3. weekly euaas returns and smoothed probabilities for regime 0 (upper panel), smoothed probabilities for regime 1 and regime 2 (lower panel) in the period 4.03.2012 – 01.10.2017 on the basis of obtained in the estimation procedure values of smoothed probabilities, the moments of process switching between particular volatility regimes were estimated. then, the euaas returns series was divided into two sub-samples generated by the low and high volatility regimes (see fig. 2) or three sub-samples generated by the regimes of low and high volatility and the spiky regime (see fig. 3). one can observe that in each case the majority of euaas returns have been assigned to the low volatility regime (respectively: 62.37% and 54.70%). however, the sub-sample connected with the high volatility regime is relatively numerous, which is a characteristic phenomenon on the co2 emission allowance market (sanin et al., 2015). in each sub-sample associated with the given volatility regime measures of profit and risk have been determined (1)–(6) (see table 3). the regime comprising turbulent changes occurring on the market of co2 emission allowances in aviation, the source of which is primarily uncertainty accompanying the changes of the eu climate policy and global regulations in the scope of pollutant emission by this sector, is characterized by much higher values of volatility and downside risk measures compared to regime 0. focusing on the neutral concept of risk, one can see that the ratio between standard deviations estimated in regime 1 and 0 amounts to almost three. having rejected the outliers in the process of risk measurement, the absolute median deviation in regime 1 is near three times higher than in regime 0. the same ratio is obtained when semi-standard deviations in high and low volatility regimes are compared, according to negative concept of risk. aircraft operators securing themselves against the euaas price risk by means of futures regime-dependent assessment of the euaa price risk dynamic econometric models 17 (2017) 129–145 141 contracts in the high volatility periods could on average lose 0.995% per week, and in the low volatility periods they could on average earn 0.621%. table 3. risk measurement for the euaas futures transactions in particular regimes statistic ms(2)-dr(4) ms(3)-dr(4) regime 0 regime 1 regime 0 regime 1 regime 2 minimum –13.246 –48.791 –8.049 –48.791 –15.214 mean 0.621 –0.995 0.710 –22.166 0.421 median 0.589 –1.017 0.692 –22.816 0.248 maximum 10.828 23.660 8.701 20.516 23.660 standard deviation 4.189 11.708 3.995 21.907 8.573 absolute median deviation 2.701 7.835 2.556 5.776 6.322 iqr 5.291 15.572 4.905 13.684 12.262 semi standard deviation 2.982 8.874 2.734 12.134 5.835 note: iqr – interquartile range, the coefficient of variation is not calculated because of negative values of expected return in some regimes. in the half of the weeks corresponding with the high volatility regime on the futures market, it was possible to lose at least 1.017%. in the half of the weeks assigned to the low volatility regime, euaas returns were not lower than 0.589%. similar conclusions can be formulated for the 3-regime markov switching model, when the estimated risk measures were compared for the high and low volatility regime. the risk accompanying the transactions concluded on the derivatives market was very high in regime 1 (21.907% according to the neutral risk concept, 12.134% according to the negative concept). in the spiky regime the volatility estimated with the means of the robust estimator which is absolute median deviation, amounts 5.776% and is significantly lower than other volatility measures. considering also the average value (–22.166%) and median (–22.816%) one can assume that the majority of observations assigned to regime 1 constitute extremely large, negative euaas returns. the results presented in table 3 indicate significant differences in the level of risk accompanying the transactions concluded on the euaas futures market, depending on the presence of the volatility regime in the given period. therefore, markov switching models may constitute a useful tool that allows to identify the moment of switching between regimes or determine the average lasting time of particular volatility regimes on the aviation allowances secondary market. conclusions decisions concerning tightening the eu climate policy in the scope of greenhouse gases emission from aviation and modifying the principles of the aneta włodarczyk dynamic econometric models 17 (2017) 129–145 142 eu ets functioning will affect competitiveness of aircraft operators performing flights within the european economic area in the years 2012–2020. therefore, it is important that the system of co2 emission risk management is adjusted to individual needs of a given aircraft operator and targeted to implement eco-innovations, making it possible not only to comply with legal regulations concerning the size of co2 emission but also to earn on the growth of fuel efficiency and transportation process optimization by the operator (ko et al., 2017). moreover, the constant control over the co2 emissions in order to the maintenance it below the upper allowed threshold and sale at an attractive price the surplus of co2 emission allowances leads to strengthening the competitive advantage of aircraft operators. by including civil aviation to the eu ets system the legislators wanted to implement the principle “polluter pays”, yet, the economic practice has shown that similarly to the energy sector, aircraft operators started gradually transfer the cost of participation in the eu ets on the customers. thanks to including the cost of co2 emission into the price of tickets, despite received free of charge share of aviation allowances, the operators recorded windfall profit. making use of this option depends on the elasticity of the demand for aviation services, offers of competitive air carriers, state regulations of air transport market in the scope of access to both domestic transports as well as on international markets (concessions, certificates) (tłoczyński, 2015; dyduch, 2013). due to the fact that aircraft operators have to make decisions on the way they use aviation allowances, the measurement of euaas price risk on the secondary market is of vital importance in this decision-making process. if the marginal profit from the sale of aviation service is higher than the market value of euaa allowances, the aviation allowances will be used to cover co2 emissions derived from the air operations (dyduch, 2013). the paper shows the changes in the euaas price risk level depending on the volatility regime in force on the aviation allowances secondary market. on the basis of estimated markov switching models the author has identified two or three volatility regimes, for each different risk measures have been determined. the high volatility regime, in which the risk accompanying the sale and purchase transactions of euaas futures was several time higher than in the low volatility regime, has been assigned to the periods when important regulations in the aviation sector were introduced. the periods assigned to the high volatility regime corresponded with such events as: the end of the first settlement period for the aviation sector in the eu ets (2012-11-25–2013-01-13), conducting the first settlement of historical co2 emissions in civil aviation for the year 2012 and publishing the decision of the european parliament and council on implementing the „stop the regime-dependent assessment of the euaa price risk dynamic econometric models 17 (2017) 129–145 143 clock” mechanism (2013-04-21, 2013-04-28–2013-06-23), publishing the regulation 421/2014 on further derogation in aviation (2014-04-06–201405-25), works on concluding a global agreement regarding implementation of the carbon offset and reduction scheme for international aviation, corsia) (2016-06-19–2016-12-25). such an approach makes it possible to measure more precisely the risk that accompanies the sale/purchase transactions of aviation allowances on the secondary market. it also allows diversification of the adopted by aircraft operators strategies of securing themselves against the risk of co2 emission allowances price depending on the price volatility regime in force. references ang, a., bekaert, g. (2002), regime switches in interest rates, journal of business & economic statistics, 20(2), 163–182, doi: http://dx.doi.org/10.1198/073500102317351930. booth, p., chadburn, r., haberman, s., james, d., khorasanee, z., plumb, r., rickayzen, b. (2005), modern actuarial theory and practice, crc press, boca raton. chin, a. t. h., zhang, p. (2013), carbon emission allocation methods for aviation sector, journal of air transport management, 28, 70–76, doi: http://dx.doi.org/10.1016/j.jairtraman.2012.12.013. commission regulation (eu) no 601/2012 of 21 june 2012 on the monitoring and reporting of greenhouse gas emissions pursuant to directive 2003/87/ec of the european parliament and of the council, http://eur-lex.europa.eu/eli/reg/2012/601/oj (12.09.2017). decision no 377/2013/eu of the european parliament and of the council of 24 april 2013 derogating temporarily from directive 2003/87/ec establishing a scheme for greenhouse gas emission allowance trading within the community, http://eurlex.europa.eu/eli/dec/2013/377(1)/oj (12.09.2017). directive 2008/101/ec of the european parliament and of the council of 19 november 2008 amending directive 2003/87/ec so as to include aviation activities in the scheme for greenhouse gas emission allowance trading within the community, http://eurlex.europa.eu/eli/dir/2008/101/oj (12.09.2017). doornik, j. a. (2013), econometric analysis with markov switching models. pcgive tm 14, vol. 4, timberlake consultants ltd., london. dyduch, j. (2013), handel uprawnieniami do emisji zanieczyszczeń powietrza (trading of air pollution emission allowances), pwe, warszawa. hamilton, j. d. (1990), analysis of time series subject to changes in regime, journal of econometrics, 45(1–2), 39–70, doi: http://dx.doi.org/10.1016/0304-4076(90)90093-9. jajuga, k. (2007), teoretyczne podstawy zarządzania ryzykiem (theoretical foundations of risk management), in jajuga k. (ed.), zarządzanie ryzykiem (risk management), pwn, warszawa. ko, y. d., jang, y. j., kim, d. y. (2017), strategic airline operation considering the carbon constrained air transport industry, journal of air transport management, 62, 1–9, doi: http://dx.doi.org/10.1016/j.jairtraman.2017.02.004. http://dx.doi.org/10.1198/073500102317351930 http://dx.doi.org/10.1016/j.jairtraman.2012.12.013 http://dx.doi.org/10.1016/0304-4076(90)90093-9 http://dx.doi.org/10.1016/j.jairtraman.2017.02.004 aneta włodarczyk dynamic econometric models 17 (2017) 129–145 144 kuziak, k. (2011), pomiar ryzyka przedsiębiorstwa: modele pomiaru i ich ryzyko (measurement of enterprise risk – measurement models and their risk), prace naukowe uniwersytetu ekonomicznego we wrocławiu, 160, seria: monografie i opracowania, 187, wrocław. lawrence, c. t., tits, a. l. (2001), a computationally efficient feasible sequential quadratic programming algorithm, siam journal of optimization, 11(4), 1092–1118, doi: http://dx.doi.org/10.1137/s1052623498344562. liu, j., chen, z. (2014), regime-dependent robust risk measures with application in portfolio selection, procedia computer science, 31, 344 – 350, doi: http://dx.doi.org/10.1016/j.procs.2014.05.277. meleo, l., nava, c. r., pozzi, c. (2016), aviation and the costs of the european emission trading scheme: the case of italy, energy policy, 88, 138–147, doi: http://dx.doi.org/10.1016/j.enpol.2015.10.008. psaradakis, z., sola, m. (1998), finite-sample properties of the maximum likelihood estimator in autoregressive models with markov switching, journal of econometrics, 86(2), 369 – 386, doi: http://dx.doi.org/10.1016/s0304-4076(98)00010-4. regulation (eu) no 421/2014 of the european parliament and of the council of 16 april 2014 amending directive 2003/87/ec establishing a scheme for greenhouse gas emission allowance trading within the community, in view of the implementation by 2020 of an international agreement applying a single global market-based measure to international aviation emissions, http://eur-lex.europa.eu/eli/reg/2014/421/oj (12.09.2017). sanin, m. e., mansanet-bataller, m., violante, f. (2015), understanding volatility dynamics in the eu-ets market, energy policy, 82, 321–331, doi: https://doi.org/10.1016/j.enpol.2015.02.024. stawicki, j. (2004), wykorzystanie łańcuchów markowa w analizie rynku kapitałowego (the use of markov chains in the analysis of the capital market), wydawnictwo umk, toruń. tłoczyński, d. (2015), rola państwa w kształtowaniu konkurencji na polskim rynku transportu lotniczego (the role of state in shaping the competition in the polish air transport market), research papers of wrocław university of economics, 401, 525–534. t r z p i o t , g . (ed.) (2010), wielowymiarowe metody statystyczne w analizie ryzyka inwestycyjnego (multidimensional statistical methods in investment risk analysis), pwe, warszawa. http://dx.doi.org/10.1137/s1052623498344562 http://dx.doi.org/10.1016/j.procs.2014.05.277 http://dx.doi.org/10.1016/j.enpol.2015.10.008 http://dx.doi.org/10.1016/s0304-4076(98)00010-4 https://doi.org/10.1016/j.enpol.2015.02.024 regime-dependent assessment of the euaa price risk dynamic econometric models 17 (2017) 129–145 145 reżimowo-zależna ocena ryzyka zmian cen unijnych uprawnień lotniczych do emisji co2 z a r y s t r e ś c i. w artykule oszacowano ryzyko zmian cen unijnych uprawnień lotniczych do emisji co2, które towarzyszy działalności operatorów statków powietrznych wykonywanej w ramach europejskiego obszaru gospodarczego, w okresach niskiej i wysokiej zmienności występujących na tym rynku. pokazano, że reżimy zmienności na rynku kontraktów futures na uprawnienia lotnicze zostały prawidłowo zidentyfikowane, zarówno dla dwustanowego, jak i trzystanowego przełącznikowego modelu markowa, a oszacowane miary ryzyka różnią się w reżimach. występowanie różnych reżimów zmienności na tym rynku można wyjaśnić modyfikowaniem zasad funkcjonowania europejskiego systemu handlu emisjami oraz wprowadzaniem zmian zarówno w polityce klimatycznej ue, jak i w globalnych regulacjach dotyczących emisji co2 przez lotnictwo międzynarodowe. s ł o w a k l u c z o w e: europejski system handlu emisjami; przełącznikowe modele markowa; ryzyko; unijne uprawnienia lotnicze do emisji co2. © 2017 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2017.010 vol. 17 (2017) 161−176 submitted november 29, 2017 issn (online) 2450-7067 accepted december 28, 2017 issn (print) 1234-3862 barbara będowska-sójka * evaluating the accuracy of time-varying beta. the evidence from poland a b s t r a c t. this paper empirically investigates various approaches to model time-varying systematic risk on the polish capital market. a plenty of methods is examined in the developed markets and the kalman filter approach is usually indicated as the best method for estimation of time-varying beta. however, there exists a gap in the studies for the emerging markets. in the paper we apply weekly data of fifteen stocks listed on the warsaw stock exchange from banking and informatics sector. the sample starts at the beginning of 2001 and ends in 2015 including the hectic crisis period. we estimate beta within few competing approaches: two mgarch models, bekk and dcc, unobserved component model, and static beta from linear regression. all beta estimates are compared in the securities market line framework. we find that unobserved component beta together with beta from dcc model have higher predictive accuracy than beta from bekk model or static beta. the beta estimates are positively correlated within the industry and negatively correlated for stocks from different sectors. finally, the prediction of beta coefficients are more accurate for stocks from banking sector than for it companies. k e y w o r d s: bekk; dcc; kalman filter; mgarch; time-varying beta. j e l classification: g15; q47. * correspondence to: barbara będowska-sójka, poznań university of economics and business, faculty of informatics and electronic economy, al. niepodległości 10, 61-857 poznań, poland, e-mail: barbara.bedowska-sojka-ue.poznan.pl. this paper was presented at forecasting financial markets and economic decision making findecon 2016 in łódź, 18th oxmetrics user conference 2016 in london and econometric research in finance workshop 2016 in warsaw. i would like to thank two anonymous reviewers for their valuable comments and suggestions. the usual disclaimer applies. 1 the other possible specifications are presented in kurach and stelmach (kurach and this paper was presented at forecasting financial markets and economic decision making findecon 2016 in łódź, 18th oxmetrics user conference 2016 in london and econometric research in finance workshop 2016 in warsaw. i would like to thank two anonymous reviewers for their valuable comments and suggestions. the usual disclaimer applies. https://orcid.org/0000-0001-5193-8304 barbara będowska-sójka dynamic econometric models 17 (2017) 161–176 162 introduction according to capital asset pricing model, capm, the beta as the measure of the systematic risk, is the only risk factor important for investors (andersen, et al., 2006). the estimation and prediction of beta value is relevant for investment decisions as well as for measuring the performance of fund managers (e.g. through the treynor ratio), betas are also strongly required in asset pricing, portfolio selection, asset allocation and risk management (choudhry and wu, 2008). moreover, investors need a good approximation of beta as it allows to measure the cost of capital. the important question in the literature is how to estimate true latent beta coefficient. in the classic capm, we assume that beta is a constant measure of systematic risk. however, a vast of the literature devoted to examination of the beta stability shows a considerable evidence against this assumption (andersen et al., 2006; brooks et al., 1998; faff et al., 2000; huang and litzenberger, 1988; jagannathan and wang, 1996; menchero et al., 2016). the static capm loses in favor of the conditional version of capm with time-varying betas (campbell et al., 1997), which accounts for the fact that betas and expected returns vary over business cycle and “depend on the nature of the information available at any given point in time” (jagannathan and wang, 1996). a great number of empirical studies use different approaches to estimate beta, all having some advantages and drawbacks. the estimates from different models usually differ substantially. the most common approaches are: linear regression where beta is assumed to be stable (dębski, et al., 2014), multivariate garch models (mgarch), that are employed to estimate the time-varying beta (brooks et al., 1998), realized betas derived from realized volatility introduced by andersen and bollerslev (1998) and developed in further works (andersen et al., 2001; andersen et al., 2010; andersen et al., 2006; hajizadeh et al., 2012), or beta estimated in the rolling window within a linear regression. another and very promising approach to estimate time-varying beta is the kalman filter technique (brooks et al., 1998; kurach and stelmach, 2014; lie et al., 2000). in few works the kalman filter is found to perform better than the garch specifications (brooks et al., 1998; lie et al., 2000). several papers examine the beta coefficients on the warsaw stock exchange. dębski et al. (2014) study the impact of sampling frequency of the beta estimates, kurach and stelmach (kurach and stelmach, 2014) focus on different behavior of sector beta that are estimated with kalman filter evaluating the accuracy of time-varying beta. the evidence from poland dynamic econometric models 17 (2017) 161–176 163 approach, while dębski et al. (2016) examine the stability of the beta parameters over bull and bear market for 134 largest companies. the purpose of our paper is to compare beta coefficients obtained from different parametric methods. we focus on data of weekly frequency for fifteen stocks quoted on the warsaw stock exchange in two sectors, banking and it. according to ftse russell country classification the warsaw stock exchange is perceived as an advanced emerging market (ftse, 2016; wyman, 2016). the stocks in our sample are listed through the relatively long period of time as for the non-developed market. in the paper we estimate time-varying beta from mgarch models and use kalman filter for unobserved component model. we examine how methods, that have been already used in the developed markets, work when applied to data from the warsaw stock exchange. two step agenda is used: first we calculate timevarying beta coefficients with different methods for each stock separately, and second we employ a procedure to asses which estimate is the closest to the true unobservable beta. this examination is done in the framework of securities market line, and as such is similar to approach presented in choudhry and wu (2008). we find that beta estimates from kalman filter together with beta estimates from dcc models have the highest accuracy. in our sample the beta estimates within one industry are highly positively correlated, while beta estimates from different sectors are characterized by negative correlation coefficients. we also find that in-sample errors are lower in case of banking companies when comparing to it stocks. the rest of the paper is as follows: in section 1 the data are described, section 2 is devoted to model specification, in section 3 we show how the different beta estimates are compared, in section 4 the empirical results are described and section 5 concludes. 1. data our sample data consists of prices of 15 stocks quoted on the wse constantly from the beginning of the 2000 till the end of 2015. they are representing two sectors: banking and informatics. these two sectors have the biggest number of stocks representatives quoted constantly in the whole period of the study. the price data are obtained from stooq database (www.stooq.pl), and they are adjusted for dividends and splits. these stocks are listed together with their full names, tickers, industry, size category and capitalization in table 1. generally, banks are big companies, whereas it stocks belongs to different size groups. for the approximation of the market barbara będowska-sójka dynamic econometric models 17 (2017) 161–176 164 portfolio we use wig index which comprises all companies listed on the warsaw stock exchange (wse) main list. in the study we also use risk free rate that is calculated as a mid-quote of 1 month wibor and wibid rates (that are the polish counterparts of libor and libid rates), table 1. the list of the stocks included in the sample company name ticker sector firm size (category) firm size (in mln euro) handlowy sa bhw banking big 2256 ing sa ing banking big 4746 mbank sa mbk banking big 3204 millenium sa mil banking big 1423 pekao sa peo banking big 7464 bz wbk sa bzw banking big 7088 bos sa bos banking medium 152 assecopol sa acp informatics big 1012 cdprojekt sa cdr informatics big 1134 comarch sa cmr informatics big 320 sygnity sa sgn informatics small 13 elzab sa elz informatics medium 54 macrologic sa mcl informatics small 16 simple sa sme informatics small 7 lark sa lrk informatics very small 3 note: the companies on the wse are categorized according to their size measured by capitalization in the following manner: ‘big’ stands for capitalization higher than 250mln of euro, ‘medium’ is in the interval (50mln, 250mln), ‘small’ is in the interval (5mln, 50mln) and ‘very small’ stands for capitalization lower than 5mln euro. the last column reports capitalization of stocks at the end of 2016. the daily stock prices, index values and wibid/wibor rates are aggregated into weekly data based on the last observation in the week. the whole sample consists of 887 weekly returns. in the further work we use the percentage logarithmic returns. the calculations and graphics are done in oxmetrics stamp7 (koopman et al., 2006), pcgive (doornik and hendry, 2006) and g@rch (laurent, 2013). 2. beta estimation in this section we briefly describe models that are used in the empirical part. the common factor for obtained measures is a time-varying feature of beta coefficient. we use two mgarch specifications and kalman filter in unobserved component models. evaluating the accuracy of time-varying beta. the evidence from poland dynamic econometric models 17 (2017) 161–176 165 2.1. mgarch specifications: scalar bekk and dcc models within multivariate garch models the conditional beta are obtained. after preliminary estimations we consider two multivariate garch specifications: scalar bekk model (engle and kroner, 1995), and dynamic conditional correlation model, dcc (andersen et al., 2006; engle, 2002). each of these models is bivariate model with two equations, one for stock return, i r , and one for market portfolio return, m r : ),0(~ ... ... 1 22110 22110 ttt mtmtmmtmmmt ititiitiiit d errr errr he         (1) in the general bekk(p,q) model the conditional covariance matrix, ht, is described in the following way:       p j jjtj q i iititit 11 '''' ehedeedcch (2) where di and ej are identity matrix multiplied by scalars. in our approach 1and1  pq . in the dynamic conditional correlation dcc model of engle (engle, 2002) the following specification is used:         p j jtjitit q i i p j j q i it tttt tttt 1111 1*1* )'()1( qzzsq qqqr drdh  (3) where ),...,,(diag 2/12/1 2 2/1 1 nttt hhhd , kt h is the conditional variance described with univariate garch models, t z is the vector of standardized kt e , 2/1 / ktktkt hez  , t r is a matrix of time-varying correlation coefficients of t z , and * t q is diagonal matrix in which elements are square roots of diagonal elements of matrix tq . in our study in both mgarch models, bekk and dcc model, in the conditional mean equations (eq.1) arma(1,0) specification are considered. in bekk model the conditional variance equations are modeled with barbara będowska-sójka dynamic econometric models 17 (2017) 161–176 166 garch(1,1), whereas in dcc model the conditional variance is modeled with gjr-garch(1,1), that accounts for possible leverage effect (andersen et al., 2007). in both models the conditional distribution of the model error terms is assumed to be student t. as the conditional beta for a stock i is described as )var(/),cov( mtmtitit rrr , both estimates of conditional covariance and conditional variance come directly from mgarch models fitted to the returns of an individual stock i r and returns of a market index m r , a proxy for the market portfolio. both the conditional variance and the conditional covariance are provided in the matrix ht. finally, the time-varying beta series are calculated from the conditional covariances in mgarch models. 2.2. unobserved component model the second approach used in the paper is a state-space representation where the kalman filter is used. we apply the unobserved component model, uc, that is considered as a multiple regression model with time-varying coefficients. this specification is based on the theory of structural time series models presented in harvey (1989), in the general form a time series is viewed as being decomposable into trend, seasonal, and cycle components. in the uc model it is assumed that near and far distant observations should not be given equal weight. for our purpose we use a local level model (random walk) with drift in the following way (harvey 1989) 1 : ttrr ttmtittit ...,,1),0(~, 2   nid , (4) tttttiit ...,,1),0(~, 2 1,    nid (5) where nid denotes normally and independently distribute, and t and t are independent variables. equation (4) is a measurement equation, whereas equation (5) is the transition equation. within this specification, any shock to asset’ beta is persistent. the kalman filter allows to obtain the time-varying beta. it should be noted that the specification of the uc model for weekly data might raise some concerns as this model assumes inter alia that the conditional variance of returns is homoscedastic (has no arch effect described in e.g. bollerslev et al. (1994). however, this effect is pretty well 1 the other possible specifications are presented in kurach and stelmach (kurach and stelmach, 2014) or będowska-sójka (2015). evaluating the accuracy of time-varying beta. the evidence from poland dynamic econometric models 17 (2017) 161–176 167 captured by bekk and dcc models. thus, if the weekly data are characterized by the arch effect, than the uc model is not appropriate 2 . finally we also obtain static beta from the ordinary least square regression as a benchmark for the comparisons of time-varying beta estimates. 3. comparisons of different betas there is no obvious benchmark for unobservable beta. therefore some proxies must be introduced when attempting to establish the relative dominance of one method over another. in a similar manner to choudhry and wu (2008) after calculation of different beta measures, we compare them on the basis of the fit to the securities market line, sml: )( ftmtitftit rrrr   (6) where ft r is a risk-free rate of return. with the estimates of time-varying betas, one easily calculates in-sample theoretical returns based on the market return and the risk-free rate of return that are actually observed. we assess the relative accuracy of time-varying beta estimates by comparing the theoretical return with the actual returns and calculating the residual,  : ititit rr ˆ (7) where it r is the actual return at time t, it r̂ is the theoretical return of stock i according to the sml, and tt ,...,1 stands for the consecutive weeks. the comparison of beta estimates are based on the standard forecasting error measures: the lower the errors, the better in-sample beta approximation. we use two standard errors used most frequently in empirical studies (menchero et al., 2016; wang, 2009): mean absolute error,    t t titi rr t mae 1 ,, ˆ1 and mean square error,    t t titi rr t mse 1 2 ,, ) ˆ( 1 . out of these two measures, mae is less sensitive to outliers. we also consider median relative absolute error, mrae, calculated as a median of the distribution of ratios b ti b tititi rrrr ,,,, ˆ/ˆ  , 2 a possible solution to this issue was proposed by rockinger and urga (2001). barbara będowska-sójka dynamic econometric models 17 (2017) 161–176 168 where b stands for benchmark model (the ols beta in our case), the last error measure is perceived in the literature as a robust comparative measure of performance (armstrong and collopy, 1992). 4. empirical results we estimated four type of models for each of fifteen stocks. due to the highly volatile measures of conditional beta within the first period, we excluded from the analysis first 52 observations and thus analysis starts from 2001. in figure 1 we show beta estimates from different methods for two stocks, each representing different sector: bhw (banking) and cdr (informatics), the remaining graphics are available on request. while the overall dynamic of beta is similar across different approaches, beta estimates from unobserved component model seem to be most smoothed and therefore most stable, whereas the beta estimates from both mgarch models are highly volatile. figure 1. the estimates of beta for bzw and cdr note: beta estimates shown in the figure are the following: bekk stands for conditional beta from mgarch scalar bekk models, dcc stands for conditional beta from mgarch dcc model and uc stands for time-varying beta from unobserved component model. evaluating the accuracy of time-varying beta. the evidence from poland dynamic econometric models 17 (2017) 161–176 169 we also calculate correlation coefficients for different beta estimates across the sample and find that these correlations are positive, medium strong and statistically significant. in case of both mgarch (bekk and dcc) models the correlation coefficient is on average 0.63, for bekk and uc models the correlation accounts for 0.68, while for dcc and uc models the correlation is equal to 0.59. table 2 reports the mean, minimum and maximum values of beta estimates from different methods. the differences in means obtained on the basis of different methods are not significantly different from each other. all ols beta coefficients are significantly different from zero. in four out of fifteen cases ols beta is not statistically different from 1. the minimum and maximum beta estimates show high variability within the sample. table 2. time varying beta estimates for polish banking and informatics stocks bekk dcc un ols beta mean min max mean min max mean min max bhw 0.7814 –0.0887 1.5478 0.7485 0.2697 1.4857 0.8161 0.0970 1.2811 0.7670 ing 0.6881 –0.0638 1.7841 0.7248 0.4115 1.6610 0.7732 0.4315 1.1505 0.7954 mbk 1.2074 0.5514 2.0564 1.1883 0.7394 2.5867 1.2191 0.7114 1.8901 1.2638 mil 1.2189 0.4439 2.0981 1.2388 0.6987 1.8624 1.2698 0.9030 1.8001 1.3310 peo 1.1592 0.3874 1.7685 1.1189 0.5660 2.0766 1.1496 0.3451 1.5487 1.1367 bzw 0.9399 –0.2699 1.8446 1.0181 0.2614 2.0313 1.0453 0.1442 1.6265 1.0875 bos 0.2717 –0.4900 1.1031 0.2742 –0.0956 0.6591 0.3536 0.0576 0.5768 0.3271 acp 0.9367 –0.0665 2.2527 1.0216 0.3337 2.5907 1.0055 0.4907 2.0023 1.0496a cdr 0.9909 –0.6677 2.6589 1.0989 0.5225 2.2277 1.1818 0.9375 1.9225 1.2213 cmr 0.8684 0.0688 1.8595 0.9418 0.3789 2.1970 0.9473 0.2939 2.1340 1.0616a sgn 0.9736 –0.0142 2.2289 1.0125 0.2495 2.6802 0.9917 0.3650 2.1843 0.9963a elz 0.4080 –0.5330 1.7864 0.4407 0.1423 0.7134 0.4427 0.3515 0.6608 0.4612 mcl 0.5738 –0.2633 1.8075 0.6836 0.2787 1.3665 0.6460 0.1155 1.2151 0.6734 sme 0.4905 –0.0393 1.6444 0.5250 0.1732 0.8998 0.6088 0.2196 1.2768 0.6607 lrk 0.7959 –1.5195 4.4349 0.9401 0.3165 3.3889 0.9343 0.2236 1.6165 0.9578a note: mean is the arithmetic average of conditional beta obtained from bekk, dcc and kalman filter models, min and max stand respectively for the minimum and maximum value of the conditional beta estimates in the sample. ols best stands for the point estimates of beta from the linear regression. all ols beta are statistically different from 0; letter a is a subscript used to show the estimates that are not statistically significantly different from 1. to evaluate beta estimates, three different measures of errors based on in-sample fit to sml (eq. 6) are employed. table 3 reports the error measures. in most cases the lowest value is observed for time-varying beta estimates from unobserved component model: mae is the lowest in all cases, whereas the mrae is the lowest in twelve out if fifteen cases. next is the bekk model with fourteen lowest forecasting errors, twelve in mse and two mrae. the linear beta obtains the lowest error measures in three cases, barbara będowska-sójka dynamic econometric models 17 (2017) 161–176 170 all of them are mse errors. the dcc model has got the only one lowest mrae. table 3. the measures of in-sample forecasting errors for different beta estimates mse mae mrae mse mae mrae mse mae mrae bhw ing mbk bekk 0.0322 0.0239 0.9816 0.0329 0.0229 0.9898 0.0359 0.0271 0.9853 dcc 0.0336 0.0241 0.9971 0.0344 0.0230 0.9915 0.0373 0.0273 0.9940 uc 0.0339 0.0236 0.9643 0.0344 0.0225 0.9873 0.0375 0.0264 0.9838 ols 0.0328 0.0243 1 0.0334 0.0232 1 0.0362 0.0277 1 mil peo bzw bekk 0.0468 0.0338 0.9954 0.0273 0.0209 0.9996 0.0326 0.0240 0.9884 dcc 0.0499 0.0341 0.9978 0.0288 0.0207 1.0000 0.0335 0.0242 0.9937 uc 0.0505 0.0334 0.9893 0.0287 0.0203 0.9920 0.0335 0.0233 0.9812 ols 0.0488 0.0347 1 0.0278 0.0208 1 0.0321 0.0253 1 bos acp cdr bekk 0.0413 0.0278 0.9983 0.0447 0.0318 0.9768 0.0725 0.0494 0.9973 dcc 0.0426 0.0274 0.9971 0.0495 0.0321 0.9830 0.0761 0.0491 0.9994 uc 0.0423 0.0274 0.9992 0.0496 0.0311 0.9753 0.0751 0.0490 0.9984 ols 0.0418 0.0277 1 0.0481 0.0324 1 0.0747 0.0496 1 cmr sgn elz bekk 0.0429 0.0308 0.9862 0.0550 0.0388 0.9998 0.0607 0.0396 0.9990 dcc 0.0454 0.0308 0.9876 0.0579 0.0387 0.9940 0.0610 0.0394 1.0007 uc 0.0454 0.0302 0.9767 0.0578 0.0380 0.9932 0.0603 0.0393 0.9986 ols 0.0443 0.0316 1 0.0566 0.0388 1 0.0602 0.0395 1 mcl sme lrk bekk 0.0679 0.0453 0.9996 0.0740 0.0504 0.9978 0.0821 0.0568 0.9909 dcc 0.0701 0.0451 1.0002 0.0779 0.0503 0.9993 0.0747 0.0565 0.9979 uc 0.0690 0.0448 0.9969 0.0778 0.0501 0.9949 0.0740 0.0558 0.9939 ols 0.0688 0.0454 1 0.0770 0.0509 1 0.0733 0.0571 1 note: ols stands for beta obtained from the linear regression estimated for the whole sample, bekk stands for conditional beta from mgarch scalar bekk models, dcc stands for conditional beta from mgarch dcc model and uc stands for time-varying beta from unobserved component model with random walk. the lowest errors are underlined. if the predictive accuracy of betas across the sectors are compared, we notice that beta estimates in banking sector have almost 1.7 lower average mse than the beta estimates in informatics sector. as the companies in the banking sector are generally bigger than these from it, the beta estimates for big stocks seem to be more accurate than estimates for medium and small stocks. in figure 2 we show the beta estimates from unobserved component model for all stocks. the behaviour of betas is similar within the industries: in case of stocks from banking sector (from bhw to bos) the beta estimates increase over the whole sample period and specially in the beginning of evaluating the accuracy of time-varying beta. the evidence from poland dynamic econometric models 17 (2017) 161–176 171 financial crisis in 2008. with respect to informatics stocks (from acp to lrk), beta estimates generally decrease. some exceptions in the overall tendency are recognized, e.g. beta in lrk is changing up and down as well as beta of bzw. the correlation matrix showing the interdependencies between beta coefficients estimated from uc model is presented in the appendix. on the one hand, the correlation matrix shows that within banking sector the correlations between beta estimates are in majority positive and statistically significant. one exception is peo, which shows negative or statistically insignificant coefficients with five out of six stocks. in case of it sector the situation is similar – the correlations for stocks from this industry are positive and statistically significant with one exception, lrk, where for four out of six stocks the correlations are negative. those two stocks, peo and lrk, are respectively the biggest and the smallest in the sample. on the other hand, the correlations between stocks from two different sectors are in most cases negative and significant. figure 2. the time-varying beta estimates from the uc model note: beta estimates shown in the figure are from unobserved component model. each graph presents the beta coefficients for a single stock. as the differences between the predictive errors are rather small we decide to use the statistic that allows for comparing accuracy of the examined methods. we employ modified diebold-mariano statistic barbara będowska-sójka dynamic econometric models 17 (2017) 161–176 172 (henceforth moddm) (harvey et al., 1997), that examines if the forecasting precision differs significantly across the methods used. it is found to perform much better than the original diebold-mariano test for different forecast horizons, as well as in cases when the forecast errors are autocorrelated or have non-normal distribution (choudhry and wu, 2009), we calculate modified diebold-mariano statistic for mse. these statistics are calculated in pairs, in which forecast errors come from one of the mgarch models, kalman filter or the ols regression beta. in table 4 we show the results of the diebold-mariano statistic for each stock and each pairs of models (model1 and model2) separately. we reject the null of equal predictive accuracy at the 5% level. the statistic has a student t distribution with 1t degrees of freedom, where t is a number of observations. in case of the negative values the errors from model 2 are lower than from model1, in case of positive values the opposite holds. table 4. the comparison of predictive accuracy of models used in the study – the modified diebold-mariano statistics model1 bekk bekk bekk dcc dcc uc model2 dcc uc ols beta uc ols beta ols beta bhw –7.4207 –7.4223 –7.3172 –1.2862 2.5595 2.7477 ing –4.7595 –4.7775 –4.5831 –0.7200 3.6377 3.2197 mbk –0.3156 –0.4811 0.2765 –0.9381 3.5396 3.1355 mil –4.1499 –4.5043 –3.7641 –0.4977 3.5911 2.0957 peo 3.3009 3.2579 4.0393 0.6722 3.2991 2.0614 bzw –1.4723 –1.5255 –0.5930 –0.2289 5.9932 3.9648 bos –9.8146 –9.4839 –9.7841 0.6072 2.3638 1.5911 acp –4.8505 –4.8918 –4.6438 –0.7892 3.4923 3.8068 cdr –7.9989 –7.8383 –7.8351 1.4541 2.2466 1.2660 cmr –5.7058 –5.4700 –5.4087 –0.1598 3.8289 2.6390 sgn –8.0543 –7.9966 –7.8688 0.7554 2.8694 2.4824 elz –8.9527 –8.8164 –8.7934 1.7822 2.0485 0.3529 mcl –8.5644 –8.8225 –8.7245 1.8249 2.8946 1.5217 sme –9.0150 –8.8529 –9.0097 0.5598 2.8379 1.9662 lrk –10.4690 –10.0370 –9.9295 0.6386 2.6865 2.4141 note: the table provides the modified diebold-mariano statistics. we compare mse from two nonnested models, model1 and model2, for each stock separately. the negative values of the statistic indicate that predictive accuracy of model2 is better than of model1. the positive values of statistics indicate the opposite. the statistics in grey font have p-values higher than α=0.05 – in such cases the predictive accuracy of two models is equal. based on the results of modified dm statistics we find that for our sample of stocks both beta from dynamic conditional correlation (dcc) models and unobserved component (uc) models provide better beta forecasts than bekk models. surprisingly, even point ols estimate of beta evaluating the accuracy of time-varying beta. the evidence from poland dynamic econometric models 17 (2017) 161–176 173 provides better forecasts than bekk model. this result is consistent for all stocks with minor exceptions: first, in case of peo stock bekk model offers better forecasts than dcc and uc models. this company is one of the biggest and more liquid among those listed on the wse. second, in case of mbk and bzw the differences between errors are not statistically significant. additionally, both dcc models and uc models have greater predictive accuracy than the point beta estimates, although in case of uc model the significant difference is observed only in 12 out of 15 stocks. we do not find any difference between forecasting accuracy of dcc model and uc model, although the latter model is usually presented as a winner in beta predictive horse races (brooks et al., 1998; choudhry and wu, 2008; lie et al., 2000). however, the result of the diebold-mariano tests might depend on the selection of the loss function, that is mse. in case of stocks listed on the warsaw stock exchange, which is the developed emerging market, the beta coefficients from kalman filter method gives as good forecasts as from mgarch dcc model. conclusion in the modern investment theory time-varying beta concept replaces the static one. in the paper we compare differently estimated beta coefficient in securities market line framework for the stocks from the banking and it sectors. these stocks have been listed constantly on the warsaw stock exchange over the period 2000–2015. first of all, we find that contrary to the previous results presented in the literature (dębski et al., 2016) the estimated beta coefficients in our sample are time-varying within the given period. we consider two mgarch model specifications, dcc and bekk model, the kalman filter technique and the estimates from linear regression models. our results show that beta estimated from the unobserved component models brings the most stable and smoothed betas. the estimates from bekk models have often the lowest mean square error, while the estimates from kalman filter in most cases offer the lowest mean absolute error as well as median relative absolute error. however, the comparison of predictive accuracy of methods used in the study shows that the errors obtained from the sml with beta estimated from bekk models are significantly bigger than from the other models. the beta estimates of both dcc model and uc model fit better in terms of the sml than the estimates of beta from ordinary least squares. we do not find the evidence for statistical difference between the predictive accuracy of dcc model and uc models. not surprisingly the barbara będowska-sójka dynamic econometric models 17 (2017) 161–176 174 correlation coefficients for beta estimates within a sect or are positive, whereas between sectors are negative. finally, when betas for stocks from different industries are compared, in-sample forecasts are more accurate for stocks from the banking sector than for stocks from the informatics sector. a natural extension of the presented study is to consider out-of-sample predictive accuracy of individual models for beta estimation. the results of such empirical exercise might be of practical interest for different groups of market professionals. references andersen, t. g., bollerslev, t. (1998), answering the skeptics: yes, standard volatility models do provide accurate forecasts. international economic review, 39(4), 885–905. andersen, t. g., bollerslev, t., christoffersen, p., diebold, f. x. (2007), practical volatility and correlation modeling for financial market risk management, risks of financial institutions, 513–548, doi: https://doi.org/10.3386/w11069. andersen, t. g., bollerslev, t., christoffersen, p. f., diebold, f. x. (2006), volatility and correlation forecasting. handbook of economic forecasting, 1, doi: https://doi.org/10.1016/s1574-0706(05)01015-3. andersen, t. g., bollerslev, t., diebold, f. x., labys, p. (2001), the distribution of realized exchange rate volatility, journal of the american statistical association, 96, 42–55, doi: https://doi.org/10.1198/016214501750332965. andersen, t. g., bollerslev, t., diebold, f. x., wu, g. (2006), realized beta: persistence and predictability, advances in econometrics, 2(05), 1–39, doi: https://doi.org/10.1016/s0731-9053(05)20020-8. andersen, t. g., bollerslev, t., meddahi, n. (2010), realized volatility forecasting and market microstructure noise, journal of econometrics, 160(1), 220–234, doi: https://doi.org/10.1016/j.jeconom.2010.03.032. armstrong, b. j. s., collopy, f. (1992), error measures for generalizing about forecasting methods: empirical comparisons by j. scott armstrong and fred collopy reprinted with permission form, international journal of forecasting, 8(1), 69–80, doi: https://doi.org/10.1016/0169-2070(92)90008-w. będowska-sójka barbara. (2015), unemployment rate forecasts: evidence from the baltic states, eastern european economics, 53(1), 57–67, doi: https://doi.org/10.1080/00128775.2015.1033236. bollerslev, t., engle, r. f., nelson, d. b. (1994), arch models. in engle, r. f. and mcfadden d. (eds.), the handbook of econometrics, 4, 2959–3038), elsevier science, amsterdam. brooks, r. d., faff, r. w., mckenzie, m. d. (1998), time-varying beta risk of australian industry portfolios: a comparison of modeling techniques, australian journal of management, 23(1), 1–22. campbell, j. y., lo, a. w., mackinlay, a. c. (1997), the econometrics of financial markets, princeton university press, chichester. choudhry, t., wu, h. (2008), forecasting ability of garch vs kalman filter method: evidence from daily uk time-varying beta, journal of forecasting, 27(8), 670–689, doi: https://doi.org/10.1002/for.1096. evaluating the accuracy of time-varying beta. the evidence from poland dynamic econometric models 17 (2017) 161–176 175 dębski, w., feder-sempach, e., świderski, b. (2014), intervalling effect on estimating the beta parameter for the largest companies on the wse, folia oeconomica stetinensia, 14(2), 270–286, doi: https://doi.org/10.1515/foli-2015-0018. dębski, w., feder-sempach, e., świderski, b. (2016), beta stability over bull and bear market on the warsaw stock exchange, folia oeconomica stetinensia, 16(1), 75–92, doi: https://doi.org/10.1515/foli-2016-0006. doornik, j. a., hendry, d. f. (2006), empirical econometric modelling using pcgive: volume i, timberlake consultants press, london. engle, r. (2002), dynamic conditional correlation – a simple class of multivariate garch models, journal of business and economic statistics, 20(3), 339–350, doi: https://doi.org/10.1198/073500102288618487. engle, r. f., kroner, k. f. (1995), multivariate simultaneous generalized arch, econometric theory, 11(1), 122–150. faff, r. w., hillier, d., hillier, j. (2000), time varying beta risk: an analysis of alternative modelling techniques, journal of business finance & accounting, 27(5–6), 523–554, doi: https://doi.org/10.1111/1468-5957.00324. ftse. (2016), ftse country classification. hajizadeh, e., seifi, a., fazel zarandi, m. h., turksen, i. b. (2012), a hybrid modeling approach for forecasting the volatility of s&p 500 index return, expert systems with applications, 39(1), 431–436, doi: https://doi.org/10.1016/j.eswa.2011.07.033. harvey, d., leybourne, s., newbold, p. (1997), testing the equality of prediction mean squared errors, international journal of forecasting, 13(2), 281–291, doi: https://doi.org/10.1016/s0169-2070(96)00719-4. huang, c. f., litzenberger, r. h. (1988), foundations for financial economics, north holland, new york. jagannathan, r., wang, z. (1996), the conditional capm and the cross-section of expected returns, journal of finance, 51(1), 3–53, doi: https://doi.org/10.2307/2329301. koopman, s. j., harvey, a. c., doornik, j. a., shephard, n. (2006), stamp 7: structural time series analyser and modeller and predictor, timberlake consultants press, london. kurach, r., stelmach, j. (2014), time-varying behaviour of sector beta risk the case of poland, romanian journal of economic forecasting, 17(1), 139–159. laurent, s. (2013), g@rch 7: estimating and forecasting arch models. timberlake consultants press, london. lie, f., brooks, r., faff, r. (2000), modelling the equity beta risk of australian financial sector companies, australian economic papers, 39(3), 301–311, doi: https://doi.org/10.1111/1467-8454.00093. menchero, j. g., nagy, z., singh, a. (2016), evaluating the accuracy of beta forecasts, the journal of portfolio management, 84–93. rockinger, m., urga, g. (2001), a time varying parameter model to test for predictability and integration in the stock markets of transition economies, journal of business & economic statistics, 19(1), 73–84, doi: https://doi.org/10.1198/07350010152472634. wang, y. h. (2009), nonlinear neural network forecasting model for stock index option price: hybrid gjr-garch approach, expert systems with applications, 36(1), 564–570, doi: https://doi.org/10.1016/j.eswa.2007.09.056. wyman, o. (2016), enhancing liquidity in emerging market exchanges, world federation of exchanges. barbara będowska-sójka dynamic econometric models 17 (2017) 161–176 176 appendix table 5 the spearman correlation matrix of beta coefficients from kalman filter approach bhw ing mbk mil peo bzw bos acp bhw 1.00 0.67 0.66 0.57 -0.32 -0.06 0.98 -0.79 ing 0.67 1.00 0.83 0.77 0.03 0.30 0.67 -0.78 mbk 0.66 0.83 1.00 0.87 0.04 0.27 0.67 -0.68 mil 0.57 0.77 0.87 1.00 -0.14 0.32 0.57 -0.65 peo -0.32 0.03 0.04 -0.14 1.00 0.19 -0.27 0.31 bzw -0.06 0.30 0.27 0.32 0.19 1.00 -0.09 -0.36 bos 0.98 0.67 0.67 0.57 -0.27 -0.09 1.00 -0.77 acp -0.79 -0.78 -0.68 -0.65 0.31 -0.36 -0.77 1.00 cdr 0.08 -0.19 0.00 0.16 -0.35 -0.59 0.10 0.27 cmr -0.73 -0.27 -0.25 -0.08 0.15 0.01 -0.73 0.55 sgn -0.51 -0.62 -0.65 -0.47 -0.21 -0.53 -0.52 0.69 elz -0.79 -0.90 -0.83 -0.78 0.04 -0.34 -0.80 0.87 mcl -0.94 -0.71 -0.68 -0.65 0.35 -0.12 -0.92 0.87 sme -0.98 -0.68 -0.70 -0.59 0.27 0.00 -0.98 0.81 lrk 0.19 0.26 0.31 0.28 -0.08 -0.26 0.23 -0.08 acp cdr cmr sgn elz mcl sme lrk bhw -0.79 0.08 -0.73 -0.51 -0.79 -0.94 -0.98 0.19 ing -0.78 -0.19 -0.27 -0.62 -0.90 -0.71 -0.68 0.26 mbk -0.68 0.00 -0.25 -0.65 -0.83 -0.68 -0.70 0.31 mil -0.65 0.16 -0.08 -0.47 -0.78 -0.65 -0.59 0.28 peo 0.31 -0.35 0.15 -0.21 0.04 0.35 0.27 -0.08 bzw -0.36 -0.59 0.01 -0.53 -0.34 -0.12 0.00 -0.26 bos -0.77 0.10 -0.73 -0.52 -0.80 -0.92 -0.98 0.23 acp 1.00 0.27 0.55 0.69 0.87 0.87 0.81 -0.08 cdr 0.27 1.00 0.23 0.54 0.15 -0.08 -0.02 0.55 cmr 0.55 0.23 1.00 0.53 0.41 0.66 0.77 0.31 sgn 0.69 0.54 0.53 1.00 0.74 0.58 0.57 0.27 elz 0.87 0.15 0.41 0.74 1.00 0.85 0.80 -0.22 mcl 0.87 -0.08 0.66 0.58 0.85 1.00 0.94 -0.21 sme 0.81 -0.02 0.77 0.57 0.80 0.94 1.00 -0.12 lrk -0.08 0.55 0.31 0.27 -0.22 -0.21 -0.12 1.00 note: the table reports spearman rank correlations for pairs of beta coefficients estimated with unobserved component model. the bolded values are statistically significant at α=0.05. © 2017 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2017.006 vol. 17 (2017) 97−114 submitted october 31, 2017 issn (online) 2450-7067 accepted december 28, 2017 issn (print) 1234-3862 adrian burda, błażej mazur , mateusz pipień * forecasting eur/pln exchange rate: the role of purchasing power parity hypothesis in estvec models  a b s t r a c t. the purpose of this paper is to verify empirical consequences of imposing various forms of purchasing power parity (ppp) within a class of smooth transition vector error correction models (estvecm) for analysis of eur/pln exchange rage. empirical importance of exponential smooth transition functions is confronted with the linear errorcorrection mechanism. a class of competing models for recursive samples are compared by the likelihood ratio test, information criteria, and out of sample forecast accuracy measures. k e y w o r d s: ppp; estvecm; cointegration; exchange rate forecasting j e l classification: c32; f31; f37 1. introduction purchasing power parity (ppp) is one of the oldest theories regarding exchange rate. its empirical importance has been investigated intensively for decades. rogoff (1996) and officer (1982) point out that ppp theory had * correspondence to: adrian burda, e-mail: adrian.marek.burda07@gmail.com; błażej mazur, department of econometrics and operations research, cracow university of economics, rakowicka 27, 31-510 kraków, e-mail: blazej.mazur@uek.krakow.pl, mateusz pipień, department of econometrics and operations research, cracow university of economics, rakowicka 27, 31-510 kraków, e-mail eepipien@cyf-kr.edu.pl.  this research was supported by the national science center, poland, based on decision number umo-2014/13/n/hs4/03593, and by the funds granted to the faculty of management, cracow university of economics within the framework of the subsidy for the maintenance of research potential. https://orcid.org/0000-0001-5096-5175 adrian burda, błażej mazur, mateusz pipień dynamic econometric models 17 (2017) 97–114 98 been initially articulated by the scholars of salamanca school in spain in the sixteenth century. modern analyses and formulations of ppp theory could be dated on early 20’s of the 20th century (e.g. cassel, 1918). nowadays ppp constitutes theoretical foundations of many long-run exchange rate models, as e.g. sticky-prices monetary models (dornbusch, 1976), flexible-price monetary model (e.g. johnson, 1978) or asset pricing models (lucas, 1982). consequently, the ppp hypothesis is also embedded in larger macroeconomic models, as e.g. in dsge models for small open economy or two economies (e.g. ca‘zorzi et al., 2017; senbeta, 2011). furthermore, ppp could be useful in forecasting exchange rate, in particular in the long-run (e.g. ca‘zorzi et al., 2016). despite of wide applications of ppp theory, empirical evidence is far from being conclusive (e.g. arize et al., 2015; kelm, 2013, p. 68). while most of the existing literature do not confirm or even reject ppp hypothesis in the short-run (arize et al., 2015), for the long-run results may vary. for example chang et al. (2010), wang (2000) or pappel (1997) reject long-run ppp, while arize et al. (2004; 2015), cheung et al. (2004), lothian & taylor (1996) found support for ppp in terms of cointegrating relationships or real exchange rate for most of their samples. however, conclusion from most of the existing research should be treated with caution due to its methodological drawbacks (for discussion see e.g., kelm, 2013, pp. 58–67). among ppp studies, positive results are reported quite frequently, when the smooth transition (str) approach is utilized, either to test stationarity of the real exchange rate (e.g. kapetanios et al., 2003; sollis, 2009; mcmillan, 2009; kelm, 2013) or to verify cointegration relationship (e.g. gefang, 2008). the aim of this study is to utilize str cointegration framework to investigate ppp hypothesis. the research is conducted for eur/pln exchange rate as an interesting example of emerging market currency. the methodology of this research is motivated by the work of gefang (2008), though the paper includes a number of extensions. firstly, we allow for str mechanism separately in selected components of the vecm model. this gives opportunity to find optimal specification, preferably linking parsimony of parametrization and good explanatory power. furthermore, in this research strong-form ppp is considered, while gefang (2008) investigated only weakform ppp. moreover, we conduct in-sample and out-of-sample analysis. the research sample covers the period between january 1999 and february 2016, while out of sample forecasts are tested until february 2017. on the one hand, it takes into account only period with one exchange rate regime, while on the other hand it has heterogeneous features as in this period forecasting eur/pln exchange rate: the role of purchasing power parity… dynamic econometric models 17 (2017) 97–114 99 poland accessed to european union, and global financial crisis as well as euro area sovereign debt crisis took place. the article is organized as follows. firstly, we provide a brief summary of ppp theory, methods of its verification and existing empirical literature. in section 3 we introduce the models used here, while section 4 contains empirical comparison of different models in terms of model fit and out of sample forecast performance. 2. purchasing power parity – general concept and empirical importance the ppp theory in all variants is rooted in the “law of the one price” (lop). it states that for any good i: (1) where is the domestic-currency price of good i at time t, is the foreign currency price of good i at time t and is the equilibrium exchange rate at time t, defined as home price of foreign currency. the relationship (1) could be expressed also for price indices: (2) where is the weight of price of good i in domestic and foreign price indices and n denotes the number of goods in domestic and foreign price indices. deviations from lop for one item does not mean deviation from lop for the whole price index, as aggregations and weighing may compensate impact of this breaches; see wdowiński (2010). thus the strong (or strict) form of ppp 1 states: (3) or, in the log-linear form: . (4) hence, the real exchange rate q can be written as: , (5) where is a constant value. empirical testing of existence of the strict ppp and law of the one price is based on assumptions that market works perfectly and any deviations from 1 alternatively, ppp models could be distinguished between satisfying and non-satisfying the long-term homogeneity restriction. adrian burda, błażej mazur, mateusz pipień dynamic econometric models 17 (2017) 97–114 100 the aforementioned relationships (4) or (5) are not persistent. thus two empirical strategies could be applied. in the first one, testing of unit root for the logarithm of the exchange rate is utilized. stationarity of the real exchange rate is interpreted as empirical support of existence of the strong-form of ppp. the second approach bases on estimation of the following long run equation: , (6) where is a stationary error term and restriction is subject to analysis. economic assumptions that guarantee the existence of the strong-form of ppp are rather restrictive. for instance, on has to assume costless spatial arbitrage (e.g. no quotas, import tariffs, transportation costs), no measurement errors, or information costs (arize et al., 2015), which implicitly is equivalent to existence of rational expectations. further conditions mentioned in the literature are: non-existence of pricing-to market (ptm) strategies, lack of nominal rigidities and lack of entry-barriers to international markets (kelm, 2013, p. 28). consequently, three general approaches have been developed in the literature to relax the assumption. in the first approach, the restriction: in (6) is relaxed and only the symmetry restriction is analysed. it reflects presence of transportation costs, other trade barriers, measurement problems (e.g. taylor, 1988; arize et al., 2015). second approach is connected with conclusion from dumas (1992), uppal (1993), sercu et al. (1995), o’connel & wei (1997) and more recently pavlidis et al. (2011) emphasizes the role of transaction costs: . (7) therefore two regimes can be identified. the first (so-called inner regime) is defined by condition , while the second regime (called outer regime) is obtained in the case . within the inner regime the real exchange rate can be described as i(1) process. within the outer regime real exchange rate should return toward inner regime. empirical testing should take into accounts existence of three regimes: the inner regime and two – positive and negative outer regimes. consequently different variants of the threshold autoregressive (tar) or smooth transition autoregressive (star) models are applied as a natural generalization of the linear scheme. similarly as in more restrictive cases above, two strategies utilizing tar or star approach has been developed. the first and the most popular one, allows for testing the unit root hypothesis of logarithm of the forecasting eur/pln exchange rate: the role of purchasing power parity… dynamic econometric models 17 (2017) 97–114 101 exchange rate against tar/star alternatives (e.g. kapetanios et al., 2003; sollis, 2009; mcmillan, 2009; månsson & sjölander 2014). the second approach, which is also applied in this study, is connected with verification of the ppp hypothesis in the multivariate framework, within threshold or smooth transition vector error correction frameworks (tvecm/stvecm), both strong-form ppp (e.g. wu & chen, 2008; nakagawa, 2010), as well as weak-form ppp (e.g. gefang, 2008). third approach is connected with a reformulation of (6) resulting in: (8) where , , – are vectors of long, medium and short run determinants of nominal exchange rate (except ppp), – vector of equilibrium coefficients, , (see kelm & bęzabojanowska, 2005; kelm, 2010). restriction = supports ppp hypothesis; see kelm (2013, p. 27). the empirical literature where ppp hypothesis is analysed is vast. in this paper only key papers are mentioned. firstly the enormous short-term volatility of the real exchange rate with the extremely slow rate at which shocks appear to damp out still misses satisfactory and sound explanation (ppp puzzle, see rogoff,1996) and that the nominal rigidities and market frictions are still unable to explain why real exchange rates (rer) deviate from the ppp level and high estimates of rers’ half-lives (3–5 years), see kelm (2017). secondly, allowing for smooth transition mechanism, seems to solve at least partially “ppp puzzle” (e.g. schnatz, 2007, norman, 2010). on the other hand, both theoretical and empirical soundness of testing procedures in many cases could be doubtful (e.g. kelm, 2013, p. 90–92). for example månsson & sjölander, (2014), and emirmahmutoglu & omay, (2014) emphasized weak power of kss (kapetanios, shin & snell) and akss (augmented kss) tests. thirdly, verification of the ppp within the cointegration framework might be difficult if the number of cointegrating vectors differs from one. empirical results of ppp testing for the polish zloty depend on the sample used in the research and on the choice of the price indices. in particular if the sample starts before 1999, the strong-form ppp is not supported, due to characteristics of the exchange rate regimes in poland before 1999 (as de facto polish zloty became free-floating). more specifically, both fixed exchange rate regime (before october 1991) and crawling peg (before may 1995) represented the nominal anchor feature (kokoszczyński, 2001). in adrian burda, błażej mazur, mateusz pipień dynamic econometric models 17 (2017) 97–114 102 terms of price indices, one should note that production price index (ppi) in manufacturing seems to be the best proxy for tradables prices (see kelm, 2013, p. 141). hence in prevailing part of the literature, strong-form ppp for the polish zloty is not supported, since the research sample contains data from mid 1990s (e.g. rubaszek & serwa; 2009; wdowiński 2010; chang & tzeng, 2011) or even include the whole transition period (arize et al., 2015). furthermore, in many studies (arize et al., 2015), at least for some specifications (rubaszek & serwa, 2009; wdowiński; 2010) several forms of weakform ppp (as proportionality restrictions) were rejected. recently for samples including only free floating exchange rate regime period the weak-form ppp is confirmed or not rejected quite frequently. surprisingly, the strongform ppp is supported only exceptionally. kelm (2013, p.188) does not reject symmetry restriction in one of specification of vecm with i(2) variables and rejects null hypothesis about unit rot against estar process for the real exchange rate. however both cases are controversial – as in vecm with i(2) variables p-value for symmetry restriction is only 0.118 and interpretation of model is not straightforward as in vecm with only i(1) variables. furthermore, results of unit root test against estar are driven by abnormal observations, mainly from the year 2008 (kelm, 2013, p.172–173). 3. econometric framework utilizing nonlinear cointegration the econometric framework applied here is designed to fulfill assumptions of existence of transaction costs. in the multivariate framework, threshold vecm (tvecm) or exponentially smooth transition vecm (estvecm) are relevant. in these cases the dynamics of the adjustment changes across regimes (inner and outer), while in the simple linear vecm adjustment is described by a linear function of the magnitude of the deviations from the long run equilibrium. the driving forces of the regime changes are governed by the observed deviations from the equilibrium through the transition function. in a tvecm, the regime changes are assumed to be discrete, whereas in an estvecm, the regimes change smoothly. the final estvecm specification used in the research is consistent with gefang (2008), with some modifications – allowing for testing the strongform ppp and more flexible specification. let , where , , and are respectively eur/pln exchange rate, domestic price index and foreign price index. asforecasting eur/pln exchange rate: the role of purchasing power parity… dynamic econometric models 17 (2017) 97–114 103 suming that cointegration relationship is common among regimes, the estvecm is described for as follows: +dt z+h=1p yt h h z)+εt (9) where , the error time ε is gaussian white noise process with and: . finally, the deterministic term contains intercept only. the dimensions of and are [ , the dimension of is [ ,while the dimensions of and are [ , with r denoting the cointegration rank. in this research it is assumed that . if strong-form ppp holds, we have: . in (9) changes of regimes are driven by past deviations from the equilibrium relationships and dynamics of changes of regimes are captured by the exponential smooth transition function proposed by teräsvirta (1994): ), (10) where the transition variable is the cointegrating combination between ln(s), ln(p) and ln( ) at time t–d 2 , c denotes the equilibrium level of cointegration relationship and γ is the smoothness parameter. higher γ induces faster transition. this function has symmetric u shape, which illustrative example for the case with c = 0 is presented on figure 1. formulas (9) and (10) allow for a set of models, varying the order of the autoregressive process, lag length of the transition variable and presence of nonlinearity in loading coefficients, autoregressive process and deterministic terms. in this research we assume that , , while different variants of non-linearity in particular element of estvecm are allowed in (9). these variants of nonlinearities could be described by zero restrictions in , . the set of competing specifications are presented in table 1. 2 when strict ppp holds it could be interpreted as real exchange rate at time t-d. adrian burda, błażej mazur, mateusz pipień dynamic econometric models 17 (2017) 97–114 104 figure 1. illustrative example for the exponential smooth transition function with different values of γ table 1. the set of competing specifications models estvec full estvec α estvec α,ξ estvec α,г estvec г,ξ estvec г estvec ξ parameters under zero restrictions none 4. empirical results we used monthly data of the nominal exchange rate eur/pln (monthly average) and ppi manufacturing indices in poland and euro area. the full sample covers the period from january 1999 to february 2016 and hence the total number of observations, including starting values, is 206. we also performed recursive estimation and prediction on the basis of expanding windows, starting from the smallest sample covering the period from january 1999 to february 2006 and ending up on the sample from january 1999 to february 2016. we estimated parameters of all competing models on the basis of maximum likelihood estimator. the strong form ppp is tested with the use of lr statistics. the model under the null hypothesis is obtained by the following restriction: . we compared the forecasting performance of models on the basis of mae and rmse summaries. also diebold and mariano (1995) test (dm) forecasting eur/pln exchange rate: the role of purchasing power parity… dynamic econometric models 17 (2017) 97–114 105 was applied to check significance of differences of generated series of forecasts from the analogous series obtained on the basis of the random walk (rw) strategy. the dm procedure was applied together with small sample corrections (hln) proposed by harvey, leybourne and newbold (1997). table 2 contains model comparison results for vec specifications and several generalisations towards smooth transition mechanism. we present logarithmic values of the likelihood calculated at ml estimates (ll) and aic, bic and hic scores, respectively. estimation is conducted for unrestricted cases and alternatively with ppp restriction imposed. the likelihood inference clearly indicates superiority of the smooth transition mechanism against simple vec construct. the greatest data support, measured by ll value, received full estvec model and some limited parameterisations with nonlinearities in г and ξ separately and jointly г, ξ. however these models seem to be too heavily parameterised and are penalised substantially by information criteria scores. among unrestricted specifications, smooth transition mechanism in parameters ξ seems to be an optimal compromise resulting with good data fit and parsimony. analyses conducted within a class of models with ppp restriction imposed show again superiority of nonlinear mechanisms. the full estvecm model receives again the highest ll value, but it is rejected by information scores as unparsimonious. among restricted cases the smooth transition construct in parameters α is supported by the data and receives the best information score. also the case with the smooth transition mechanism in parameters г receives attention. in the next step we analysed statistical significance of parameters of estr function and restriction that guarantees ppp effect. on the basis of expanding data window described above we performed a sequence of appropriate lr tests. on figure 2 and 3 we present fractions of analysed subsamples where estr coefficients were significant (figure 2) and where strong form of ppp restriction was not rejected (figure 3). decisive and more importantly time invariant inference about significance of underlying nonlinearities is impossible for heavy parameterised models. full estvec model and these limitations with smooth transition parameters imposed on pairs of groups of parameters – α and г, г and ξ, α and ξ perform worse and there are subsamples that do not support statistical significance of estr coefficients. the model estvec-ξ, performs the best receiving in all analysed subsamples statistically significant estr construct at 0.1 and 0.05 significance level. rejection of the strong form of ppp are analysed in subsamples on figure 3. in this element of analyses simple linear vecm is also considered. adrian burda, błażej mazur, mateusz pipień dynamic econometric models 17 (2017) 97–114 106 we report surprisingly good performance of vecm specification and also relatively worse results given models with smooth transition component. table 2. log-likelihood and information criteria obtained for all competing specifications in case of the whole sample model ll aic bic hic unrestricted specifications vec 2267.2 –4500.3 –4443.9 –4477.5 estvec full 2282.3 –4496.6 –4383.8 –4450.9 estvec α 2274.8 –4505.6 –4432.6 –4476.1 estvec α,ξ 2277.8 –4505.6 –4422.7 –4472.1 estvec α,г 2280.0 –4498.0 –4395.1 –4456.4 estvec г,ξ 2282.6 –4503.2 –4400.3 –4461.6 estvec г 2283.4 –4510.9 –4418.0 –4473.3 estvec ξ 2280.6 –4517.1 –4444.1 –4487.6 models with strong-form ppp restriction imposed vec 2265.1 –4500.3 –4450.5 –4480.1 estvec full 2280.5 –4497.1 –4390.9 –4454.1 estvec α 2272.6 –4505.1 –4438.8 –4478.3 estvec α,ξ 2273.3 –4500.6 –4424.3 –4469.8 estvec α,г 2277.8 –4497.7 –4401.5 –4458.8 estvec г,ξ 2278.9 –4499.7 –4403.5 –4460.8 estvec г 2277.9 –4503.7 –4417.5 –4468.8 estvec ξ 2267.4 –4494.7 –4428.4 –4467.9 figure 2. fraction of subsamples, where estr coefficients are significant on given significance level, according to lr test we also compare competing specifications with respect to the forecasting power. for the whole set of observations we generated 1, 3, 6 and 12 month-ahead forecasts. in table 3 we put rmse values normalised to rw forecasting eur/pln exchange rate: the role of purchasing power parity… dynamic econometric models 17 (2017) 97–114 107 strategy. we also present p-values (in italics) of the hln-dm test against rw for quadratic loss function. simple vecm specification exhibits much better forecasting performance in the short term. for one month-ahead case none of analysed smooth transition models generate better forecasts. also in case of this horizon all competing specifications are much better than random walk and generate lower rmse. however only simple vecm and estvecmξ generate forecasts significantly different than rw case at the significance level of 0.1. in case of unrestricted models specification estvec with smooth transition mechanism in parameters ξ outperform other models for long term forecasting. for the case of 12 months ahead forecasts simple vecm is worse than rw and differences in point forecasts are statistically insignificant. for a set of restricted models we report relatively good forecasting performance in smooth transition class except the full case and cases with st mechanism in α, jointly α,г and ξ. vecm model produces much better forecasts than st class and outperforms rw case. figure 3. fraction of subsamples, where strong-form ppp restrictions are not rejected given significance level, according to lr test analysing statistical significance of differences in forecasting performance (against rw) on the basis of hln-dm test, in the set of unrestricted models, we can only be sure that data support poor forecasting performance in case of some estvec specifications. simple vecm model generates better forecasts that rw, but with differences to rw being statistically insignificant in case of all analysed horizons. in case of restricted models only the case of one month horizon exhibit significant differences from rw stratadrian burda, błażej mazur, mateusz pipień dynamic econometric models 17 (2017) 97–114 108 egy in all cases. the strongest data support in favour of significance of differences is attached to simple vecm model. table 3. rmse for the whole sample and p-values (in italics) for hln-dm test against rw for quadratic loss function relative rmse (rw=1) – log(fx) unrestricted models horizon vecm estvec α estvec ξ estvec г estvec α,ξ estvec г,ξ estvec α,г,ξ estvec full 1m. 0.917 0.958 0.933 0.948 0.962 0.992 0.961 0.983 0.1 0.3 0.1 0.2 0.3 0.5 0.3 0.4 3m. 0.955 1.049 0.965 0.996 1.055 1.068 1.102 1.065 0.3 0.7 0.4 0.5 0.7 0.7 0.8 0.8 6m. 0.997 1.142 0.937 0.965 1.117 1.080 161.531 1.153 0.5 1.0 0.2 0.3 0.9 0.2 0.2 1.0 12m. 1.014 1.265 0.889 0.921 1.201 76.448 >1e+10 1.321 0.7 0.9 0.06 0.1 0.9 0.8 0.8 1.0 relative rmse (rw=1) – log(fx) restricted models horizon vecm estvec α estvec ξ estvec г estvec α,ξ estvec г,ξ estvec α,г,ξ estvec full 1m. 0.900 0.918 0.916 0.900 0.925 0.931 0.928 0.933 0.07 0.1 0.1 0.07 0.1 0.1 0.1 0.1 3m. 0.923 0.969 0.950 0.923 0.956 0.946 0.962 0.969 0.3 0.4 0.3 0.3 0.4 0.3 0.4 0.4 6m. 0.949 0.997 0.958 0.949 0.946 0.932 1.036 0.969 0.3 0.5 0.3 0.3 0.3 0.3 1.0 0.4 12m. 0.936 1.049 0.944 0.936 0.989 0.913 176.208 1.015 0.3 0.9 0.3 0.3 0.5 0.3 0.8 0.5 the forecasting performance of both (restricted and unrestricted) class of models for subsamples is presented in tables 4 and 5. table 4 presents rmse relative to rw for a group of unrestricted specifications, while table 5 shows results for models with strong ppp restriction imposed. the sequence of subsamples were split into two sets, the first one covers the subsamples ending ad 2011:03 to 2016:02 and the second one contains these ending at 2011:03 to 2016:02. just like in case of the whole sample, we report in all tables p-values of the hln-dm test against rw for the quadratic loss function. again, in case of a set of unrestricted models estvecm with smooth transition function in ξ provides much better forecasts for longer horizon in both series of subsamples. in short term forecasting this model also beats simple vecm in second set of subsamples (2011:03 – 2016:02). simple vecm performs relatively better than rw only in case of first series of subsamples. forecasting eur/pln exchange rate: the role of purchasing power parity… dynamic econometric models 17 (2017) 97–114 109 analysing results presented in table 5 we report relatively good forecasting power of estvecm specification in the long term. also simple vecm provide the best forecasts in case of one month ahead and three month-ahead horizon. table 4. rmse for the sequential forecasting comparison out of sample periods for unrestricted models and p-values (in italics) for hln-dm test against rw for quadratic loss function relative rmse (rw=1) – log(fx) –for subsamples: 2006:03–2011:02 horizon vecm estvec α estvec ξ estvec г estvec α,ξ estvec г,ξ estvec α,г,ξ estvec full 1m. 0.904 0.948 0.926 0.945 0.955 1.005 0.959 0.983 0.1 0.3 0.2 0.2 0.3 0.5 0.3 0.4 3m. 0.917 1.022 0.937 0.972 1.022 1.069 1.120 1.037 0.3 0.6 0.3 0.4 0.6 0.7 0.8 0.6 6m. 0.988 1.126 0.929 0.954 1.087 1.091 174.947 1.137 0.4 1.0 0.2 0.3 1.0 0.8 0.8 1.0 12m. 1.010 1.249 0.886 0.910 1.172 79.682 >1e+10 1.314 0.6 1.0 0.04 0.1 1.0 0.9 0.9 1.0 relative rmse (rw=1) – log(fx) –for subsamples: 2011:03–2016:02 horizon vecm estvec α estvec ξ estvec г estvec α,ξ estvec г,ξ estvec α,г,ξ estvec full 1m. 0.955 0.979 0.946 0.954 0.977 0.953 0.965 0.982 0.3 0.4 0.2 0.2 0.4 0.2 0.3 0.4 3m. 1.085 1.136 1.060 1.079 1.165 1.066 1.031 1.163 0.8 0.9 0.8 0.9 0.9 0.8 0.6 1.0 6m. 1.056 1.225 0.971 1.022 1.271 1.009 1.188 1.241 0.8 1.0 0.4 0.6 1.0 0.5 1.0 0.9 12m. 1.056 1.434 0.892 1.014 1.487 1.013 >5e+5 1.385 0.7 0.9 0.2 0.5 0.8 0.5 0.8 0.8 however, only a few analysed cases generate forecasts significantly different compared to the rw-based ones. relatively good-performing simple vecm model for one month ahead horizon as well as estvec-ξ and estvec-г for 12 months ahead differs significantly from the rw case for unrestricted models and the first set of subsamples; see table 4. in this case the best forecasting performance, obtained for 12 month horizon in estvec-ξ model receives also the strongest data evidence against rw forecasts as tested by hln-dm quadratic score. we report not so strong statistical significance in case of restricted models; see table 5. only in case of short term forecasts p-values are not greater than 0.1 making weak data evidence in favour of significant differences of generated forecasts from rw case. adrian burda, błażej mazur, mateusz pipień dynamic econometric models 17 (2017) 97–114 110 table 5. rmse for the sequential forecasting comparison out of sample periods for models with strong-form ppp restriction and p-values (in italics) for hlndm test against rw for quadratic loss function relative rmse (rw=1) – log(fx) –for subsamples: 2006:03–2011:02 horizon vecm estvec α estvec ξ estvec г estvec α,ξ estvec г,ξ estvec α,г,ξ estvec full 1m. 0.886 0.905 0.907 0.886 0.909 0.927 0.919 0.919 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 3m. 0.882 0.928 0.909 0.882 0.900 0.912 0.921 0.921 0.2 0.3 0.3 0.2 0.3 0.3 0.3 0.3 6m. 0.928 0.973 0.924 0.928 0.896 0.892 1.032 0.925 0.2 0.4 0.3 0.2 0.2 0.2 0.9 0.3 12m. 0.905 1.038 0.894 0.905 0.955 0.859 183.662 0.981 0.2 0.8 0.2 0.2 0.3 0.2 0.9 0.4 relative rmse (rw=1) – log(fx) –for subsamples: 2011:03–2016:02 horizon vecm estvec α estvec ξ estvec г estvec α,ξ estvec г,ξ estvec α,г,ξ estvec full 1m. 0.938 0.954 0.943 0.938 0.963 0.939 0.955 0.969 0.2 0.3 0.2 0.2 0.3 0.2 0.3 0.3 3m. 1.062 1.108 1.087 1.062 1.134 1.062 1.102 1.128 0.8 0.9 0.9 0.8 1.0 0.9 0.9 1.0 6m. 1.067 1.130 1.136 1.067 1.186 1.134 1.062 1.188 0.8 1.0 1.0 0.8 1.0 1.0 0.9 1.0 12m. 1.246 1.175 1.399 1.246 1.300 1.386 1.131 1.337 0.8 1.0 0.9 0.8 1.0 1.0 0.9 1.0 conclusions in this paper we investigated ppp hypothesis for the polish zloty in a multivariate dynamic econometric model. we confront the explanatory power of relatively broad class of models utilizing certain nonlinear cointegration concepts. we depart from a simple vecm framework into the stvecm class and verify empirical importance of generalizations (and hence the strength of statistical evidence in favor of ppp hypothesis). the in-sample results of model comparison by information criteria (based on full dataset) deliver no clear-cut conclusions, though in general the strongest support is allocated to two specifications: a linear vecm model with ppp restriction and an unrestricted estvec ξ specification. for a more detailed analysis we consider an out-of-sample expandingwindow recursive forecasting experiment with two verification windows, one for the period 2006–2011 and one for 2011–2016. the results are the following: in short forecast horizons (1–3 months ahead), the imposition of forecasting eur/pln exchange rate: the role of purchasing power parity… dynamic econometric models 17 (2017) 97–114 111 ppp improves the forecasting performance across all the specifications and verification windows under consideration. the winning specification is a linear vec model (with ppp). however, for longer forecasting horizons (6–12 months-ahead), the results are more complicated. in general, estvecm specifications involving smooth transition mechanism for the ξ parameters provide the best performance. there are though important differences between the two verification periods. in the first period, the estvecm-ξ model with ppp imposed is an overall winner for the 12-months ahead forecasts, although some other models involving smooth transition for ξ (with or without ppp) also offer similar performance. in the second period, the imposition of ppp restrictions results in a drop in longhorizon forecasting performance of the models mentioned above. in general, all the winning specifications mentioned above outperform simple rw-type forecasts. hence, our main results are the following. on the one hand, for the short-term prediction, a linear vecm with ppp seems to be a preferable tool. on the other hand, as longer horizons are involved, the need for nonlinear dynamics (as in estvecm-ξ) becomes more evident. as to the validity of ppp, the empirical support (based on long-horizon performance) is however time-inhomogeneous – it seems that more recent observations provide evidence against the ppp restrictions of the form considered here. references arize, a. c., malindretos, j., ghosh, d. (2015), purchasing power parity-symmetry and proportionality: evidence from 116 countries, international review of economics & finance, 37, 69–85, doi: http://dx.doi.org/10.1016/j.iref.2014.11.014. arize, a. c., malindretos, j., nippani, s. (2004), variations in exchange rates and inflation in 82 countries: an empirical investigation, the north american journal of economics and finance, 15(2), 227–247, doi: http://dx.doi.org/10.1016/j.najef.2003.12.002. bęza-bojanowska, j., macdonald m. r. (2009), the behavioral zloty/euro equilibrium exchange rate, nbp working papers 55 http://www.nbp.pl/publikacje/materialyistudia/55en.pdf. cassel, g. (1918), abnormal deviations of international exchanges, economic journal, 28, 413–415, doi: http://dx.doi.org/10.2307/2223329. ca' zorzi, m. ,muck, j., rubaszek, m. (2016). real exchange rate forecasting and ppp: this time the random walk loses. open economies review. 27. 1–25. doi: http://dx.doi.org/10.1007/s11079-015-9386-4. ca’zorzi, m., kolasa, m., & rubaszek, m. (2017), exchange rate forecasting with dsge models, journal of international economics, 107, 127–146, doi: http://dx.doi.org/10.1016/j.jinteco.2017.03.011. chang, t., tang, d.-p., liu, w.-c., lee, c.-h. (2010), purchasing power parity for 15 comesa and sadc countries: evidence based on panel suradf tests, applied economics letters, 17(17), 1721–1727, http://www.nbp.pl/publikacje/materialyistudia/55en.pdf http://dx.doi.org/10.2307/2223329 adrian burda, błażej mazur, mateusz pipień dynamic econometric models 17 (2017) 97–114 112 doi: http://dx.doi.org/10.1080/13504850903153775. diebold, f. x., mariano, r. s. (1995), comparing predictive accuracy, journal of business & economic statistics, 13(3), 253, doi: http://dx.doi.org/10.2307/1392185. dornbusch, r. (1976), expectations and exchange rate dynamics, journal of political economy, 84(6), 1161–1176, doi: http://dx.doi.org/10.1086/260506. dumas, b. (1992), dynamic equilibrium and the real exchange rate in a spatially separated world, review of financial studies, 5(2), 153–180, doi: http://dx.doi.org/10.1093/rfs/5.2.153. emirmahmutoglu, f., omay, t. (2014), reexamining the ppp hypothesis: a nonlinear asymmetric heterogeneous panel unit root test, economic modelling, 40, 184–190, doi: http://dx.doi.org/10.1016/j.econmod.2014.03.028. gefang, d. (2008), investigating nonlinear purchasing power parity during the post-bretton woods era – a bayesian exponential smooth transition vecm approach. advances in econometrics, 471–500, doi: http://dx.doi.org/10.1016/s0731-9053(08)23014-8. harvey, d., leybourne, s., newbold, p. (1997), testing the equality of prediction mean squared errors, international journal of forecasting, 13(2), 281–291, doi: http://dx.doi.org/10.1016/s0169-2070(96)00719-4. johnson, h. g. (1976). the monetary approach to balance-of-payments theory, in j. a. frenkel, & h. g. johnson (eds.), the monetary approach to the balance of payments (pp. 147–167). london: george allen & unwin ltd. kapetanios, g., shin, y., snell, a. (2003), testing for a unit root in the nonlinear star framework, journal of econometrics, 112, 359–373, doi: http://dx.doi.org/10.1016/s0304-4076(02)00202-6. kelm, r. (2010), model behawioralnego kursu równowagi złotego do euro w okresie styczeń 1996 – czerwiec 2009 r., bank i kredyt, 41, 21–42. kelm, r. (2013), kurs złoty/euro: teoria i empiria, wyd. uł, łódź. kelm, r. (2017), the purchasing power parity puzzle and imperfect knowledge: the case of the polish zloty, central european journal of economic modelling and econometrics, 9, 1–27. kelm, r., j. bęza-bojanowska (2005), polityka monetarna i fiskalna a odchylenia realnego kursu złoty/euro od kursu równowagi 1995:01–2004:06, bank i kredyt, 36, 4–19. kokoszczyński, r. (2001), from fixed to floating: other country experiences: the case of poland, paper to be presented at the imf seminar exchange rate regimes: hard peg or free floating?, washington, dc, march 19–20. lothian, j. r., taylor, m. p. (1996), real exchange rate behavior: the recent float from the perspective of the past two centuries, journal of political economy, 104(3), 488–509, doi: http://dx.doi.org/10.1086/262031. lucas, r. e. (1982), interest rates and currency prices in a two-country world, journal of monetary economics, 10(3), 335–359, doi: http://dx.doi.org/10.4159/harvard.9780674067851.c6. månsson, k., sjölander, p. (2014), testing for nonlinear panel unit roots under cross-sectional dependency — with an application to the ppp hypothesis, economic modelling, 38, 121–132, doi: http://dx.doi.org/10.1016/j.econmod.2013.12.013. mcmillan, d. (2009), the confusing time-series behaviour of the real exchange rates: are asymmetries important?, journal of international financial markets, institutions and money, 19, 692–711, doi: http://dx.doi.org/10.1016/j.intfin.2008.12.002. nakagawa, h. (2010), investigating nonlinearities in real exchange rate adjustment: threshold cointegration and the dynamics of exchange rates and relative prices, journal of international money and finance, 29, 770–790, http://dx.doi.org/10.1086/260506 http://dx.doi.org/10.1016/s0731-9053(08)23014-8 http://dx.doi.org/10.1016/s0304-4076(02)00202-6 https://doi.org/10.1016/j.econmod.2013.12.013 forecasting eur/pln exchange rate: the role of purchasing power parity… dynamic econometric models 17 (2017) 97–114 113 doi: http://dx.doi.org/10.1016/j.jimonfin.2010.03.002. norman s., (2010), how well does nonlinear mean reversion solve the ppp puzzle, journal of international money and finance, 29, 919–937, doi: http://dx.doi.org/10.1016/j.jimonfin.2010.01.009. o’connell, p., wei, s. j. (1997), the bigger they are, the harder they fall: how price differences across u.s. cities are arbitraged, nber working paper, 6089. officer, l.,h. (1982), purchasing power parity and exchange rates: theory, evidence and relevance greenwich, ct: jai press. papell, d. h. (1997), searching for stationarity: purchasing power parity under the current float, journal of international economics, 43(3–4), 313–332, doi: http://dx.doi.org/10.1016/s0022-1996(96)01467-5. pavlidis e.g., paya i., peel d.a., (2011), real exchange rates and time-varying trade costs, journal of international money and finance, 30, 1157–1179, doi: http://dx.doi.org/10.1016/j.jimonfin.2011.06.004. rogoff k. (1996), the purchasing power puzzle, journal of economic literature, 34(2), 647–668. rubaszek, m., serwa d. (2009), analiza kursu walutowego, c.h. beck, warszawa. schnatz, b. (2007), is reversion to ppp in euro exchange rates non-linear? international economics and economic policy, 4(3), 281–297, doi: http://dx.doi.org/10.1007/s10368-007-0091-7. senbeta, s.,r. (2011), a small open economy new keynesianvmodel for a foreign exchange constrained economy, mpra paper, 29996, https://mpra.ub.uni-muenchen.de/ 29996/1/mpra_paper_29996.pdf. sercu, p., uppal, r., van hulle, c. (1995), the exchange rate in the presence of transaction costs: implications for tests of purchasing power parity, the journal of finance, 50, 1309–1319, doi: http://dx.doi.org/10.1111/j.1540-6261.1995.tb04060.x. sollis, r. (2009), a simple unit root test against asymmetric star nonlinearity with an application to real exchange rates in nordic countries, economic modelling, 26, 118–125, doi: http://dx.doi.org/10.1016/j.econmod.2008.06.002. taylor, m. p. (1988), an empirical examination of long-run purchasing power parity using cointegration techniques, applied economics, 20(10), 1369–1381, doi: http://dx.doi.org/10.1080/00036848800000107. teräsvirta, t. (1994), specification, estimation and evaluation of smooth transition autoregressive models, journal of the american statistical association, 89, 208–218, doi: http://dx.doi.org/10.1080/01621459.1994.10476462. uppal, r. (1993), a general equilibrium model of international portfolio choice, journal of finance, 48, 529–553, doi: http://dx.doi.org/10.1111/j.1540-6261.1993.tb04726.x. wang, p. (2000), testing ppp for asian economies during the recent floating period, applied economics letters, 7, 545–548, doi: http://dx.doi.org/10.1080/13504850050033355. wdowiński, p. (2010), modele kursów walutowych, wyd. uł, łódź. wu, j.l., chen, p.d. (2008), a revisit on dissecting the ppp puzzle: evidence from a nonlinear approach, economic modelling, 25, 684–695, doi: http://dx.doi.org/10.1016/j.econmod.2007.10.011. http://dx.doi.org/10.1016/j.jimonfin.2010.03.002 http://dx.doi.org/10.1016/j.jimonfin.2010.01.009 http://dx.doi.org/10.1016/s0022-1996(96)01467-5 http://dx.doi.org/10.1016/j.jimonfin.2011.06.004 http://doi.org/10.1007/s10368-007-0091-7 http://dx.doi.org/10.1111/j.1540-6261.1995.tb04060.x http://dx.doi.org/10.1111/j.1540-6261.1993.tb04726.x http://dx.doi.org/10.1016/j.econmod.2007.10.011 adrian burda, błażej mazur, mateusz pipień dynamic econometric models 17 (2017) 97–114 114 weryfikacja hipotezy parytetu sił nabywczej dla kursu walutowego eur/pln w ramach wektorowych modeli korekty błędu z funkcją wygładzonego przejścia (estvecm) z a r y s t r e ś c i. celem artykułu jest ocena empirycznych konsekwencji narzucenia hipotezy ppp w formie mocnej (ang. strong-form) dla kursu eur/pln przy wykorzystaniu wybranych modeli kointegracji nieliniowej, to jest modeli estvec. zasadność wykładniczej funkcji przejścia dla mechanizmu korekty błędu jest testowana w odniesieniu do liniowego modelu vec. konkurencyjne modele są porównywane zarówno pod względem dopasowania wewnątrz próby, jak i zdolności predyktywnych. wyniki wspierają mechanizm wygładzonego przejścia w składniku deterministycznym. żaden z modeli estvecm nie generuje systematycznie lepszych prognoz niż liniowy model vecm s ł o w a k l u c z o w e parytet siły nabywczej; estvecm; kointegracja; prognozowanie kursu walutowego. microsoft word 00_tresc.docx dynamic econometric models vol. 10 – nicolaus copernicus university – toruń – 2010 małgorzata doman poznań university of economics liquidity and market microstructure noise: evidence from the pekao data† a b s t r a c t. the availability of ultra-high frequency data justifies the use of a continuous-time approach in stock prices modeling. however, this data contain, apart from the information about the price process, a microstructure noise causing a bias in the realized volatility. this noise is connected with all the reality of trade. in the paper we separate the microstructure noise from the price process and determine the noise to signal ratio for the estimates of the realized volatility in the case of the shares of the polish company pekao s.a. the results are used to discover the optimal sampling frequency for the realized volatility calculation. moreover, we check the linkages between the noise and some liquidity measures. k e y w o r d s: market microstructure, volatility, realized variance, liquidity, stock market, trading volume, high frequency data. 1. introduction continuous-time econometric models are becoming now a standard tool for describing the financial market dynamics. they correspond well to the theoretical models of financial mathematics and can be quite easy estimated due to the availability of ultra-high frequency data. it seems natural that tick-by-tick data are the most useful in the context of continuous-time models. however, it is not the all truth. this kind of data contains apart from the useful information about the price process a noise which, for instance, causes a bias in the daily realized volatility estimates. the sources of the noise are connected with the reality of trade. dealing with continuous-time models, we make many assumptions that are not satisfied in the real market. they concern time, price process, and market mechanism. the departures of the observed process from these assumptions † this work was financed from the polish science budget resources in the years 2007-2010 as the research project nn 111 1256 33. the author would like to thank an anonymous referee for useful comments and suggestions. małgorzata doman 6 are very often connected with the so-called market microstructure effects. the most known factors of market microstructure are liquidity, nonsynchronous trade, bid-ask spread, discrete-valued price, irregular time intervals between trades, and existence of diurnal pattern (tsay, 2000). usually these effects depend on legal regulations, market electronic systems, and traders knowledge and behavior. volatility is one of the most important parameter in risk management, derivative pricing and portfolio allocation. nowadays, one of the most popular and promising estimator of daily volatility is the daily realized variance (andersen, bollerslev, 1998; barndorff-nielsen, shephard, 2002). it is calculated as a sum of squared intraday returns and so it depends on the chosen frequency of observations. the frequency of intraday data should be high enough to capture as much as possible of available information and small enough to avoid including a noise into the realized variance estimates. it seems rather obvious that the problem of separating the noise from the „true price” process is of great importance for quality of the daily volatility estimates. in this connection, the most significant microstructure phenomenon is liquidity. the microstructure noise is usually weaker for very liquid shares. the presented analysis applies the aït-sahalia and yu (2009) approach to separate the microstructure noise from the price process in the case of shares of the polish company pekao s.a. basing on the noise estimates, we determine the noise to signal ratio where the signal is the realized volatility. as a result of the analysis we obtain the optimal sampling frequency for the realized volatility calculation. since liquidity is considered to be the crucial factor determining the noise level, we try to determine the dependencies between the noise to signal ratio and chosen liquidity measures. moreover, we apply the signal to noise ratio to compare the strength of market microstructure effects observed in the analyzed pekao data with that reported from more developed stock markets. 2. realized volatility and market microstructure noise we consider a daily log-price process ))(ln()( tpty = where t is measured in days. then the logarithmic returns are given by formula )()(),( htytyhtr −−= , and the daily realized variance (volatility) (andersen, bollerslev, 1998; barndorff-nielsen, shephard, 2002) is defined as ,)),1(()( /1 1 2∑ = +−= h j t hjhtrhrv (1) where h denotes time between two consecutive observations. the daily volatility 2tσ of a financial instrument is defined as the conditional variance of its daily return given the set of information 1−ωt available on day 1−t , i.e. liquidity and market microstructure noise: evidence from the pekao data 7 ).|))|((( 1 2 1 2 −− ωω−= ttttt rereσ (2) thus the volatility is an unobservable variable. the realized variance (1) is a possible estimator of it. in the following discussion we assume that )(ty is described by the following stochastic differential equation ).()()()( tdwtdtttdy σμ += (3) here )(tw denotes a brownian motion, )(tσ is an instantaneous volatility and )(tμ is a drift function. in such a framework an ideal ex post measure of the daily volatility 2tσ is the integrated variance .)()( 1 2∫ − = t t duutiv σ (4) from the quadratic variation theory it follows that .0 if ,)()( 1 2 →→ ∫ − hduuhrv t t t σ (5) it means that in absence of market microstructure noise the realized variance is a consistent estimator of the integrated variance. following aït-sahalia and yu (2009), we assume that the observed price tx is a sum of the “true price” ty and the microstructure noise tε : ,ttt yx ε+= (6) and we are interested in determining the daily volatility 2tσ of the ty basing on discrete observations obtained in moments tn =δδ ,,,0 … . the model given by (6) is deep-rooted in the market microstructure theory. many authors consider the noise tε as a result of bid-ask spread (roll, 1984; huang stoll, 1996), transaction costs (huang, stoll, 1996; chan, lakonishok, 1997), discrete price changes (gottlieb, kalay, 1985). manganelli (2005) and aït-sahalia and yu (2009) associate the noise with the low liquidity level. the framework of the presented investigation is based on the hasbrouck (1993) model according to which the standard deviation of tε is a total measure of the market quality. in the following empirical analysis our main goal is to separate the microstructure noise from the fundamental price and evaluate the share of noise in observed values of the daily realized variance. we can use this result to determine the frequency for intraday returns allowing to minimize bias in the realized variance estimates. moreover, we try to discover the dependencies bemałgorzata doman 8 tween liquidity and microstructure noise by modeling dependence of the later on a variety of liquidity measures. from now on we assume that the conditional mean of the return process is equal to 0. it means that (3) reduces to ( ) ( ) ( ).σ=dy t t dw t (7) aït-sahalia, mykland and zhang (2005) showed that in the parametric case this model is equivalent to that with constant σ . if 0=tε , i.e. if no microstructure noise is present, the observed log returns 1− −= ii xxri ττ are i.i.d. ).,0( 2δσn the daily realized volatility is then the maximum likelihood estimator for 2σ and ).2,0())(( 42 δ⎯⎯ →⎯−δ ∞→ σσ nrvt n (8) in such a case the best estimates of volatility are obtained for the smallest possible δ (aït-sahalia, yu, 2009; aït-sahalia, mykland, zhang, 2005). the situation changes in presence of the microstructure noise. assume now that the noise tε is i.i.d. with mean 0 and variance a. thus the observed logreturns process is ma(1) ,)( 111 iii uuwwyyr iiiiii ηεεσ ττττττ +≡−+−=−= −−− (9) with ),0.(i.i.d~ 2γiu , 22 2)var( ari +δ=σ and 2)cov( ari −= . the above dependencies form a theoretical framework for the empirical analysis presented in section 4. 3. the data we consider the polish bank pekao s.a. stock returns. the period under scrutiny is from august 8, 2006 to february 13, 2009. the tick-by-tick data are provided by stooq.pl. table 1. number of observations in the considered frequencies type of observations number of observations transactions 361 314 tick by tick 325 177 5 minute 53 520 10 minute 27 160 daily 629 the analysis was performed for 10, 5, 2 and 1 minute observations and for the duration returns which are calculated from transaction data. the time between the as equal t for th turns. tab considere sented in figure 1. d 4. empi the s volatility structure noise in noise to s ns noise what degr structure yu, 2009 friction. the n crostructu types of r ta liquidity and m e closing of t to 0. he sake of pl ble 1 contain ed frequencie figure 1. daily returns o rical resu steps of the of the funda noise tε fo the daily re signal ratio (n var(sig var(no sr = e to signal ra ree the obse theory it is 9) because in next part of t ure noise on regressions. t 0 1 1−= + +tc c x market microstr the stock ex lace, we show ns the inform es. the plot of pekao s.a. lts presented a amental price or each cons ealized volat nsr) from th . )gnal oise) atio is a mea rved signal h s often used n some sense the investiga liquidity. t the first one ,ν+ t ructure noise: change and w here only mation about showing the . period: augu analysis are e process tx idered day t ility estimat he following asure commo has been cor d as a mark e it allows to ation deals w o determine e is of the for evidence from its opening n the results fo the number e dynamics o ust 8, 2006 to as follows. and the var t. to evalua tes we calcu g formula only used in rrupted by no ket quality m o evaluate th with depende the possible rm the pekao data next day is c or 5 and 10 m of observati of daily retur february 13, first we est riance ta of t ate the partiti ulate for eac science to q oise. in mark measure (aï he level of t ence of the m e linkages w a 9 considered minute reions in the rns is pre 2009 timate the the microion of the ch day the (10) quantify to ket microït-sahalia, the market market miwe run two (11) 10 and captu variance ns allows us quidity. t (v), the d during a d the e presented contains m noise and 5 minute result is s 2009). table 2. m ta tσ nsr figure 2. ures the imp ta . the seco 0 1sr = +t c c x s to establish the consider daily mean tr day (dtn). estimates of t d in figure 2 mean values d noise to si returns. the similar to th mean and stand the realized (black line) r m act of liquid ond one, 1 ,ν− +t tx h the connec red liquidity ransaction vo the realized v . figure 3 sh and standar ignal ratio. e mean level hat observed dard deviation frequency mean standard devia mean standard devia mean standard devia d volatility esti returns małgorzata dom dity (measure ctions betwee measures ar olume (dmt volatility bas hows the plo rd deviations the lowest l of noise to in develope n of noise and y ation ation ation imates based man ed by a vari en the noise re logarithm tv), and the n sed on 5 and ot of corresp of the daily t values of n signal ratio ed stock mar d realized vola 5 min 0.1123 0.0953 2.4584 1.4047 0.3381 0.2784 on 10 minute iable tx ) on to signal rat ms of the dail number of tra d 10 minute r ponding nois y volatility, v noise are ob is about 3/1 rkets (aït-sa atility 1 3 0 3 0 4 2 7 1 1 0 4 0 (grey line) an n the noise (12) tio and lily volume ansactions returns are e. table 2 variance of btained for 3 and this ahalia, yu, 10 min 0.1468 0.1408 2.2690 1.3096 0.3314 0.2977 nd 5 minute liquidity and market microstructure noise: evidence from the pekao data 11 figure 3. the market microstructure noise estimates for the realized volatility estimates based on 10 minute (grey line) and 5 minute (black line) returns the results of analysis on the connections between the microstructure noise and liquidity are presented in table 3. table 3. parameter estimates for regressions (11) and (12) ta explanatory variable 5 min 10 min 1c 2r 1c 2r log(dnt) 0.012 (0.008) 0.01 0.042 (0.011) 0.03 log(mdtv) -0.005 (0.009) 0.004 0.004 (0.0129) 0.001 log(v) 0.006 (0.006) 0.002 0.0278 (0.008) 0.02 nsr explanatory variable 5 min 10 min 1c 2r 1c 2r log(dnt) -0.095 (0.019) 0.1 -0.066 (0.021) 0.02 log(mvtd) -0.037 (0.028) 0.03 0.017 (0.030) 0.0005 log(v) -0.072 (0.015) 0.03 -0.037 (0.016) 0.01 surprisingly, the obtained estimates show rather weak connections between the both measures of the noise level and the considered liquidity measures. in the case of 10 minute returns there exists a positive and significant, though not very strong, dependence of the strength of noise and the number of transactions during a day, and the transaction volume. the expectations were that these dependencies should be negative (the higher liquidity, the lower noise). as con0,0 0,2 0,4 0,6 0,8 1,0 1,2 1 61 121 181 241 301 361 421 481 541 601 małgorzata doman 12 cerns the noise to signal ratio, a significant negative dependence on the daily number of transaction is in agreement with our early conjecture, but the results concerning the remaining liquidity measures are unexpected. it seems that in the case of pekao s.a. the measures based on trading volume are not good liquidity measures. some explanation of this fact can be derived from the plots in figures 4–5, which show a typical dynamics of returns in days with high and low level of the noise. during the days with high noise to signal ratio the tick-bytick returns exhibit a very regular pattern caused probably by market makers activity. the high values of volume are presumably connected with this spurious trade. on the other hand, during the days with the noise to signal ratio close to zero the dynamics of the returns is irregular and strong, which is characteristic for the days with high activity of uninformed traders. so, the conclusion is that in the case of analyzed equities the microstructure noise is to a large extent connected with the market makers activity. figure 4. tick-by-tick returns with the noise to signal ratio equal to 0.91 observed on may 13, 2008 figure 5. tick-by-tick returns with the noise to signal ratio equal to 0.03 observed on september 7, 2006 -0,5 -0,3 -0,1 0,1 0,3 0,5 1 41 81 121 161 201 241 281 321 361 401 441 481 521 561 -0,5 -0,3 -0,1 0,1 0,3 0,5 1 41 81 121 161 201 241 281 liquidity and market microstructure noise: evidence from the pekao data 13 5. conclusions due to the availability of ultra-high frequency data, a continuous-time approach to modeling the stock markets dynamics is still becoming more popular. in fact, many of fruitful research areas in financial econometrics are based on this methodology and use the realized variance as an estimator of true volatility. the daily realized variance is calculated as a sum of squared intraday returns. however, the estimates of volatility obtained in such a way are usually biased due to the presence of the market microstructure noise in the observed data. the market microstructure effects include all the phenomena connected with the reality of the trade that usually contradict the continuous-time model assumptions. in the paper we considered the quotations of the polish stock company pekao s.a. and attempted to separate the market microstructure noise from the observed daily realized variance process. our main findings are as follows. the best volatility estimates are obtained for 5-minute returns. the market microstructure noise is to a large extent connected with market makers activity. the analyzed liquidity measures (volume, mean volume of transaction, number of transactions during a day) poorly explain the market microstructure noise. the mean level of the noise to signal ratio in the case of the pekao data is comparable to that observed in developed markets. this result seems to support the opinion about good quality of market regulations and procedures on the warsaw stock exchange. references aït-sahalia, y., yu, j. (2009), high frequency market microstructure noise estimates and liquidity measures, annals of applied statistics, 3, 422–457. aït-sahalia, y., mykland, p. a., zhang, l. (2005), how often to sample a continuous-time process in the presence of market microstructure noise, review of financial studies 18(2), 351–416. andersen, t. g., bollerslev, t. (1998), answering the skeptics: yes, standard volatility models do provide accurate forecasts, international economic review, 39, 885–905. andersen, t. g., bollerslev, t., diebold, f. x., ebens, h. (2001), the distribution of realized stock return volatility, journal of financial economics, 61, 43–76. barndorff-nielsen, o. e., shephard, n. (2002), econometric analysis of realised volatility and its use in estimating stochastic volatility models, journal of the royal statistical society, 64, series b, 253–280. chan, l., lakonishok, j. (1997), institutional equity trading costs: nyse versus nasdaq, journal of finance, 52, 713–735. gottlieb, g., kalay, a. (1985), implications of the discreteness of observed stock prices, journal of finance, 40, 135–153. hasbrouck, j. (1993), assessing the quality of a security market: a new approach to transaction cost measurement, review of financial studies, 6, 191–212. huang, r., stoll, h. (1996), dealer versus auction markets: a paired comparison of execution costs on nasdaq and the nyse, journal of financial economics,41 (3), 313–357. manganelli, s. (2005), duration, volume and volatility impact of trades, journal of financial markets, 8, 377–399. małgorzata doman 14 roll, r. (1984), a simple model of the implicit bid–ask spread in an efficient market, journal of finance, 39, 1127–1139. tsay, r. s. (2002), analysis of financial time series, wiley series in probability and statistics, john wiley& sons, new york. płynność a szum mikrostruktury rynku na przykładzie notowań spółki pekao z a r y s t r e ś c i. dostępność danych giełdowych o bardzo wysokiej częstotliwości stanowi argument za stosowaniem do opisu dynamiki cen akcji modeli z czasem ciągłym. jednak dane takie zawierają oprócz informacji na temat procesu ceny także szum mikrostruktury rynku, którego obecność powoduje obciążenie oszacowań zmienności. szum ten jest związany z rzeczywistymi warunkami, w jakich odbywa się handel. w pracy dokonano oszacowania szumu mikrostruktury rynku w zmienności zrealizowanej cen akcji spółki pekao sa oraz wyliczono stosunek sygnału do szumu. wyniki badań wskazują, że optymalna częstotliwość wyliczania stóp zwrotu przy wyznaczaniu zmienności zrealizowanej to częstotliwość pięciominutowa, a obserwowany stosunek sygnału do szumu jest na poziomie zbliżonym do obserwowanego na rozwiniętych rynkach giełdowych. ponadto, przeprowadzona została analiza powiązań pomiędzy wybranymi miarami płynności a poziomem szumu mikrostruktury rynku. s ł o w a k l u c z o w e: mikrostruktura rynku, zmienność, wariancja zrealizowana, płynność, rynek giełdowy, wolumen obrotu, dane wysokiej częstotliwości. © 2017 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2017.002 vol. 17 (2017) 19−39 submitted june 6, 2017 issn (online) 2450-7067 accepted december 4, 2017 issn (print) 1234-3862 goksu aslan * the effects of income inequality and redistribution in democracies: a dynamic panel data approach  a b s t r a c t. in this paper, the simultaneous effects of the inequality and redistribution on economic growth are tested for the whole sample and for a subset of democratic countries, following sys-gmm estimation on a panel dataset over a period from 1960 to 2010. overall, net inequality has a negative significant effect on subsequent 5 years for both samples, while redistribution impact is only significant in democracies. the findings are consistent with the fact that governments tend to significantly redistribute more in democracies. k e y w o r d s: democracy; economic growth; inequality; redistribution. j e l classification: c23; e62; o40. introduction the effects of income inequality and redistribution are complex and controversial. income inequality may affect economic growth both negatively and positively. furthermore, income inequality and redistribution may interact such that higher inequality may be associated with higher redistribution and this may create pressure on economic growth. this paper handles the simultaneous effects of net inequality and redistribution on economic growth through a dynamic panel data framework and tries to show that inequality is harmful while redistribution is not for the economic growth at least in democracies. the main aim of the paper is to show that in democracies redis * correspondence to: goksu aslan, university of messina, department of economics, piazza pugliatti, 1, 98122 messina, italy, e-mail: goksuaslan@goksuaslan.com.  this work was financed by the university of messina. mailto:goksuaslan@goksuaslan.com goksu aslan dynamic econometric models 17 (2017) 19–39 20 tribution has the growth-enhancing effect on economic growth and this effect is absolutely larger than the negative effect of income inequality. for this purpose, first, the whole sample and then the democratic sample is particularly observed. the analyses have been carried out in dynamic panel data frameworks over a period from 1960–2010 for 137 countries of which 112 are democracies. as for the results of the statistical tests for difference in covariance and in mean, there exist differences between democratic and non-democratic samples. due to the endogeneity and reverse causality problem between regressors and the dependent variable, system generalized method of moments estimation, henceforth sys-gmm is used. in order to provide the comparisons of the models, ordinary least squares (ols) and within group (wg) estimations have also been reported. the results show that net inequality has a negative significant impact on economic growth in both samples, and that redistribution has a positive significant impact on economic growth in the democratic sample. in democracies, the effect of redistribution is larger than the impact of income inequality in absolute terms. according to the estimation results for the whole sample, a rise in the gini of net income by 10 points leads to a decrease in the growth of real gdp per capita in the range of 0.74–1.10%. however, redistribution impact remains insignificant for the whole sample. in democracies, both net inequality and redistribution impacts are significant. in democracies, net inequality has a negative effect which is strictly significant. a rise in the gini of net income by 10 points leads to a decrease in the growth of real gdp per capita in the range of 0.84–1.13%. on the other hand, a rise in the redistribution by 10 points leads to an increase in the growth of real gdp per capita in the range 1.27–1.4%. in countries where income is distributed relatively equally, economic growth benefits from the positive impact of variables related to redistribution. income inequality, in its different forms, such as gini index or income shares of deciles of the population, may be harmful to economic growth. policies reducing inequality, such as higher tax policies, may be also destructive for the economic growth. these effects could differ to the extent of democracy. another important issue is that how income inequality change before and after taxes and transfers. if inequality after taxes and transfer, in other words, net inequality, is quite lower than inequality before taxes and transfer; this means that redistributive policies are implemented in the favor of the poor. inequality effects on economic growth may be handled as before and after taxes, or with the implementation of the effects of redistributive policies the effects of income inequality and redistribution in democracies dynamic econometric models 17 (2017) 19–39 21 which may differ to the extent of democracy levels of countries. in order to achieve comparable results for inequality-redistribution-growth relationships, net inequality is used in the analyses. section 1 discusses the literature and finds that most authors have reached a similar conclusion. section 2 devises a new statistical model whereby that conclusion can be confirmed or rejected. section 3 presents the econometric results. the last section evaluates the results warrant any change in the hitherto conventional conclusion. 1. literature review in the inequality and economic growth literature, both negative and positive effects of inequality are mentioned. inequality which can undermine progress in health and education, causes investment-reducing political and economic instability and undercuts the social consensus required to adjust in the face of shocks. inequality is also said beneficial for growth, by fostering aggregate savings (kuznets, 1955; kaldor, 1956), by promoting high return projects (rosenzweig and binswanger, 1993), by stimulating research and development (foellmi and zweimüller, 2006). some inequality is integral to the effective functioning of a market economy and the incentives needed for investment and growth. as for the okun’s (1975) big-trade-off hypothesis, a treatment for inequality as redistributive policies may be also worse for growth than disease itself by creating disincentives and inefficiencies. inequality is harmful for growth by promoting expensive fiscal policies (perotti, 1993; alesina and rodrik, 1994; persson and tabellini, 1994); by inducing an inefficient state bureaucracy (acemoglu et al., 2011), by hampering human capital formation (bénabou, 1996); or by undermining the legal system (glaeser et al., 2003). the paper by barro (2000) which shows a negative significant inequality impact in developing economies, applying simultaneous equation models and brings complex relationship between inequality and the fertility rate as the negative impact of inequality on economic growth is only significant when the fertility rate is omitted. voitchovsky (2005), applying a sys-gmm estimator, shows a negative impact of the gini coefficient. in connection with the fact that inequality and especially redistribution may have different effects on economic growth, in the following section these effects are discussed in the light of leading studies. goksu aslan dynamic econometric models 17 (2017) 19–39 22 1.1. democracy and inequality the literature on the relationship between democracy and inequality has controversial findings. sirowy and inkeles (1990) find that the existing evidence suggest the level of political democracy as measured at one point of time tends not to be widely associated with lower levels of income inequality. they suggest that there may be evidence in favor of the relevance of the history of democracy for inequality. muller (1988) finds a negative correlation between inequality and the numbers of years a country has been democratic. simpson (1990), burkhart (1997) and gradstein and justman (1999b) find a nonlinear reduced form relationship between democracy and inequality where at both low and high levels of democracy, inequality is lower, and it is higher at intermediate levels of democracy. however, the impact of the history of democracy identified in the models that do not include fixed effects, it will capture the impact of these omitted fixed effects. this is a special case of the difficulty of identifying duration dependence and unobserved heterogeneity. (acemoglu et.al, 2009) recent studies by li et.al (1998), rodrik (1999), and scheve and stasavage (2010) use better data and exploited the time as well as cross-sectional dimensions. li et. al (1998) find that an index of civil liberties is negatively correlated with inequality, such that greater civil liberties are associated with lower inequality, using a pooled ols. rodrik (1999) finds a positive correlation between freedom house of polity iii measure of democracy and both average real wages in manufacturing and the share of wages in national incomes, showing in both a crosssection and a panel of countries using country fixed effects. also, he finds an evidence that political competition and participation at large are important parts of the mechanisms via which democracy worked. scheve and stasavage (2009) use a long-run panel data from 1916 to 2000 for 13 oecd countries with country-specific effects. they find a significant positive correlation between the universal suffrage dummy and the share of income accruing to people between the 90th and 99th percentiles of the income distribution divided by the share accruing to the people above the 99th percentile. lee (2005) uses an unbalanced panel data with random effects, covering 64 countries for the period from 1970 to 1994. he finds a significant positive correlation between the size of government measured by tax revenues as a percentage of gdp and democracy, suggesting that for large enough levels of government, democracy reduces inequality. the effects of income inequality and redistribution in democracies dynamic econometric models 17 (2017) 19–39 23 1.2. democracy and redistribution meltzer and richard (1981) find that an expansion of democracy should lead to greater tax revenues and redistribution. gil et.al (2004) find no correlation between tax revenues and government spending measures and democracy, using a cross-sectional specification. aidt et.al (2006) and aidt and jensen (2009b) observe the impact of democratization measured by the proportion of adults who could vote in a cross-sectional panel. aidt et.al (2006) observe a cross-sectional panel of 12 west european countries over the period 1830–1938. aidt and jensen (2009b) find positive effects of suffrage on government expenditure as percentage of gdp and also tax revenues as a percentage of gdp, using a cross-national panel of 10 west european countries over the period 1860–1938. democracy is expected to change not only the amount of tax revenues, but also what taxes were used for. one might expect democracies to move towards more progressive taxation (acemoglu et.al, 2015). aidt and jensen (2009b) find that suffrage expansion leads to lower direct taxes and higher indirect taxes. aidt and jensen (2009a) find a nonlinear relationship between the introduction of an income tax and suffrage where an expansion of the franchise starting from very restrictive levels reduces the probability that an income tax will be introduced, but also that this probability increases at higher levels of the franchise. scheve and stasavage (2010, 2012) find no correlation between democracy and either tax progressivity or the rate of capital taxation, using a long-run approach of the oecd countries. lindert (1994) finds an impact of democracy on various types of social spending in a panel data over the period 1880–1930, stating that “there was so little social spending of any kind before the twentieth century mainly because political voice so restricted.” huber and stephens (2012) find the story of democracy which is measured by the cumulative years a country has been democratic since 1945 is positively correlated with education spending, health spending, and social security and welfare spending, observing a panel dataset for latin america over the period 1970–2007. kaufman and segura-ubiergo (2001) find that democracy which is measured by dichotomous measured introduced by przeworski et.al (2000) is positively correlated with government expenditure on health and education. brown and hunter (1999) also show that democracies have greater social spending than autocracies, using democracy measured by przeworski et.al (2000). persson and tabellini (2003) find some evidence that democracy measured by the gastil index and the polity score, has positive effects on government expenditure and government revenues, and social security spending as percentage of gdp. an additional focus of the democgoksu aslan dynamic econometric models 17 (2017) 19–39 24 racy-redistribution literature is based on whether if female enfranchisement has an additional or differential impact on government taxation or spending. results from lindert (1994) showing that financial enfranchisement has an independent impact on social spending are consistent with aidt and dallal (2008) for a later period. lott and kenny (1999) find that the expansion of women’s voting rights in the united states between 1870 and 1940 is associated with increases in per capita government revenues and expenditures. miller (2008) shows that female suffrage increases health spending and let to significant falls in infant mortality. aidt and jensen (2013) provide an identification strategy for the endogeneity of the democracy. they argue that revolutionary threat measured by revolutionary events in other countries is a viable instrument for democracy in a panel of west european countries, building on the theoretical ideas of acemoglu and robinson (2000, 2006) and aidt and jensen (2011). they find that democracy measured by the extent of suffrage has a robust positive effect on government spending. acemoglu et.al (2015) explain that the expectation in the literature has been that democracy should increase redistribution and reduce inequality, and why this expectation may not be borne out in the data because democracy may be captured or constrained, because democracy may cater to the wishes of the middle class, or because democracy may be simultaneously open up new economic opportunities to the previously excluded, contributing to economy inequality. they find that democratization increases government taxation and revenue as fractions of gdp, confirming the prediction of the standard meltzer-richard model. they do not find a robust evidence that democracy reduces inequality. 2. methodology 2.1. the model the starting point is the so-called solow-swan model which is developed by islam (1995). this approach allows including unobservable individual country and period-specific effects. the model is estimated as below: ' , 5 0 1 , 5 2 3it i t i t it it it i it ty y y n r                  (1) where ity – log of real gdp per capita of country i for time t, , 5i ty  – log of real gdp per capita of country i for time t–5, , 5it i ty y  – 5-year average growth rate of country i for time t, itn – initial level of net inequality of country i for time t, itr – initial level of redistribution of country i for time t, the effects of income inequality and redistribution in democracies dynamic econometric models 17 (2017) 19–39 25 i – unobserved time invariant country specific effects, t – time effect of period t, it – idiosyncratic error term, it – additional control variables, including investment share, population growth and total education. 1 2.2. the data the dataset includes 5-year average observations for 137 countries over a period from 1960 to 2010. the countries and periods are restricted to the availability of data. the data comes from penn world table (pwt) 2 version 8.0, the standardized world income inequality database (swiid) version 5.0, polity iv project (2014) and berg et.al (2014). as regard as several income inequality measures, gini index is most widely used among them. income shares of the deciles or quantiles of the population are also highly informative on how income distributed within population. redistribution is calculated as the difference between market income and net income that is equal the sum of taxes and transfers. 3 the standardized world income inequality database (swiid) which was introduced in 2008 has provided income inequality data that seek to maximize comparability while providing the broadest possible coverage of countries and years. (solt, 2014) the polity iv dataset covers all major, independent states in the global system over the period 1800–2013 (i.e., states with a total population of 500,000 or more in the most recent year; currently 167 countries). the "polity score" captures this regime authority spectrum on a 21-point scale ranging from –10 (hereditary monarchy) to +10 (consolidated democracy). the polity scores can also be converted into regime categories in a suggested three-part categorization of "autocracies" (–10 to –6), "anocracies" (–5 to +5), and "democracies" (+6 to +10). (center for systemic peace, 2014) 1 ,i t s n  and ,i t s r  – lagged terms of independent variables are used as instrument variables. 2 the penn world table (pwt) is a database with information on relative levels of income, output, inputs and productivity, with country and period coverage depending on the release. pwt version 8.0 which is released in 2013 is a database with information on relative levels of income, output, inputs and productivity, covering 167 countries between 1950 and 2011. 3 market income is defined as amount of money coming into the household, excluding any government cash or near-cash benefits, the so-called “pre-tax, pre-transfer income”. gross income is the sum of market income and government transfer payments; it is “pre-tax, post-transfer income”. net income, in turn, is gross income minus direct taxes: ‘post-tax, post-transfer income”. solt (2009) goksu aslan dynamic econometric models 17 (2017) 19–39 26 as regard as the inclusion of control variables, a higher saving rate can sustain a higher level of output per capita as capital accumulation per individual also increases. in turn, the higher rate of saving is associated the richer country, in contrast with the higher rate of population growth. augmented solow growth model associates higher saving rate to higher growth rate of real gdp per capita, higher n+g+δ to lower growth rate of real gdp per capita, with g+δ constant and equal to 0.05 across countries. in the analyses, following mankiw et al. (1992), g+δ is taken to be equal to 0.05 assuming this value to be the same for all countries. human capital, particularly as attained through education, to economic progress (lucas, 1988; mankiw, romer and weil, 1992). an abundance of well-educated people goes along with a high level of labor productivity. it also implies larger numbers of more skilled workers and greater ability to absorb advanced technology from developed countries. the level and distribution of educational attainment also have impact on social outcomes, such as child mortality, fertility, education of children, and income distribution (barro and lee, 1994; de gregorio and lee, 2002; breierova and duflo, 2004; cutler et al., 2006). 1.2. the estimator since the right-hand-side variables are typically endogenous and measured with error, and there are omitted variables, estimating growth regressions becomes problematic. in the presence of omitted variables, least squares parameter estimates are biased (bond et.al, 2001). applying ols creates the problem that is correlated with the fixed effects in the error term, which causes dynamic panel bias (nickell, 1981). the difference and system generalized method of moments estimators, hence diff-gmm and sys-gmm respectively, are designed for dynamic panel analysis where current realizations of the dependent variable influenced by past ones in the cases where large n, small t; there may be arbitrarily distributed fixed individual effects. this argues against cross-section regressions which must essentially assume fixed effects away, and in favor of a panel setup where variation over time can be used to identify parameters. other problem is to have endogeneity while observing the effects of inequality on economic growth, because of the unavailability of time-variant exogenous regressors. following the diff-gmm and sys-gmm estimators would be appropriate approach if the instruments are correctly used. the diff-gmm and sysgmm estimators assume that some regressors may be endogenous; the idiosyncratic disturbances (those apart from the fixed effects) may have individual specific patterns of heteroskedasticity and serial correlation; the idiosynthe effects of income inequality and redistribution in democracies dynamic econometric models 17 (2017) 19–39 27 cratic disturbances are uncorrelated across individuals; some regressors can be predetermined but not strictly exogenous. finally, because the estimators are designed for general use, they do not assume that good instruments are available outside the immediate dataset. in effect, it is assumed that the only available instruments are “internal” – based on lags of the instrumented variables. however, the estimators do allow inclusion of external instruments (roodman, 2009). comparing to diff-gmm estimator, the sys-gmm estimator has many advantages as in variables which are random-walk or close to be random-walk (baum, 2006). by using diff-gmm, differencing variables within groups will remove any variable that is constant. sysgmm produces more efficient and precise estimates compared to diffgmm, by improving precision and reducing the finite sample bias (baltagi, 2008). while working with unbalanced panel as in the panel dataset used in this paper, diff-gmm approach would be weak in filling gaps (roodman, 2006, p.20). the sys-gmm estimator is unbiased and most efficient if there are endogenous and predetermined regressors. by construction, the residuals of the differenced equation should possess serial correlation, but if the assumption of serial independence in the original errors is warranted, the differenced residuals should not exhibit significant ar(2) behaviour. if a significant ar(2) statistic is encountered, the second lags of endogenous variables will not be appropriate instruments for their current values (baum, 2013). 3. estimations the countries have been grouped into two groups to the extent of their regime authorities, such that countries having polity iv score greater than 5 defined as democracies and otherwise defined as non-democracies. the analyses cover 137 countries of which 112 are democracies first, the test is conducted to compare means and covariance matrices of these two groups. table 1. test of equality of covariance matrices across two samples test test statistic prob. modified lr chi2 231.2813 box f(12,18884415.1) 19.18 prob>f= 0.000 box chi2 230.11 prob>chi2 0.000 lr chi-squared, box’s f, box’s chi-squared statistics, and associated pvalues for the equality of covariance matrices are reported in table 1. according to the test of equality covariance matrices, the covariance matrices goksu aslan dynamic econometric models 17 (2017) 19–39 28 could not be considered as equal. assuming heterogeneous covariance matrices, the samples, the equality of means is tested. table 2. test of equality of means of two samples test test statistic prob. lr chi2(3) 68.36 prob>chi2 0.000 lr chi-squared statistic and associated p-values for the equality of means are reported in table 2. the null hypothesis of equal means for the samples is rejected. the means of two groups could not be considered as equal. table 3. the impact of inequality and redistribution on economic growth sysgmm estimation – the whole sample models (1) (2) (3) (4) (5) baseline baseline + controls baseline + controls baseline + controls baseline + controls ln(initial income) –0.0163** –0.0186** –0.0217*** –0.0264*** –0.0207*** (0.0070) (0.0079) (0.0072) (0.0083) (0.0079) net inequality –0.0930** –0.1036** –0.0462 –0.0859** –0.0735* (0.0423) (0.0410) (0.0404) (0.0354) (0.0429) redistribution 0.0831 0.0701 0.0603 0.0313 0.0624 (0.0623) (0.0706) (0.0692) (0.0495) (0.0698) ln(i) 0.0181 0.0332** 0.0120 (0.0158) (0.0150) (0.0234) ln(n) –0.0810** –0.0063 (0.0384) (0.0365) ln(total education) 0.0429 (0.0317) ln (s) – ln (n+g+δ) 0.0368** (0.0177) constant 0.1914*** 0.1612** 0.2658** 0.1680** 0.1775** (0.0725) (0.0764) (0.1069) (0.0843) (0.0707) observations 676 676 676 623 676 number of countries 137 137 137 118 137 hansen 0.5643 0.4586 0.6022 0.4530 0.4629 difference-in-hansen 0.761 0.681 0.610 0.384 0.650 ar1 0.0000 0.0000 0.0001 0.0002 0.0001 ar2 0.1195 0.1897 0.1248 0.1868 0.1655 number of instruments 58 58 58 58 58 notes: the table reports sys-gmm estimations. all regressions include time dummies; second lags in gmm instruments; robust standard errors in parentheses; ar(1) is the p-value on the test for the first order serial correlation in the differenced error terms; ar(2) is the p-value on the test for the second order serial correlation in the differenced error terms; hansen denotes the p-value on the test for the validity of the full instrument set; difference-in-hansen denotes the p-value for the validity of the set of levelequation. ***, **, * denote significance at the 1, 5, and 10 % levels, respectively. the effects of income inequality and redistribution in democracies dynamic econometric models 17 (2017) 19–39 29 the results from the sys-gmm estimations are presented in table 3 for the whole sample. the results from the ols and wg estimations are documented in appendix in table 2.1a and 2.1b respectively for comparison. for the whole sample, net inequality has a negative significant impact on economic growth. a rise in the gini of net income by 10 points leads to a decrease in the growth of real gdp per capita in the range of 0.74–1.10%. however, redistribution impact remains insignificant for the whole sample. the results from the sys-gmm estimations for democracies are presented in table 4. the results from the ols and wg estimations are documented in appendix in table 3.1a and 3.1b respectively for comparison. table 4. the impact of inequality and redistribution on economic growth sysgmm estimation – sub-group of democracies models (1) (2) (3) (4) (5) baseline baseline + controls baseline + controls baseline + controls baseline + controls ln(initial income) –0.0276*** –0.0278*** –0.0333*** –0.0275*** –0.0292*** (0.0078) (0.0080) (0.0104) (0.0081) (0.0081) net inequality –0.1134** –0.1126*** –0.1222** –0.0888* –0.0877** (0.0479) (0.0400) (0.0614) (0.0473) (0.0393) redistribution 0.1408** 0.1265** 0.1645** 0.0934 0.1314** (0.0588) (0.0567) (0.0739) (0.0659) (0.0569) ln(i) 0.0398* 0.0507** 0.0561*** (0.0225) (0.0199) (0.0160) ln(n) –0.0091 0.0111 (0.0626) (0.0712) ln(total education) 0.0292 (0.0322) ln (s) – ln (n+g+δ) 0.0432* (0.0222) constant 0.3026*** 0.1809* 0.2081 0.0526 0.2524*** (0.0773) (0.0925) (0.1359) (0.1727) (0.0738) observations 503 503 503 458 503 number of countries 112 112 112 97 112 hansen 0.1103 0.2671 0.3195 0.4615 0.3522 difference-in-hansen 0.515 0.343 0.431 0.493 0.465 ar1 0.0001 0.0003 0.0004 0.0005 0.0005 ar2 0.0561 0.1456 0.1541 0.2579 0.1157 number of instruments 59 59 59 59 59 notes: the table reports sys-gmm estimations. all regressions include time dummies; second lags in gmm instruments; robust standard errors in parentheses; ar(1) and ar(2) are the p-values on the test for the first and second order serial correlation in the differenced error terms respectively; hansen denotes the p-value on the test for the validity of the full instrument set; difference-in-hansen denotes the p-value for the validity of the set of level-equation. ***, **, * denote significance at the 1, 5, and 10 % levels, respectively. goksu aslan dynamic econometric models 17 (2017) 19–39 30 from the column 1 to 4, the standard control variables; and in column 5 only difference of logarithms of saving rate and population growth rate, technological progress, capital depreciation rate (ln (s) – ln (n+g+δ)) are added into the model. all models include time-specific dummies. the obtained results show that net inequality and redistribution have significant effects on economic growth in democratic countries. net inequality has always a negative significant effect, only its effect vanishes with the addition of the control variables. in the first model which includes only initial income, net inequality, and redistribution, both variables are significant. thus, a rise in the gini of net income by 10 points leads to a decrease of 1.13% in the growth of real gdp per capita. on the other, a rise in the redistribution by 10 points leads to an increase of 1.4% in the growth of real gdp per capita. by adding investment share and population growth as control variables, the impacts of both variables are still significant and in the same direction as in the first model. in column 4, with the addition of total education as a control variable, redistribution impact becomes insignificant, but its coefficients still maintain positive sign. however, in the model which includes this control variable, net inequality has a significant negative effect, while redistribution has a significant positive effect. a rise in the gini of net income by 10 points leads to a decrease of 0.877% in the growth of real gdp per capita. on the other hand, a rise in the redistribution by 10 points leads to an increase of 1.3% in the growth of real gdp per capita. by construction, the residuals of the differenced equation should possess serial correlation, but if the assumption of serial independence in the original errors is warranted, the differenced residuals should not exhibit significant ar(2) behavior. if a significant ar(2) statistic is encountered, the second lags of endogenous variables will not be appropriate instruments for their current values (baum, 2013). as expected, in the sys-gmm estimation, there exists first-order serial correlation and no second-order correlation. the only uncomfortable p-value associated with second-order correlation arises in the baseline model; nevermore, there is still no evidence for second-order serial correlation at the 5% significance level. in the models estimated with control variables, there is no evidence for second-order serial correlation at the 1, 5, and 10 % levels. the null hypothesis that overidentifying restrictions are valid cannot be rejected as to the p-value of hansen test. the null hypothesis of the joint validity of the subset of instruments cannot be rejected. the preferred specifications, the p-values on the difference-in-hansen test are typically above 0.45 (halter et al. 2014) an additional check for the dpd estimates’ validity is that the estimated coefficient of the lagged dependent variable lies between the values obtained the effects of income inequality and redistribution in democracies dynamic econometric models 17 (2017) 19–39 31 from the ols and wg estimations, as suggested by bond (2002). this is confirmed by the baseline model as ols=–0.0041>sys-gmm= –0.0276>wg=–0.0703. the results obtained from the ols and wg estimations which are presented in the appendix, show that the estimated significance and influence direction of the inequality and the redistribution is approximately the same regardless of the estimation method used. overall, net inequality impact is still significant and negative on the growth of real gdp per capita. instead, the impact of redistribution appears also negative but not significant. this might be related that the least squares parameter estimates are biased since the omitted variable is correlated with one of the regressors. the sys-gmm estimator is unbiased and most efficient if there are endogenous predetermined regressors. conclusions the estimation results could be interpreted as the net inequality has a negative significant impact on economic growth of the countries. this impact is lower for the whole sample than in democracies. instead, there is no evidence that redistribution is significant for the whole sample. according to the sys-gmm results for the whole sample, a rise in the gini of net income by 10 points leads to a decrease in the growth of real gdp per capita in the range of 0.74–1.10%. however, the redistribution remains insignificant for the whole sample. in democratic countries, both net inequality and redistribution impacts are significant. in democracies, net inequality has a negative effect which is strictly significant, as supported by also ols and wg estimations. a rise in the gini of net income by 10 points leads to a decrease in the growth of real gdp per capita in the range of 0.84–1.13%. on the other hand, a rise in the redistribution by 10 points leads to an increase in the growth of real gdp per capita in the range 1.27–1.4%. in democracies, redistribution has a positive significant impact on economic growth. the findings are consistent with the fact that in democracies governments tend to significantly redistribute more. the findings are also consistent with the fact that institutions matter as it has been shown in acemoglu et.al (2009) that democracy has redistributive effects. the results show that redistribution is not bad for growth when tested simultaneously with net inequality. especially in democracies, policymaker should decide in favor of redistribution since redistribution may not always create inefficiencies in the growth rate of the output per capita. goksu aslan dynamic econometric models 17 (2017) 19–39 32 references acemoglu, d., robinson, j. (2000), why did the west extend the franchise? democracy, inequality, and growth in historical perspective, q. j. econ., 115, 1167–1199, doi: http://dx.doi.org/10.1162/003355300555042. acemoglu, d., robinson, j. (2006), economic origins of dictatorship and democracy, new york, ny: cambridge university press, doi: http://dx.doi.org/10.1017/cbo9780511510809. acemoglu, d., johnson, s., robinson, j., yared, p. (2009), reevaluating the modernization hypothesis, j. monet. econ., 56, 1043–1058, doi: http://dx.doi.org/10.1016/j.jmoneco.2009.10.002. acemoglu, d., naidu, s., restrepo, p., robinson, j. (2015), democracy, redistribution and inequality, in a. b. atkinson, handbook of income distribution, vol 2b. chapter 21 (pp. 1885–1996). amsterdam: elsevier, doi: http://dx.doi.org/10.1016/b978-0-444-59429-7.00022-4. acemoglu, d., ticchi, d., vindigni, a. (2011), emergence and persistence of inefficient states, j. eur. econ. assoc., 9(2), 177–208, doi: http://dx.doi.org/10.1111/j.1542-4774.2010.01008.x. aidt, t., & dallal, b. (2008), female voting power: the contribution of women’s suffrage to the growth of social spending in western europe (1869–1960), public choice, 134(3–4), 391–417, doi: http://dx.doi.org/10.1007/s11127-007-9234-1. aidt, t., dutta, j., & loukoianova, e. (2006), democracy comes to europe: franchise expansion and fiscal 1830–1938, eur. econ. rev., 50(2), 249–283, doi: http://dx.doi.org/10.1016/j.euroecorev.2004.07.004. aidt, t., & jensen, p. (2009a), the taxman tools up: an event history study of the introduction of the personal income tax in western europe, 1815–1941, j. public econ., 93, 160–175, doi: http://dx.doi.org/10.1016/j.jpubeco.2008.07.006. aidt, t., & jensen, p. (2009b), tax structure, size of government, and the extension of the voting franchise in western europe, 1860–1938, int. tax public fin., 16(3), 362–394, doi: http://dx.doi.org/10.1007/s10797-008-9069-9. aidt, t., & jensen, p. (2011), workers of the world unite! franchise extensions and the threat of revolution in europe, cambridge working papers in economics, 1102, faculty of economics, university of cambridge, 1820–1938. aidt, t., & jensen, p. (2013), democratization and the size of government: evidence from the long 19th century, public choice, 157(3–4), 511–542, doi: http://dx.doi.org/10.1007/s11127-013-0073-y. alesina, a., rodrik, d. (1994), distributive politics and economics growth, q. j. econ., 109, 465–490, doi: http://dx.doi.org/10.2307/2118470. baltagi, b. (2008), forecasting with panel data, journal of forecasting, 27(2), 153–173. barro, r. j. (2000), inequality and growth in a panel of countries, journal of economic growth, springer, 5(1), 5–32, doi: http://dx.doi.org/10.1023/a:1009850119329. barro, r. j., lee, j. (1994), sources of economic growth, carnegie-rochester conference series on public policy, elsevier, 40(1), 1–46, doi: http://dx.doi.org/10.1016/0167-2231(94)90002-7. baum, c. (2006), an introduction to modern econometrics using stata, stata press books, statacorp lp, number imeus january. baum, c. (2013), implementing new econometric tools in stata, mexican stata users' group meetings, stata users group. http://dx.doi.org/10.1162/003355300555042 http://dx.doi.org/10.1017/cbo9780511510809 http://dx.doi.org/10.1016/j.jmoneco.2009.10.002 http://dx.doi.org/10.1016/b978-0-444-59429-7.00022-4 http://dx.doi.org/10.1111/j.1542-4774.2010.01008.x http://dx.doi.org/10.1007/s11127-007-9234-1 http://dx.doi.org/10.1016/j.euroecorev.2004.07.004 http://dx.doi.org/10.1016/j.jpubeco.2008.07.006 http://dx.doi.org/10.1007/s10797-008-9069-9 http://dx.doi.org/10.1007/s11127-013-0073-y http://dx.doi.org/10.2307/2118470 https://ideas.repec.org/a/kap/jecgro/v5y2000i1p5-32.html https://ideas.repec.org/s/kap/jecgro.html https://ideas.repec.org/s/kap/jecgro.html http://dx.doi.org/10.1023/a:1009850119329 https://ideas.repec.org/a/eee/crcspp/v40y1994ip1-46.html https://ideas.repec.org/s/eee/crcspp.html https://ideas.repec.org/s/eee/crcspp.html http://dx.doi.org/10.1016/0167-2231(94)90002-7 the effects of income inequality and redistribution in democracies dynamic econometric models 17 (2017) 19–39 33 bénabou, r. (1996), inequality and growth, cepr discussion papers, 1450, c.e.p.r. discussion papers. berg, a., ostry, j., tsangarides, c. (2014), redistribution, inequality, and growth, imf staff discussion note, february. bond, s.r. (2002), dynamic panel data models: a guide to micro data methods and practice, portuguese economic journal, 1(2), 141–162, doi: http://dx.doi.org/10.1007/s10258-002-0009-9. bond, s., anke, h., temple, j. (2001), gmm estimation of empirical growth models, cepr discussion papers, 3048. breierova, l., duflo, e. (2004), the impact of education on fertility and child mortality: do fathers really matter less than mothers?, nber working papers, 10513, national bureau of economic research, inc. brown, d., hunter, w. (1999), democracy and social spending in latin america, 1980–92, am. polit sci. rev., 93(4), 779–790, doi: http://dx.doi.org/10.2307/2586112. burkhart, r. (1997), comparative democracy and income distribution: shape and direction of the causal arrow, j. polit., 59(1), 148–164, doi: http://dx.doi.org/10.2307/2998219. cutler, d., deaton, a., lleras-muney, a. (2006), the determinants of mortality, journal of economic perspectives, american economic association, 20(3), 97–120, doi: http://dx.doi.org/10.1257/jep.20.3.97. de gregorio, j., lee, j., (2002), education and income inequality: new evidence from crosscountry data, review of income and wealth, international association for research in income and wealth, 48(3), 395–416, doi: http://dx.doi.org/10.1111/1475-4991.00060. foellmi, r., & zweilmueller, j. (2006), structural change and the kaldor facts of economic growth, meeting papers, 342, society for economic dynamics. gil, r., mulligan, c., sala-i-martin, x. (2004), do democracies have different public policies than nondemocracies?, j. econ. perspect., 18, 51–74. glaeser, e. (2003), introduction to "the governance of not-for-profit organizations, nber chapters, 1–44. gradstein, m., justman, m. (1999b), the democratization of political elites and the decline in inequality in modern economic growth, in breizes, e.s., temin, p. (eds), elites, minorities and economic growth, amsterdam: elsevier, halter, d., oechslin, m., zweimüller, j. (2014), inequality and growth: the neglected time dimension, journal of economic growth, 19(1), 81–104, doi: http://dx.doi.org/10.1007/s10887-013-9099-8. huber, e., stephens, j. (2012), democracy and the left: social policy and inequality in latin america, university of chicago press, chicago, il, doi: http://dx.doi.org/10.7208/chicago/9780226356556.001.0001. islam, n. (1995), growth empirics: a panel data approach, the quarterly journal of economics, 110(4), 1127–70, doi: http://dx.doi.org/10.2307/2946651. kaldor, n. (1956), alternative theories of distribution, review of economic studies, 23, 83–100. kaufman, r., segura-ubiergo, a. (2001), globalization, domestic politics, and social spending in latin america: a time-series cross-section analysis, 1973–97, world polit., 53(4), 553–587, doi: http://dx.doi.org/10.1353/wp.2001.0016. kuznets, s. (1955), economic growth and income inequality, american economic review, 65, 1–28. lee, c.-s. (2005), income inequality, democracy, and public sector size, am. sociol. rev., 70(1), 158–181, doi: http://dx.doi.org/10.1177/000312240507000108. http://dx.doi.org/10.1007/s10258-002-0009-9 https://ideas.repec.org/p/nbr/nberwo/10513.html https://ideas.repec.org/p/nbr/nberwo/10513.html https://ideas.repec.org/s/nbr/nberwo.html http://dx.doi.org/10.2307/2586112 http://dx.doi.org/10.2307/2998219 https://ideas.repec.org/a/aea/jecper/v20y2006i3p97-120.html https://ideas.repec.org/s/aea/jecper.html https://ideas.repec.org/s/aea/jecper.html http://dx.doi.org/10.1257/jep.20.3.97 https://ideas.repec.org/a/bla/revinw/v48y2002i3p395-416.html https://ideas.repec.org/a/bla/revinw/v48y2002i3p395-416.html https://ideas.repec.org/s/bla/revinw.html http://dx.doi.org/10.1111/1475-4991.00060 http://dx.doi.org/10.1007/s10887-013-9099-8 http://dx.doi.org/10.7208/chicago/9780226356556.001.0001 http://dx.doi.org/10.2307/2946651 http://dx.doi.org/10.1353/wp.2001.0016 http://dx.doi.org/10.1177/000312240507000108 goksu aslan dynamic econometric models 17 (2017) 19–39 34 li, h., squire, l., zou, h. (1998), explaining international and intertemporal variations in income inequality, econ. j., 108(1), 26–43, doi: http://dx.doi.org/10.1111/1468-0297.00271. lindert, p. (1994), the rise of social spending, 1880–1930, explor. econ. hist., 31(1), 1–37, doi: http://dx.doi.org/10.1006/exeh.1994.1001. lott jr., j., kenny, l. (1999), how dramatically did women’s suffrage change the size and scope of government?, j. polit. econ., 107(6), 1163–1198, doi: http://dx.doi.org/10.1086/250093. lucas, r. j. (1988), on the mechanics of economic development, journal of monetary economics, elsevier, vol. 22(1), 3–42, doi: http://dx.doi.org/10.1016/0304-3932(88)90168-7. mankiw, n.g, romer, d., weil, d.n. (1992), a contribution to the empirics of economic growth, the quarterly journal of economics, oxford university press, 107(2), 407–437, doi: http://dx.doi.org/10.2307/2118477. meltzer, a., & richard, s. (1981), a rational theory of the size of government, j. polit. econ., 89, 914–927, doi: http://dx.doi.org/10.1086/261013. miller, g. (2008), women’s suffrage, political responsiveness, and child survival in american history, q. j. econ., 123(3), 1287–1327, doi: http://dx.doi.org/10.1162/qjec.2008.123.3.1287. muller, e. (1988), democracy, economic development, and income inequality, am. sociol. rev., 53(1), 50–68, doi: http://dx.doi.org/10.2307/2095732. nickell, s. (1981), biases in dynamic models with fixed effects, econometrica, 49(6), 1417–1426, doi: http://dx.doi.org/10.2307/1911408. okun, a. (1975), equality and efficiency: the big trade-off, brookings institution press. penn world table, version 8.0. (2013), retrieved from rug.nl/research/ggdc/data/pwt. perotti, r. (1993), political equilibrium, income distribution, and growth, review of economic studies, 60(4), 755–776, doi: http://dx.doi.org/10.2307/2298098. persson, t., tabellini, g. (2003), the economic effects of constitutions, mit press, cambridge. phillips, p., sul, d. (2003), dynamic panel estimation and homogeneity testing under cross section dependence, econometrics journal, 6, 217–259, doi: http://dx.doi.org/10.1111/1368-423x.00108. polityiv project. (2014), retrieved from systemicpeace.org/polityproject.html. przeworski, a., alvarez, m., cheibub, j., limongi, f. (2000), democracy and development: political institutions and material well-being in the world, 1950–1990, new york, ny: cambridge university press, doi: http://dx.doi.org/10.1017/cbo9780511804946. rodrik, d. (1999), democracies pay higher wages, q. j. econ., 114, 707–738, doi: http://dx.doi.org/10.1162/003355399556115. roodman, d. (2006), how to do xtabond2: an introduction to "difference" and "system" gmm in stata, working papers, 103, center for global development, doi: http://dx.doi.org/10.2139/ssrn.982943. roodman, d. (2009), how to do xtabond2: an introduction to difference and system gmm. the stata journal, 9(1), 86–136. rosenzweig, m., binswanger, h. (1993), wealth, weather risk and the composition and profitability of agricultural investments, economic journal, 103(416), 56–78, doi: http://dx.doi.org/10.2307/2234337. scheve, k., stasavage, d. (2009), institutions, partisanship, and inequality in the long run, world polit., 61, 215–253, doi: http://dx.doi.org/10.1017/s0043887109000094. http://dx.doi.org/10.1111/1468-0297.00271 http://dx.doi.org/10.1006/exeh.1994.1001 https://econpapers.repec.org/scripts/redir.pf?u=http%3a%2f%2fdx.doi.org%2f10.1086%2f250093;h=repec:ucp:jpolec:v:107:y:1999:i:6:p:1163-1198 https://ideas.repec.org/a/eee/moneco/v22y1988i1p3-42.html https://ideas.repec.org/s/eee/moneco.html https://ideas.repec.org/s/eee/moneco.html http://dx.doi.org/10.1016/0304-3932(88)90168-7 https://ideas.repec.org/a/oup/qjecon/v107y1992i2p407-437..html https://ideas.repec.org/a/oup/qjecon/v107y1992i2p407-437..html https://ideas.repec.org/s/oup/qjecon.html http://dx.doi.org/10.2307/2118477 http://dx.doi.org/10.1086/261013 http://dx.doi.org/10.1162/qjec.2008.123.3.1287 http://dx.doi.org/10.2307/2095732 http://dx.doi.org/10.2307/1911408 http://dx.doi.org/10.2307/2298098 http://dx.doi.org/10.1111/1368-423x.00108 http://dx.doi.org/10.1017/cbo9780511804946 http://dx.doi.org/10.1162/003355399556115 http://dx.doi.org/10.2139/ssrn.982943 http://dx.doi.org/10.2307/2234337 http://dx.doi.org/10.1017/s0043887109000094 the effects of income inequality and redistribution in democracies dynamic econometric models 17 (2017) 19–39 35 scheve, k., stasavage, d. (2010), the conscription of wealth: mass warfare and the demand for progressive taxation, inter. organ., 64, 529–561, doi: http://dx.doi.org/10.1017/s0020818310000226. scheve, k., & stasavage, d. (2012), democracy, war, and wealth: lessons from two centuries of inheritance taxation, am. polit. sci. rev., 106(1), 82–102, doi: http://dx.doi.org/10.1017/s0003055411000517. simpson, m. (1990), political rights and income inequality: a cross-national test, am. sociol. rev., 55(5), 682–693. sirowy, l., inkeles, a. (1990), the effects of democracy on economic growth and inequality: a review, stud. comput. inter. develop., 25, 126–57, doi: http://dx.doi.org/10.1007/bf02716908. solt, f. (2009), standardizing the world income inequality database, social science quarterly, 90(2), 231–242, doi: http://dx.doi.org/10.1111/j.1540-6237.2009.00614.x. solt, f. (2014), swiid version 5.0., retrieved from the standardized world income inequality database. stiglitz, j. (2012), the price of inequality: how today's divided society endangers our future, w. w. norton & company. thewissen, s. (2013), is it the income distribution or redistribution that affects growth?, socio-economic review, 12(1), 545–571, doi: http://dx.doi.org/10.1093/ser/mwt019. voitchovsky, s. (2005), does the profile of income inequality matter for economic growth? journal of economic growth, 10(3), 273–296, doi: http://dx.doi.org/10.1007/s10887-005-3535-3. wpływ nierówności dochodowych i redystrybucji na wzrost w krajach demokratycznych: dynamiczny model panelowy s t r e s z c z e n i e. w artykule analizuje się jednoczesny wpływ nierówności dochodowych i redystrybucji na wzrost dla całej próby oraz dla podgrupy demokratycznych krajów, wykorzystując estymator sys-gmm na danych panelowych dla okresu od 1960 do 2010. ogólnie, nierówności dochodowe mają negatywny istotny wpływ na wzrost w kolejnych 5 latach w obydwu grupach podczas gdy wpływ redystrybucji jest istotny tylko w krajach demokratycznych. wyniki są spójne z faktem, że poziom redystrybucji jest co do zasady wyższy w krajach pod rządami demokratycznymi. s ł o w a k l u c z o w e: demokracja; nierówność; redystrybucja; wzrost gospodarczy http://dx.doi.org/10.1017/s0020818310000226 http://dx.doi.org/10.1017/s0003055411000517 http://dx.doi.org/10.1007/bf02716908 http://dx.doi.org/10.1111/j.1540-6237.2009.00614.x http://dx.doi.org/10.1093/ser/mwt019 http://dx.doi.org/10.1007/s10887-005-3535-3 goksu aslan dynamic econometric models 17 (2017) 19–39 36 appendix box 1. country list angola, argentina, armenia, australia, austria, azerbaijan, burundi, belgium, burkina faso, bangladesh, bulgaria, bahamas, bosnia and herzegovina, belarus, bolivia, brazil, barbados, botswana, central african, republic, canada, switzerland, chile, china, ivory coast, cameroon, colombia, cape verde, costa rica, cyprus, czech republic, germany, djibouti, denmark, dominican republic, algeria, ecuador, egypt, spain, estonia, ethiopia, finland, fiji, france, gabon, united kingdom, georgia, ghana, guinea, the gambia, guinea-bissau, greece, guatemala, hong kong, honduras, croatia, haiti, hungary, indonesia, india, ireland, iran, iceland, israel, italy, jamaica, jordan, japan, kazakhstan, kenya, kyrgyzstan, cambodia, republic of korea, laos, st. lucia, sri lanka, lesotho, lithuania, luxembourg, latvia, morocco, madagascar, mexico, macedonia, mali, malta, mongolia, mauritania, mauritius, malawi, malaysia, namibia, niger, nigeria, nicaragua, netherlands, norway, nepal, new zealand, pakistan, panama, peru, philippines, papua, new, guinea, poland, puerto rico, portugal, paraguay, moldova, romania, russia, rwanda, senegal, singapore, sierra leone, el salvador, slovak republic, slovenia, sweden, swaziland, thailand, tajikistan, turkmenistan, trinidad and tobago, tunisia, turkey, taiwan, tanzania, uganda, ukraine, uruguay, united, states, uzbekistan, venezuela, vietnam, yemen, south africa, zambia. table 1.1. descriptive statistics for the whole sample variable number of obs. mean sd min. max. log of income per capita 1672 8.31719 1.295737 5.270576 11.58735 average growth of income per capita 987 0.02216 0.031082 –0.103573 0.202936 net inequality 1065 0.38007 0.104694 0.162734 0.673439 redistribution 1057 0.07043 0.068471 –0.105736 0.303942 log of initial income 987 8.45723 1.264281 5.191837 11.20154 log of investment share 988 3.04982 0.445171 1.086359 4.303144 log of population growth 1065 1.86565 0.204923 –0.484576 2.652895 log of total education 947 1.79102 0.57002 –0.969471 2.572392 table 1.2. descriptive statistics for democracies variable number of obs. mean sd min. max. log of income per capita 1258 8.537148 1.307467 5.398523 11.58735 average growth of income per capita 574 0.0238848 0.0281755 –0.0888746 0.2029362 net inequality 636 0.3518494 0.1004397 0.1672558 0.6734387 redistribution 634 0.0867381 0.0704751 –0.1056635 0.3039419 log of initial income 574 9.079349 1.064366 5.650741 11.20154 log of investment share 574 3.092598 0.3372506 1.193682 4.303144 log of population growth 636 1.790123 0.1790554 0.4518194 2.271904 log of total education 559 2.046362 0.4070422 0.1049389 2.572392 the effects of income inequality and redistribution in democracies dynamic econometric models 17 (2017) 19–39 37 table 1.3. descriptive statistics for non-democracies variable number of obs. mean sd min. max. log of income per capita 414 7.64882 0.9974434 5.270576 10.81616 average growth of income per capita 413 0.0197546 0.0346132 –0.1035728 0.1386531 net inequality 429 0.421903 0.0966637 0.1627338 0.6551682 redistribution 423 0.0459772 0.0572967 –0.1057364 0.2690889 log of initial income 413 7.592597 0.980364 5.191837 10.64515 log of investment share 414 2.990511 0.5565421 1.086359 4.168078 log of population growth 429 1.977622 0.1891687 –0.4845762 2.652895 log of total education 388 1.423142 0.5705287 –0.9694707 2.379649 table 2.1a. the impact of inequality and redistribution on economic growth ols estimation – the whole sample models (1) (2) (3) (4) (5) baseline baseline + ctrls baseline + ctrls baseline + ctrls baseline + ctrls ln(initial income) –0.0005 –0.0030** –0.0048*** –0.0076*** –0.0046*** (0.0012) (0.0012) (0.0013) (0.0015) (0.0012) net inequality –0.0523*** –0.0520*** –0.0347** –0.0306** –0.0385*** (0.0155) (0.0136) (0.0142) (0.0137) (0.0135) redistribution –0.0209 –0.0136 –0.0167 –0.0129 –0.0156 (0.0203) (0.0214) (0.0207) (0.0219) (0.0209) ln(i) 0.0209*** 0.0224*** 0.0226*** (0.0031) (0.0029) (0.0030) ln(n) –0.0294** –0.0200* (0.0116) (0.0115) ln(total education) 0.0095*** (0.0031) ln (s) – ln (n+g+δ) 0.0232*** (0.0028) constant 0.0418*** –0.0008 0.0602** 0.0504* 0.0452*** (0.0131) (0.0131) (0.0290) (0.0269) (0.0114) observations 979 979 979 882 979 notes: the table reports ols estimations. all regressions include time dummies; robust standard errors in parentheses. ***, **, * denote significance at the 1, 5, and 10 % levels, respectively. goksu aslan dynamic econometric models 17 (2017) 19–39 38 table 2.1b. the impact of inequality and redistribution on economic growth wg estimation – the whole sample models (1) (2) (3) (4) (5) baseline baseline + ctrls baseline + ctrls baseline + ctrls baseline + ctrls ln(initial income) –0.0378*** –0.0071*** –0.0077*** –0.0114*** –0.0099*** (0.0061) (0.0016) (0.0016) (0.0022) (0.0016) net inequality 0.0115 –0.0475*** –0.0385** –0.0242* –0.0287** (0.0262) (0.0156) (0.0152) (0.0146) (0.0131) redistribution 0.0500* 0.0154 0.0124 –0.0054 0.0066 (0.0282) (0.0210) (0.0205) (0.0219) (0.0214) ln(i) 0.0224*** 0.0233*** 0.0249*** 0.0227*** (0.0036) (0.0034) (0.0050) (0.0031) ln(n) –0.0205 –0.0331*** –0.0170 (0.0131) (0.0090) (0.0130) ln(total education) 0.0127*** 0.0121*** (0.0041) (0.0035) constant 0.2948*** 0.0191 0.0577* 0.0918*** 0.0522* (0.0464) (0.0160) (0.0334) (0.0250) (0.0304) observations 979 979 979 514 882 number of groups 164 164 164 101 133 notes: the table reports wg estimations. all regressions include time dummies and country specific fixed effects; robust standard errors in parentheses. ***, **, * denote significance at the 1, 5, and 10 % levels, respectively. table 3.1a. the impact of inequality and redistribution on economic growth ols estimation – sub-group of democracies models (1) (2) (3) (4) (5) baseline baseline + ctrls baseline + ctrls baseline + ctrls baseline + ctrls ln(initial income) –0.0041** –0.0060*** –0.0079*** –0.0102*** –0.0075*** (0.0019) (0.0020) (0.0017) (0.0022) (0.0019) net inequality –0.0501** –0.0553*** –0.0288 –0.0212 –0.0406** (0.0213) (0.0197) (0.0206) (0.0160) (0.0196) redistribution –0.0172 –0.0139 –0.0152 –0.0080 –0.0140 (0.0225) (0.0217) (0.0208) (0.0202) (0.0211) ln(i) 0.0195*** 0.0222*** 0.0241*** (0.0049) (0.0049) (0.0051) ln(n) –0.0418*** –0.0301*** (0.0079) (0.0079) ln(total education) 0.0120*** (0.0038) ln (s) – ln (n+g+δ) 0.0249*** (0.0040) constant 0.0683*** 0.0298 0.1082*** 0.0786*** 0.0692*** (0.0213) (0.0224) (0.0252) (0.0240) (0.0195) observations 572 572 572 514 572 notes: the table reports ols estimations. all regressions include time dummies; robust standard errors in parentheses. ***, **, * denote significance at the 1, 5, and 10 % levels, respectively. the effects of income inequality and redistribution in democracies dynamic econometric models 17 (2017) 19–39 39 table 3.1b. the impact of inequality and redistribution on economic growth wg estimation – sub-group of democracies models (1) (2) (3) (4) (6) baseline baseline + ctrls baseline + ctrls baseline + ctrls baseline + ctrls ln(initial income) –0.0703*** –0.0094*** –0.0105*** –0.0114*** –0.0106*** (0.0093) (0.0025) (0.0019) (0.0022) (0.0022) net inequality –0.0059 –0.0714*** –0.0462** –0.0242* –0.0559*** (0.0426) (0.0202) (0.0226) (0.0146) (0.0196) redistribution –0.0231 –0.0064 –0.0075 –0.0054 –0.0057 (0.0290) (0.0243) (0.0233) (0.0219) (0.0237) ln(i) 0.0232*** 0.0249*** 0.0249*** (0.0053) (0.0050) (0.0050) ln(n) –0.0405*** –0.0331*** (0.0126) (0.0090) ln(total educatinon) 0.0127*** (0.0041) ln (s) – ln (n+g+δ) 0.0275*** (0.0042) constant 0.6184*** 0.0541** 0.1252*** 0.0918*** 0.0975*** (0.0717) (0.0238) (0.0273) (0.0250) (0.0216) observations 572 572 572 514 572 number of groups 122 122 122 101 122 notes: the table reports wg estimations. all regressions include time dummies and country specific fixed effects; robust standard errors in parentheses. ***, **, * denote significance at the 1, 5 and 10 % levels, respectively. microsoft word 00_tresc.docx © 2013 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2013.009 vol. 13 (2013) 163−174 submitted september 30, 2013 issn accepted december 30, 2013 1234-3862 joanna małgorzata landmesser* decomposing the gender gap in average exit rate from unemployment a b s t r a c t. in the paper, we analyse the exit rates from unemployment, taking into account gender differences. the process of leaving the unemployment state was examined for each sex separately using the parametric hazard models. the objective was to present a decomposition of inequalities between men and women when leaving unemployment. the application of the modified oaxaca-blinder decomposition technique allowed us to isolate the factors explaining the observed inequalities. we found, that the gender gap is explained almost exclusively by differences in the effects of men’s and women’s characteristics. k e y w o r d s: duration of unemployment, parametric hazard models, gender gap, oaxacablinder decomposition. j e l classification: j16, j64. introduction different levels of the economic activity between women and men are often analyzed in the economic literature. it is noted that men are more frequently associated with the labor market, while women, to a lesser extent (due to their involvement in the family career). the review of the various aspects related to the activity of women and men in the labor market is provided, for example, by altonji, blank (1999). numerous empirical studies tend to focus on the gender wage gaps (blinder, 1973; oaxaca, 1973; beblo et al., 2003). the findings of these * correspondence to: joanna landmesser, department of econometrics and statistics, warsaw university of life sciences, nowoursynowska 166, 02-787 warszawa, poland, e-mail: joanna_landmesser@sggw.pl. joanna małgorzata landmesser dynamic econometric models 13 (2013) 163–174 164 studies show that males earn substantially higher wages than females. the part of each wage differential is due to differences in „objective” characteristics such as education and work experience. however, a partial differential remains even when male-female differences in these traits are controlled for. also the occupational segregation by gender is discussed in the literature (petrongolo, 2004). the studies show that women have a weaker position in the labor market. women are often discriminated against even if their qualifications are higher than for men. different attitudes of men and women are also reflected in the opportunities to take employment. gender is a significant factor affecting the movements between labor market states. it is an important variable in determining the chances of finding a job. many studies have shown that women have a lower probability of finding a new job (katz, meyer, 1990b), especially on a permanent basis (edin 1989), and are exposed to more frequent periods without work (steiner, 1989; jensen, westergard-nielsen, 1990). women are at a disadvantage even if their job-search activity is higher than for men. analyses focused on the description of leaving the unemployment state, which were carried out in poland so far, disregard the issue of gender behaviour. they generally do not consider that the effect of different factors may depend on gender. in empirical models, gender only shifts the dependent variable and there are not estimated separate equations for men and women (see for example malarska (2007)). the opposite approach propagate gonzalo, saarela (2000), ollikainen (2003), tansel, taşçi (2010) or landmesser (2008), who estimated separate hazard models for both sexes. the present study is an analysis of exit rates from unemployment, taking into account gender differences. in the paper, the process of leaving the unemployment state was examined for each sex separately using the tools of duration analysis – the parametric hazard models. the results confirmed that the influence of explanatory variables on the chance to exit the unemployment depends on the sex of the individual. however, the main objective of the study was to perform a decomposition of inequalities between men and women when leaving unemployment. the quantitative dimensions of the various causes of unequal chances by taking a job are not well known, but the oaxaca-blinder microeconometric decomposition technique applied allowed us to isolate the factors explaining the inequalities. decomposing the gender gap in average exit rate from unemployment dynamic econometric models 13 (2013) 163–174 165 1. the analysis method the study comprises the duration of time a person spent in the unemployment state (t). modeling of such a variable requires the use of survival analysis tools. constructed hazard models are suitable for the analysis of censored observations and for studying the influence of individual characteristics on the chances of leaving the unemployment state (cf. lancas ter, 1979). the basic function describing the distribution of the duration times is the survival function [ ] )(1pr)( tfttts −=>= , which expresses the probability of survival beyond a certain point in time t (cf. kalbfleisch, prentice, 2002). the hazard function (risk, intensity function) is the ratio of the density function and the survival function. it is the limit of probability that the episode is completed during the time interval [t, t+dt] given that it has not been completed before the moment t, for dt→0: [ ] . pr lim )( )( )( 0→ dt ttdtttt ts tf th dt ≥+<≤ == (1) the hazard rates describe the intensity of the transition from one state to another. using the parametric proportional hazards models, it is possible to specify hazard as a function of time and some explanatory variables: ).exp()()( 0 βxx jj thth ′= (2) models in this class differ in their assumptions about the baseline hazard )(0 th . the courses of hazard functions can take many forms, from monotonic to non-monotonic. to model the monotonic hazard, the weibull distribution with two parameters ),( pw λ is often used. the weibull distribution described originally the dispersion of the fatigue life of materials (weibull, 1939). it is also used by researchers modeling the duration of unemployment time (due to the decreasing intensity while leaving the state). for a variable ),(~ pwt λ survival and hazard functions are defined as )exp()( ptts λ−= and λ1)( −= pptth , for 0>λ , 0>p and 0≥t . the estimated parameter p is called a shape parameter. if p = 1, then the hazard function is constant; if p > 1, it monotonically increases, while for p < 1 the hazard monotonically decreases. the form of the weibull hazard model after taking the parameterization )exp( βx′=λ is as follows: joanna małgorzata landmesser dynamic econometric models 13 (2013) 163–174 166 ).exp()( 1 βxx j p j ptth ′= − (3) in order to describe the process of leaving the state of unemployment the weibull proportional hazard models were estimated for both men and women separately. this was followed by the analysis of inequalities observed using a modification of the oaxaca-blinder decomposition technique. the idea of oaxaca-blinder decomposition can be applied whenever we need to explain the differences between two comparison groups. let mw xx , be the characteristics of men and women, respectively, and let mw ββ , be the returns to these characteristics and φ the average level of dependent variable in the econometric model (e.g. the average hazard rate). the popular oaxaca-blinder decomposition for the gender gap at the aggregate level is as follows (oaxaca, 1973; blinder, 1973): ].)()([])()([)()( mwwwmmmwmmww βxβxβxβxβxβx '''''' φ−φ+φ−φ=φ−φ (4) the equation (4) is based on one group’s characteristics and the estimated coefficients of another group’s equation. the first term on the right hand side of the equation expresses the difference of the potentials of both groups (women and men). if the characteristics x used to estimate an econometric model exhausted all the factors affecting the chance to leave the unemployment state, it is possible to assume that the second term on the right hand side of this equation represents the amount of discrimination. this expression is the result of differences in the estimated parameters, and so in the „prices” of individual characteristics of men and women. blinder argued that „the latter sum […] exists only because the market evaluates differently the identical bundle of traits if possessed by different demographic groups” (blinder, 1973, pp. 438–439). the contribution of the differences in characteristics and coefficients of individual variables (the detailed decomposition) can be easily found when linear equations are used. the methodology proposed by yun (2004) provides a way to apply the oaxaca-blinder decomposition to a non-linear function for both aggregate and detailed decompositions. a modification made by yun allowed the use of the above concepts to determine the contribution of each explanatory variable in weibull regression to the total difference. the formula given by yun has the following form (as given in ortega masagué, 2008): decomposing the gender gap in average exit rate from unemployment dynamic econometric models 13 (2013) 163–174 167 ],)()([])()([ )()( 11 mwww k i mmmw k i x mmww ii ww βxβxβxβx βxβx '''' '' φ−φ+φ−φ= =φ−φ ∑∑ = δ = δ β (5) where: mmw m i m i w i x xx w i βxx ')( )( − − =δ β and ,1 1 =∑ = δ k i xi w )( )( ' mww m i w i w ixw i ββx − − =δ ββ β and ,1 1 =∑ = δ k i i w β k is the number of explanatory variables in the model, ,w mx x the mean levels of characteristics for men and women, respectively. the detailed decomposition methodology proposed by yun is the decomposition of differences in the first moment, i.e. differences in the mean value of the variable of interest. the method „does not depend on the functional form as long as the dependent variable is a function of a linear combination of independent variables and the function is once differentiable” (yun, 2004, p. 275). in order to obtain a proper weight yun evaluated the value of the function using mean characteristics and used a first order taylor expansion. 2. the empirical data unemployment is a problem frequently connected with specific regions. in our research work we try to analyze in detail the situation in district słupsk in north poland (voivodeship pomorskie). in this region, many inhabitants of rural areas were previously employed in the state-owned agricultural farms. however, a large number of these farms went bankrupt after the political system change in 1989. the former employees, women and men, turned out to be the most helpless social group in poland. therefore, it seems desirable to analyze the situation of the inhabitants of this region on the labor market over the past 20 years. the study was conducted using individual data of people registered as unemployed in the district labor office in słupsk. the selected sample consisted of 4 372 people registered in the office from january 1990 to august 2007. there were 2 203 women and 2 169 men randomly selected (women constituted 50.4% and men 49.6% of the selected sample). the data joanna małgorzata landmesser dynamic econometric models 13 (2013) 163–174 168 about each person took the multiepisode form and contained a detailed history of the office customers. at the data basis it was established how long unemployment episodes lasted (in days) or how long they are still going on (in the case of censored episodes). each person could register many times in the labor office in his or her history. therefore, for 4 372 examined people a total of 10 118 episodes of being unemployed was noted (10% of these episodes were censored) (see table 1). in the case of women, censored episodes constituted larger share of total episodes than it was for men. this was due to the fact that women had on average longer episodes of unemployment than men. for example, uncensored episodes for women lasted on average 413.1 days and for men 271.2 days. the information on the average duration in the unemployment state during a single episode is also presented in the table 1. table 1. the number and the average duration of unemployment episodes number of episodes all persons women men total 10118 100% 4786 100% 5332 100% censored 1007 10% 664 13.9% 343 6.4% uncensored 9111 90% 4122 86.1% 4989 93.6% average duration of episode (in days) all persons women men total 406.5 523.2 301.7 censored 1049 1207 745.5 uncensored 335.4 413.1 271.2 on average, a woman registered in the labor office 2.55 times and a man 3.02 times (more often, but for shorter periods). the average age of women at the beginning of the unemployment episode was 31.99 years old, while for men it was 33.66 years old. detailed information on the age of registrants, their level of education and place of residence are shown in the table 2. women registered in the labor office were usually younger than men, more frequently they had higher or secondary education level than in the case of men, they rarely had vocational education level. they are characterized by a higher proportion of residence in the city. episodes of unemployment were also examined for the payment of unemployment benefits, training benefits and social security benefits. it was found that men, more often than women received unemployment benefits, while women were often assigned training allowances and social security benefits. decomposing the gender gap in average exit rate from unemployment dynamic econometric models 13 (2013) 163–174 169 table 2. the structure of unemployment episodes by selected characteristics characteristics all persons women men age-group 17–24 years old 3068 30.3% 1512 31.6% 1556 29.2% 25–34 years old 2755 27.2% 1353 28.3% 1402 26.3% 35– 44 years old 2408 23.8% 1200 25.1% 1208 22.7% 45–54 years old 1742 17.2% 699 14.6% 1043 19.6% over 55 years old 145 1.4% 22 0.5% 123 2.3% education level tertiary 767 7.6% 517 10.8% 250 4.7% vocational secondary 1957 19.3% 1170 24.4% 787 14.8% general secondary 710 7.0% 516 10.8% 194 3.6% basic vocational 3100 30.6% 1240 25.9% 1860 34.9% lower second. or primary 3584 35.4% 1343 28.1% 2241 42.0% place of residence town 5079 50.2% 2551 53.3% 2528 47.4% village 5039 49.8% 2235 46.7% 2804 52.6% the information obtained from the database of the labor office allowed to establish a set of potential explanatory variables in the models describing the intensity of leaving the unemployment state. most of them are dichotomous variables, such as: − „gender” (number 1 coded the male sex), − set of variables for the five age categories („age1724”, „age2534”, „age3544”, „age4554”, „age55over”), − set of variables for the education level („tertiary”, „vocational secondary”, „general secondary”, „basic vocational”, „lower secondary or primary”), − „married” (1, if a person is not of free marital status), − „town” (1, if the person lives in the city), − „disabled” (1, if a person is disabled), − „unemployment benefits” (1 for those who receive unemployment benefits), − „training benefits” (1 for those who receive training benefits), − „social security” (1 for social security beneficiaries from polish zakład usług społecznych). in addition, the models used information about the consecutive unemployment episode number for the person (the variable called „episode_nr”). 3. the results of empirical analysis in order to analyse the distribution of unemployment durations, first the nonparametric kaplan-meier survival function s(t) separately for men and women was plotted (figure 1 (a)). joanna małgorzata landmesser dynamic econometric models 13 (2013) 163–174 170 (a) (b) figure 1. nonparametric kaplan-meier estimator for the survival function (a) and plots of the hazard functions in weibull model (b) the survival curve for men indicates a lower likelihood of continuation in the state of unemployment than for women (see figure 1(a)). there was a statistical significance of differences in the courses of the survival functions for both sexes. then, based on the total sample, the parametric weibull proportional hazard model for the chance to leave the unemployment state was estimated (the parameter estimates of this model are listed in table 3, part (a)). for example, the interpretation of the parameter by the variable "gender" is as follows: an opportunity to leave the unemployment state in the case of men is about 58.2% higher than in the case of women (exp (0.459) = 1.582). a positive value of the parameter βk means that a one unit change in the k-th explanatory variable results in the relative increase of hazard. a negative value is associated with the relative decrease of hazard, respectively. we conclude, that the chances of leaving unemployment rise among men, people at a younger age, better educated, married, living in the city, with the next in turn episode without a job, and they decrease with a disability and the fact of receiving unemployment and training benefits or social security payments. figure 1 (b) shows two graphs of weibull hazard functions derived from the estimated model for "gender" = 0 and "gender" = 1. these functions decrease (as p < 0) and at any time women are characterized by the lower intensity of leaving the unemployment than men. the next step was to estimate two weibull hazard models for time spent in unemployment state for men and women separately (the estimation results are presented in table 3, part (b) and (c)). these models differ in parameter estimates standing by the relevant variables. compared to the reference group (women and men aged 55 years and over, respectively) younger wom0. 00 0. 25 0. 50 0. 75 1. 00 s ur vi va l f un ct io n 0 2000 4000 6000 analysis time gender = 0 gender = 1 kaplan-meier survival estimates 0 .0 02 .0 04 .0 06 .0 08 h az ar d fu nc tio n 0 2000 4000 6000 analysis time gender=1 gender=0 weibull regression decomposing the gender gap in average exit rate from unemployment dynamic econometric models 13 (2013) 163–174 171 en are more likely to exit from unemployment than young men. for women the positive effect of higher education is stronger than for men. marriage intensifies the exit from unemployment among men. the level of the corresponding parameter for women shows a decrease in employment opportunities. among the remaining parameters, attention should be paid to the coefficients which appear by the variables associated with financial benefits. the negative effect of unemployment benefits and training allowances on the opportunities to leave unemployment for women is smaller than for men, but the impact of social security payments are stronger for women. apart from that, the estimated values of the parameter p indicate a steeper decline in employment opportunity for women over time. table 3. the results of the weibull model estimation for the whole sample (a), for women only (b) and for men only (c) variable all persons (a) women (b) men (c) βia exp(βia) βiw exp(βiw) βim exp(βim) gender 0.459 *** 1.582 – – – – – – age1724 0.633 *** 1.884 1.213 *** 3.362 0.713 *** 2.039 age2534 0.526 *** 1.692 1.208 *** 3.348 0.502 *** 1.653 age3544 0.410 *** 1.508 1.185 *** 3.269 0.293 *** 1.340 age4554 0.324 *** 1.382 1.096 *** 2.993 0.217 ** 1.243 tertiary 0.571 *** 1.769 0.769 *** 2.157 0.355 *** 1.427 vocational secondary 0.316 *** 1.372 0.425 *** 1.529 0.211 *** 1.235 general secondary 0.348 *** 1.416 0.428 *** 1.534 0.275 *** 1.317 basic vocational 0.168 *** 1.183 0.263 *** 1.300 0.113 *** 1.120 married 0.115 *** 1.121 -0.099 *** 0.906 0.287 *** 1.332 town 0.095 *** 1.100 0.057 * 1.058 0.128 *** 1.136 disabled -0.275 *** 0.759 -0.153 * 0.858 -0.321 *** 0.726 episode_nr 0.069 *** 1.072 0.098 *** 1.104 0.051 *** 1.052 unemploy. benefits -0.290 *** 0.748 -0.231 *** 0.793 -0.372 *** 0.690 training benefits -0.613 *** 0.542 -0.554 *** 0.575 -0.670 *** 0.512 social security -0.698 *** 0.497 -0.787 *** 0.455 -0.425 *** 0.654 cons -5.503 *** 0.004 -6.148 *** 0.002 -5.104 *** 0.006 p 0.777 *** – 0.770 *** – 0.797 *** – number of episodes 10118 4786 5332 lnl -17804.816 -8356.835 -9357.641 note: *** significant at 1%; ** significant at 5%; * significant at 10%. however, estimates of the parameters obtained in the models (b) and (c) are difficult to compare to each other, due to different empirical subsamples. thus, subsequently observed inequalities between women and men in leaving the unemployment were decomposed using the modified oaxaca-blinder technique. the results on an aggregated basis are presented in table 4 (the aggregation consisted in an accumulation of effects for related variables). joanna małgorzata landmesser dynamic econometric models 13 (2013) 163–174 172 table 4. decomposition of the gender gap in hazard rates from unemployment (results of modified oaxaca-blinder decomposition technique) value % observed differential (total) -0.001507 100 value % value % characteristics –0.000100 6.63 returns –0.001407 93.37 age-group –0.000113 7.50 age-group –0.000953 63.28 education level –0.000388 25.76 education level –0.000450 29.89 married 0.000124 –8.24 married 0.000134 –8.91 town –0.000036 2.41 town 0.000051 –3.39 disabled –0.000016 1.09 disabled –0.000008 0.54 episode_nr 0.000115 –7.63 episode_nr –0.000158 10.48 benefits 0.000215 –14.26 benefits –0.000022 1.48 note: the factors, that cause diversity of chances on the labor market to the greatest extent, are in bold. there is a negative difference between the mean values of the hazard function for men and women (–0.001507), meaning that women typically have lower chances of leaving the unemployment state than men. decomposition, which was carried out, made it possible to isolate the factors explaining the inequality observed to a different extent. it turns out that the differences in exit rates are only in the 6.63 percent explained by the individual characteristics of women and men (vectors xw and xm). the gender gap in the chances of exit from unemployment recognized in this way comes from the fact that women are different from men due to certain characteristics relevant in the labor market. the effect of the different education levels of women and men can be noticed. even though women on average are better educated than men, they have rarely technical education, which results in a smaller probability of employment than in the case of men. the differences in transition rates from unemployment are reduced by more frequent training allowances and social security benefits for women. however, gender inequalities examined should be assigned in the majority – in 93.37% – to the coefficients βw and βm of estimated hazard models (rather than to the differentiation of individuals characteristics). we find out that people with the same characteristics, if they are of different sexes, have various chances of exiting unemployment. the gender gap in unemployment rates is therefore explained not by differences in the characteristics of men and women but by differences in the labor market returns to their characteristics. a different "evaluation" of men’s and women’s characteristics is a major cause of the inequality. it can be assumed that employers favor men. women are discriminated against in the labor market. different opportunities are mainly due to prejudices associated with the woman's age decomposing the gender gap in average exit rate from unemployment dynamic econometric models 13 (2013) 163–174 173 (generally it is believed that younger women are less involved in work for family reasons) and her education level. conclusions the analysis conducted shows that the differences in the intensity of leaving unemployment are explained by individual characteristics of women and men only in a limited way. to a much greater extent these differences can be attributed to the „valuations” of men’s and women’s characteristics carried out by the labor market. inequalities observed in opportunities for men and women would probably occur even if the characteristics of both groups were identical. the calculated differences result from the parameter values, and thus it is important that all the parameters in the models have the statistical properties expected. therefore, for a correct inference – using the method presented in the article – the careful selection of variables for hazard models is very important. references altonji, j. g., blank, r. m. (1999), race and gender in the labor market, in ashenfelter, o., card, d. (eds.), handbook of labor economics, volume iii, north-holland, amsterdam, doi: http://dx.doi.org/10.1016/s1573-4463(99)30039-0. beblo, m., beninger, d., heinze, a., laisney, f. (2003), methodological issues related to the analysis of gender gaps in employment, earnings and career progression, european commission. blinder, a. s. (1973), wage discrimination: reduced form and structural estimates, journal of human resources, 8(4), 436–455, http://dx.doi.org/10.2307/144855. edin, p. a. (1989), unemployment duration and competing risks: evidence from sweden, scandinavian journal of economics, 91(4), 639–653, doi: http://dx.doi.org/10.2307/3440211. gonzalo, m. t., saarela, j. (2000), gender differences in exit rates from unemployment: evidence from a local finnish labour market, finnish economic papers, 13(2), 129–139. jensen, p., westergård-nielsen, n. (1990), temporary layoffs, in hartog, j., ridder, g., theeuwes, j. (eds.), panel data and labour market studies, north-holland, amsterdam. kalbfleisch, j., prentice, r. (2002), the statistical analysis of failure time data, second edition, john wiley and sons, new york, doi: http://dx.doi.org/10.1002/9781118032985. katz, l. f., meyer, b. d. (1990), unemployment insurance, recall expectations, and unemployment outcomes, quarterly journal of economics, 105(4), 973–1002, doi: http://dx.doi.org/10.2307/2937881. lancaster, t. (1979), econometric methods for the duration of unemployment, econometrica, 47, 939–956, doi: http://dx.doi.org/10.2307/1914140. joanna małgorzata landmesser dynamic econometric models 13 (2013) 163–174 174 landmesser, j. (2008), analiza aktywności ekonomicznej kobiet wiejskich z wykorzystaniem ekonometrycznych modeli hazardu (the analysis of economic activity of women in rural areas using the econometric hazard models), roczniki naukowe stowarzyszenia ekonomistów rolnictwa i agrobiznesu (annals of the polish association of agricultural and agribusiness economists), 1/x, 233–239. malarska, a. (2007), diagnozowanie determinantów bezrobocia w polsce nieklasycznymi metodami statystycznymi. analiza empiryczna na podstawie danych bael, (diagnosing the determinants of unemployment in poland with non-classical statistical methods. the empirical analysis based on lfs-data), wydawnictwo biblioteka, łódź. oaxaca, r. l. (1973), male-female wage differentials in urban labor markets. international economic review, 14, 693–709, doi: http://dx.doi.org/10.2307/2525981. ollikainen, v. (2003), the determinants of unemployment duration by gender in finland, vatt discussion papers, 316, helsinki. ortega masagué, a. c. (2008), gender gaps in unemployment rates in argentina, económica, la plata, 54(1–2), 161–202. petrongolo, b. (2004), gender segregation in employment contracts, discussion paper 4303, cepr, doi: http://dx.doi.org/10.1162/154247604323068032. steiner, v. (1989), causes of recurrent unemployment an empirical analysis, empirica, 16, 53–65, doi: http://dx.doi.org/10.1007/bf00924940. tansel, a., taşçi, h. m. (2010), hazard analysis of unemployment duration by gender in a developing country: the case of turkey, iza discussion paper, 4844, bonn, doi: http://dx.doi.org/10.1111/j.1467-9914.2010.00480.x. weibull, w. (1939), a statistical theory of the strength of materials, ingeniörs vetenskaps akademiens, handlingar, 151, stockholm. yun, m. (2004), decomposing differences in the first moment, economic letters, 82, 275–280, doi: http://dx.doi.org/10.1016/j.econlet.2003.09.008. dekompozycja nierówności płciowych w przeciętnych stopach wyjścia ze stanu bezrobocia z a r y s t r e ś c i. w pracy przeprowadzono analizę stóp wyjścia ze stanu bezrobocia, uwzględniającą zróżnicowanie płciowe. proces opuszczania stanu bezrobocia badano dla obu płci osobno wykorzystując parametryczne modele hazardu. głównym celem było dokonanie rozkładu nierówności między kobietami i mężczyznami podczas opuszczania bezrobocia. zastosowana zmodyfikowana mikroekonometryczna technika dekompozycji oaxaca-blindera pozwoliła na wyodrębnienie czynników wyjaśniających zaobserwowane nierówności. otrzymano, że nierówności płciowe są wyjaśniane w większości przez różniące się „wyceny” cech kobiet i mężczyzn dokonywane przez rynek. s ł o w a k l u c z o w e: czas trwania w bezrobociu, parametryczne modele hazardu, nierówności płciowe, dekompozycja oaxaca-blindera. © 2019 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2019.005 vol. 19 (2019) 85−96 submitted december 7, 2019 issn (online) 2450-7067 accepted december 28, 2019 issn (print) 1234-3862 dorota witkowska, piotr kuźnik  does fundamental strength of the company influence its investment performance? a b s t r a c t. the aim of our research is to find out whether the fundamental strength of the company affects its investment performance. the research is provided for 27 non-financial companies listed on the warsaw stock exchange in the years 2012–2017. these companies belong to the stock indexes wig20 and mwig40 portfolios. the obtained results show that the proposed synthetic measure makes it possible to estimate the fundamental strength of listed companies, and the correlation between values of the constructed measure and rates of return is positive but usually statistically insignificant. k e y w o r d s: capital market; fundamental analysis; taxonomic measure; investment performance. j e l classification: : c1, g11 introduction in the process of making investment decisions, investors use different supporting tools such as fundamental and technical analysis. the former requires taking into account a number of factors that are particularly important in assessing the current economic and financial condition of the company and to consider the environment in which analyzed firm operates. fundamental analysis involves assessing a firm’s equity value based on the analysis of published financial statements and other information without  correspondence to: dorota witkowska, university of johannesburg, college of business and economics, d ring 414, apk, johannesburg, south africa, e-mail: mariaw@uj.ac.za. dorota witkowska, piotr kuźnik dynamic econometric models 19 (2019) 85–96 86 reference to the prices at which a company’s securities trade in the capital markets (bauman, 1996, p.1). this analysis attempts to measure a security's intrinsic value by examining related economic and financial factors including the balance sheet, strategic initiatives, microeconomic and macroeconomic indicators, together with consumer behavior. fundamental analysis is usually used to find long-term opportunities to invest. studies that employ fundamental analysis to forecast earnings and future stock returns include ou and penman (1989a, b), ou (1990), greig (1992), stober (1993), kerstein, kim (1995) seng, hancock (2012), muhammad, gohar (2018) and bintara, tanjung (2019), among others. a modern approach to assessing the economic and financial condition of enterprises is applying the concept of fundamental strength of a company which bases on multidimensional comparative analysis methods. these methods allow to construct aggregated measures on the basis of many different variables, describing the condition of the company. in other words, to examine the state of the enterprise, its major economic and financial factors such as financial liquidity, level of debt, management efficiency, profitability, etc. are taken into account. the first proposal to measure the fundamental power of the enterprise was so-called taxonomic measure of investment attractiveness (tmai) proposed by tarczyński (1994), and further developed by tarczyński (2002) and tarczyńska-łuniewska (2013). tmai is an application of a synthetic measure of development constructed by hellwig (1968), which contains diagnostic variables describing financial situation of the company. there have been many attempts to construct taxonomic measures which have been used: (1) to evaluate the state of enterprises, e.g. kompa (2019), (2) to select companies for the investment portfolio construction, e.g. staszak (2017), (3) to find relation between financial condition of companies and their performance, e.g. juszczyk (2015). application of taxonomic measures to different purposes requires usage of different variables to their construction. this study aims to find our if the fundamental strength of the company affects its investment performance. fundamental strength of the company is measured by taxonomic measure which is constructed using 15 financial indicators evaluated for 27 non-financial companies listed on the warsaw stock exchange in the years 2012–2017. these companies are classified as big and medium size firms since they have been included in the stock indexes wig20 and mwig40 portfolios. investment efficiency is measured by annual logarithmic rates of return. does fundamental strength of the company influence its investment performance? dynamic econometric models 19 (2019) 85–96 87 investigation consists of several stages. in the first one synthetic measures of development are evaluated for all analyzed companies and years. in the second stage, annual rates of return for each company and the years 2012– 2017 are calculated. in the third stage, the hypothesis that fundamental strength of companies influences their investment performance is verified applying regression functions. 1. data and methodology the research concerns companies listed on the warsaw stock exchange which constantly belonged to the stock indexes wig20 and mwig40 in the period from 31.12.2012 to 31.12.2017. however, companies: − without financial statements in the analyzed period, − with negative equity or zero sales revenues, − excluded from trading on the warsaw stock exchange since 2017, − from the sectors defined as: banks, insurance and finance were excluded from investigation. therefore, an analysis (that is carried out according to the above assumptions) made it possible to qualify to research 27 non-financial companies. as it was already mentioned, investigation is provided in several steps. in the first one synthetic measures of development are evaluated for all analyzed companies and years. these measures base on financial coefficients which are evaluated for the end of each year under consideration, using data from balance sheets and annual financial reports provided by selected companies. taxonomic measure of investment attractiveness (tmaiit) is calculated for the i-th company in the t-th year as (łuniewska and tarczyński, 2004, p. 43): 𝑇𝑀𝐴𝐼𝑖𝑡 = 1 − 𝑑𝑖𝑡 𝑑𝑡̅̅ ̅+2𝑆𝑑𝑡 (1) where: 𝑑𝑖𝑡 – the distance (from the benchmark) of the i-th company (i = 1, 2,…, n) in the t-th period of time (t = 1, 2,…, t), 𝑑𝑡̅̅ ̅ – the average of distances 𝑑𝑖𝑡 in time t, 𝑆𝑑𝑡 – the standard deviation of distances 𝑑𝑖𝑡 in time t. euclidean distance in m-dimensional space is defined as: 𝑑𝑖𝑡 = √ ∑ (𝑧𝑗𝑡 𝑖 −𝑧𝑗𝑡 0 )2𝑚𝑗=1 𝑚 (2) where: 𝑧𝑗𝑡 𝑖 – is the standardized variable describing the j-th feature (j = 1, 2,…, m) in the i-th company in time t, 𝑧𝑗𝑡 0 – is the value of the j-th variable of the benchmark in time t, the benchmark is defined for each year and described by dorota witkowska, piotr kuźnik dynamic econometric models 19 (2019) 85–96 88 m variables. standardization of all variables used for the measure construction is provided according to the formula: 𝑧𝑗𝑡 𝑖 = 𝑥𝑗𝑡 𝑖 −�̅�𝑗𝑡 𝑆(𝑥𝑗𝑡) (3) where: 𝑥𝑗𝑡 𝑖 – observation of the j-th variable in the i-th company in the t-th year, �̅�𝑗𝑡 , 𝑆(𝑥𝑗𝑡 ) – average and standard deviation of the j-th variable in the tth year, respectively. the benchmark used in the formula (2) might be either real or hypothetical object. since it is difficult to determine a company which will be the pattern for others, the hypothetical object is usually used. such benchmark is constructed from m variables as maximal value of stimulants and minimal values of de-stimulants i.e.: 𝑧𝑗𝑡 0 = { min 𝑧𝑗𝑡 𝑖 𝑖𝑓 𝑥𝑗𝑡 𝑖 𝜖𝐷 max 𝑧𝑗𝑡 𝑖 𝑖𝑓 𝑥𝑗𝑡 𝑖 𝜖𝑆 (4) where: d, s, – are sets of de-stimulants and stimulants, respectively. stimulants are variables whose rise in quantity indicates an increase of economic and financial standing of the enterprise whereas de-stimulants are variables with the opposite direction of influence. in the second stage, annual logarithmic rates of return are calculated, according to the formula: 𝑅𝑖𝑡 = 𝑙𝑛 ( 𝑦𝑖𝑡 𝑦𝑖𝑡0 ) (5) where 𝑦𝑖𝑡, 𝑦𝑖𝑡0 – quotations of share price of the i-the company on the last and the first day of warsaw stock exchange quotation in the t-the year (t = 2012, 2013,…,2017), respectively. in the third stage, the hypothesis that fundamental strength of companies influences their investment performance is verified, applying pearson correlation coefficients and regression functions. in other words, the relations between logarithmic annual rates of return from shares of considered companies and values of synthetic measure tmai (for current and lagged dependencies) are estimated. 2. evaluation of companies based on taxonomic measure to evaluate the fundamental strength of companies, it is necessary to apply numerous indicators that are of particular importance when assessing the current state of the enterprise and its further development prospects. in the does fundamental strength of the company influence its investment performance? dynamic econometric models 19 (2019) 85–96 89 construction of the aggregated measures, the selection of diagnostic variables is extremely important since it determines the quality of evaluation and signals the proper functioning of companies, taking into account their economic and financial situation (tarczyński and łuniewska, 2004). table 1. tmai values of selected companies for the years 2012–2017 company tmai value 2012 2013 2014 2015 2016 2017 average amrest 0.15 0.13 0.18 0.11 0.11 0.08 0.13 assecopol 0.36 0.39 0.36 0.19 0.24 0.20 0.29 bogdanka 0.30 0.31 0.27 0.11 0.27 0.38 0.27 boryszew 0.14 0.15 0.18 0.10 0.16 0.15 0.15 budimex 0.23 0.35 0.12 0.19 0.27 0.29 0.24 ccc 0.27 0.32 0.20 0.15 0.11 0.19 0.21 cdproject 0.45 0.32 0.35 0.28 0.41 0.34 0.36 ciech –0.04 0.18 0.19 0.15 0.24 0.25 0.16 cyfrplsat 0.27 0.35 0.18 0.14 0.16 0.16 0.21 echo 0.21 0.29 0.36 0.35 0.33 0.23 0.29 enea 0.24 0.42 0.36 0.14 0.23 0.21 0.27 eurocash 0.07 0.12 0.08 0.05 0.05 –0.06 0.05 gtc 0.09 –0.10 –0.04 0.18 0.16 0.27 0.09 intercars 0.16 0.21 0.20 0.11 0.14 0.11 0.15 kernel 0.32 0.26 0.12 0.11 0.22 0.22 0.21 kety 0.28 0.34 0.29 0.18 0.25 0.22 0.26 kghm 0.42 0.37 0.23 0.01 0.06 0.13 0.20 kruk 0.22 0.06 0.10 0.13 0.16 0.18 0.14 lotos 0.09 0.10 0.14 0.05 0.15 0.20 0.12 lpp 0.28 0.32 0.24 0.14 0.17 0.23 0.23 netia 0.08 0.15 0.18 0.08 0.09 0.04 0.10 orangepl 0.16 0.17 0.18 0.11 0.06 0.07 0.13 oribs 0.34 0.44 0.46 0.23 0.38 0.31 0.36 pge 0.28 0.31 0.37 0.11 0.23 0.18 0.25 pgnig 0.25 0.32 0.33 0.21 0.25 0.25 0.27 pknorlen 0.18 0.18 0.13 0.14 0.24 0.27 0.19 tauronpe 0.23 0.25 0.23 0.09 0.05 0.09 0.16 note: bold letters denote the state treasury companies. synthetic measures of development (tmai) are constructed employing 15 financial indicators belonging to four groups: 1. profitability ratios: return on assets ratio (roa), return on equity ratio (roe) and return on sales ratio (ros); 2. liquidity ratios: current ratio, acid-test ratio (quick ratio) and cash ratio; dorota witkowska, piotr kuźnik dynamic econometric models 19 (2019) 85–96 90 3. efficiency (activity) ratios: average collection period, average payment period, fixed asset turnover ratio, asset turnover ratio and inventory turnover ratio; 4. leverage (debt) ratios: debt ratio, debt to ebitda ratio, interest coverage ratio and long-term debt to equity ratio. obtained tmai values for considered companies in the years 2012–2017 are presented in table 1. according to (łuniewska and tarczyński, 2006, p. 95), the level of synthetic measures for companies with strong foundational and being attractive in terms of investment is determined by tmai in the range of 0.3– 0.5. analysing values of constructed measure in table 1, it may be noticed that in each of the audited periods there are several companies in good economic and financial condition, for which tmai values equal 0.3 and above. namely for rounded measure values, there are 15 such firms in 2013, 11 companies in 2012, 9 enterprises in 2014, 6 companies in 2017, 5 enterprises in 2016, and 2 firms in 2015 (i.e. there are only 48 such cases, and among them only in 33 cases tmai is bigger than 0.3). it means that the majority of companies under study are characterized by weak economic and financial results in the years 2015–2017 (i.e. in more than 70% of all analysed cases). among analysed companies only four of them can be classified into the group of the best companies, i.e. cdprojekt in all analysed years, orbis together with bogdanka in five years and echo in four years. ten enterprises are characterized by values of the synthetic measure below 0.3 in all years of investigation, which means low investment attractiveness and fundamental strength. other companies show low values of tmai in three and more years (among six considered years). cdprojekt obtains the highest value of taxonomic measure of investment attractiveness in 2012 (tmai= 0.45) and keeps the tmai value above 0.3 in five following years, that proves its strong financial situation and proper management. the second place in 2012, in terms of investment attractiveness, belongs to kghm company, which in the following years does not perform well, and keeps the last place in the constructed ranking for 2015. the years 2013 and 2014 are favourable for orbis, which seems to be of the best financial standing, and the values of aggregate measures equal 0.44 and 0.46, respectively. tmai values for orbis in the considered period 2012– 2017, are regularly in the range between 0.3 and 0.5, with the exception of 2015, when the measure decreases to 0.23. definitely unfavourable results in 2013 and 2014, in terms of tmai, belong to gtc, which is the weakest among all companies in both years. in does fundamental strength of the company influence its investment performance? dynamic econometric models 19 (2019) 85–96 91 2015, echo reaches the highest tmai level, while in the following year the first place is taken by cdprojekt again, whereas tauronpe turns out to be the least attractive investment. bogdanka is the highest ranked company in the ranking in 2017, while eurocash characterizes by the lowest tmai level among all companies. orbis and cdprojekt have the highest average values of tmai which for both companies over the six examined years are above 0.36. these two firms should be classified as companies with strong fundamental and being attractive for investment. other companies achieve much worse results thus their investment attractiveness is at an average level. it is worth mentioning that among the state treasury companies, enea from the energy sector and pgnig, belonging to the wig-oil&gas industry index, keep the highest place in the created ranking of companies. the least level of taxonomic measure of investment attractiveness is observed for eurocash. the company fares by far the worst in the period under review, reaching an average tmai of 0.05. it is worth mentioning that the tmai negative values presented in table 1 are irrelevant since values of the measure depend on the normalization formula (1), which is the special case of the formula presented by tarczyński (1994, p. 177). 3. rates of return of analyzed listed companies annual logarithmic rates of return, calculated for all companies, are presented in table 2. it is visible, that average annual rates of return in the years 2015–2017 are below 20% (only 14% in 2015) whereas they are over 20% in three first years of investigation (25% in 2014). the rates of return of companies listed on the warsaw stock exchange were characterized by high volatility in the period 2012–2017. among the selected companies, there are those that systematically generated positive returns on the capital employed by investors, e.g. budimex, intercars, ccc, kęty, kruk, orbis. the shares of ccc brought on average 31.66% of profit per year. in the case of kruk, the return was 30.43%, while investors obtained 17.92% on shares of orbis. the highest rate of return in 2012 is recorded by lpp (83.76%), while orangepl generated the lowest value i.e. 25.01% losses. cdprojekt achieved the highest rate of return among all selected enterprises in 2013 (104%). the main factor was the high sales of the popular game series translating into high and positive financial flows of the company. cdprojekt was keeping the leader position in 2016 and 2017, and also achieved the highest average rate of return over the entire period (48.75%). dorota witkowska, piotr kuźnik dynamic econometric models 19 (2019) 85–96 92 ciech company was successful in 2014 and 2015 (70.23%). the highest losses were generated by gtc (i.e. 32.37% in 2014) and bogdanka (i.e. 101.18% in 2015). other raw material and energy companies, e.g. kghm, enea, pge and tauronpe, did not avoid losses in 2015. taking into consideration average returns obtained by analyzed companies, it is visible that cdproject keeps the first place (48.75% average annual rate of return) and is followed by ccc (31.66%), amrest (30.97%) and kruk (30.42%). orbis, which is a leader in the tmai classification, achieved average returns of 17.92% over the entire period but it did not generate losses in any year. there are 7 companies which generated negative average annual rates of returns. among them orangepl had the lowest average rate of return from shares in all years 2012–2017. on average, it generated losses of 12.96% per year. table 2. logarithmic rate of return for the years 2012–2017 company annual rate of return (%) 2012 2013 2014 2015 2016 2017 average amrest 40.83 –6.66 10.54 63.23 45.29 32.58 30.97 assecopol –1.96 6.55 16.45 15.74 0.44 –14.86 3.73 bogdanka 29.06 –3.32 –21.56 –101.18 73.85 –2.18 –4.22 boryszew –4.73 –21.51 13.63 –18.10 53.31 14.07 6.11 budimex 5.51 69.28 15.74 35.08 6.40 12.63 24.11 ccc 47.23 48.27 14.72 5.49 39.66 34.57 31.66 cdproject 16.06 104.00 –4.91 28.24 85.73 63.41 48.75 ciech 24.43 33.55 35.27 70.23 –33.54 –1.40 21.42 cyfrplsat 19.60 18.78 18.37 –11.82 16.40 2.31 10.61 echo 40.15 28.27 4.95 –2.01 74.11 –4.83 23.44 enea –11.57 –12.03 14.98 –26.48 –17.35 20.79 –5.28 eurocash 39.92 10.14 –20.72 26.53 –18.88 –37.26 –0.05 gtc 15.43 –28.43 –32.37 29.97 13.70 20.78 3.18 intercars 6.48 79.09 17.25 6.15 15.64 11.29 22.65 kernel –2.88 –56.15 –29.10 54.98 30.20 –28.09 –5.17 kety 37.53 45.30 30.72 14.22 27.23 12.59 27.93 kghm 69.46 –39.28 –4.04 –50.67 39.68 19.32 5.75 kruk 1.78 63.04 28.45 46.83 31.93 10.48 30.42 lotos 54.04 –15.04 –22.13 5.72 34.84 42.85 16.71 lpp 83.76 69.27 –20.80 –25.99 2.88 45.55 25.78 netia –20.04 20.81 13.56 6.47 –7.37 24.27 6.28 orangepl –25.01 –15.46 –11.49 –17.98 –12.79 4.96 –12.96 oribs 3.63 9.87 12.16 36.50 19.80 25.55 17.92 pge –0.78 –7.22 17.19 –27.97 –21.41 13.08 –4.52 pgnig 24.45 0.94 –11.49 17.48 12.35 14.08 9.63 pknorlen 33.25 –15.58 21.06 35.14 26.01 24.42 20.72 tauronpe –5.23 –3.62 18.35 –52.41 –1.05 6.78 –6.19 note: bold letters denote the state treasury companies. does fundamental strength of the company influence its investment performance? dynamic econometric models 19 (2019) 85–96 93 4. relationship between fundamental strength of companies and their investment performance supporters of fundamental analysis claim that profits from capital investments can be achieved by investing in companies characterized by good economic and financial conditions. assuming that the taxonomic measure of investment attractiveness (1) properly describes financial standing of companies and annual rate of return (5) is a good measure of firm performance, we apply pearson correlation coefficients and linear regression functions to verify the existence of positive relationship between both phenomena. the research is conducted for the following 16 relations between: (1) values of average returns and average values of synthetic measures, (2)–(7) values of rates of return in the years: 2012, 2013, 2014, 2015, 2016, 2017 and values of synthetic measures in the same years, (8) values of rates of return in the whole period 2012–2017 and values of synthetic measures in the six-years period, (9) values of rates of return in the five-years period 2013–2017 and values of tmai in the period 2012–2016 (i.e. tmai is lagged by one year), (10)–(14) values of rates of return in the period 2013, 2014, 2015, 2016, 2017 and values of synthetic measures lagged by one, (15) values of rates of return in 2012–2017 and values of tmai in the sixyears period for the companies with the highest average annual returns i.e. cdproject, ccc, amrest and kruk, (16) values of rates of return in 2012–2017 and values of tmai in the sixyears period for companies with the highest values of taxonomic measure i.e. cdproject, bogdanka, orbis and echo. values of pearson correlation coefficients and characteristics of the regression models are presented in table 3. based on the results in table 3, it is visible that the relation between fundamental strength of the company and its performance is positive (except for the lagged tmai in 2015) but usually statistically insignificant. only correlations between both phenomena, observed for the whole period of analysis and rates of return from 2016 for both current and lagged values of tmai, are statistically significant. pearson correlation coefficient obtained for four companies which were selected as the ones with the highest values of taxonomic measure is quite high i.e. 0.31 but it is not statistically significant. in general, values of pearson coefficient are low and do not excide 0.36, therefore also determination coefficients of regression functions are not bigger than 0.13. dorota witkowska, piotr kuźnik dynamic econometric models 19 (2019) 85–96 94 does fundamental strength of the company influence its investment performance? dynamic econometric models 19 (2019) 85–96 95 table 3. relations between tmai and rates of return. number of the relation 1 2 3 4 5 6 7 8 constant number of observations 27 27 27 27 27 27 27 162 correlation pearson coefficients 0.26 0.09 0.17 0.27 0.28 0.36 0.31 0.22 regression functions constant a 0.03 0.14 0.01 –0.06 –0.16 –0.02 0.00 0.00 slope factor b 0.50 0.21 0.53 0.46 1.53 1.15 0.69 0.63 t-statistics ta 0.34 1.22 0.05 –0.69 –0.95 –0.17 0.03 0.00 t-statistics tb 1.35 0.45 0.86 1.38 1.46 1.94 1.64 2.91 𝑅2 0.07 0.01 0.03 0.07 0.08 0.13 0.10 0.05 number of the relation 9 10 11 12 13 14 15 16 number of observations 135 27 27 27 27 27 24 24 correlation pearson coefficients 0.00 0.11 0.24 –0.27 0.33 0.18 0.15 0.31 regression functions constant a 0.12 0.06 –0.05 0.27 0.00 0.06 0.28 –0.29 slope factor b 0.00 0.37 0.38 –0.94 1.41 0.39 0.38 1.56 t-statistics ta 2.03 0.34 –0.57 1.62 0.01 0.65 1.73 –0.84 t-statistics tb 0.01 0.54 1.25 –1.40 1.74 0.89 0.71 1.52 𝑅2 0.00 0.01 0.06 0.07 0.11 0.03 0.02 0.09 note: bold numbers denote significant relations at the significance level 0.05. conclusions an ongoing assessment of the company's operations is necessary to company management and providing development perspectives. for this purpose, the state of the enterprise is examined in terms of its economic and financial condition. in our research we applied linear ordering method to evaluate the fundamental strength of the company. in majority of research, ratios describing financial liquidity, level of debt, management efficiency and profitability are taken into account to construct taxonomic measure of investment attractiveness, and such financial indicators were used in this study. the aim of our research was to find out if the fundamental strength of the company affects its investment performance. in order to achieve that goal, we constructed the synthetic measure determining the fundamental strength of public companies that are characterized by good economic and financial condition and market value. then we checked if the statistically significant relation between values of aggregated measure and annual rates of return exists. dorota witkowska, piotr kuźnik dynamic econometric models 19 (2019) 85–96 96 the obtained results show that in all cases (but one) correlation between taxonomic measures and logarithmic rates of return is positive. however, statistically significant relationship between tmai values and the rates of return from the shares of the analyzed public companies is observed only for the whole period of investigation 2012–2017 and for 2016 for both current and lagged relations. in other words, the statement, that fundamental strength of companies affects their investment performance, seems to be confirmed although our study also shows that there are other factors influencing rates of returns and correlation between both phenomena is usually not strong. these results are consistent with study (juszczyk, 2015) although in this research different set of companies, considered periods and variables were applied for the taxonomic measure construction. however, lack of significant relations for lagged tmai values (except one year) shows weak forecasting properties of constructed synthetic measure which cannot be used in the initial selection of companies for the investment portfolio. our results contradict (staszak, 2017) who obtained promising results applying constructed by him tmai to investment portfolio determination. but in his research, portfolios were built using only companies being leaders in the rankings of considered companies. taking that fact in consideration, we notice the similarity to our results since correlation between fundamental strength and returns evaluated for four companies, selected as the most attractive for investors, is relatively high. references bauman, m.p. (1996). a review of fundamental analysis research in accounting. journal of accounting literature, 15, 1–33. bintara, r., tanjung p. r. s. (2019). analysis of fundamental factors on stock return, international journal of academic research in accounting, finance and management sciences, 9(2), 49–64. http://dx.doi.org/10.6007/ijarafms/v9-i2/6029 greig, a. (1992). fundamental analysis and subsequent stock returns. journal of accounting and economics, 15, 413-442. http://dx.doi.org/10.1016/0165-4101(92)90026-x hellwig, z. (1968). zastosowanie metody taksonomicznej do typologicznego podziału krajów ze względu na poziom ich rozwoju oraz zasoby i strukturę wykwalifikowanych kadr, przegląd statystyczny, 4. juszczyk, m. (2015). powiązanie kondycji finansowej spółek giełdowych określonej syntetycznym miernikiem atrakcyjności inwestowania (tmai) z kształtowaniem się kursów ich akcji, zeszyty naukowe szkoły głównej gospodarstwa wiejskiego, ekonomika i organizacja gospodarki żywnościowej nr 111, 81–95. http://dx.doi.org/10.22630/eiogz.2015.111.36 kerstein, j., & kim, s. (1995). the incremental information content of capital expenditures. the accounting review, 70(3), 513–526. http://dx.doi.org/10.22630/eiogz.2015.111.36 does fundamental strength of the company influence its investment performance? dynamic econometric models 19 (2019) 85–96 97 kompa, k. (2019). zmiany w kierownictwie spółek giełdowych a zmiany sytuacji finansowej, w: śliwicki a. (ed.) zarządzanie w warunkach ryzyka, oficyna wydawnicza sgh, warszawa, 187–206. łuniewska, m., tarczyński, w. (2006). metody wielowymiarowej analizy porównawczej na rynku kapitałowym, wydawnictwo naukowe pwn. muhammad, s., & gohar, a. (2018). the relationship between fundamental analysis and stock returns based on the panel data analysis; evidence from karachi stock exchange (kse), research journal of finance and accounting, 9(3), 84–96. ou, j.a. (1990). the information content of nonearnings accounting numbers as earnings predictors. journal of accounting research, 28(1), 144–163. http://dx.doi.org/10.2307/2491220 ou, j.a., & penman, s.h. (1989a). financial statement analysis and the prediction of stock returns. journal of accounting and economics, 11, 295–329. http://dx.doi.org/10.1016/0165-4101(89)90017-7 ou, j.a., & penman, s.h. (1989b). accounting measurement, price-earnings ratio, and the information content of security prices. journal of accounting research, 27, supplement, 111–144. http://dx.doi.org/10.2307/2491068 seng, d., & hancock, j. r. (2012). fundamental analysis and the prediction of earnings, international journal of business and management, 7(3), 32–46. http://dx.doi.org/10.5539/ijbm.v7n3p32 staszak, m. (2017). eksperymentalna ocena efektywności portfela fundamentalnego dla spółek z indeksu wig20 za lata 2004 –2016, metody ilościowe w badaniach, tom xviii/4, 672–678. http://dx.doi.org/10.22630/mibe.2017.18.4.62 stober, t.l. (1993). the incremental information content of receivables in predicting sales, earnings, and profit margins. journal of accounting, auditing and finance, 8, 447–473. http://dx.doi.org/10.1170/0148558x9300800406 tarczyńska–łuniewska, m. (2013). definition and nature of fundamental strengths. actual problems of economics, 2(1), 15–23. tarczyński, w. (1994). taksonomiczna miara atrakcyjności inwestycji w papiery wartościowe. przegląd statystyczny, 3, 275–300. tarczyński, w. (2002). fundamentalny portfel papierów wartościowych, polskie wydawnictwo ekonomiczne, warszawa. tarczyński, w., & łuniewska, m. (2004). dywersyfikacja ryzyka na polskim rynku kapitałowym, wydawnictwo placet, warszawa. http://dx.doi.org/10.2307/2491220 http://dx.doi.org/10.1016/0165-4101(89)90017-7 http://dx.doi.org/10.2307/2491068 http://dx.doi.org/10.5539/ijbm.v7n3p32 http://dx.doi.org/10.22630/mibe.2017.18.4.62 introduction 1. data and methodology 2. evaluation of companies based on taxonomic measure 3. rates of return of analyzed listed companies 4. relationship between fundamental strength of companies and their investment performance conclusions references dem_2019_29to40 © 2019 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2019.002 vol. 19 (2019) 29−40 submitted september 11, 2019 issn (online) 2450-7067 accepted december 17, 2019 issn (print) 1234-3862 mitra lal devkota* impact of export and import on economic growth: time series evidence from india a b s t r a c t. this paper examines the cointegration and causal relationships between export, import, and economic growth in india using quarterly data from 1996:q2 to 2019:q2. stationarity properties of the time series data are investigated using augmented dickey fuller (adf) and phillips-perron (pp) unit root tests, and the existence of cointegrating relationship is studied using johansen’s cointegration test. finally, the causal relationships between the variables are examined using vector error correction model (vecm). the results show that, under both tests, the time series variables are non-stationary at their levels and are stationary at their first differences. the johansen’s cointegration test shows the existence of a long run equilibrium relationship among the variables. the results from the vecm indicate that there is a unidirectional causal relationship running from economic growth to import in india. this implies that with an increase in the income of the nation, the nation’s spending will increase, and some of the spending will be on import. k e y w o r d s: export, import, gross domestic product, causality, cointegration, india j e l classification: c22; e00; e44 introduction over the past several decades, there has been a great debate among the researchers on three hypotheses in trade and development literature. some empirical studies (michaely, 1977; bhagwati, 1978; balassa, 1978; tang et *correspondence to mitra lal devkota, university of north georgia, department of management and marketing, mike cottrell college of business, 82 college circle, dahlonega, ga 30597, united states, e-mail: mldevkota@ung.edu mitra lal devkota dynamic econometric models 19 (2019) 29–40 30 al., 2015, among others) argue in favor of exports led growth (elg) hypothesis, which states that exports make a significant contribution to economic growth of an economy (i.e. the flow of causality is from exports to economic growth). they claim that the countries exporting a large amount of their production grow more rapidly than others do. a number of empirical studies (krugman, 1984; bhagwati, 1988; oxley, 1993; sharma and dhakal, 1994; ghatak and price, 1997, among others) have shown the possibility of growth led exports (gle) in which the flow of causality is from economic growth to exports. the third alternative is that of imports led growth (ilg) in which the flow of causality is from imports to the economic growth (awokuse, 2008). growth in imports can help in transfer of growth, enhancing foreign trade research and development knowledge from developed to the developing countries (mazumdar, 2001). however, the flow of causality depends on the time period of the data under study, frequency of the data such as monthly, quarterly, and yearly, and the methodology employed in the study. this paper differs from the existing literature on the study of cointegration and causality relationships between export, import, and economic growth in india in several ways. first, the existing studies have used either annual or monthly time series data covering comparatively a shorter period of time (guntukula, 2018; jyoti kumari, 2014). we have used quarterly time series data that cover a longer period of time period and include recent data (1996:q2 to 2019:q2). the study not including the recent data will not reflect the current economic development of the country. second, most of these studies suffer from some weaknesses in the adopted methodologies. for instance, the augmented dickey fuller (adf) method used for testing the stationarity of the time series data is widely criticized for its low power and size properties. we have supplemented this by including phillips perron unit root test in addition to the adf test. phillips-perron test is nonparametric, i.e., it does not require one to select the level of serial correlation as in adf. in addition, the phillipsperron test is robust to general forms of heteroscedasticity in the error term and the user does not have to specify a lag length for the test in regression. finally, to the best of our knowledge, no previous studies have employed the granger causality test based on vecm framework to examine both short and long run causality relationships between export, import, and the economic growth in india.1 we tested the long run causality between the variables 1 some studies such as mishra (2012) and malhotra and meenu (2009), among others, only study the impact of import on gdp, but exclude the impact of export on gdp. on the other hand, studies such as guntukula (2018) study the impact of export and import on gdp, but fail to examine both shortand long-run causal relationships between the variables under vecm framework. impact of export and import on economic growth: time series evidence… dynamic econometric models 19 (2019) 29–40 31 through the statistical significance of the error correction term (ect) by a ttest. on the other hand, we tested the short run causality through the significance of the lags of each explanatory variable by a wald chi-squared test. the rest of the paper is organized as follows. a review of previous empirical studies is carried out in section 1. a detailed description of the data and the variables used in the study are presented in section 2. the econometric methodology used in the study and discussion of the empirical results are presented in section 3. the last section concludes the paper. 1. literature review a number of time series and cross-sectional methodologies have been used for testing the causality and cointegration relationships between export, import, and economic growth. these studies have used data from both developed and developing countries. in this section, we review a selected number of empirical studies from a plethora of research articles in the area. alam, uddin, and taufique (2009) investigated the existence of gravity theory for the import of bangladesh with its eight major trading partner countries india, china, singapore, japan, hong kong, south korea, usa, and malaysia using data from 1985 to 2003 in panel approach. they found that the gravity theory is consistent with the imports of bangladesh, that is, the geographical distance of bangladesh with its partner countries has significant impacts on its imports. they further document that the gdp of the major partner countries had significant positive impact on the import of bangladesh, and that the population of these countries has a mixed impact on imports of bangladesh. din (2004) investigated the export-led growth hypothesis for the five largest economies-nepal, bangladesh, india, sri lanka, and pakistan of the south asian region using a multivariate time-series framework. the author finds an interesting result in view of their increasing outward orientation and adoption of export promotion policies as part of their growth strategies in these south asian countries. while controlling for imports, the results showed a bidirectional causality between exports and output growth in bangladesh, india, and sri lanka in the short-run. the study also found long-run equilibrium relationships between exports, imports, and output growth for bangladesh and pakistan. however, the study found no evidence of a long-run relationship between the variables for india, nepal, and sri lanka. kogid et. al (2011) investigated the causality and cointegration relationships between the economic growth and the import in malaysia using annual mitra lal devkota dynamic econometric models 19 (2019) 29–40 32 time series data from 1970 to 2007. their results show no evidence of a cointegrating relationship between economic growth and import in malaysia. however, they find the evidence of a bidirectional causality between the economic growth and import. they further document that import could indirectly contribute to economic growth, and economic growth could directly contribute to import. tagavhi (2012) investigated the relationship between export, import, and economic growth in iran for the period between 1962 and 2011. the empirical results show the existence of a long run relationship between the variables under consideration. the results further show that, in the long run, export had positive relationship with economic growth, and import had a negative relationship with economic growth. guntukula (2018) investigated the relationships between export, import, and economic growth in india using monthly time series data from april 2005 to march 2017. the study found a long run relationship between export, import, and economic growth for the study period. the study also found the evidence of a bidirectional causality between exports and economic growth, and between imports and economic growth, confirming the existence of a bidirectional causality between exports and economic growth. similarly, jyoti kumari (2014) investigated the relationship between export, import, and economic growth using annual data for india for the period post liberalization from 1991–92 to 2012–13. the author found that import must be supported by export to have continuous growth in the economy. she further documents that export has positive effect on economic growth by keeping import fixed, while import has negative effect on economic growth by keeping export fixed. ramos (2001) investigated the granger causality between exports, imports, and economic growth in portugal using annual data over the period 1865–1998. the empirical investigation finds no evidence of a unidirectional causality relation between export, import, and output growth. they found the evidence of feedback relationships between export and output growth and import and output growth. but they found no significant causality between import and export growth. 2. data our analysis uses the seasonally adjusted quarterly time series data for india for the period between 1996:q2 to 2019:q2. the variables used in the study are real export, real import, and real gdp in local currency (i.e. indian rupees), which are then expressed in natural logarithms. we have transformed impact of export and import on economic growth: time series evidence… dynamic econometric models 19 (2019) 29–40 33 the export, import and gdp into natural logarithms because this transformation is the most commonly used variance stabilizing tool for variables that have wide range (weisberg, 1980). we take the gross domestic product (gdp) as the proxy for economic growth because gdp is essentially an indicator of aggregate economic activity of a country. these data are obtained from economic research database of federal reserve bank of st. louis. statistical software packages r and eviews are used for the econometric analyses of the data. figure 1. export, import and gdp for india:1996–2019 table 1. summary statistics for export, import, and gdp export import gdp min 26.61 26.61 28.84 1st qu 27.34 27.41 29.41 median 28.61 28.77 30.16 mean 28.44 28.58 30.18 3rd qu 29.55 29.66 30.95 max 29.91 30.09 31.54 std. dev skewness kurtosis jarque-bera p-value 1.078 –0.262 1.617 0.014 1.107 0.063 1.648 0.028 0.836 –0.268 1.571 0.011 time series plots for the logarithmic values of export, import and gdp are shown in figure 1. the plot indicates that the time series data for each of the variables have fairly strong upward trends. this gives anecdotal evidence that 26 27 28 29 30 31 32 96 98 00 02 04 06 08 10 12 14 16 18 export gdp import mitra lal devkota dynamic econometric models 19 (2019) 29–40 34 each of the time series data tend to move together. summary statistics for export, import, and gdp indicate that these variables have means equal to 28.44, 28.58, 30.18 with associated standard deviations of 1.078, 1.107, 0.836, and coefficient of variation 0.0379, 0.0387, 0.0277 respectively. the jarque-bera test statistic for each variable has a p-value greater than 0.01. this implies that each of the variables’ distributions are normal. this study includes the maximum range of the time series data available to the author available at the time of analysis. 3. methodology and empirical results existence of cointegrating relationships and the directions of causalities (if any) between export, import and gdp can be decided only by empirical investigation. as such, we first study the time series properties of the data by using augmented dickey fuller (adf) and phillips-perron (pp) unit root tests. we then use johansen’s cointegration test to examine the existence of long run equilibrium relationship between the variables and to find the number of cointegrating vectors. finally, we study the direction of causality and model the short and long run causal relationships between the variables by using granger causality test under vector error correction model (vecm) framework. 3.1. unit root test for testing stationarity the initial step in the time series data analysis involves testing the presence of unit root of each of the variables. for this purpose, we have used augmented dickey fuller unit root test (dickey and fuller; 1979, 1981) and phillips-perron unit root test (phillips and perron, 1988) at the levels and first differences of the variables. table 2. unit root test results variable adf-test phillips-perron test levels first difference levels first difference gdp 0.25 [0.975] −8.01 [0]* 0.17 [0.970] −8.17 [0]* export −1.22 [0.662] −10.66 [0]* −1.38 [0.589] −10.74 [0]* import −1.45 [0.553] −9.63 [0]* −1.59 [0.483] −9.68 [0]* note: given figures are the test statistics and their respective p-values are inside the brackets. * denotes the rejection of the null hypothesis of non-stationarity of the variables at α = 0.01 level of significance. impact of export and import on economic growth: time series evidence… dynamic econometric models 19 (2019) 29–40 35 the results of both the adf test and pp test for the levels and first differences of the variables are shown in table 2. these results suggest that all the variables are non-stationary in their levels. however, the unit root test applied in the first difference of these variables indicate that each of the series is stationary in their first differences. thus, we conclude that each of the export, import, and gdp are integrated of order one, i.e. i(1). 3.2 cointegration analysis once establishing the stationarity of the variables, we proceed to the next step of our analysis, which is to investigate the existence of long run equilibrium relationship between the variables and to determine the number of cointegrating vectors. for this, we employ the johansen (1988, 1991, 1992) and johansen and juselius (1990) maximum likelihood cointegration technique. this technique is based on granger’s (1981) ecm representation. to determine the number of cointegrating vectors, we employed both the available likelihood ratio tests. these are trace test (λ-trace) and maximum eigenvalue test (λ-max). for both the directions, 1 lag was used (as determined by lrt, fpe, aic, sic, and hq criteria). the results for both the trace test and maximum eigenvalue tests are reported in table 3. table 3. johansen cointegration test results (trace and max. eigenvalue) null hypotheses λ!"#$% stat 5% critical value p-value λ&#' stat 5% critical value p-value r = 0 34.48 29.98 0.0134* 25.69 21.13 0.0106* r ≤ 1 8.78 15.49 0.3857 7.43 14.26 0.4394 r ≤ 2 1.35 3.84 0.2449 1.35 3.84 0.2449 note: rstands for the hypothesized number of cointegrating relationships between export, import, and gdp; h!: r = 0 means that there is no cointegrating relationship betwen export, import and gdp. * indicates the rejection of the respective null hypotheses at α = 0.05 level of significance; the cointegration model is based on the vector autoregression model (var) with 1 lag as determined by the lrt, fpe, aic, sic, and hq criteria; the critical values for trace and max-eigenvalue statistics are calculated by eviews (10). the results indicate that both the tests unanimously identified one cointegrating relationship among the export, import, and gdp at α = 0.05 level of significance. thus, we conclude that there is one cointegrating relationship between the export, import, and gdp for india for the sample period. in other words, there exists a long run equilibrium relationship between the export, import, and gdp for india. mitra lal devkota dynamic econometric models 19 (2019) 29–40 36 3.3 granger causality and vector error correction model (vecm) the empirical results from the johansen and juselius test for cointegration in 3.2 indicate that there is a cointegrating relationship between export, import, and gdp. this also means that a long run equilibrium relationship exists between the variables. we next move on to conduct the granger causality test (engle and granger, 1987) under vecm framework. this test is a statistical procedure used to determine if one time series is helpful in forecasting another. vecm includes lags of the dependent variables, in addition to its own lags (upadhyaya, nag and franklin jr, 2018). in addition to indicating the direction of causality amongst the variables, the vecm also allows one to distinguish between short-run and long-run granger causality relationships because it can capture both the short-run dynamics between time series and their long-run equilibrium relationship (masih and masih, 1996; devkota and panta (2018)). table 4. granger causality results from vector error correction model (vecm) response 𝜒(statistic 𝐸𝐶𝑇 ∆export ∆import ∆gdp ∆export – 0.15 0.12 1.33 ∆import 0.04 – 6.68** –2.93** ∆gdp 1.464 0.004 – 1.35 note: ∆export, ∆import and ∆gdp denote the first differences of the logarithmic values of the export, import, and the gdp respectively, (**) denotes the rejection of the null hypothesis at the 1% level of significance. number of lags was selected as identified by using the lrt, hq, aic, sc, and hq criteria. we tested the long-run causality through the statistical significance of each of the error correction term (ect) by an individual t-test, and the shortrun granger causality through the significance of the lags of each explanatory variable by a wald 𝜒3test. a variable 𝑋4 is said to cause another variable 𝑌4 in the granger sense if the one step ahead forecast of 𝑌4 in the regression model improves the quality of the model and/or forecasts by taking into account the historical values of 𝑋4 (see, din, 2004; syczewska, 2014; and osinska, 2011, for details). the results for both the long run and short run granger causalities are reported quantitatively in table 4, and then qualitatively in table 5. the results indicate that the coefficient of the error correction term of import variable is negative and statistically significant at the 1% level of significance. this implies that there are long run causalities running from gdp and export to import. also, the chi-square statistic for the causality from gdp to import is statistically significant at the 1% level of significance. this suggests that there is a short run granger causality running from gdp to import. it means that both the gdp and export granger cause import in india. impact of export and import on economic growth: time series evidence… dynamic econometric models 19 (2019) 29–40 37 table 5. causality results based on vector error correction model causality short-run long-run direction of causality from to export gdp no no none unidirectional unidirectional gdp export no no import gdp no no gdp import yes** yes** export import no yes** import export no no note: ** denotes the rejection of the null hypothesis at the 1% level of significance; number of lags in the vecm was determined using the lrt, fpe, aic, sic, and hq criteria. the interpretations of the results of our study are straightforward. we have evidence of unidirectional causality running from gdp to import. this implies that an increase in economic growth increases power to purchase foreign goods and services. similarly, we have evidence of unidirectional causality running from export to import. a possible explanation could be that when a country exports goods and services, it sells them to a foreign market which brings in money into the country. this money can then be used to purchase goods and services for the country. the diagnostic testing of the estimated model (with the gdp as the dependent variable and export and import as independent variables) is performed using the residual analysis based on breusch-godfrey serial correlation lm test and the results are shown in table 6. under this test, the null hypothesis states that there is no serial autocorrelation of residuals. the chi-squared test statistic of 1.858151 with a p-value of 0.3949 suggests that the null hypothesis of no serial correlation is not rejected at the 1% level of significance, and thus confirms the adequacy of the model. table 6. breusch-godfrey serial correlation lm test results f-statistic 0.875485 prob f(2,84) 0.4204 obs*r-squared 1.858151 prob chi-squared (2) 0.3949 conclusions and discussion the present study investigated the causal relationships between the export, import, and gdp for india using the seasonally adjusted quarterly time series data for the period between 1996:q2 and 2019:q2. the empirical results suggest that each of the export, import and gdp are non-stationary in their levels and are stationary in their first differences. in addition, the johansen cointegration test suggests the existence of a long run equilibrium relationship between the variables. finally, the granger causality mitra lal devkota dynamic econometric models 19 (2019) 29–40 38 test based on vecm framework suggests the existence of unidirectional causal relationships from gdp to import, and from export to import. thus, the sample data suggest that none of the elg, gle, and ilg hypotheses are valid for india. our finding of existence of long run equilibrium relationships between export, import, and economic growth for india is in line with guntukula (2018) for india, din (2004) for bangladesh and pakistan, and moroke and manoto (2015) for south africa. however, our finding is contrary to din (2004), who found no evidence of a long run equilibrium relationship between these variables for india, nepal, and sri lanka. also, our finding of the rejection of the elg hypothesis is consistent to mishra (2011) for india. however, this finding is contrary to the finding of moroke and manoto (2015) for south africa, awokuse (2007) for bulgaria, and guntukula (2018) for india. in addition, our finding of the rejection of the ilg hypothesis is contrary to the finding of moroke and manoto (2015) for south africa and awokuse (2007) for poland. finally, our findings of the rejection of the gle hypothesis is contrary to the finding of guntukula (2018) for india and awokuse (2007) for bulgaria. we recognize, however, that this study only examined the impact of export and import on economic growth, thereby ignoring the myriad of other factors that also may affect the economic growth (zang and baimbridge, 2012). in addition, as sharma and dhakal (1994) mention, due to some limitations of the causality tests employed in this study, our results should be viewed with caution. finally, as ratanapakorn and sharma (2007) rightly argue, one should use at least 30 years of data to run a cointegration test. unfortunately, the unavailability of data led us to use only 24 years of quarterly data. similar to those authors, our objective is to investigate a shorter timeperiod, and hence, the results should be viewed with some degree of caution. references awokuse, t. o. (2008). trade openness and economic growth: is growth export-led or import-led? applied economics, 40(2), 161–173. alam, m., uddin, g., & taufique, k. (2009). import inflows of bangladesh: the gravity model approach. international journal of economics and finance (issn 1916-971x), 1(1), 131–139. balassa, b. (1978). exports and economic growth: further evidence. journal of development economics, 5(2), 181–189. bhagwati, j. n. (1978). appendix to "anatomy and consequences of exchange control regimes". in anatomy and consequences of exchange control regimes, nber, 219–221. bhagwati, j. n. (1988). export-promoting trade strategy: issues and evidence. the world bank research observer, 27–57. impact of export and import on economic growth: time series evidence… dynamic econometric models 19 (2019) 29–40 39 devkota, m. l., & panta, h. (2018). an inquiry into the effect of the interest rate, gold price, and the exchange rate on stock exchange index: evidence from nepal. dynamic econometric models, 18, 49–65. https://doi.org/10.12775/dem.2018.003 dickey, d. a., & fuller, w. a. (1979). distribution of the estimators for autoregressive time series with a unit root. journal of the american statistical association, 74(366a), 427–431. dickey, d.a., & fuller, w.a. (1981). likelihood ratio statistics for autoregressive time series with a unit root. econometrica, 1057–1072. din, m. u. (2004). exports, imports, and economic growth in south asia: evidence using a multivariate time-series framework. the pakistan development review, 105–124. engle, r. f., & granger, c. w. (1987). co-integration and error correction: representation, estimation, and testing. econometrica, 251–276. https://doi.org/10.2307/1913236 ghatak, s., & price, s. w. (1997). export composition and economic growth: cointegration and causality evidence for india. weltwirtschaftliches archiv, (h. 3), 538–553. granger, c. w. (1981). some properties of time series data and their use in econometric model specification. journal of econometrics, 16(1), 121–130. https://doi.org/10.1016/0304-4076(81)90079-8 guntukula, r. (2018). exports, imports and economic growth in india: evidence from cointegration and causality analysis. university of hyderabad, india, 2(615), 221–230. johansen, s. (1988), statistical analysis of cointegration vectors, journal of economic dynamics and control, 12, 231–254, https://doi.org/10.1016/0165-1889(88)90041-3 johansen, s. (1991), estimation and hypothesis testing of cointegration vectors in gaussian vector autoregressive models, econometrica, 59,1551–1580. https://www.jstor.org/stable/2938278 johansen, s. (1992). determination of cointegration rank in the presence of a linear trend. oxford bulletin of economics and statistics, 54(3), 383–397. https://doi.org/10.1111/j.1468-0084.1992.tb00008.x johansen, s. & juselius, k. (1990), maximum likelihood estimation and inference on cointegration—with applications to the demand for money, oxford bulletin of economics and statistics, 52, 169–210. https://doi.org/10.1111/j.1468-0084.1990.mp52002003.x kogid, m., mulok, d., ching, k. s., lily, j., ghazali, m. f., & loganathan, n. (2011). does import affect economic growth in malaysia. empirical economics letters, 10(3), 297–307. krugman, p. (1984). import protection as export promotion. monopolistic competition and international trade. kumari, j. (2014). export, import and economic growth in india: a study. the indian economic journal, 62(3), 1157–1168. malhotra, m., & meenu (2009). imports-growth relationship in india: causality analysis. indian journal of economics, 90(356), 33–46. masih, a. m., & masih, r. (1996). energy consumption, real income and temporal causality: results from a multi-country study based on cointegration and error-correction modelling techniques. energy economics, 18(3), 165–183. mazumdar, j. (2001). imported machinery and growth in ldcs. journal of development economics, 65(1), 209–224. michaely, m. (1977). exports and growth: an empirical investigation. journal of development economics, 4(1), 49–53. mishra, p. k. (2011). the dynamics of relationship between exports and economic growth in india. international journal of economic sciences and applied research, 4(2), 53–70. mitra lal devkota dynamic econometric models 19 (2019) 29–40 40 mishra, p. k. (2012). the dynamics of the relationship between imports and economic growth in india. south asian journal of macroeconomics and public finance, 1(1), 57–79. moroke, n. d., & manoto, m. (2015). how applicable is export-led growth and import-led growth hypotheses to south african economy? the vecm and causality approach. journal of governance and regulation, 4(2), 15–25. http://dx.doi.org/10.22495/jgr_v4_i2_p2 osińska, m. (2011), on the interpretation of causality in granger sense, dynamic econometric models. http://dx.doi.org/10.12775/dem.2011.009 oxley, l. (1993). cointegration, causality and export-led growth in portugal, 1985. economics letters, 43(2), 163–166. phillips, p. c., & perron, p. (1988). testing for a unit root in time series regression. biometrika, 75(2), 335–346. ramos, f. f. r. (2001). exports, imports, and economic growth in portugal: evidence from causality and cointegration analysis. economic modelling, 18(4), 613–23. ratanapakorn, o., & sharma, s. c. (2007). dynamic analysis between the us stock returns and the macroeconomic variables. applied financial economics, 17(5), 69– –377. https://doi.org/10.1080/09603100600638944 sharma, s. c., & dhakal, d. (1994). causal analyses between exports and economic growth in developing countries. applied economics, 26(12), 1145–1157. syczewska, e. m. (2014), the eurpln, dax and wig20: the granger causality tests before and during the crisis, dynamic econometric models, 14, 93–104, http://dx.doi.org/10.12775/dem.2014.005 taghavi, m., goudarzi, m., masoudi, e., & gashti, h. p. (2012). study on the impact of export and import on economic growth in iran. journal of basic and applied scientific research, 2(12), 12787–12794. tang, c. f., lai, y. w., & ozturk, i. (2015). how stable is the export-led growth hypothesis? evidence from asia's four little dragons. economic modelling, 44, 229–235. upadhyaya, k. p., nag, r., and mixon jr, f. g. (2018). stock market capitalization and the macroeconomics of transition economies: the case of india. dynamic econometric models, 35–47. http://dx.doi.org/10.12775/dem.2018.002 weisberg, s. (2005). applied linear regression (vol. 528). john wiley & sons. zang, w., & baimbridge, m. (2012). exports, imports and economic growth south korea and japan: a tale of two economies. applied economics, 44(3), 361–372. © 2017 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2017.004 vol. 17 (2017) 59−80 submitted december 4, 2017 issn (online) 2450-7067 accepted december 27, 2017 issn (print) 1234-3862 michal bernardelli, mariusz prochniak, bartosz witkowski * the application of hidden markov models to the analysis of real convergence  a b s t r a c t. this paper employs hidden markov models and the viterbi path to analyze the process of real convergence. such an approach combines the analysis of cyclical and incomelevel convergence. twelve macroeconomic variables in the sample of 28 eu countries observed in the 1995–2016 period are within the scope of the study. the results indicate, among others, the existence of real convergence of poland toward the remaining eu countries in terms of the levels of gdp per capita at ppp and gdp growth rates, with a short-run period of divergence during the global crisis. k e y w o r d s: catching-up; convergence; hidden markov model; european union; viterbi path. j e l classification: c61, e32, o47, o52. introduction cyclical convergence and income-level convergence are usually tested separately in empirical studies. however, these are interrelated phenomena because business cycles are very closely linked with economic growth. that * correspondence to: mariusz prochniak, warsaw school of economics, collegium of world economy, department of economics ii, al. niepodleglosci 162, 02-554 warszawa, poland, e-mail: mproch@sgh.waw.pl; michal bernardelli, warsaw school of economics, collegium of economic analysis, institute of econometrics, al. niepodleglosci 162, 02-554 warszawa, poland, e-mail: mbernard@sgh.waw.pl; bartosz witkowski, warsaw school of economics, collegium of economic analysis, institute of econometrics, al. niepodleglosci 162, 02-554 warszawa, poland, e-mail: bwitko@sgh.waw.pl.  the research project has been financed by the national science centre, poland (project number 2015/19/b/hs4/00362). michal bernardelli, mariusz prochniak, bartosz witkowski dynamic econometric models 17 (2017) 59–80 60 is why in order to obtain a full picture of output fluctuations and dynamics, it is necessary to apply a tool which would allow to test and assess simultaneously both cyclical and income-level convergence. hidden markov models (hmm) are used as such. the application of the method that combines the hmm and the viterbi path in the analysis of real convergence is the main value added of the paper. the holistic approach involves the use of the baum-welch algorithm, viterbi algorithm, and monte carlo simulations, which fills in the gap existing in the literature. in the study, the procedure based on hmm is applied in order to assess the character and the rate of real convergence in europe. while applied to the gdp time series, this approach encompasses both the cyclical and income-level convergence. the results allow us to verify the conformity of business cycles between various economies as well as the equalization of income levels between countries. additionally, time series of some other macroeconomic variables are analyzed as a kind of robustness check of the proposed procedure as a tool to analyze the dynamics of different macroeconomic factors. the analysis covers the 28 eu countries and the 1995–2016 period. the main results are plotted for poland (its convergence toward the other 27 eu countries). the main results for germany are also reported in the form of a robustness check. germany is selected as the additional country due to three reasons. first, it is the biggest eu economy. second, it is the main trade partner for poland – a fact, which is even more important in the time periods around the big financial crisis (see sledziewska and witkowski, 2012). third, poland’s macroeconomic performance depends deeply on the situation in germany due to very close links between both countries caused e.g. by large flows of the factors of production (labor and capital). the paper is organized as follows. after the introduction, the theoretical background of real convergence is described. the concept of the hmm and the viterbi path are discussed in the next section. the macroeconomic variables included in the study and the method of estimation are then presented, which are followed by the empirical results. the last section concludes. 1. real convergence various ways of defining the process of convergence as well as many methods of testing a particular convergence hypothesis are proposed in the literature (see, e.g., islam, 2003) in general, nominal and real convergence are the two types that can be outlined. the nominal convergence means the tendency of nominal variables, like price levels, interest rates, or exchange the application of hidden markov models to the analysis of real convergence dynamic econometric models 17 (2017) 59–80 61 rates to level up. the real convergence refers to variables given in real terms (mainly real output or income) and can be divided into cyclical convergence and income-level convergence. the cyclical convergence means the tendency of different economies toward the conformity of business cycles while the income-level convergence means the equalization of the gdp per capita levels. this general classification has been extended in a number of ways, including the types and definitions of convergence, as well as the methods of verifying of a given concept of convergence. for example, the new concept of growth cycle has been developed to analyze cyclical fluctuations of many western economies where gdp has not dropped for many years (see, e.g., zarnowitz and ozyildirim, 2006). unlike the classical business cycle where recession means the absolute fall in gdp, if the growth cyclicality is observed, the gdp rises during both the contractionary and the expansionary phase. however, during the contractionary period the growth rate of gdp is less than the trend, while in the expansionary period the gdp growth rate exceeds the long term trend. various definitions of cyclical fluctuations, as well as a variety of methods to extract the trend and to find peaks and troughs in business activity, implied that there are many quantitative methods to cope with output fluctuations and cyclical conformity. furthermore, there is still much room to develop new methods and concepts, including those based on the hmm approach. similarly, there is no unique definition of income-level convergence. the existence of β convergence means that less developed countries (with lower gdp per capita) grow faster than the more developed ones. on the other hand, the σ convergence is observed when the differences of the income levels between countries (measured e.g. by the standard deviation of the logarithm of the gdp per capita levels) decrease over time (barro and sala-i-martin, 2003). the latest studies on convergence in the eu (including the papers on the catching-up process between the cee countries and western europe) take into account the effects of the global crisis and the crisis in the eurozone (kaitila, 2013; dauderstadt, 2014; nenovsky and tochkov 2014; simionescu 2014; forgo and jevcak, 2015). many of these recent analyses prove that the convergence process decelerated after the crisis, indicating even some divergence tendencies. there are also studies that suggest the existence of convergence clubs in the eu (e.g. borsi and metiu, 2013; monfort et al., 2013; gligoric, 2014). the book by jozwik (2017) presents the analysis of convergence at the national and regional levels, focusing on institutional changes due to systemic transformation, economic integration, michal bernardelli, mariusz prochniak, bartosz witkowski dynamic econometric models 17 (2017) 59–80 62 and the cohesion policy. prochniak and witkowski (2014) as well as matkowski et al. (2016a) show the analysis of β and σ convergence between the cee11 and eu15 in the 1993–2015 period. they conclude that the catching-up process was not continuous, showing some breaks and divergence episodes. the most intensive convergence was observed in the years 2000–2007, just before and after the eu’s major enlargement. the other recent studies on convergence include, among others, papers by batog (2013) and grzelak and kujaczynska (2013). the review of the literature, focused on the studies published in the last years, shows the necessity to check carefully the time stability of the catching-up process along with structural breaks. such an analysis is carried out in this study. 2. hidden markov models (hmm) a number of methods of analyzing convergence have been developed. some of them have purely mathematical background, others involve the expert insight. in this paper, the hmm is used. its concept is often identified with the name of hamilton (1989), however it was present in the literature at least since the 60s of the previous century, long before the first articles by hamilton. for the comprehensive description and characteristics of these models, one can refer to cappe et al. (2005), while in this paper we discuss very briefly only those of the main definitions and ideas behind the hmm models, markov chains (mc) and viterbi paths, which are used extensively in the following sections of the article. we concentrate rather on the idea behind those terms, restricting to basic definitions and notation necessary to understand the proposed method of analysis of the real convergence. let be a discrete stochastic process satisfying the following conditions:  the unobservable process is a homogenous mc with a finite state space s,  conditionally on the process the observations are independent, and for each t the conditional distribution of depends only on . the which fulfills the above conditions is referred to as a hidden markov model (hmm). in macroeconomic applications often has univariate or multivariate gaussian distribution. in that we refer to a resulting hmm as to the normal hmm. hmm are widely used in the areas, where the pattern recognition is explored. therefore, the most common use of hmm is in speech or handwritthe application of hidden markov models to the analysis of real convergence dynamic econometric models 17 (2017) 59–80 63 ing recognition, as well as in the cases when gesture or voice patterns need to be obtained. hmm is also a basic tool in bioinformatics, for example in the dna sequencing process. in macroeconomics applying hmm is one of the methods of business cycles synchronization analysis and turning points identification. this approach, however, should be treated rather as a whole class of models, due to the possibility of choosing a form of an observable and unobservable component. for this reason, a huge variety of types of models were being under study over the years – see hamilton (1994) or koskinen and oeller (2004) for a comprehensive review. unlike the classical markov models, where all states are visible and the markov model is defined only by the transition probabilities, in hmm the states are unobservable and need to be calculated based on another observable time series. therefore, besides the transition probabilities, the parameters of the probability distribution related to each state are also present. the problem which needs to be solved is to find the unobservable path of states. the deterministic algorithm of finding the parameters of the hmm which is used was described by baum et al. (1970) and is known under the name of the baum-welch algorithm. however, knowing the model parameters does not solve the stated problem as the state of an unobservable mc still remains to be estimated. there are few alternatives of performing this estimation. one of them are the smoothed probabilities given by hamilton (1994). second are the filtered probabilities, see chauvet and hamilton (2005) or harding and pagan (2002). in both those approaches, the most likely estimation of the state of the hidden mc at the given moment is chosen on the basis of those probabilities. so the states on the path of mc are estimated locally, which could be inefficient. however, an alternative, which makes use of the global decoding exists: instead of a single point of time, the whole period covered by the analysis is taken under consideration. the path of states being the result of this approach is called the viterbi path. elements of the viterbi path are calculated with the use of the viterbi algorithm, descried by viterbi (1967). the viterbi algorithm together with the baum-welch algorithm provide the deterministic procedure of transforming the time series into the most probable path of states. unfortunately, the results strongly depend on the initial values and can be far from optimal. therefore, to increase the chance of finding the globally optimal solution, the computations are performed repeatedly with the same set of data and different initial values. this approach is usually referred to as monte carlo simulations. depending on a number of factors (bernardelli, 2013), computation can be quite timeconsuming. michal bernardelli, mariusz prochniak, bartosz witkowski dynamic econometric models 17 (2017) 59–80 64 in this paper, due to the length of the input time series, only the hmms with two-element state space are considered. therefore the state space has the form of s={0,1} and the interpretation of states is as follows: 0 is associated with the periods of relatively good conditions and 1 is associated with a worse situation. we restrict ourselves to the analysis of normal hmm. thus the observable component yt, which corresponds to economic time series under the analysis, must satisfy the conditions and . (1) we additionally assume that μ0 < μ1. to have the same order of states in each considered case (state 1 is associated with a greater mean value), which in turn allows to compare viterbi paths for different pairs of countries. theoretically, the state space could be extended to the case of more than two states. however, larger state space would obviously involve longer computation time and often cause problems with the numerical stability of results. the combination of the viterbi path and hmm with three and four states are rather uncommon in the macroeconomic literature with a series of articles by bernardelli and dedys in 2012 (e.g. bernardelli and dedys, 2012) serving as an exemption. the authors explore the discussed method in order to describe the business cycle synchronization and identify turning points. however, the application of the hmm method and the viterbi path to the analysis of real convergence is absent in the literature. the discussed method has several advantages as compared with the standard β and σ convergence tests. first of all, we do not assume a priori both the number and the timing of turning points (structural breaks). turning points are identified on the basis of the behavior of individual indicators and, based on this, convergence and divergence periods are identified. second, convergence and divergence periods are identified for each pair of countries individually. third, the concepts of income-level and cyclical convergence are integrated. as a result, even if the obtained empirical results in this case are intuitive and comply with expectations of most economists, this should be viewed as a confirmation of method’s proper functioning. 3. data and method of estimation the empirical analysis includes 12 macroeconomic variables: a) gdp per capita at purchasing power parity (ppp, constant 2011 international $) [gdppc_ppp], b) growth rate of total real gdp (%) [g_gdp], c) cpi inflation (%) [inf], the application of hidden markov models to the analysis of real convergence dynamic econometric models 17 (2017) 59–80 65 d) unemployment rate (%) [une], e) household final consumption expenditure (constant 2010 us$) [cons_usd], f) household final consumption expenditure growth rate (%) [cons_g], g) government final consumption expenditure (constant 2010 us$) [gov_usd], h) government final consumption expenditure growth rate (%) [gov_g], i) foreign trade balance (current us$) [nx_usd], j) foreign trade balance (% of gdp) [nx_gdp], k) domestic credit provided by financial sector (% of gdp) [cred_byfin], l) bank nonperforming loans (% of total gross loans) [nonp_loans]. the variables are taken from the international monetary fund and world bank databases (imf, 2017; world bank, 2017). the first two variables represent the level of gdp per capita at ppp and the real gdp growth rate. these are the two basic variables that are used in the studies on real convergence. the results for these two variables allow us to assess the cyclical and income-level convergence on the basis of the hmm analysis. given that one of the aims of this study is to check the appropriateness of the hmm algorithm in the analysis of the dynamics of various macroeconomic variables, the list has been extended by a few other time series. firstly, both inflation and unemployment are included as important variables from the point of view of the wellbeing of the society and the standard of living. variables which are components of the gdp (household consumption, government expenditure, and net exports) are also analyzed. those are considered both in levels or as growth rates (except net exports which are taken as the level and percentage of gdp). finally, the two variables that represent the stability and development of the financial sector are included. those are very important in particular in the period which followed the global crisis and the crisis in the euro area when financial turbulences largely influenced the real economy. the analysis covers the 28 eu countries observed in the 1995–2016 period. in each of the cases the series are annual. in the case of missing observations, the calculations include a shorter period or lower number of countries. this study focuses on poland. that is why the detailed results for poland against the remaining eu countries are presented throughout the major part of the paper. it means that the results show – for each individual variable – the comparison of poland with each of the 27 remaining eu countries (in the figures – as average values). michal bernardelli, mariusz prochniak, bartosz witkowski dynamic econometric models 17 (2017) 59–80 66 table 1 shows the evolution of selected indicators for poland. for the sake of conciseness, we do not present the remaining indicators for poland and the other countries. for comparison purposes and as a form of robustness check of the proposed method at the end of the paper selected results for germany are included. germany is the biggest economy in the eu. in 2016, its gdp constituted 21.2% of the eu28’s gdp at current exchange rates and 20.0% of the eu28’s gdp at purchasing power parity (european commission, 2016). hence, it is interesting to compare the viterbi path for poland’s gdp with the analogous path for german gdp. the details of the applied method of the analysis are the same both in the case in poland and germany and are provided in detail in the remaining part of this section. germany is compared with each of the other 27 eu countries. table 1. the evolution of selected indicators for poland gdppc_ppp g_gdp inf une cred_byfin nonp_loans 1995 11300 6,7 27,9 13,3 29,4 – 1996 11976 6,2 19,9 12,3 30,9 – 1997 12741 7,1 14,9 11,2 32,0 10,5 1998 13324 5,0 11,8 10,6 33,4 10,5 1999 13944 4,5 7,3 13,1 35,9 13,3 2000 14732 4,3 10,1 16,1 34,3 15,5 2001 14920 1,2 5,5 18,2 38,9 18,6 2002 15232 1,4 1,9 19,9 38,9 21,1 2003 15785 3,6 0,8 19,6 40,2 21,2 2004 16606 5,1 3,5 19,0 39,0 14,9 2005 17194 3,5 2,1 17,7 38,4 11,0 2006 18268 6,2 1,0 13,8 42,9 7,4 2007 19563 7,2 2,5 9,6 47,8 5,2 2008 20392 3,9 4,2 7,1 63,5 2,8 2009 20953 2,6 3,5 8,2 61,6 4,3 2010 21771 3,7 2,6 9,6 63,2 4,9 2011 22850 5,0 4,3 9,6 65,9 4,7 2012 23218 1,6 3,7 10,1 64,1 5,2 2013 23555 1,3 0,9 10,3 67,2 5,0 2014 24346 3,3 0,0 9,0 71,0 4,8 2015 25323 3,7 –0,9 7,5 73,2 4,3 2016 – 3,1 –0,6 6,3 – 4,4 source: imf, 2017; world bank, 2017. the procedure used in the empirical analysis explores the concept hmm and viterbi path described in the previous section. in order to get the reliable results, monte carlo simulations are used. the initial values for the baumwelch algorithm are chosen randomly with the use of independent and identhe application of hidden markov models to the analysis of real convergence dynamic econometric models 17 (2017) 59–80 67 tically distributed draws from the univariate distribution. the number of draws used for parameters estimation of the time series being under study was set to 1000. in order to choose the best model, three criteria are taken into account:  akaike's information criterion (aic),  bayesian information criterion (bic),  the log likelihood value. the procedure used in the analysis can be described in the following steps (poland is used as an example for the clarity of description). 1. for each of the 12 variables and for each of the 27 eu countries (all except poland), the time series of differences are constructed as ,~ c t pl t c t vvv  (2) where t = 1995, 1996, …, 2016 and c refers to one of the eu countries. 2. the estimation of the hmm parameters is performed with the use of the baum-welch algorithm. the resulting estimates are used to find the viterbi path. state 0 on that path identifies a year of greater similarity in terms of the variable under analysis, whereas state 1 indicates a divergence between countries (poland and the country c). 3. for each year, averages of the states of viterbi paths for all countries for the given variable are calculated. the value of 0 means perfect convergence while the year when the average equals 1 means the period of undisputable divergence between poland and other eu countries. this kind of approach allows to both analyse a pair of countries separately, as well as to consider the real convergence between the chosen country and the group of other countries jointly. plots presented in the figures and summaries in the tables visualize both of those possibilities. the discussed procedure is followed for the cases of poland and germany, however, it may be used to determine the convergence of the other countries as well. this study is the initial application of the hmm and the viterbi path and, for the sake of conciseness, none of the advanced algorithms in calculating reference values were employed. in further studies, it is possible to extend the analysis by considering weighted averages or assessing convergence toward certain subgroups of countries (their clusters). 4. empirical analysis the results of the empirical analysis for poland are presented in figures 1–12 and tables 2–3. the figures show the averages of the states of the viterbi paths calculated with the use of the estimated parameters of the michal bernardelli, mariusz prochniak, bartosz witkowski dynamic econometric models 17 (2017) 59–80 68 hmm for the differences between the annual values of a given variable for poland and each of the other eu countries. lower values (closer to 0) indicate the existence of real convergence (greater similarity in terms of a given variable) while higher values (closer to 1) – real divergence (bigger differences of a given variable). the detailed results in the form of the states of the viterbi path are given in tables 2 and 3. for a given year, the hmm parameters were estimated for the differences in the values between poland and each other eu country separately. the value of 1 is assigned when the calculated probabilities for being in certain state – on the basis of the viterbi algorithm – are relatively large as compared with the other years in the whole period. on the other hand, when the difference is relatively low, a 0 value is assigned. hence, value 1 indicates real divergence (bigger differences between countries) while value 0 can be interpreted as a real convergence (greater similarity). the functions plotted in figures 1 and 2 are the arithmetic averages of the values provided in the respective columns of tables 2 and 3. figure 1 shows the viterbi path of the gdp per capita at ppp. the results indicate a clear-cut cyclical convergence before the beginning of the global crisis, that is from 1995 to 2008. in 2009, the differences between the gdp per capita at ppp in poland and the other eu countries increased. this was caused by the economic and financial crisis. the global crisis led to the recession in all the eu countries, except poland, which resulted in the significant change of the earlier convergence trends. this tendency is visible on the basis of the hmm method – the average values of the states of viterbi paths rose in 2009 and 2010, indicating real divergence. since 2011, gdp per capita levels between poland and the other eu countries have converged in terms of the viterbi path but the process has not been so intensive as in the first part of the analyzed period. these results are in line with some other studies that confirm the existence of divergence tendencies in europe in the last years (see, e.g., mucha, 2012; stanisic, 2012; borsi and metiu, 2013; monfort et al., 2013). for example, the study by matkowski et al. (2016b) showed – with the use of the σ convergence concept – that in 2009 and 2010, income differences among the 26 eu countries increased. the application of hidden markov models to the analysis of real convergence dynamic econometric models 17 (2017) 59–80 69 figure 1. cyclical convergence of gdp per capita at ppp [gdppc_ppp] between poland and the other eu countries figure 2 illustrates the results for gdp growth rates. the curve plotted in figure 2 shows a sharp increase in the values of averages of the states of the viterbi paths in the year 2009. this means that a large rise in differences between gdp growth rates in poland and the other eu countries was observed during the crisis. it was caused by the fact that in 2009 poland was the only country that recorded the increase in gdp while all the other eu countries noted a recession. this atypical behavior which consisted in the difference between poland and the remaining eu countries was confirmed by the results attained from the proposed procedure based on the hmm as reflected by the increase in the average values of the states of viterbi paths for the pair of poland and other eu countries for 2009. figure 2. cyclical convergence of gdp growth rates [g_gdp] between poland and the other eu countries the results for inflation rates are shown in figure 3. the tendency of the viterbi paths is declining throughout the whole 1995–2016 period. this out0,30 0,40 0,50 0,60 0,70 0,80 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 0,30 0,40 0,50 0,60 0,70 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 2 0 1 6 michal bernardelli, mariusz prochniak, bartosz witkowski dynamic econometric models 17 (2017) 59–80 70 come points to a regular fall in differences in inflation rates between poland and the other eu countries. a significant convergence of inflation is in line with the economic theory and official statistics – along with the further openness of the economy and integration with the eu, there took place nominal convergence in terms of price levels and inflation rates. clearly the combination of the baum-welch and viterbi algorithm, yielded economically justified results not only in terms of gdp convergence but also the inflation convergence. figure 3. cyclical convergence of inflation rates [inf] between poland and the other eu countries figure 4. cyclical convergence of unemployment rates [une] between poland and the other eu countries the results for unemployment rates (figure 4) indicate the strengthening of the similarity of unemployment rates in the last years. this may be caused by the fact that the global crisis and the crisis in the euro area both were the factors leading to the increase of unemployment in many countries. moreover, the official statistics for real economies often do not support the okun’s 0,20 0,30 0,40 0,50 0,60 0,70 0,80 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 2 0 1 6 0,35 0,40 0,45 0,50 0,55 0,60 0,65 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 2 0 1 6 the application of hidden markov models to the analysis of real convergence dynamic econometric models 17 (2017) 59–80 71 law, meaning that economic growth in real countries needn’t lead to the fall in unemployment. that is why the results for the convergence in unemployment rates are different compared with those for gdp. figures 5 and 6 show the results for consumption (in terms of levels and growth rates, respectively). consumption constitutes the largest portion of gdp. as we can see, the viterbi paths for both the level and growth rate of consumption are quite similar to the respective paths for gdp per capita levels and gdp growth rates. it reinforces the appropriateness of the hmm method as a tool to analyze real convergence. given these outcomes, the results are unlikely to be a coincidence. additionally, significant changes in consumption pattern may also confirm the appropriateness of the keynesian consumption function where consumption depends mainly on current disposable income. the permanent income hypothesis, according to which short-run fluctuations in income do not influence the level of consumption, is unlikely to be supported by this study. figure 5. cyclical convergence of household consumption [cons_usd] between poland and the other eu countries the results for convergence of government expenditures are shown in figure 7 (the levels) and 8 (growth rates). unlike private consumption that behaves very similarly to the total output, the averages of the states of viterbi paths for government consumption are different. in terms of levels, there is a tendency toward decreasing cross-country differences. as regards the growth rates, the results reveal large fluctuations from one year to another. this is in line with the economic theory, including the keynesian cross model, according to which the government purchases of goods and services are autonomous, that is independent of income. the level of government spending depends on the economic policy performed by a given country. the increasing convergence in terms of the level of government consumption may be also caused by the fact that, after the eu accession, poland re0,35 0,40 0,45 0,50 0,55 0,60 0,65 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 michal bernardelli, mariusz prochniak, bartosz witkowski dynamic econometric models 17 (2017) 59–80 72 ceived a lot of aid funds from the european union (it was the main recipient of eu funds from the 2007–2013 budget). figure 6. cyclical convergence of household consumption growth rates [cons_g] between poland and the other eu countries figure 7. cyclical convergence of government consumption [gov_usd] between poland and the other eu countries data for net exports (figures 9 and 10) indicate the increase in differences of the foreign trade balance between the countries. it is likely to be related to the different specialization of individual countries and a different reaction to various external shocks as well as different involvement in international flows of goods, services, assets, and labor. greater differences in foreign trade balance may also be caused by the fact that the intraregional trade within the eu is substantial. in such a case, if a country increases its exports, another country’s import must be increased. this hampers the convergence tendency of foreign trade balances. 0,30 0,40 0,50 0,60 0,70 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 0,30 0,40 0,50 0,60 0,70 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 the application of hidden markov models to the analysis of real convergence dynamic econometric models 17 (2017) 59–80 73 figure 8. cyclical convergence of government consumption growth rates [gov_g] between poland and the other eu countries figure 9. cyclical convergence of foreign trade balance [nx_usd] between poland and the other eu countries figure 10. cyclical convergence of foreign trade balance to gdp ratio [nx_gdp] between poland and the other eu countries 0,30 0,40 0,50 0,60 0,70 0,80 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 0,20 0,30 0,40 0,50 0,60 0,70 0,80 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 0,30 0,40 0,50 0,60 0,70 0,80 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 michal bernardelli, mariusz prochniak, bartosz witkowski dynamic econometric models 17 (2017) 59–80 74 the results for the financial sector development are mixed. on the one hand, the volume of domestic credit provided by the financial sector (% of gdp) shows the tendency toward divergence throughout the analyzed period (figure 11). on the other hand, figure 12 indicates that the volume of nonperforming loans (% of total loans) exhibits the tendency toward convergence. the latter outcome results from the fact that the extent of nonperforming loans depends on the situation in global markets. after the economic and financial crisis as well as the crisis in the euro area the majority of eu countries noticed a considerable rise in nonperforming loans. if the volume of nonperforming loans rises simultaneously in both countries, it means greater convergence. figure 11. cyclical convergence of domestic credit provided by financial sector [cred_byfin] between poland and the other eu countries figure 12. cyclical convergence of bank nonperforming loans [nonp_loans] between poland and the other eu countries the results presented in tables 2 and 3 indicate that there is no common regularity as regards catching-up toward the subgroups of countries (central0,20 0,30 0,40 0,50 0,60 0,70 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 0,30 0,40 0,50 0,60 0,70 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 2 0 1 6 the application of hidden markov models to the analysis of real convergence dynamic econometric models 17 (2017) 59–80 75 eastern or western europe). in each subperiod (the 1990s, 2000s, and 2010s), there were countries both from western europe as well as centraleastern europe which poland converged to or poland diverged from. it reflects the fact that there are many countries poland cooperates with and the character of bilateral relations is different for different partners. hence, it is possible to reveal convergence with some countries but divergence with another ones. this outcome is also in line with the hypothesis of club convergence – some studies suggest the necessity to divide world countries into clusters in the frame of convergence analysis (see, e.g., battisti and parmeter, 2013). table 2. viterbi paths for gdp per capita at ppp [gdppc_ppp] between poland and the individual eu countries 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 western europe at 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 be 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 dk 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 fi 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 fr 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 de 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 gr 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 ie 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 it 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 lu 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 nl 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 pt 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 es 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 se 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 uk 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 central-eastern europe bg 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 hr 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 cz 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 ee 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 hu 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 lv 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 lt 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ro 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 sk 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 si 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 cy 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 mt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 note: 0 indicates convergence in differences; 1 indicates a divergence in differences. michal bernardelli, mariusz prochniak, bartosz witkowski dynamic econometric models 17 (2017) 59–80 76 table 3. viterbi paths for gdp growth rates [g_gdp] between poland and the individual eu countries 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 2 0 1 6 western europe at 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 be 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 dk 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 fr 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 de 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 gr 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 ie 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 it 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 lu 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 nl 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 pt 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 es 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 se 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 uk 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 central-eastern europe bg 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 hr 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 cz 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ee 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 hu 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 lv 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 lt 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ro 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 sk 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 si 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 cy 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 mt 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 note: 0 indicates convergence in differences; 1 indicates a divergence in differences. finland was excluded from calculations due to having the time series that does not satisfy the conditions of the hmm procedure. the viterbi paths for gdp per capita levels and gdp growth rates for germany have been estimated for both comparison and robustness check. those are illustrated in figures 13 and 14. in general, the results for germany are quite similar to those for poland (although some differences appear). this is economically justified as germany is poland’s main trading partner and polish economy vastly depends on the development of germany. the theoretical structural model implies that the growth rate of the polish economy should approximately follow the growth of output in germany. the results are partly in line with this view. the viterbi path for gdp per capita in germany shows quite strong real convergence toward the 27 eu countries the application of hidden markov models to the analysis of real convergence dynamic econometric models 17 (2017) 59–80 77 at the beginning of the analyzed period and an evident real divergence afterwards (around the global crisis). the viterbi path for gdp growth rates in germany shows a peak in 2009 meaning that in this year the highest differences appeared. the latter outcome is the same as in poland. figure 13. cyclical convergence of gdp per capita at ppp [gdppc_ppp] between germany and the other eu countries figure 14. cyclical convergence of gdp growth rates [g_gdp] between germany and the other eu countries conclusions the analysis confirms that the procedure involving the use of the hmm, the viterbi path and joining the paths using averages, is a good tool to analyze real convergence. it focuses on different aspects of catching-up as compared with the standard approaches and should be treated as complementary rather than substitutive. the majority of the results are economically justified. 0,40 0,45 0,50 0,55 0,60 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 0,35 0,45 0,55 0,65 0,75 1 9 9 5 1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 2 0 1 6 michal bernardelli, mariusz prochniak, bartosz witkowski dynamic econometric models 17 (2017) 59–80 78 in terms of the gdp per capita at ppp, the results indicate a clear-cut real convergence of poland and the other eu countries before the beginning of the global crisis, that is from 1995 to 2008. in 2009, the differences between gdp per capita at ppp increased due to the economic and financial crisis. as regards gdp growth rates, there was a sharp increase in the values of averages of the states of the viterbi paths in the year 2009, meaning that during the crisis there was observed a large rise in differences between gdp growth rates in poland and the other eu countries, while prior to the global crisis and afterwards there was evident real convergence. considering other variables, the results indicate, among others, a clearcut nominal convergence in inflation years between poland and the other eu countries throughout the whole analyzed period. hmm seems to be an effective method of analyzing the macroeconomic time series. besides the turning point identification and synchronization of the business cycles, the proposed procedure should be considered as new, powerful method that could be extensively explored also in the real convergence studies. references barro, r., sala-i-martin, x. (2003), economic growth, the mit press, cambridge – london. batog, j. (2013), analiza krancowej pionowej konwergencji dochodowej typu β w krajach unii europejskiej w latach 1993–2010, studia i prace wydzialu nauk ekonomicznych i zarzadzania, 31, 39–47, http://www.wneiz.pl/nauka_wneiz/sip/sip31-2013/sip-31-39.pdf (22.12.2017). battisti, m., parmeter, ch.f. (2013), clustering and polarization in the distribution of output: a multivariate perspective, journal of macroeconomics, 35, 144–162, doi: http://dx.doi.org/10.1016/j.jmacro.2012.10.003. baum, l.e., petrie, t., soules, g., weiss, n. (1970), a maximization technique occurring in the statistical analysis of probabilistic functions of markov chains, the annals of mathematical statistics, 41(1), 164–171, doi: http://dx.doi.org/10.1214/aoms/1177697196. bernardelli, m. (2013), nieklasyczne modele markowa w analizie cykli koniunktury gospodarczej w polsce [non-classical markov models in the analysis of business cycles in poland], roczniki kolegium analiz ekonomicznych sgh, 30, 59–74, http://rocznikikae.sgh.waw.pl/p/roczniki_kae_z30_04.pdf (22.12.2017). bernardelli, m., dedys, m. (2012), ukryte modele markowa w analizie wyników testu koniunktury gospodarczej [hidden markov models in analysis of results of business tendency surveys], prace i materialy instytutu rozwoju gospodarczego sgh, 90, 159–181, http://yadda.icm.edu.pl/yadda/element/bwmeta1.element.desklight5b844cbc-6bea-4ab6-95fd-52f6e5e158ac/c/irg_pim_90_r07tr.pdf (22.12.2017). borsi, m.t., metiu, n. (2013), the evolution of economic convergence in the european union, deutsche bundesbank discussion paper, no. 28/2013, https://www.bundesbank.de/redaktion/en/downloads/publications/discussion_paper _1/2013/2013_08_13_dkp_28.pdf?__blob=publicationfile (10.02.2017). the application of hidden markov models to the analysis of real convergence dynamic econometric models 17 (2017) 59–80 79 cappe, o., moulines, e., ryden, t. (2005), inference in hidden markov models, springer series in statistics, springer, doi: http://dx.doi.org/10.1007/0-387-28982-8. chauvet m., hamilton j.d., (2005), dating business cycle turning points, nber working paper series, working paper 11422, http://www.nber.org/papers/w11422 (20.02.2017), doi: http://dx.doi.org/10.3386/w11422. dauderstadt, m. (2014), convergence in crisis: european integration in jeopardy, friedrich ebert stiftung, international policy analysis, berlin, http://library.fes.de/pdf-files/id/ipa/11001.pdf (22.12.2017). european commission (2016), statistical annex of european economy, autumn 2016, https://ec.europa.eu/info/files/statistical-annex-european-economy-autumn-2016_en (17.02.2017). forgo, b., jevcak, a. (2015), economic convergence of central and eastern european eu member states over the last decade (2004–2014), european economy discussion paper 001, https://ec.europa.eu/info/sites/info/files/file_import/dp001_en_2.pdf (22.12.2017). gligoric, m. (2014), paths of income convergence between country pairs within europe, economic annals, 59(201), 123–155, doi: http://dx.doi.org/10.2298/eka1401123g. grzelak, a., kujaczynska, m. (2013), real convergence of the european union member states – evaluation attempt, management, 17, 393–404, https://www.degruyter.com/downloadpdf/j/manment.2013.17.issue-1/manment-20130028/manment-2013-0028.pdf (22.12.2017). hamilton, j.d. (1989), a new approach to the economic analysis of non-stationary time series and business cycle. econometrica, 57, 357–384. hamilton, j.d. (1994), time series analysis. princeton university press, princeton, new jersey. harding, d., pagan, a., (2002), a comparison of two business cycle dating methods, journal of economic dynamics and control, 27, 1681–1690, doi: http://dx.doi.org/10.1016/s0165-1889(02)00076-3. imf (2017), world economic outlook database, october 2016 (updated 16 january 2017), https://www.imf.org/external/pubs/ft/weo/2016/02/weodata/index.aspx (30.01.2017). islam, n. (2003), what have we learnt from the convergence debate?, journal of economic surveys, 17(3), 309–362, doi: http://dx.doi.org/10.1111/1467-6419.00197. jozwik, b. (2017), realna konwergencja gospodarcza panstw czlonkowskich unii europejskiej z europy srodkowej i wschodniej. transformacja, integracja i polityka spojnosci, wydawnictwo naukowe pwn, warszawa. kaitila, v. (2013), convergence, income distribution and the economic crisis in europe, etla working paper no. 14, https://www.etla.fi/en/publications/convergence-income-distribution-economic-crisiseurope/ (22.12.2017). koskinen, l., oeller, l.e. (2004), a classifying procedure for signaling turning points, journal of forecasting, 23, 197–214, doi: http://dx.doi.org/10.1002/for.905 matkowski, z., prochniak, m., rapacki, r. (2016a), real income convergence between central eastern and western europe: past, present, and prospects, ekonomista, 6, 853–892, http://www.ekonomista.info.pl/openaccess/?rok=2016&nr=6&strona=853&sw=1354 (22.12.2017). matkowski, z., prochniak, m., rapacki, r. (2016b), income convergence in poland vis-à-vis the eu: major trends and prospects, in weresa m.a. (ed.), poland. competitiveness michal bernardelli, mariusz prochniak, bartosz witkowski dynamic econometric models 17 (2017) 59–80 80 report 2016. the role of economic policy and institutions, warsaw, world economy research institute, sgh warsaw school of economics. monfort, m., cuestas, j.c., ordonez, j. (2013), real convergence in europe: a cluster analysis, economic modelling, 33, 689–694, doi: http://dx.doi.org/10.1016/j.econmod.2013.05.015. mucha, m. (2012), mechanizm dywergencji gospodarczej w strefie euro [mechanism of economic divergence in the euro area], ekonomista, 4, 487–498. nenovsky, n., tochkov, k. (2014), transition, integration and catching up: income convergence between central and eastern europe and the european union, mondes en developpement, 3(167), 73–92, https://www.cairn.info/revue-mondes-en-developpement-2014-3-page-73.htm (22.12.2017). prochniak, m., witkowski, b. (2014), on the stability of catching up process among the old and new eu member states: implications from bayesian model averaging, eastern european economics, 52(2), 5–27, doi: http://dx.doi.org/10.2753/eee0012-8775520201. simionescu, m. (2014), testing sigma convergence across eu-28, economics and sociology, 7(1), 48–60, doi: http://dx.doi.org/10.14254/2071-789x.2014/7-1/5. sledziewska, k., witkowski, b. (2012), swiatowy kryzys gospodarczy a handel miedzynarodowy, ekonomista, 4, 427–448, http://www.ekonomista.info.pl/openaccess/?rok=2012&nr=4&strona=427&sw=1024 (22.12.2017). stanisic, n. (2012), the effects of the economic crisis on income convergence in the european union, acta oeconomica, 62(2), 161–182, doi: http://dx.doi.org/10.1556/aoecon.62.2012.2.2. viterbi, a. (1967), error bounds for convolutional codes and an asymptotically optimum decoding algorithm, ieee transactions on information theory, 13, 260–269, doi: http://dx.doi.org/10.1109/tit.1967.1054010. world bank (2017), world development indicators database, http://databank.worldbank.org (30.01.2017). zarnowitz, v., ozyildirim, a. (2006), time series decomposition and measurement of business cycles, trends and growth cycles, journal of monetary economics, 53(7), 1717–1739, doi: http://dx.doi.org/10.1016/j.jmoneco.2005.03.015. zastosowanie ukrytych modeli markowa w analizie realnej konwergencji z a r y s t r e ś c i. artykuł przedstawia zastosowanie ukrytych modeli markowa i ścieżki viterbiego do badania realnej konwergencji (zbieżności). takie podejście łączy analizę konwergencji cyklicznej i dochodowej. badanie obejmuje 28 krajów ue, okres 1995–2016 oraz 12 zmiennych makroekonomicznych. wyniki pokazują m.in. realną zbieżność polski do pozostałych krajów ue w kategoriach poziomów pkb per capita wg psn oraz stóp wzrostu pkb, z krótkim okresem dywergencji podczas kryzysu globalnego. s ł o w a k l u c z o w e: konwergencja; ścieżka viterbiego; ukryty model markowa; unia europejska; zbieżność. microsoft word 00_tresc.docx dynamic econometric models vol. 10 – nicolaus copernicus university – toruń – 2010 blanka łęt poznań university of economics dynamics of multivariate return series of u.s. automotive stock companies in conditions of crisis† a b s t r a c t. this article contains an analysis of dynamic interrelations between log-returns series of three automotive companies listed on the new york stock exchange: gm, f and dai. we consider two periods: before and during crisis. we apply diagbekk model and we calculate dynamic conditional correlations. as a result of our research we found that in conditions of crisis there were strong connections between considered stock companies. k e y w o r d s: diagbekk model, dynamic conditional correlation. 1. introduction general motors, ford and chrysler, known as the detroit’s big three, are the major companies of american automotive industry. they have in common not only strong worldwide position but also problems, among others, with high labor costs as a result of activities of united auto workers. this inevitably results in high prices of cars offered, which can be afforded by fewer and fewer potential customers. general motors, ford and chrysler face such problems for several years. very high oil prices, soaring in the period from january 2007 to mid-2008, also resulted in fewer big three car sales, because of high fuel costs. credit crunch, due to the prevailing economic crisis, caused deeper and deeper problems of the big three. dismal financial performance of these companies inevitably resulted in a weakening listing on the new york stock exchange1. † this work was financed from the polish science budget resources in the years 2007-2010 as the research project nn 111 1256 33. 1 chrysler is not listed on nyse. from 1998 to mid-may 2007, chrysler were part of daimlerchrysler ag, later the shares were held by an investment fund cerberus. at the end of april 2009, declared bankruptcy, fiat hold 20% of his shares. blanka łęt 44 this paper contains an analysis of daily log-returns series of three automotive companies listed on the new york stock exchange: gm (general motors), f (ford motor company) and dai (daimler ag, for mid-may 2007 as daimlerchrysler). the goal of this paper is to investigate and describe dependencies between them. the research is made for two periods separately: before and during crisis. this allows us to observe the changes that have taken place. 2. methodology let ),...,,( ,,2,1 ′= tnttt rrrr denote a multivariate time series of returns with the following decomposition ,ttt yμr += (1) where: )|( 1−φ= ttt e rμ is conditional mean, 1−φt is the information set available at time 1−t . conditional expected value )|( 1−φ= ttt e rμ can be modeled by varma models (tsay, 2002): ,)()( 0 tt bb aθφμφ += (2) where: 0φ is n-dimensional vector, p p bbb φφiφ −−−= ...)( 1 and q q bb θθιθ −−−= ...)( 1 are two nn × matrix polynomials, qb is back-shift operator: ,,, qtiti qb −= aa { }ta is a sequence of serially uncorrelated random vectors with mean zero and covariance matrix .σ a general multivariate garch model for ty is given by equation ,2/1 ttt εhy = (3) where: tε is n-dimensional i.i.d. process with zero mean and identity covariance matrix, 2/1 th is a nn × matrix satisfying ( ) ,2/12/1 ttt hhh =′ 0y =φ − )|( 1tte and .)|( 1 tttte hyy =φ′ − dynamics of multivariate return series of u.s. automotive stock companies… 45 specific mgarch (multivariate garch) model is described by parameterization of positive definite covariance matrix th . the bekk model have been proposed by engle and kroner (1995). the following equation defines bekk(p,q,k) model: , 1 1 1 1 00 ∑∑ ∑∑ = = = = −−− ′+′′+′= k k q i k k p i ikitikikititikt ghgayyacch (4) where: ikac ,0 and ikg are matrices of dimension nn × but 0c is upper triangular matrix, k term determines generality of the model. on this model positivity of matrix th can be easily imposed. to reduce number of parameters one can impose a diagonal bekk model (diagbekk) where matrices ika and ikg are diagonal. consequently, the generality of the model decreases. we can write equation for the simplest diagbekk model with 1=k for garch(1,1) model in following way: .1111111111100 ghgayyaccη −−− ′+′′+′= tttt (5) 3. empirical analysis this paper contains an analysis of daily log-returns series of three automotive companies listed on the new york stock exchange: general motors (nyse:gm), ford motor company (nyse:f) and daimler ag (nyse: dai). there are some indications that the crisis in the automobile industry connected with the current economic crisis began in mid-2008. in our analysis, a start date for crisis in the automotive industry is set on 1 july 2008. an investigation is made for two periods separately: before crisis (january 3, 2007 to june 30, 2008) and during crisis (july 1, 2008 to may 5, 2009). we use following designations for time-series: gm_1, f_1, dai_1 and gm_2, f_2, dai_2 respectively. figure 1 presents the plots of daily log-returns series for investigated companies. one can see that in mid-2008 dynamics of examined series has changed. there is a significant increase in volatility. this is most strongly marked in the period from mid-september to the end of december. this was the period of greatest uncertainty in financial markets: september 15 – the collapse of lehman brothers, which caused a crash on the american stock exchange, early october – events in iceland. in addition, due to rapid increase in oil prices (by mid-2008) and credit crunch, car manufacturers have noted the next drop in sales. general motors, along with chrysler, has started efforts to receive aid from the american governments because of the threat of bankruptcy. these events coincide in time with periods of increased volatility in the exblanka łęt 46 amined series. december 19, 2008, u.s. president has approved financial help for general motors and chrysler. from that moment, we can observe some sedation, which lasts only until mid-february 2009. this corresponds to the moment when general motors and chrysler have asked the government for more financial support (february 18). figure 1. gm, f and dai – daily log-returns. period 3.01.2007 – 5.05.2009 we calculate the most important descriptive statistics of the return series for first and second period separately. the results are contained in table 1. table 1. descriptive statistics of the return series min mean max std. deviation skewness kurtosis gm_1 -0.114 -0.003 0.096 0.029 0.096 1.164 gm_2 -0.373 -0.009 0.301 0.097 -0.281 1.749 f_1 -0.113 -0.001 0.111 0.026 0.012 2.167 f_2 -0.288 -0.001 0.259 0.072 0.088 3.120 dai_1 -0.076 0.000 0.079 0.020 -0.032 1.417 dai_2 -0.165 -0.002 0.199 0.053 0.050 1.219 clear differences between log-returns series of the investigated companies have been noted based on chart already. these observations are confirmed by calculated descriptive statistics. in all cases during the crisis, there is stronger volatility and extreme values – both positive and negative – are higher. the changes of skewness vary for each series. we observe that log-return series of gm had positive skew in the first period and during the crisis it changed to negative. negative returns of general motors occurred more often because of the many problems of this company and necessity of government help. return series of daimler ag behaved conversely. ford was characterized by positive skewness and during the crisis asymmetry was stronger. more frequent positive dynamics of multivariate return series of u.s. automotive stock companies… 47 returns for ford probably resulted from a better perception of the manufacturer by investors – company did not ask for government aid despite the prevailing situation. in all cases series were leptokurtic. table 2. parameter estimates for the fitted diagbekk model for two periods: before (first period) and during crisis (second period). df is the degree of freedom for student’s t error distribution first period second period estimate p-value estimate p-value 01φ -0.0019 0.1685 02φ -0.0005 0.6900 03φ 0.0004 0.6621 11,0c 0.0143 0.0000 0.0428 0.0126 12,0c 0.0207 0.0000 0.0171 0.0007 13,0c 0.0033 0.0000 0.0169 0.0000 22,0c 0.0040 0.2478 0.0183 0.0130 23,0c 0.0043 0.0118 0.0041 0.4872 33,0c 0.0000 0.7009 0.0000 0.1715 11,11g 0.8511 0.0000 0.8154 0.0000 22,11g 0.5137 0.0000 0.9081 0.0000 33,11g 0.9541 0.0000 0.9200 0.0000 11,11a 0.1685 0.0626 0.3839 0.0260 22,11a -0.2554 0.0054 0.2056 0.0703 33,11a -0.1260 0.0209 -0.2426 0.0000 df 5.3920 0.0000 we applied diagbekk model to characterize the dynamics of multivariate time series and changing dependencies between examined companies before and during the crisis. the results are presented in table 2. we fit diagbekk model with normal distribution in the first period and student’s t in the second. figure 2 presents the plots of dynamic conditional correlations from model fitted for the first period. in all cases there were only positive conditional correlations between examined series. the strongest dependency was between general motors and ford. the mean of conditional correlations between them equals 0.67, the lowest value equals 0.12, the highest: 0.78. the mean of conditional correlations between gm and dai equals 0.67, the lowest value equals 0.12, the highest: 0.78. conditional correlations between stock returns of ford and daimler were at the blanka łęt 48 lowest level with mean 0.35, minimum 0.11, maximum 0.48 and characterized by strongest dynamics. figure 2. dynamic conditional correlations. period 3.01.2007 – 30.06.2008 it can be concluded that companies belonging to the u.s. producers of the detroit three (gm, f) are much more strongly linked to each other. in the period before the crisis, linkages with companies of the big three (gm, f) were not high despite the fact that daimler was connected to may 2007 with chrysler. figure 3 presents the plots of dynamic conditional correlations during the crisis. once again, the strongest dependency was between general motors and ford motor company. the mean of conditional correlations between them equals 0.62 (small decrease, compared with the last period), the lowest value equals 0.43, the highest: 0.8. strength of connections between gm, f and dai increased in condition of crisis. the mean of conditional correlations between general motors and daimler equals 0.44, the lowest value equals -0.33, the highest: 0.82. conditional correlations between ford and daimler were as follows: mean 0.47 (increase), minimum -0.24, maximum 0.76. the most interesting behavior occur in october, 2008. strength of dependencies between general motors and ford increased until october 14, to the value 0.8. then conditional correlations decreased rapidly to the value 0.44 (october, 31). quite the contrary in those days was behaviour of conditional correlations of general motors – daimler and ford – daimler. interestingly, they changed the sign from positive to negative, which remained so far one week starting october, 10 and it reached a value close to -0.3. perhaps it was a reaction to earlier events in iceland, which shook the financial markets. a similar phenomenon, but with less power changes also occurred in late november and december. on november 28, 2008 conditional correlations of gendynamics of multivariate return series of u.s. automotive stock companies… 49 eral motors ford increased to a value of 0.77, while the conditional correlations of general motors daimler and ford daimler fell almost to zero. this was perhaps the result of the events associated with the efforts of general motors for government aid because of the threat of bankruptcy. since the beginning of 2009 dynamics of the analyzed dependencies reduced. figure 3. dynamic conditional correlations. period 1.07.2008 – 5.05.2009 we conclude that the strongest linkages were between general motors and ford motor company. it turned out that the estimated strength of linkages between them was high all the time, but decreased slightly during the crisis. this is an interesting phenomenon, because one would expect a completely different perception by investors of the two companies during the crisis, because of the much larger gm's financial problems that led to the threat of bankruptcy. the strength and dynamics of linkages between companies in the u.s. automotive market and daimler rose during the crisis. numerous problems faced by car manufacturers, have contributed to increase the relationship between gm, f and dai. in time of crisis this may be an indication for investors, who should take into account the presence of strong dependencies between the companies belonging to the same industry. references bauwens, l., laurent, s., rombouts, j. (2006), multivariate garch models: a survey, journal of applied econometrics, 21, 79–109. doornik, j. a. (2007), ox 5 – an object-oriented matrix programming using ox, timberlake consultants, london. engle, r., kroner, k. f. (1995), multivariate simultaneous generalized arch, econometric theory , 11, 122–150. blanka łęt 50 laurent, s. (2007), g@rch 5, estimating and forecasting arch models, timberlake consultants press, london. leonhardt, d. (2008), $73 an hour: adding it up, the new york times, www.nytimes.com/ /2008/12/10/business/worldbusiness/10iht-10leonhardt.18542483.html (10.12.2008). mccullagh, d. (2008), big three bailout? not so fast, cbs news, www.cbsnews.com/stories/ /2008/11/12/politics/otherpeoplesmoney/main4595068.shtml (12.11.2008). osińska, m. (2006), ekonometria finansowa (financial econometrics), pwe, warszawa. tsay, r. s. (2002), analysis of financial time series, john wiley&sons, new york. dynamika wielowymiarowych szeregów czasowych notowań spółek amerykańskiego rynku motoryzacyjnego w warunkach kryzysu z a r y s t r e ś c i. w artykule przeprowadzono analizę dynamiki powiązań pomiędzy szeregami zwrotów logarytmicznych trzech spółek rynku motoryzacyjnego notowanych na nowojorskiej giełdzie: gm, f i dai. badanie przeprowadzone zostało dla dwóch okresów: przed i w czasie kryzysu. dopasowano model diagbekk, uzyskując oszacowania dynamicznych korelacji warunkowych. wyniki badania wskazują na występowanie w czasie kryzysu silnych powiązań pomiędzy badanymi spółkami. s ł o w a k l u c z o w e: model diagbekk, dynamiczna korelacja warunkowa. microsoft word 00_tresc.docx © 2013 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2013.007 vol. 13 (2013) 127−143 submitted november 8, 2013 issn accepted december 30, 2013 1234-3862 joanna górna, karolina górna, elżbieta szulc* analysis of β-convergence. from traditional cross-section model to dynamic panel model a b s t r a c t. the aim of the paper is to discuss the course of development of methodology of economic convergence analyses, which points up the necessity of taking into consideration spatial connections among regions in regional growth models. it presents empirical models of β-convergence concerning the economic growth of european regions using various methodological conceptions. in the paper the models offered by spatial econometrics are recommended. the empirical data refer to per capita gdp across the european union regions at a nuts-2 level over the period 1995–2009 (annual data). k e y w o r d s: economic convergence, spatial effects, connectivity matrix, spatial panel models. j e l classification: c52. introduction the empirical analyses of β-convergence described in the growth literature can be divided into two parts. the first part contains the analyses which use cross-sectional data, while the second one refers to the analyses based on pooled time series and cross-sectional data. among other things the works of mankiw, romer and weil (1992), barro and sala-i-martin (1995) can be found within the classic literature on cross-sectional data models of convergence, while islam (1995) is a representative for the second part of the growth literature. the theoretical framework for these analyses of growth * correspondence to: elżbieta szulc, nicolaus copernicus university, department of econometrics and statistics, gagarina 13a, 87-100 toruń, poland, e-mail: eszulc@umk.pl. joanna górna, karolina górna, elżbieta szulc dynamic econometric models 3 (2013) 127–143 128 and convergence is the neo-classical solow-swan model (solow, 1956; swan, 1956). the solow-swan model also forms the base of many modern analyses in this domain. the spatial and spatio-temporal econometrics points out that for more precise explanation of economic growth it is necessary to take into consideration spatial connections among economics as the connected economies’ income levels are interdependent. elhorst, piras and arbia (2010) emphasize that the hypothesis that the relative location of an economy affects economic growth has been underpinned by theoretical extensions of the solow-swan model and confirmed by numerous and vast empirical analyses using mostly cross-sectional data. exemplifying literature is as follows: le gallo, ertur and baumont (2003), lópez-bazo, vayá and artis (2004), abreu, de groot and florax (2005), rey and janikas (2005), arbia (2006), bode and rey (2006), fingleton and lópez-bazo (2006), ertur and koch (2007), rey and le gallo (2009). up to now there have been fewer empirical analyses which support the hypothesis using panel data. the examples are: badinger, müller and tondl (2004), elhorst, piras and arbia (2010). this paper presents a review of fundamental conceptions of verifying the hypothesis of β-convergence, starting with the traditional regional crosssectional data model, through the models which include the spatial connections among the regions and the panel data models without the spatial effects, concluding on the spatial panel data models. the considerations presented are a continuation of the previous works by górna, górna and szulc (2013, 2014). the contents of the successive sections of the paper are as follows: in section 2 the subject and range of the investigation are defined as well as the aim of the paper and the research hypothesis are formulated. section 3 characterizes the data used in the investigation. section 4 presents the methodology. in this section the theoretical models of β-convergence in formulation of the cross-section regressions and the regressions for the pooled time series and cross-sectional data are presented. moreover, in section 4 the diagnostic tests for verification of the empirical models are pointed out. the results of the analysis are presented in section 5. recapitulation contains final conclusions and indicates further investigations. 1. subject and range of the investigation the paper concerns the phenomenon of economic convergence. it presents changes of spatial differentiation of per capita incomes across the euanalysis of β-convergence. from traditional cross-section model to… dynamic econometric models 3 (2013) 127–143 129 ropean union regions at a nuts-2 level over the period 1995–2009. the question is about validity of economic convergence hypothesis in the area of the european countries in the investigated period. in the paper the methods of verification of the hypothesis are considered. as a result of the analysis the empirical models of β-convergence concerning the economic growth of the european regions were obtained. to achieve it various methodological conceptions, especially the models offered by spatial econometrics, have been used. the attention was paid to the so-called absolute β-convergence (baumol, 1986; de long, 1988; arbia, 2006). in the constructed models of convergence apart from per capita gdp in the initial (basic) period no additional variable explaining the state of economies was taken into consideration. on the so-called conditional βconvergence in classic version, see e.g. bal-domańska (2010, 2011). on the contrary elhorst, piras and arbia (2010) is an example of the spatial approach. the aim of the investigation is to show that the empirical spatial models of convergence for cross-sectional data as well as the spatial panel data models have better statistical properties then the models which ignore the spatial and spatio-temporal connections among the regions. in addition, they allow more precise interpretation of the model parameters. the models presented are used to verify the hypothesis that relative location of a region affects the economic growth rate of the region. 2. data in the investigation the data on per capita gdp for 261 regions of 27 european countries were used. the data refer to the period of 1995–2009, i.e. to 15 years. they describe the per capita gdp spatial distributions and dynamics of incomes in the european union and come from eurostat data -base (ec.europa.eu/eurostat/). all calculations were prepared with the use of r (versions 2.7.2 and 3.0.1). figure 1 presents the spatial distribution of per capita gdp values (expressed by log terms) at the beginning of 1995–2009 period, i.e. in the year 1995 (figure 1a) and the analogical distribution in the year 2009 (figure 1b). it can be noticed that the spatial regularities of the distributions did not change in the period investigated. however, comparing the spatial trends, which have been fitted to the data and presented in figure 2, it can be seen that the surface of the trend for 2009 is flatter than the surface for 1995. this fact seems to confirm the supposition that economic convergence of the european regions occurs in the investigated period. joanna górna, karolina górna, elżbieta szulc dynamic econometric models 3 (2013) 127–143 130 figure 1. distribution of the per capita gdp in the european regions: (a) in the year 1995, (b) in the year 2009 figure 2. trend surfaces of per capita gdp for the european regions: (a) in the year 1995, (b) in the year 2009 then, figure 3 compares the spatial distribution of per capita gdp in the year 1995 with the growth rates of gdp across the regions during the period 1995–2009. this comparison shows that the poorer, at the beginning, regions have faster growth rates than the richer ones. thus, the economic convergence of the regions in the period considered is probable. this conclusion is conformable to the one formulated above. moreover, the next figure 4 which presents the surfaces of regions’ per capita gdp (expressed by log terms) and of regions’ per capita gdp growth rates (also expressed by log terms), seems to substantiate the supposition that the convergence is possible as well. the tendencies in figure 4 in parts a) and b) are inverted. additionally, figure 5 presents the average annual growth rate of gdp across the analysis of β-convergence. from traditional cross-section model to… dynamic econometric models 3 (2013) 127–143 131 regions of the eu over the investigated period (the map of spatial distribution of the growth rates and the trend surface of them respectively). figure 3. distributions of the per capita gdp: (a) in the year 1995, (b) growth rates during the period 1995–2009, in the european regions figure. 4. spatial distributions – trend surfaces: (a) of regions’ (log) per capita gdp in the year 1995, (b) of regions’ (log) per capita gdp growth rates during the period 1995–2009 joanna górna, karolina górna, elżbieta szulc dynamic econometric models 3 (2013) 127–143 132 figure 5. map and trend surface of average annual growth rate of gdp across the european regions 3. methodology the values of per capita gdp in the area established are treated as realizations of spatial stochastic process ( )iz s , where: [ ]iii yx ,=s – location coordinates on the plane, i = 1, 2, …, n – spatial units (regions) or of spatiotemporal stochastic process ( )tz i ,s , where i – as above and t = 1, 2, …, t – the successive years in the period considered. the cross-sectional data or the pooled time series and cross-sectional data are used. each observation is connected with the established location on the plane within some structure of connections among the spatial units. the structure is quantified by spatial connectivity matrix w. the matrix w has as many rows and columns as there are the regions. each row of the matrix contains non-zero elements in columns which correspond to the connected regions (the so-called neighbours), according to the established criterion (wij ≠ 0). furthermore, the given region cannot be connected to itself, i.e. it cannot be a neighbour of itself, so wij = 0 for all i = j. thus, the diagonal elements of w are zeros. the successive specifications of β-convergence models are considered. the classical model of β-convergence using data in cross-section takes the form: [ ] .lnln 1 1 ii i it gdp gdp gdp εβα ++=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ (1) analysis of β-convergence. from traditional cross-section model to… dynamic econometric models 3 (2013) 127–143 133 the spatial cross-sectional data model of β-convergence which contains the spatially lagged dependent variable (spatial lag model – slm) can be written as follows: [ ] .lnlnln 1 1 1 i j jt ij iji i it gdp gdp wgdp gdp gdp ερβα + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ++=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∑ ≠ (2) model (2) belongs to the class of the spatial autoregressive models (sar). in turn, the cross-section model of β-convergence with the spatially autocorrelated error component (spatial error model – sem) takes the form: [ ] ,lnln 1 1 ii i it gdp gdp gdp ηβα ++=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ij ij iji w εηλη += ∑ ≠ . (3) the use of cross-section regressions (1)–(3) for the investigation of economic convergence is connected with the loss of the information on variability of economies in time. this also means that in the analysis the individual economies’ features are omitted. besides, the model (1) ignores the spatial connections among the economies which appear important for describing the economic growth (see empirical characteristics of models (2)–(3)). below, the spatial models of β-convergence for pooled time series and cross-sectional data (tscs) are considered. including into the model tscs a spatial component leads to the following specifications: 1. the spatial autoregressive model (sar_pooled) [ ] .lnlnln 1 1 1 it jt jt ij ijit it it gdp gdp wgdp gdp gdp ερβα + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ++=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −≠ − − ∑ (4) 2. the model with spatial autoregressive residuals (se_pooled) [ ] ,lnln 1 1 itit it it gdp gdp gdp ηβα ++=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − .itjt ij ijit w εηλη += ∑ ≠ (5) pooled spatial and temporal data create the so-called panel data. methodology of the panel data analysis suggests including into the model individual or/and time effects which can be taken into consideration as fixed – and then they can be estimated – or as random becoming the part of the error component. as much as in the cross-sectional data growth models also in the panel data models more and more frequently the spatial connections among the joanna górna, karolina górna, elżbieta szulc dynamic econometric models 3 (2013) 127–143 134 regions are taken into account. it is caused by the opinion that the growth rate of any region is connected with the growth rates of its neighbours. the spatial panel models for verification of β-convergence hypothesis are as follows: 1. the spatial autoregressive panel model with individual fixed effects (the spatial autoregressive fixed-effects model) (sar_fe_ind) [ ] .lnlnln 1 1 1 it jt jt ij ijiti it it gdp gdp wgdp gdp gdp ερβα + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ++=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −≠ − − ∑ (6) 2. the spatial autoregressive panel model with individual and time fixed effects (sar_fe_two-way) [ ] .lnlnln 1 1 1 it jt jt ij ijitti it it gdp gdp wgdp gdp gdp ερβγα + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ +++=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −≠ − − ∑ (7) 3. the spatial error panel model with individual fixed effects (se_fe_ind) [ ] ,lnln 1 1 ititi it it gdp gdp gdp ηβα ++=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − .itjt ij ijit w εηλη += ∑ ≠ (8) 4. the spatial error panel model with individual and time fixed effects (se_fe_two-way) [ ] ,lnln 1 1 ititti it it gdp gdp gdp ηβγα +++=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − .itjt ij ijit w εηλη += ∑ ≠ (9) 5. the spatial autoregressive panel model with individual random effects (sar_re_ind) [ ] ,lnlnln 1 10 1 it jt jt ij ijit it it gdp gdp wgdp gdp gdp ζρβα + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ++=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −≠ − − ∑ (10) itiit εαζ += , or ittiit εγαζ ++= (in the case se_re_two-way). 6. the spatial error panel model with individual random effects (se_re_ind) [ ] ,lnln 10 1 itit it it gdp gdp gdp ζβα ++=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − (11) analysis of β-convergence. from traditional cross-section model to… dynamic econometric models 3 (2013) 127–143 135 itiit ηαζ += , itjt ij ijit w εηλη += ∑ ≠ , and in the model with individual and time random effects (se_re_two-way), ittiit ηγαζ ++= . economic convergence is said to be confirmed by the data if the estimates of β coefficients in models (1)–(11) are negative and statistically significant. furthermore, if parameters ρ in models: (2), (4), (6), (7), (10) and parameters λ in models: (3), (5), (8), (9), (11) are significantly different from zero, then in the convergence process the spatial connections among economies are important, and the hypothesis that the rate of growth of any economy is related to that of its neighbours is confirmed. as a result of explicit including the components of spatial dependence into the economic growth model is the evaluation of the convergence phenomena on the ground of β parameter estimated better than in the traditional approach. then the estimate of the parameter will not be influenced by omitting the dependence and it will more precisely reflect the influence of the per capita gdp in the basic period on the growth rate of incomes. in order to evaluate the quality of the empirical models in the investigation the following tools were used: the moran test (moran’s i) for spatial independence of the residuals, the lagrange multiplier tests (lmlag, lmerr) and their robust versions (rlmlag, rlmerr) as spatial dependence diagnostics, the likelihood ratio test (lr) for testing the significance of the spatial dependence, the breusch-pagan heteroskedasticity test, the chow test for spatial invariance of β-convergence parameters and for verifying the need of including into the spatial panel models the fixed effects, the lagrange multiplier tests to verify the spatial interactions and the random effects, the hausman test for choosing between the fixed-effects (fe) model and the random-effects (re) model (on the tools see e. g. arbia, 2006; millo and piras, 2012; mutl and pfaffermayr, 2011; baltagi et al., 2003; suchecki (ed.), 2012). 5. results the successive tables presented below contain the information on the usefulness of various methodological conceptions expressed by the model specifications presented in section 4. the part of the information which refers to the models estimated on the ground of cross-sectional data is also presented in the works: górna, górna and szulc (2013, 2014), which unlike this paper are limited to the β-convergence analysis by using cross-section regressions. joanna górna, karolina górna, elżbieta szulc dynamic econometric models 3 (2013) 127–143 136 table 1 contains the results of estimation and verification of three models: the linear regression model, i.e. the traditional model without the spatial effects, the spatial autoregressive model (sar) and the spatial error model (se). table 1. the results of estimation and verification of the cross-sectional data models of β-convergence linear regression spatial autoregressive model spatial error model parameters α β ρ λ 3.8739 (0.0000) –0.3535 (0.0000) − − 2.8602 (0.0000) –0.2618 (0.0000) 0.2860 (0.0000) − 3.6383 (0.0000) –0.3269 (0.0000) − 0.4548 (0.0000) goodness of fit adjusted r2 aic 0.7642 –166.71 − –191.86 − –195.33 heteroskedasticity breusch-pagan test 11.0877 (0.0009) 13.1741 (0.0003) 4.0292 (0.0450) autocorrelation of residuals moran test 5.4531 (0.0000) 2.0380 (0.0416) –0.0946 (0.0753) spatial dependence lr lmlag lmerr rlmlag rlmerr − 26.3535 (0.0000) 29.7572 (0.0000) − − 27.1540 (0.0000) − − 4.0923 (0.0431) − 30.6230 (0.0000) − − − 7.4940 (0.02654) speed of convergence half-life 0.0291 23.8477 0.0202 34.2531 0.0264 26.2654 note: numbers in brackets refer to the p-values. diagnostics for the models considered suggest that the classical model is the worst of them. this result is conformable to our anticipation because the analysis of β-convergence. from traditional cross-section model to… dynamic econometric models 3 (2013) 127–143 137 assumptions of the model, especially the same variance in the space and independence across residuals for all regions, are usually unrealistic in practice. in this case, the breusch-pagan statistic is significant, leading to rejecting the model assumption of homoskedasticity. in addition, on the basis of the moran’s i test it is necessary to state that the hypothesis of independence of the traditional model residuals should be rejected. as the moran test does not admit an explicit alternative hypothesis opposed to the null, the lagrange multiplier tests (lm) were used (see table 1). the lm tests for the linear model used consider the spatial lag model (spatial autoregressive) and the spatial error model as alternatives (lmlag and lmerr, respectively). table 1 reports the results of using the robust tests (rlmlag, in which h0: ρ = 0 under the assumption that λ ≠ 0 and rlmerr, where h0: λ = 0 under the assumption that ρ ≠ 0) as well. subsequently, the significance of the spatial effects in sar and se models using the likelihood ratio test (lr) was confirmed. taking into account the spatial connections across the european regions at a nuts-2 level over the period 1995–2009, in the β-convergence models has removed the problem of autocorrelation of the residuals (at the level of significance γ = 0.01). however, the problem of variance heteroskedasticity has remained, especially in the case of the spatial autoregressive model. the spatial heteroskedasticity can be caused by omitting the factor responsible for systematic spatial variability. in this connection in górna, górna and szulc (2014) the additional analysis searching for the spatial regimes was performed. for this purpose the considered area of the regions has been divided into two sub-areas (see figure 6). to justify the division the chow test for verifying the spatial changeability of β parameters was used (see also arbia, 2006, p. 133). table 2 contains the results of the test. table 2. results of the test for spatial invariance of the β-convergence parameters chow test linear regression spatial lag model spatial error model values of test 8.2701 12.1347 18.9393 p-value 0.0407 0.0069 0.0003 the hypothesis of constancy of parameters in β-convergence models estimated in the cited investigation should be rejected. this leads to identification of spatial regimes and means that in the considered area as a whole there are differentials in relationships between regional growth rate and initial per capita gdp. the models of β-convergence according to the sub-areas considered were estimated and verified and the differentiation of the paramejoanna górna, karolina górna, elżbieta szulc dynamic econometric models 3 (2013) 127–143 138 ters of β-convergence as well as of properties of the empirical models obtained were stated (for the details, see górna, górna and szulc, 2014). figure 6. classification of the eu regions within the two spatial regimes the classical model estimated with the use of the pooled time series and cross-sectional data like the classical cross-section regression does not satisfy the fundamental criteria of statistical verification. for the purpose of spatial effects verification in the models for pooled time series and cross-sectional data the lagrange multiplier tests analogical to those applied in the case of the cross-section regressions are used. the results of them have confirmed the need of re-specifications towards the models with the spatial effects (see table 3). moreover, the significance of the effects with the aid of the lr test was confirmed. according to the methodology of panel data modeling in the investigation the reasonableness of including into the spatial models the fixed or/and the random effects was considered. for this purpose the chow test (the spatial model for pooled tscs data vs. the spatial panel model with fixed effects) and the lm tests (to verify spatial interactions and random effects) were used. the results of the chow test have pointed out the statistical significance of the fixed effects in the spatial autoregressive panel model but not in the panel spatial error model. on the basis of the joint lm test, which tested the hypothesis: h0: λ = 2 ασ = 0 under the alternative that at least one component was not zero, it was stated that the null should be rejected. analysis of β-convergence. from traditional cross-section model to… dynamic econometric models 3 (2013) 127–143 139 table 3. results of estimation and verification of β-convergence models for pooled time series and cross-sectional data linear regression spatial autoregressive model spatial error model parameters α β ρ λ 0.3558 (0.0000) –0.0326 (0.0000) − − 0.3132 (0.0000) –0.0299 (0.0000) 0.4044 (0.0000) − 0.3334 (0.0000) –0.0303 (0.0000) − 0.4250 (0.0000) goodness of fit adjusted r2 aic 0.1439 –9442 − –9974.6 − –9981.7 heteroskedasticity breuch-pagan test 184.5378 (0.0000) 243.7477 (0.0000) 250.5849 (0.0000) autocorrelation of residuals moran test 26.7685 (0.0000) –2.1504 (0.0158) –3.333 (0.0304) spatial dependence lr lmlag lmerr rlmlag rlmerr − 702.5570 (0.0000) 713.3021 (0.0000) − − 534.61 (0.0000) − − 12.5044 (0.0004) − 541.73 (0.0000) − − − 23.2494 (0.0000) speed of convergence half-life 0.0331 20.82 0.0304 22.73 0.0308 22.43 note: numbers in brackets refer to the p-values. next, in the investigation the so-called marginal lm tests for verification of the hypotheses: h0: λ = 0 (under the assumption that 2 ασ = 0) and h0: 2 ασ =0 (assuming λ = 0), were used respectively. in both of the cases the result of the test has pointed out the lack of the bases for rejecting the null. therefore, the conditional lm tests, which are more useful tests in this framework, because they test for one effect, and are robust against the other (h0: λ = 0 joanna górna, karolina górna, elżbieta szulc dynamic econometric models 3 (2013) 127–143 140 assuming 2ασ = 0 or 2 ασ ≠ 0; h0: 2 ασ = 0 with λ = 0 or λ ≠ 0), confirmed that the interactions of spatial and random effects were possible. the hausman test was used to choose between the fe and re models. it has suggested the choice of the spatial panel model with the fixed effects. table 4. selected characteristics of spatial panel models parameter sar_fe_ind sar_re_ind se_fe_ind se_re_ind estimate of parameter statistic t estimate of parameter statistic t estimate of parameter statistic t estimate of parameter statistic t α β ρ λ − –0.064 0.340 − − –16.08 19.42 − 0.319 –0.030 0.344 − 26.23 –24.10 20.24 − − –0.083 − 0.349 − –16.13 − 19.62 0.335 –0.030 − 0.359 26.57 –23.50 − 19.60 speed of convergence half-life 0.0658 10.4832 0.0308 22.4255 0.0865 7.9733 0.0309 22.3506 chow test f 121.4659 (0.0000) –331.9926 (1.0000) hausman test 74.5562 (0.0000) lm tests joint lm (random effects and spatial autocorrelation as alternative hypothesis): 460.0374; p-value=0.0000 marginal lm: lm1 (random effects as alternative hypothesis): –0.0005; p-value =1 lm2 (spatial autocorrelation as alternative hypothesis): 0.0064; p-value=0.9949 conditional lm (assuming 02 ≥ασ ; spatial autocorrelation as alternative hypothesis): 24.5265; p-value=0.0000 note: numbers in brackets refer to the p-values. conclusions on the ground of the analysis the following conclusions can be formulated: 1. the fact of including the components of spatial dependence (the spatially lagged dependent variable or the spatial autoregressive error component) in the economic convergence model is justified and valid for the analyses of per capita incomes across the regions investigated and of income changes in time. 2. as a result of that it is possible to define the influence of the neighbour connections on the economic growth, the estimate of parameter β is more precise, and properties of the model are better. analysis of β-convergence. from traditional cross-section model to… dynamic econometric models 3 (2013) 127–143 141 3. like in the growth models for the cross-sectional data in the panel data models it is necessary to take into consideration the spatial connections among the regions. 4. the dynamic panel models with the spatial effects are the natural extension of cross-section regressions as the tools of verifying hypothesis of economic convergence. in further investigations the models of convergence with the additional explanatory variables and also with these variables spatially lagged are to be considered. additionally, more diagnostic tests for the spatial panel models are expected to be used. references abreu, m., de groot, h. l. f., florax, r. j. g. m. (2005), space and growth: a survey of empirical evidence and methods, région et développment, 21, 13–14, doi: http://dx.doi.org/10.2139/ssrn.631007. arbia, g. (2006), spatial econometrics. statistical foundations and applications to regional convergence, springer-verlag, berlin heidelberg. badinger, h., müller, w. g., tondl, g. (2004), regional convergence in the european union (1985–1999): a spatial dynamic panel analysis, regional studies, 38(3), 241–253, doi: http://dx.doi.org/10.1080/003434042000211105. bal-domańska, b. (2010), the application of dynamic panel data models in the analysis of conditional convergence, in pociecha, j. (ed.), data analysis methods in economic research, studia i prace uniwersytetu ekonomicznego w krakowie, 11, 107–123. bal-domańska, b. (2011), ekonometryczna identyfikacja β-konwergencji regionów szczebla nuts 2 państw unii europejskiej (econometric identification of β-convergence of the nuts 2 level regions of the european union states), in suchecka, j. (ed.), ekonometria przestrzenna i regionalne analizy ekonomiczne (spatial econometrics and regional economic analyses), acta universitatis lodziensis, folia oeconomica, 253, 9–23. baltagi, b. h., song, s. h., koch, w. (2003), testing panel data regression models with spatial error correlation, journal of econometrics, 117(1), 123–150, doi: http://dx.doi.org/10.1016/s0304-4076(03)00120-9. barro, r. j., sala-i-martin, x. (1995), economic growth, new york: mcgraw-hill. baumol, w. j. (1986), productivity growth, convergence and welfare: what the long-run data show, american economic review, 76(5), 1072–1085. bode, e., rey, s. j. (2006), the spatial dimension of economic growth and convergence, papers in regional science, 85(2), 171–176, doi: http://dx.doi.org/10.1111/j.1435-5957.2006.00073.x. de long, j. b. (1988), productivity growth, convergence and welfare: comment, american economic review, 78(5), 1138–1154. elhorst, p., piras, g., arbia, g. (2010), growth and convergence in multiregional model with space-time dynamics, geographical analysis, 42(3), 338–355, doi: http://dx.doi.org/10.1111/j.1538-4632.2010.00796.x. joanna górna, karolina górna, elżbieta szulc dynamic econometric models 3 (2013) 127–143 142 ertur, c., koch, w. (2007), growth, technological interdependence and spatial externalities: theory and evidence, journal of applied econometrics, 22(6), 1033–1062, doi: http://dx.doi.org/10.1002/jae.963. fingleton, b., lópez-bazo, e. (2006). empirical growth models with spatial effects, papers in regional science 85(2), 177–198, doi: http://dx.doi.org/10.1111/j.1435-5957.2006.00074.x. górna, j., górna, k., szulc, e. (2013a), a β-convergence analysis of european regions. some re-specifications of the traditional model, in papież, m. and śmiech, s. (eds.), proceedings of the 7th professor aleksander zeliaś international conference on modelling and forecasting of socio-economic phenomena. cracow: foundation of the cracow university of economics, 73–81, http://www.konferencjazakopianska.pl/proceedings/pdf/gorna_gorna_szulc.pdf. górna, j., górna, k., szulc, e. (2014), a β-convergence analysis of european regions. some re-specifications of the traditional model (extended version), forthcoming. islam, n. (1995), growth empirics: a panel data approach, the quarterly journal of economics, 110(4), 1127–1170, doi: http://dx.doi.org/10.2307/2946651. le gallo, j., ertur, c., baumont, c. (2003), a spatial econometric analysis of convergence across european regions, 1980–1995, in fingleton, b. (ed.), european regional growth. advances in spatial science. springer-verlag, berlin heidelberg, 99–129, doi: http://dx.doi.org/10.1007/978-3-662-07136-6_4. lópez-bazo, e., vayá, e., artis, m. (2004), regional externalities and growth: evidence from european regions, journal of regional science, 44(1), 43–73, doi: http://dx.doi.org/10.1111/j.1085-9489.2004.00327.x. mankiw, n. g., romer, d., weil, d. (1992), a contribution to the empirics of economic growth, the quarterly journal of economics, 107(2), 407–437, doi: http://dx.doi.org/10.2307/2118477. millo, g., piras, g. (2012), splm: spatial panel data models in r, journal of statistical software, 47(1), 2–38. mutl, j., pfaffermayr, m. (2011), the hausman test in a cliff and ord panel model, econometrics journal, 14(1), 48–76, doi: http://dx.doi.org/10.1111/j.1368-423x.2010.00325.x. rey, s. j., janikas, m. j. (2005), regional convergence, inequality, and space, journal of economic geography, 5(2), 155–176, doi: http://dx.doi.org/10.1093/jnlecg/lbh044. rey, s. j., le gallo, j. (2009), spatial analysis of economic convergence, in mills, t. c. and patterson, k. (eds.), palgrave handbook of econometrics, volume ii: applied econometrics. palgrave macmillan, new york, 1251–1290. solow, r. m. (1956), a contribution to the theory of economic growth, the quarterly journal of economics, 70(1), 65–94, doi: http://dx.doi.org/10.2307/1884513. suchecki, b. (ed.) (2012), ekonometria przestrzenna ii. modele zaawansowane (spatial econometrics ii. advanced models), wydawnictwo c.h.beck, warszawa. swan, t. w. (1956), economic growth and capital accumulation, economic record, 32 (2), 334–361, doi: http://dx.doi.org/10.1111/j.1475-4932.1956.tb00434.x. analysis of β-convergence. from traditional cross-section model to… dynamic econometric models 3 (2013) 127–143 143 analiza β-konwergencji. od tradycyjnego modelu przekrojowego do dynamicznego modelu panelowego z a r y s t r e ś c i. celem artykułu jest omówienie kierunku rozwoju metodologii badania konwergencji gospodarczej, wskazującego na potrzebę uwzględniania w modelach wzrostu regionów powiązań przestrzennych między nimi. artykuł prezentuje empiryczne modele β-konwergencji dotyczące wzrostu gospodarczego regionów europy, otrzymane przy wykorzystaniu różnych koncepcji metodologicznych. w artykule rekomenduje się modele z zakresu ekonometrii przestrzennej. dane empiryczne dotyczą pkb per capita w regionach nuts-2 27 państw europejskich, będących członkami unii europejskiej. zakres czasowy analizy obejmuje lata 1995–2009 (dane roczne). s ł o w a k l u c z o w e: konwergencja gospodarcza, efekty przestrzenne, macierz sąsiedztwa, przestrzenny model panelowy. microsoft word dem_2018_99to114.docx © 2018 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2018.006 vol. 18 (2018) 99−114 submitted november 25, 2018 issn (online) 2450-7067 accepted december 27, 2018 issn (print) 1234-3862 rumiana górska* decomposition of sovereign cds spread using the concept of factorization a b s t r a c t. sovereign cds (credit default swap) is a derivative that provides insurance of repayment of the government’s loans and may be considered as a market indicator of the insolvency risk of a country. the aim of the study is to identify factors affecting the sovereign cds spreads of selected european countries for the period from 2008 to 2016. factor analysis shows that there are two common factors that have explained about 92% of the variation of the cds spreads. next, the decomposition of the spreads presents the influence of these factors on cds spreads of surveyed countries. k e y w o r d s: cds spread; factor analysis; global risk; insolvency risk. j e l classification: g14; g15. introduction cds (credit default swap) is a derivative that provides insurance of repayment of the loan. one of the parties of the transaction receives a specified remuneration and agrees to pay the debt to the other side of the transaction, if the debtor is unable to pay it back. sovereign cds is a derivative that provides insurance of repayment of the government’s loans. it is often considered as a market indicator of the insolvency risk of a country. the cds contract price (or cds spread) is calculated as a percentage of the debt, payable annually. the higher the risk of repayment of debt, the higher the insurance premium (risk premium) required by the issuer of the cds contracts and the higher the * correspondence to: rumiana górska, warsaw school of economics, collegium of economic analysis, institute of econometrics, 6/8 madalinskiego street, 02-513 warsaw, poland, e-mail: rgorska@sgh.waw.pl. rumiana górska dynamic econometric models 18 (2018) 99–114 100 cds spread. cds spreads for each country therefore reflect the confidence in the issuer of the sovereign bonds. variation in the levels of cds quotes for each country shows the differences in the market assessment of the risk of investments in treasury securities of this country. for example, on 29.11.2013 sovereign cds spreads for five-year bonds in basis points was: for poland 82.48; germany 23.56; spain 153.5; portugal 341.4. investors holding bonds of these countries can insure themselves against the insolvency of the given country, paying annually a percentage of the value of the bond (for poland 0.8248%, germany 0.2356 %, spain 1.535 %, portugal 3.414 %). cds quotes for greece in march 2012 (during the greek debt crisis) reached 25422.80 basis points. sovereign cds spread depends both on the so-called economic fundamentals (i.e. macroeconomic variables such as level of indebtedness, debt structure, the cost of servicing the debt, debt to gdp relation, the rate of gdp growth etc.), as well as global factors (global financial market conditions, market liquidity, investor expectations etc.). the aim of this study is to identify factors affecting the sovereign cds spreads of selected european countries: germany, france, great britain, italy, spain, portugal, czech republic, hungary and poland for the period from 2008 to 2016. 1. literature review publications on the issues related to the information capacity of cds spreads securing repayment of government debt focus on different aspects of the problem. some authors (varga, 2008; komarek et al., 2013; arce et al., 2013; coudert and gex, 2013) examined the relationship between the cds market and the government bonds market and came to the conclusion that prices in the market cds and yields in the treasury bond market reflect the same information about the credit risk of that country. differences in the levels of spreads in the pricing of risk in both markets were associated with the following factors: counterparty risk, the lack of liquidity or liquidity problems, transaction costs, flight-to-quality and the debt repurchase decisions by the european central bank. moreover, the cds market preceded the bond market in the pricing of risk. in addition, during periods of turbulence in the financial markets, the role of the cds markets increases. another group of publications includes studies of the relationship between the cds spread and selected macroeconomic variables, market indicators or information about political events. coronado et al. (2012) examined the relationship between the cds market and the stock market in eight european countries for the period 2007–2010. the first of these markets represented the credit risk of the country and the second, the market risk. the authors came to decomposition of sovereign cds spread using the concept of factorization dynamic econometric models 18 (2018) 99–114 101 the conclusion that the stock market leads the cds market until 2010, the time of the first appearance of the debt crisis. following that period, the reaction of the cds market quickened and new information was included in the quotations of cds for the first time. aizenman et al. (2013) examined the pricing of risk for selected european countries based on their fiscal situation (debt, deficit, taxes) for the years 2005–2010. they considered two groups of countries: the central eu countries (germany, france, great britain) and the peripheral eu (greece, ireland, italy, portugal, and spain). in 2010 (a period of increased risk) the risk of peripheral countries was valued at a higher level than was consistent with their fiscal situation. the authors explained this effect through the mechanism of the pessimistic self-fulfilling forecasts – the expectations being that the adaptation of the peripheral countries to the new situation will be more difficult. a panel study showed that the situation of public finances and other economic variables are important determinants of a country’s risk. büchel (2013) examined the effect of public information and announcements made by the european central bank governing council members, european union officials and national representatives in the period 2009–2011 to the cds and bond markets. information from the representatives of germany, france and the european union authorities had an immediate effect on both markets, while the expression of representatives of smaller countries had no influence. the third group of publications relates to the contagion effect (risk transfer) between countries. kalbaska and gątkowski (2012) identified the increased risk of contagion between european countries after 2007 on the basis of studying the dynamics of the cds market in the period 2005–2010. spain, ireland, and greece had the biggest impact on the destabilization of the cds markets. the most resilient country to contagion was the united kingdom, and the least resistant, portugal. the aforementioned studies were conducted using econometric methods, such as the analysis of stationarity and cointegration, vecm, var and garch models, irf analysis, and granger causality tests. the studies, closest in terms of the methodology of the presented here study was made by badaou et al. (2013) and fabozzi et al. (2016). these authors used a factor analysis framework to decompose cds spreads. badaou et al. (2013) identified the following components: the insolvency of the country (55.6% of the variation) and the liquidity of the market (44.32% of the variation). the authors concluded that increases in cds spreads observed in times of crisis are mainly caused by the surge of liquidity, and not by an increase in the intensity of the risk of insolvency. fabozzi et al. (2016) decompose cds spreads using independent component analysis (a technique similar to principal component analysis) of so-called “old” eu members. rumiana górska dynamic econometric models 18 (2018) 99–114 102 in poland, the issues of sovereign cds was discussed by kliber (2011), (2016) and bieńkowski et al. (2011). kliber (2011) explored the relationship between cds spreads for three countries – poland, czech republic, and hungary. the study showed that there is a variable relationship between these instruments. during the hungarian crisis, the relationship between prices of polish and hungarian cds and that of the czech and hungarian cds weakened, and during the greek crisis, they rapidly increased. this means that global phenomena play a greater role than regional ones. kliber (2016) also verified the impact of the ban on uncovered sovereign cds trade in europe on the interdependencies between the sovereign cds market and other sectors of financial markets. the analysis of two european markets: swedish and hungarian shows that relationships of financial markets with the sovereign cds was much weaker in the case of safe and developed swedish market, so it is less prone to crisis transmission than hungarian market. tests made by bieńkowski et al. (2011) on the susceptibility of the polish currency market to internal and external instability showed, that the cds spread belongs to the factors that cause the greatest disruption in the polish currency market (the other factors are variables associated with the trade balance and the presence of a crisis in the peripheral european countries). this paper extends the literature by presenting the decomposition of the cds spread of different european countries, representing both “old” and “new” eu members, as well as the “peripheral” and the “central” countries using factor analysis and the factorization framework. 2. methodology in general, the factor analysis idea is based on the presentation of a large number of observable, correlated variables in terms of a potentially lower number of unobservable, "hidden" variables, called factors. observable variables are modeled as a linear combination of a small number orthogonal factors that are sources of common variability of the primary variables and a unique variance of each variable. common factors may be interpreted as a source of a systematic risk and random component reflects the unsystematic (specific) risk. factor analysis is often used as a data reduction method or a data classification method. the first goal of the presented research is to identify common risk factors affecting cds spreads of the countries surveyed, as well as factors specific to each country. factor analysis as a method allowing a reduction in a large number of interrelated variables down to a few factors, without a significant loss of information contained therein seems to be a suitable for the purpose. each primary variable is presented as a linear combination of common and specific factors in the following way: decomposition of sovereign cds spread using the concept of factorization dynamic econometric models 18 (2018) 99–114 103 𝑍"# = 𝑤##𝐹"# + 𝑤#(𝐹"( + ⋯+ 𝑤#*𝐹"* + 𝑉"# … 𝑍", = 𝑤,#𝐹"# + 𝑤,(𝐹"( + ⋯+ 𝑤,*𝐹"* + 𝑉"# (1) … 𝑍"= 𝑤-#𝐹"# + 𝑤-(𝐹"( + ⋯+ 𝑤-*𝐹"* + 𝑉" where: zti – standardized i-th variable at a moment t, i =1,2,…,n, ftl – l-th common factor at a moment t, l=1,2,…,k, wil – factor loading l-th common factor to i-th variable, vti – specific factor for the i-th variable at a moment t, n – number of primary variables, k – number of common factors. variance of the each variable is decomposed according to the following formula: 𝑣𝑎𝑟(𝑍,) = 𝑤,# ( + 𝑤,( ( + ⋯+ 𝑤,* ( + 𝜓, ( = 1, (2) where 𝜓, ( – variance corresponding to specific factor. correlation between p-th and q-th variables can be obtained as follows: 𝑟56 = ∑ 𝑤5,𝑤6, * ,8# , (3) where rpq – correlation coefficient between p-th and q-th variable. in matrix notation, a set of primary variables as present in the following way: 𝐙 = 𝐅𝐖𝐓 + 𝐕, (4) where: z – primary variables matrix (t × 𝑛), f – factor values matrix (t × 𝑘), w – factor loadings matrix (n × 𝑘), v – specific factors matrix (t × 𝑛). the decomposition of the covariance matrix of the primary variables is conducted as follows: 𝐑 = 𝐖𝐖𝐓 + 𝚿, (5) where: r – variance-covariance matrix of a primary set of variables, 𝚿 – covariance matrix for specific factors. one of the ways to define the number of common factors is based on eigenvalues of variance-covariance matrix after factors extracted. in this study the minimum value of eigenvalues to be retained is 1. rumiana górska dynamic econometric models 18 (2018) 99–114 104 after common factors are identified, factor loadings are calculated. the next step is the decomposition of the cds spread and measurement of the impact of the risk factors using the concept of factorization. the idea of factorization was introduced by ho (1999), who used it for the decomposition of the total rate of return in the model capm for the bond market. cds spreads are presented according to the following formula: 𝑆", = 𝑤#,𝐹"# + 𝑤(,𝐹"( + ⋯+ 𝑤*,𝐹"* + 𝑉", for i=1,2,…,n (6) where: sti – cds spread of a given country at a moment t, ftl – l-th common factor at a moment t, l=1,2,…,k, vti – standardized specific factor at a moment t, n – number of primary variables, k – number of common factors. all variables are standardized. proposed two-step methodology allows to measuring of the response of individual country to the change of common risk factors and obtaining a historical decomposition of cds spreads. 3. characteristic of the data empirical analysis is performed for a set of 9 primary variables – these are cds spreads for the following countries: germany, france, great britain, italy, spain, portugal, czech republic, hungary and poland for the period from 2008 to 2016 year. the data have daily frequency. the factor analysis as well the decomposition of the cds spreads does not involve any macroeconomic data. the purpose is to identify factors ruling the evolution of spreads only based on the data concerning spreads. table 1 presents descriptive statistics of the data. table 1. descriptive statistics of the data country mean median min max standard deviation de 24,88 20,94 6,64 92,50 16,67 fr 49,43 38,02 11,25 171,56 32,46 gb 48,48 42,82 11,66 165,00 27,70 it 162,05 121,39 39,50 498,66 100,22 es 154,34 113,97 37,00 492,07 100,51 pt 343,22 219,23 37,00 1521,50 315,49 cz 80,22 69,24 33,00 350,00 46,87 hu 271,31 248,00 76,00 661,24 122,96 pl 119,31 92,78 43,50 421,00 66,50 note: de – germany, fr – france, gb – great britain, it – italy, es – spain, pt – portugal, cz – czech republic, hu – hungary, pl – poland. decomposition of sovereign cds spread using the concept of factorization dynamic econometric models 18 (2018) 99–114 105 table 2 reports correlation coefficients matrix. table 2. correlation coefficients matrix country de fr gb it es pt cz hu pl de 1,00 0,80 0,90 0,56 0,55 0,51 0,92 0,86 0,83 fr 0,80 1,00 0,55 0,86 0,85 0,85 0,69 0,57 0,81 gb 0,90 0,55 1,00 0,33 0,33 0,22 0,92 0,90 0,75 it 0,56 0,86 0,33 1,00 0,93 0,87 0,51 0,41 0,76 es 0,55 0,85 0,33 0,93 1,00 0,86 0,47 0,37 0,73 pt 0,51 0,85 0,22 0,87 0,86 1,00 0,42 0,27 0,63 cz 0,92 0,69 0,92 0,51 0,47 0,42 1,00 0,96 0,87 hu 0,86 0,57 0,90 0,41 0,37 0,27 0,96 1,00 0,83 pl 0,83 0,81 0,75 0,76 0,73 0,63 0,87 0,83 1,00 cds spreads are highly correlated, thus factor analysis is suitable statistical procedure for this data set. it allows a reduction of the data dimension and identification the interrelationships among the primary variables and latent, common factors. 3. results at the first step of the study factor analysis revealed that two common factors have explained about 92% of the variation of the cds spreads. these factors are unobservable, latent variables, but they may have some economic interpretation. figure 1 presents common factors, labeled as f1 and f2. figure 1. common factors affecting cds spreads the first factor reached the maximum values at the turn of 2008–2009 years during the global financial crisis. next, this factor increased in 2011. a hypothesis that this factor reflects a global risk will be investigated. -3 -2 -1 0 1 2 3 4 2008 2009 2010 2011 2012 2013 2014 2015 f1 f2 rumiana górska dynamic econometric models 18 (2018) 99–114 106 financial crisis from 2008 year influences the financial markets worldwide and contributed to the european sovereign-debt crisis in 2010 year. it can be expected that the first factor has influenced cds spreads of all of the countries surveyed. the second factor has different characteristics. it reached maximum values in 2011–2012 years when the eurozone sovereign debt crisis exacerbated. this crisis has started in late 2009 and remained a critical factor at the sovereign credit market. prior to the crisis the credit risk of eurozone countries was considered to be very low; however, after the onset of the crisis, the credit risk has increased dramatically. it can be expected that second factor has influenced the countries with the biggest debt problems. next question is how these two common factors affect cds spreads of the counties. factor loadings obtained during the factor analysis show the influence of the factors on the cds spreads. in order to obtain a clearer picture of these influences orthogonal varimax rotation was applied. varimax rotation is orthogonal rotation that maximizes the sum of the variances of the squared loadings. it is achieved if any given variable has a high loading on a single factor but low loadings on the remaining factors. factor loadings for each primary variable after varimax rotation are shown in table 3. table 3. rotated factor loadings, share of explained variation and unique variances country common factors uniqueness f1 f2 gb 0.9519 0.1111 0,0816 cz 0.9504 0.1691 0,0681 pl 0.9361 0.3025 0,0322 de 0.8647 0.4088 0,0852 hu 0.7303 0.5972 0,1100 fr 0.4754 0.8308 0,0838 es 0.2220 0.9204 0,1036 it 0.2468 0.9216 0,0898 pt 0.1367 0.9211 0,1329 variation explained 4.3214 3.8914 share of variation 0.4802 0.4324 0,0874 note: cz – czech republic, pl – poland, gb – great britain, de – germany, hu – hungary, fr – france, es – spain, it – italy, pt – portugal. factor loadings higher than 0,8 is hig. first common factor (f1) affects in the highest degree great britain, czech republic, poland, and germany. second common factor (f2) affects mostly portugal, italy, and spain. hungary and france are under the influence of both common factors. unique variance of the countries’ cds is between 3,22% (poland) and 13,29% (portugal). as cds spreads of nine european countries evolved under the influence of two common factors, it is important to gave some economic explanation of decomposition of sovereign cds spread using the concept of factorization dynamic econometric models 18 (2018) 99–114 107 these factors. interpretation of the first common factor could be facilitated by comparing it to the volatility index (vix), known as “fear index”. the vix index is designed to produce a measure of constant, 30-day expected volatility of the u.s. stock market, derived from real-time, mid-quote prices of s&p500 index call and put options. on a global basis, it is one of the most recognized measures of volatility, widely reported by financial media and closely followed by a variety of market participants as a daily market indicator. figure 2 presents first common factor (f1) and volatility index. correlation coefficient between two variables equals 0,75. figure 2. first common factor and index vix second common factor affects at the largest extent countries with a highest indebtedness problems during the eurozone debt crisis. table 4 reports government debt to gdp ratio for the surveyed countries. countries are sorted in ascending order (in 2012 year). table 4. government debt to gdp ratio (percentage) country 2008 2009 2010 2011 2012 2013 2014 2015 2016 cz 28,7 34,1 38,2 39,8 44,5 45 42,2 40 36,8 pl 46,3 49,4 53,1 54,1 53,7 55,7 50,3 51,1 54,2 gb 35,4 50,1 64,6 71,4 75,1 78,6 80,5 82,9 82,6 hu 71,6 77,8 80,5 80,7 78,2 76,6 76,6 77,7 76 de 65,1 72,6 81 78,7 79,9 77,5 74,7 71 68,2 es 39,5 52,8 61 69,5 85,7 95,5 100,4 99,4 99 fr 68 78,9 81,6 85,2 89,5 92,3 94,9 95,6 96,6 it 102,4 112,5 115,4 116,5 123,3 129 131,8 131,5 132 pt 71,7 83,6 96,2 111,4 126,2 129 136 128,8 129,9 -2 0 2 4 6 8 2008 2009 2010 2011 2012 2013 2014 2015 f1 vix rumiana górska dynamic econometric models 18 (2018) 99–114 108 countries with the highest government debt to gdp ratio after 2012 year are portugal and italy. countries with the lowest ratio are czech republic and poland. in summary, based on the factor loadings and additional analysis, we can interpret two common factors as the global risk factor, and the eurozone insolvency risk factor. these findings are in line with results presented by ang and longstaff (2013). they used the cds for us treasuries, us states, and major eurozone countries. they found that the systemic risk component is related to global financial factors such as the vix index. in the pricing model, they proposed the use of two types of credit events, systemic and sovereignspecific. presented in this paper results show that the source of volatility for cds spreads shifted from the global risk factor in 2008 to the eurozone insolvency risk factor in 2011. the final step of presented research is a historical decomposition of cds spreads, using the factorization framework. it allows to investigate which factor and to what extent affects the cds spread of the given country. consequently, it enables identification of investors’ perception of the risk connected with investment in sovereign debt securities of this country. historical decomposition is obtained using the equation (6), for k=2 common factors. standardized cds spread of each country is presented as a sum of three components, which indicate influence of two common factors and one specific to each country factor. figure 3. decomposition of cds spread of great britain -2 -1 0 1 2 3 4 5 2008 2009 2010 2011 2012 2013 2014 2015 f1 f2 specific sum gb decomposition of sovereign cds spread using the concept of factorization dynamic econometric models 18 (2018) 99–114 109 figures 3 to 6 show decomposition of cds spreads for the first group of counties, correlated to the first common factor – great britain, czech republic, poland and germany. figure 4. decomposition of cds spread of czech republic figure 5. decomposition of cds spread of poland -2 -1 0 1 2 3 4 5 2008 2009 2010 2011 2012 2013 2014 2015 f1 f2 specific sum cz -2 -1 0 1 2 3 4 5 2008 2009 2010 2011 2012 2013 2014 2015 f1 f2 specific sum pl rumiana górska dynamic econometric models 18 (2018) 99–114 110 figure 6. decomposition of cds spread of germany figure 7. decomposition of cds spread of portugal historical decomposition of cds spreads of these countries revealed that cds spread of these countries evolve mainly due to the influence of the first factor. the influence of the second factors is lower. but at the period of -2 -1 0 1 2 3 4 5 6 2008 2009 2010 2011 2012 2013 2014 2015 f1 f2 specific sum de -2 -1 0 1 2 3 4 2008 2009 2010 2011 2012 2013 2014 2015 f1 f2 specific sum pt decomposition of sovereign cds spread using the concept of factorization dynamic econometric models 18 (2018) 99–114 111 insolvency crisis (2011 year) poland and czech republic are more influenced by the second factor than great briatan and germany. figures 7, 8 and 9 present the decomposition of the cds spread for the second group of countries, the most affected by the second common factor – portugal, italy and spain. figure 8. decomposition of cds spread of italy figure 9. decomposition of cds spread of spain -2 -1 0 1 2 3 4 2008 2009 2010 2011 2012 2013 2014 2015 f1 f2 specific sum it -2 -1 0 1 2 3 4 2008 2009 2010 2011 2012 2013 2014 2015 f1 f2 specific sum es rumiana górska dynamic econometric models 18 (2018) 99–114 112 the cds spreads of these countries are influenced mainly by the second factor. and last, france and hungary – countries that are under the influence of both common factors. figures 10 and 11 show the decomposition of cds spread of these countries. figure 10. decomposition of cds spread of france figure 11. decomposition of cds spread of hungary -2 -1 0 1 2 3 4 2008 2009 2010 2011 2012 2013 2014 2015 f1 f2 specific sum fr -2 -1 0 1 2 3 4 2008 2009 2010 2011 2012 2013 2014 2015 f1 f2 specific sum hu decomposition of sovereign cds spread using the concept of factorization dynamic econometric models 18 (2018) 99–114 113 historical decomposition revealed the evidence that the sovereign cds spreads are under combined influence of two common factors and one specific factor – time-varying and different for each country. summarising, first factor (f1) may be interpreted as a global risk factor, appearing during the crisis in 2008. it affected most of the countries, but in the larger extent influenced cds spreads of great britain, czech republic, poland and germany. the second factor (f2) can be interpreted as an insolvency risk factor, it appeared in 2011–2012 years and influenced mainly the countries, which had debt problems – portugal, spain, italy and to a lesser extent france and hungary. comparative analysis shows that cds spread of poland’s government debt reacts similarly to those of czech republic, great britain, and germany. this leads to the conclusion that poland treasury securities are considered by investors in a similar way as tree mentioned-above countries. conclusions the presented analysis confirms that sovereign cds spread is a measure of risk and it depends both on global risk factors (global financial market conditions, investor expectations etc.) as well as macroeconomic fundamentals (eg. level of indebtedness of the country). the factor analysis shows that two common factors explained about 92% of the variation of the cds spreads of the surveyed countries. the results of decomposition of cds spreads confirm that great britain, poland, czech republic, and germany are countries most influenced by the first common factor, interpreted as a global risk factor, appearing during the crisis in 2008. portugal, italy, and spain are countries under the impact of the second common factor interpreted as an insolvency risk factor appearing in 2011–2012 years. france and hungary are the countries influenced by both common factors. references ang, a., longstaff, f. a. (2013), systemic sovereign credit risk: lessons from the u.s. and europe, journal of monetary economics, 60(5), 493–510, doi: http://dx.doi.org/10.1016/j.jmoneco.2013.04.009. aizenman, j., hutchison, m., jinjarak, y. (2013), what is the risk of european sovereign debt defaults? fiscal space, cds spreads and market pricing of risk, journal of international money and finance, 34, 37–59. arce, o., mayordomo, s., peña, j. (2013), credit-risk valuation in the sovereign cds and bonds markets: evidence from the euro area crisis, journal of international money and finance, 35, 124–145. badaoui, s., cathcart, l., el-jahel, l. (2013), do sovereign credit default swaps represent a clean measure of sovereign default risk? a factor model approach, journal of banking and finance, 37(7), 2392–2407. rumiana górska dynamic econometric models 18 (2018) 99–114 114 bieńkowski, w., gawrońska-nowak, b., grabowski, w. (2011) podatność polskich rynków finansowych na niestabilności wewnętrzne i zewnętrzne, materiały i studia nbp. zeszyt nr 258, nbp. (the vulnerability of polish financial markets to internal and external instabilities, materials and studies of the national bank of poland. no. 258, nbp.) büchel, k. (2013), do words matter? the impact of communication on the piigs' cds and bond yield spreads during europe's sovereign debt crisis, european journal of political economy, 32, 412–431, doi: http://dx.doi.org/10.1016/j.ejpoleco.2013.08.004. coronado, m., corzo, t., lazcano, l. (2012), a case for europe: the relationship between sovereign cds and stock indexes, frontiers in finance and economics, 9(2), 32–63. coudert, v., gex, m. (2013), the interactions between the credit default swap and the bond markets in financial turmoil, review of international economics, 21(3), 492–505. fabozzi, f. j., giacometti, r., tsuchida, n. (2016), factor decomposition of the eurozone sovereign cds spreads, journal of international money and finance, elsevier, 65(c), 1–23, doi: http://dx.doi.org/10.1016/j.jimonfin.2016.03.003. ho, t. s. y. (1990), strategic fixed income investment, dow jones-irwing homewood, illinois kalbaska, a., gątkowski, m. (2012), eurozone sovereign contagion: evidence from the cds market (2005–2010), journal of economic behavior and organization, 83(3), 657–673. kliber, a. (2011), sovereign cds instruments in central europe – linkages and interdependence, dynamic econometric models, 11, 111–128. kliber, a. (2016), impact of the ban on uncovered scds trade on the interdependencies between the cds market and other sectors of financial markets. the case of safe and developed versus risky and developing european markets, comparative economic research, 19(1), 77–99, doi: http://dx.doi.org/10.1515/cer-2016-0005. komarek, l. komarkova, z., lesanovska, j., (2013) analysis of sovereign risk market indicators: the case of the czech republic, czech journal of economics and finance, 63(1), 5–24. varga, l. (2008), the information content of hungarian sovereign cds spreads, mnb occasional papers, 78, 9–24. dekompozycja spreadów kontraktów cds na obligacje skarbowe przy zastosowaniu koncepcji faktoryzacji z a r y s t r e ś c i. cds (credit default swap) na obligacje skarbowe to instrument pochodny, który stanowi ubezpieczenie spłaty długu rządowego i może być uważany za rynkowy wskaźnik ryzyka niewypłacalności danego kraju. celem badania jest identyfikacja czynników wpływających na spready cds wybranych krajów europejskich w okresie od 2008 do 2016 r. na podstawie przeprowadzonej analizy czynnikowej stwierdzono, że istnieją dwa wspólne czynniki, które wyjaśniają około 92% zmienności spreadów cds. następnie, dekompozycja spreadów cds pokazuje wpływ tych czynników ryzyka na spready cds badanych krajów. s ł o w a k l u c z o w e: cds spread; analiza czynnikowa; ryzyko globalne; ryzyko niewypłacalności. microsoft word dem_2018_67to79.docx © 2018 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2018.004 vol. 18 (2018) 67−79 submitted november 23, 2018 issn (online) 2450-7067 accepted december 19, 2018 issn (print) 1234-3862 joanna olbryś * the non-trading problem in assessing commonality in liquidity on emerging stock markets** a b s t r a c t. the purpose of this study is to explore commonality in liquidity on seven small emerging cee stock markets in the czech republic, hungary, slovakia, slovenia, lithuania, estonia, and latvia, in the context of serious problems with stock illiquidity. the number of companies that reveal a substantial non-trading problem is large. a modified version of the amihud measure is utilized as daily liquidity proxy for stocks. the ols-hac method and the garch-type models are employed to infer the patterns of commonality in liquidity. no reason has been found to support intra-market commonality in liquidity on each investigated stock exchange. k e y w o r d s: cee; commonality in liquidity; garch; hac; non-trading problem. j e l classification: c32; c58; g15; o57. introduction investors prefer assets that are liquid, therefore stock market liquidity is of important concern to many investors. commonality in liquidity means that financial asset liquidity changes over time, and that these time variations are ruled by a significant common component in the liquidity across assets or market liquidity. the existence of commonality suggests the assumption that there exists at least one common factor that simultaneously influences liquidity of all stocks in a market. * correspondence to: joanna olbryś, bialystok university of technology, faculty of computer science, 45a wiejska street, 15-351 białystok, poland, e-mail: j.olbrys@pb.edu.pl. ** this work was financed by the grant from the national science centre, poland, no. 2016/21/b/hs4/02004. joanna olbryś dynamic econometric models 18 (2018) 67–79 68 chordia et al. (2000) have been the first that analysed commonality in liquidity on a stock exchange. the authors explored the u.s. market and the literature concerning commonality in liquidity for the u.s. stock market has emerged in the last years, e.g (kamara et al., 2008; korajczyk and sadka, 2008; kang and zhang, 2013). commonality in liquidity has been investigated for other individual equity markets in the world. among others, brockman and chung (2006) explored the stock exchange of hong kong, fabre and frino (2004) studied the australian stock exchange, kempf and mayston (2008) analysed the frankfurt stock exchange, pukthuanthong-le and visaltanachoti (2009) assessed the stock exchange in thailand, foran et al. (2015) examined the london stock exchange, narayan et al. (2015) tested the chinese stock exchanges (in shanghai and shenzhen), miralles marcelo et al. (2015) studied the euronext lisbon stock exchange, sensoy (2016) investigated the turkish stock market, syamala et al. (2017) analysed the indian stock market, and olbryś (2018) explored the polish stock exchange in warsaw. in general, the empirical results on different stock markets in the world are not homogeneous. a number of studies that concern commonality in liquidity for a group of international financial markets is rather limited. for example, brockman et al. (2009) applied methodology of chordia et al. (2000) to examine commonality in liquidity on 47 stock exchanges. they documented the pervasive role of commonality in liquidity within individual exchanges and they found evidence of a distinct, global component in bid/ask spreads and depths across exchanges. karolyi et al. (2012) explored cross-country commonality in liquidity using daily data for individual stocks from 40 developed and emerging countries. wang (2013) investigated the impact of a set of common factors on liquidity variations in 12 asian equity markets. bai and qin (2015) analysed commonality in liquidity on 18 emerging markets. the authors pointed out that liquidity co-movements across emerging markets have a strong geographic component. among others, bekaert et al. (2007) stress that liquidity is more critical for emerging than developed markets. moreover, it is a well–known fact that a non–trading effect induces potentially serious biases in various statistical measures of asset returns (e.g. nowak and olbryś, 2016). we refer to non– trading as a lack of transactions over a particular period when a stock exchange is open for trading. the lack of transactions means that the volume (in items) is equal to zero. empirical analyses of a number of zeros in daily volume for companies listed on seven small cee stock exchanges in the czech republic, hungary, slovakia, slovenia, lithuania, estonia, and latvia reveal that the non-trading problem is crucial on these markets. therefore, in this paper we analyse intra-market liquidity co-movements in the context of the non-trading problem in assessing commonality in liquidity… dynamic econometric models 18 (2018) 67–79 69 problems with asset illiquidity. the research hypothesis that there is no commonality in liquidity on each investigated cee stock market, taken separately, is tested. a modified version of the amihud (2002) measure is used as daily liquidity proxy, in the period from january 2, 2012 to december 30, 2016. we utilize the research design of chordia et al. (2000), but we employ the olshac method (newey and west, 1987) and the garch-type models (if necessary) to infer the patterns of commonality in liquidity. in the market models of liquidity, positive and statistically significant slope coefficients are especially desired, because they indicate intra-market co-movements in liquidity in the same direction, and therefore confirm commonality in liquidity. however, the regressions provide no evidence of commonality in liquidity on the cee stock markets because positive and statistically significant coefficients are scarce. therefore, no reason has been found to support commonality in liquidity on each investigated stock exchange. the empirical findings are homogeneous for all investigated markets. to the best of the author’s knowledge, the results concerning commonality in liquidity on the cee stock exchanges in the context of non-trading problems are novel and have not been presented in the literature thus far. the remainder of the study is organized as follows. section 1 specifies the methodological background concerning the measurement of commonality in liquidity. section 2 describes the data and discusses serious problems in illiquidity on the cee emerging stock markets. section 3 presents some empirical results of testing for commonality in liquidity on the investigated markets. conclusion covers the main findings. table 1. nomenclature pse prague stock exchange (the czech republic) bse budapest stock exchange (hungary) bsse bratislava stock exchange (slovakia) ljse ljubljana stock exchange (slovenia) nasdaq vilnius stock exchange in vilnius (lithuania) nasdaq tallinn stock exchange in tallinn (estonia) nasdaq riga stock exchange in riga (latvia) 1. assessing commonality in liquidity the literature contains a number of methods for assessing commonality in liquidity. to identify common determinants of liquidity, the classical market model of liquidity, introduced by chordia et al. (2000) has been most frequently used. in this research, the modified version of classical market model of liquidity, including the dimson (1979) correction for daily data is applied: 𝐷𝐿#,% = 𝛼# + 𝛽#,*+𝐷𝐿,,%*+ + 𝛽#,-𝐷𝐿,,% + 𝛽#,.+𝐷𝐿,,%.+ + 𝜀#,%, (1) joanna olbryś dynamic econometric models 18 (2018) 67–79 70 where 𝐷𝐿#,% for stock i is the change in liquidity variable l from trading day t–1 to t, i.e. 𝐷𝐿% = 01*0123 0123 . according to the dimson procedure, the 𝐷𝐿,,%*+, 𝐷𝐿,,%, and 𝐷𝐿,,%.+ are the lagged, concurrent, and leading changes is a cross-sectional average of the liquidity variable l, respectively. the dimson correction enables us to accommodate the problem of nonsynchronous trading effects (campbell et al., 1997). in computing the ‘market’ liquidity proxy 𝐿,, stock i is excluded, so the explanatory variables in model (1) are slightly different for each stock’s time series regression. chordia et al. (2000) stress that changes are examined rather than levels because the interest is fundamentally in discovering whether liquidity moves. based on model (1), positive and statistically significant slope coefficients are especially desired, as they indicate intra-market co-movements in liquidity and therefore confirm commonality in liquidity. in other words, they inform about liquidity co-movements in the same direction. in this study, a modified version of the amihud (2002) measure, 𝑀𝐴𝑚𝑖ℎ#,9 given by eq. (2), is utilized as liquidity/illiquidity proxy: 𝑀𝐴𝑚𝑖ℎ#,9 = : 𝑙𝑜𝑔>1 + @ab,c@ db,c e ,𝑤ℎ𝑒𝑛 𝑉#,9 ≠ 0 0,𝑤ℎ𝑒𝑛 𝑉#,9 = 0 , (2) where: 𝑟#,9 is the simple rate of return of stock i on day d, 𝑉#,9 is the trading volume of stock i on day d. we follow karolyi et al. (2012), but our method is slightly different, because they use return and volume in local currency, and finally multiply the result by negative one to obtain a variable that is increasing alongside with liquidity of individual stock. moreover, the value of daily amihud measure (2) is defined to be equal to zero when the total volume of daily trading, in the denominator, is equal to zero. the second condition in definition (2) is consistent with analogous conditions for other daily liquidity/illiquidity proxies (olbrys and mursztyn, 2018). furthermore, to avoid numerical problems, the daily values of the estimator (2) are rescaled by multiplying by 102. in the literature, the amihud measure is usually calculated for stock i for each month, e.g. (goyenko et al., 2009; olbryś, 2014; foran et al., 2015; fong et al., 2017; będowska-sójka, 2018). in this paper, we estimate daily time series of the amihud proxy (2). for each stock, the model (1) is initially estimated by using the ols method and the robust hac estimates (newey and west, 1997). however, the newey and west method may not fully correct for the influence problems the non-trading problem in assessing commonality in liquidity… dynamic econometric models 18 (2018) 67–79 71 introduced by the arch effect. for this reason, the estimation of the model (1) as a garch-type model is appropriate. to test for the arch effect, the test of engle (1982) with the lagrange multiplier (lm) statistic is employed. in this research, the garch(p, q) model is utilized. according to the literature, the lower order garch(p, q), p, q = 1, 2, models are used in most applications (tsay, 2010). the garch(p, q) models are compared and selected by the akaike (aic) and schwarz (sc) information criteria. the garch(p, q) model is given by eq. (3): 𝐷𝐿#,% = 𝛼# + 𝛽#,*+𝐷𝐿,,%*+ + 𝛽#,-𝐷𝐿,,% + 𝛽#,.+𝐷𝐿,,%.+ + 𝜀#,%, 𝜀#,% = 𝑧#,%ohi,t, 𝑧#,%~𝑁(0,1), hi,t = 𝑎#,+ ∑ 𝑎#,v𝜀#,%*v wx vy+ + ∑ 𝑏#,[hi,t-l, \ [y+ (3) where: 𝑎#,> 0,𝑎#,v ≥ 0,𝑘 = 1,…,𝑞,𝑞 > 0,𝑏#,[ ≥ 0,𝑙 =,…,𝑝,𝑝 ≥ 0, 𝜀#,% is the innovation in a linear regression with 𝑉(𝜀) = 𝜎w, ℎ#,% is the variance function, and remaining notation like in eq. (1). the parameters of garch(p, q) models are almost invariably estimated via maximum likelihood (ml) or quasi-maximum likelihood (qml) (bollerslev and wooldridge, 1992) methods, which bring up the subject of a suitable choice for the conditional distribution of innovation. our primary interest is the estimating the conditional mean equation, but hamilton (2008) stresses that having a correct description of the conditional variance can still be quite important. by incorporating the observed features of the heteroskedasticity into the estimation of the conditional mean, substantially more efficient estimates of the conditional mean can be obtained. 2. data description and serious problems with asset illiquidity in this research, daily data for seven small emerging stock markets from the central and eastern european countries, namely the czech republic, hungary, slovakia, slovenia, lithuania, estonia, and latvia, are used. data is coming from bloomberg under the license agreement between bloomberg and bialystok university of technology (the grant no. 2016/21/b/hs4/ 02004). the data set contains the opening, high, low, and closing prices, as well as volume for a security over each trading day, in the period from january 2, 2012 to december 30, 2016. specifically, the database holds 1252 (for the pse), 1240 (for the bse), 1244 (for the bsse), 1245 (for the ljse), 1245 (for the nasdaq vilnius), 1251 (for the nasdaq tallinn), and 1242 (for the nasdaq riga) trading days, respectively. the warsaw stock exchange (wse) is not included in the research because it is large compared to the other joanna olbryś dynamic econometric models 18 (2018) 67–79 72 cee stock exchanges. for comparison, at the end of 2016 the total number of listed stocks was equal to: 881 (wse), 23 (pse), 41 (bse), 71 (bsse), 37 (ljse), 34 (nasdaq vilnius), 17 (nasdaq tallinn), and 32 (nasdaq riga). it is commonly known fact that a large number of the cee stock markets listed companies reveal a substantial non-trading problem. therefore, to mitigate this problem, we excluded the stocks that exhibited extraordinarily many non-traded days during the whole sample period. specifically, because the analysed cee stock markets were extremely illiquid, the basic condition concerning the maximum number of non-traded days for these markets was equal to 373, which constituted about 30% of all trading days. table 2. serious problems with stock illiquidity on the cee stock markets – the reduction of the number of companies (january 2012 – december 2016) stock exchange index market cap. eur billion, dec 2016 the initial number of companies the number of companies after the reduction due to a stock illiquidity (< 10% zeros in daily volume) the number of companies after the reduction due to a stock illiquidity (≈ 30% zeros in daily volume) pse px 22.19 23 10 10 bse bux 21.27 41 13 18 bsse sax 5.28 71 0 3 ljse sbi-top 5.00 37 6 10 nasdaq vilnius omxv 3.50 34 7 15 nasdaq tallinn omxt 2.29 17 8 12 nasdaq riga omxr 0.80 32 3 7 note: the fourth column presents the number of all companies listed on each stock exchange. the fifth column reports the number of companies that exhibited less than 125 zeros in daily volume, which constituted about 10% of all trading days. the last column contains final number of companies (less than 373 zeros in daily volume, which constituted about 30% of all trading days during the whole sample period). table 2 reports brief information about the reduction of the number of companies on the investigated markets that was caused by serious non-trading problems. the cee stock exchanges are presented in order of decreasing value of the market index capitalization (in eur billion) at the end of 2016. for comparison, the wig20 index market capitalization at the end of december 2016 was equal to 130.99 (eur billion). the wig20 index consists of 20 major and most liquid companies in the warsaw stock exchange main list. therefore, as mentioned above, the wse is not included in the study because it is not a small stock market. the non-trading problem in assessing commonality in liquidity… dynamic econometric models 18 (2018) 67–79 73 as one can observe in table 2, the number of zeros in daily volume was tremendously high for the bsse-traded companies. the total number of stocks on the bsse was equal to 71, while only 3 out of them met basic condition. finally, 10 (prague), 18 (budapest), 3 (bratislava), 10 (ljubljana), 15 (vilnius), 12 (tallinn), and 7 (riga) companies were contained in the data set. table 3 presents information about all companies in an alphabetical order according to the company’s full name. table 3. companies contained in the database (january 2012 – december 2016) stock exchange companies 1 prague stock exchange (the czech republic) cetv, cez, foreg, komb, pegas, rbag, tabak, telec, unipe, vig 2 budapest stock ex-change (hungary) any, appeninn, pannonia, elmu, emasz, estmedia, fhb, gspark, mtelekom, mol, nutex, opimus, otp, pannergy, plotinus, raba, richt, zwack 3 bratislava stock ex-change (slovakia) sln1, tmr, vub 4 ljubljana stock exchange (slovenia) cicg, grvg, iekg, krkg, lkpg, melr, petg, posr, tlsg, zvtg 5 nasdaq vilnius (lithuania) avg1l, apg1l, grg1l, knf1l, lnr1l, lna1l, lgd1l, ptr1l, pzv1l, rsu1l, sab1l, teo1l, vlp1l, vbl1l, zmp1l 6 nasdaq tallinn (esto-nia) arc1t, mrk1t, tveat, blt1t, eeg1t, hae1t, ncn1t, oeg1t, prf1t, sfg1t, tal1t, tkm1t 7 nasdaq riga (latvia) grd1r, bal1r, gze1r, lsc1r, olf1r, saf1r, vss1r note: the companies are presented in an alphabetical order according to the company’s full name. 3. some empirical results of commonality in liquidity on the cee stock markets in the first step, we detected with the adf-gls test (elliott et al., 1996) or adf test (dickey and fuller, 1981) whether the analysed daily time series are stationary. using daily data, we utilized a maximum lag equal to five and then removed lags until the last one was statistically significant (adkins, 2014). we proved that the unit-root hypothesis can be rejected at the 5% significance level for all time series used in the study. in order to reduce the effects of possibly spurious outliers, we ‘winsorized’ the data by using the 1st and 99th percentiles for each time series, e.g. (korajczyk and sadka, 2008; kamara et al., 2008). in the next step, we employed the ols method with the hac covariance matrix estimator to estimate the parameters of the model (1). in total, 75 models for seven stock markets were estimated, comprising 10 models for the pse, 18 models for the bse, 3 models for the bsse, 10 models for the ljse, 15 joanna olbryś dynamic econometric models 18 (2018) 67–79 74 for the nasdaq vilnius, 12 for the nasdaq tallinn, and 7 for the nasdaq riga. table 4a. testing for market-wide commonality in liquidity on four cee stock markets (the pse, bse, bsse, and ljse) in the whole sample period from january 2, 2012 to december 30, 2016 pse (10 models) bse (18 models) bsse (3 models) ljse (10 models) o ls -h ac 9 m od el s g ar ch 1 m od el o ls h ac 14 m od el s g ar ch 4 m od el s o ls h ac 3 m od el s g ar ch 0 m od el s o ls h ac 10 m od el s g ar ch 0 m od el s intercept 𝛼# significant all all all all all – all – mean 2.424 2.333 3.843 5.310 median 2.129 1.604 1.287 5.059 concurrent 𝛽#, mean 0.043 –0.002 0.0002 0.023 median 0.011 –0.002 0.0002 0.028 positive significant 2/9 0/1 0/14 0/4 0/3 – 0/10 – positive insignificant 3/9 0/1 5/14 0/4 3/3 – 7/10 – negative significant 2/9 0/1 3/14 1/4 0/3 – 1/10 – negative insignificant 2/9 1/1 6/14 3/4 0/3 – 2/10 – lag 𝛽#,*+ mean –0.007 –0.002 0.0001 –0.005 median –0.020 –0.002 0 0.001 positive significant 0/9 0/1 1/14 0/4 0/3 – 0/10 – positive insignificant 4/9 0/1 1/14 3/4 1/3 – 5/10 – negative significant 2/9 0/1 6/14 0/4 1/3 – 2/10 – negative insignificant 3/9 1/1 6/14 1/4 1/3 – 3/10 – lead 𝛽#,.+ mean –0.019 0.002 –0.0004 –0.001 median –0.032 –0.001 –0.0003 –0.016 positive significant 0/9 1/1 1/14 0/4 0/3 – 0/10 – positive insignificant 2/9 0/1 6/14 1/4 1/3 – 4/10 – negative significant 2/9 0/1 4/14 1/4 1/3 – 4/10 – negative insignificant 5/9 0/1 3/14 2/4 1/3 – 2/10 – note: notation like in table 1. for each stock, daily proportional changes in individual stock liquidity variables were regressed in time-series on the changes of an equally weighted cross-sectional average of the liquidity variable for all stocks in the sample, excluding the dependent variable stock. empirical results revealed that the ols-hac method turned out to be appropriate for the estimation of model (1) in the case of all companies from the bsse, ljse, and nasdaq riga exchanges. in other cases, the garch(p, q), p, q = 1, 2, models (3) were the non-trading problem in assessing commonality in liquidity… dynamic econometric models 18 (2018) 67–79 75 estimated. the number of lags p, q was selected on the basis of the aic and sc information criteria. the summarized cross-sectional estimation results of the models (1) and (3) are presented in tables 4a–4b. these tables report basic statistics and the proportion of positive significant, positive insignificant, negative significant, and negative insignificant coefficients (at 10% significance level), for each stock market, separately. the empirical findings are worth comments. the regressions provide weak evidence of commonality in liquidity on the investigated markets because positive and statistically significant coefficients are scarce or not present at all. for example, positive and statistically significant concurrent coefficients constitute 2/10 (pse), 0/18 (bse), 0/3 (bsse), 0/10 (ljse), 0/15 (nasdaq vilnius), 0/12 (nasdaq tallinn), and 0/7 (nasdaq riga), respectively. the evidence concerning lag and lead coefficients is very similar. moreover, the proportions of negative and statistically significant concurrent, lag, and lead coefficients are even greater for all markets, which informs about liquidity co-movements in the opposite direction. conclusion the main goal of this paper was to explore the existence of commonality in liquidity patterns on seven small cee emerging stock markets in the czech republic, hungary, slovakia, slovenia, lithuania, estonia, and latvia, in the context of serious problems with stock liquidity. due to the non-trading effect, the data was filtered out, and the reduction of the number of companies on the investigated markets was crucial. the reduction was tremendously high for the bsse-traded companies. the total number of stocks on the bsse was equal to 71, while only 3 out of them were contained in the database. the most liquid bsse-company (vub) exhibited 295 non-traded days during the whole sample period, which constituted about 24% of all trading days. the sample covered a period from january 2, 2012 to december 30, 2016. a modified version of the amihud (2002) measure was used as daily liquidity proxy for stocks. we followed the research design of chordia et al. (2000), but we employed the ols method with the hac covariance matrix estimation and the garch-type models to infer the patterns of intra-market commonality in liquidity on the cee stock markets. according to the literature, positive and statistically significant slope coefficients in the estimated models are especially desired, as they indicate intra-market co-movements in liquidity in the same direction, and therefore confirm commonality in liquidity. joanna olbryś dynamic econometric models 18 (2018) 67–79 76 table 4b. testing for market-wide commonality in liquidity on three cee stock markets (the nasdaq-omx group) in the whole sample period from january 2, 2012 to december 30, 2016 nasdaq vilnius (15 models) nasdaq tallinn (12 models) nasdaq riga (7 models) o ls -h ac 14 m od el s g ar ch 1 m od el o ls h ac 11 m od el s g ar ch 1 m od el o ls h ac 7 m od el s g ar ch 0 m od el s intercept 𝛼# significant all all all all all – mean 1.334 2.078 2.332 median 0.996 2.001 2.029 concurrent 𝛽#, mean 0.003 0.009 0.009 median 0.003 0.006 0.001 positive significant 0/14 0/1 0/11 0/1 0/7 – positive insignificant 9/14 1/1 6/11 1/1 4/7 – negative significant 1/14 0/1 2/11 0/1 1/7 – negative insignificant 4/14 0/1 3/11 0/1 2/7 – lag 𝛽#,*+ mean –0.007 –0.012 –0.006 median –0.009 –0.011 –0.007 positive significant 0/14 0/1 0/11 0/1 0/7 – positive insignificant 3/14 0/1 2/11 0/1 3/7 – negative significant 5/14 0/1 5/11 0/1 2/7 – negative insignificant 6/14 1/1 4/11 1/1 2/7 – lead 𝛽#,.+ mean 0.000 –0.012 –0.010 median 0.004 –0.009 –0.015 positive significant 0/14 0/1 0/11 0/1 0/7 – positive insignificant 9/14 0/1 1/11 0/1 2/7 – negative significant 2/14 0/1 3/11 0/1 1/7 – negative insignificant 3/14 1/1 7/11 1/1 4/7 – note: notation like in table 1. in general, our estimation results do not support the existence of commonality in liquidity effects on all investigated markets, because positive and statistically significant coefficients are scarce or not present at all. the results are consistent with the literature regarding other emerging small markets in the world. a possible direction for further investigation could be to study marketwide co-movements in liquidity on the cee stock exchanges by utilizing different daily liquidity proxy. moreover, another possible direction could be to identify components of liquidity on the cee stock markets by using methods based on principal component analysis. the non-trading problem in assessing commonality in liquidity… dynamic econometric models 18 (2018) 67–79 77 references adkins, l.c. (2014), using gretl for principles of econometrics. 4th ed.version 1.041. amihud, y. (2002), illiquidity and stock returns: cross-section and time-series effects, journal of financial markets, 5(1), 31–56, doi: http://dx.doi.org/10.1016/s1386-4181(01)00024-6. bai, m., qin, y. (2015), commonality in liquidity in emerging markets: another supply-side explanation, international review of economics & finance, 39, 90–106, doi: http://dx.doi.org/10.1016/j.iref.2015.06.005. bekaert, g., harvey, c.r., lundblad, c. (2007), liquidity and expected returns: lessons from emerging markets, review of financial studies, 20(6), 1783–1831, doi: http://dx.doi.org/10.1093/rfs/hhm030. będowska-sójka, b. (2018), the coherence of liquidity measures. the evidence from the emerging market, finance research letters, 27, 118–123, doi: https://doi.org/10.1016/j.frl.2018.02.014. bollerslev, t., wooldridge, j.m. (1992), quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances, econometric reviews, 11, 143–179, doi: http://dx.doi.org/10.1080/07474939208800229. brockman, p., chung, d.y. (2006), index inclusion and commonality in liquidity: evidence from the stock exchange of hong kong, international review of financial analysis, 15(4–5), 291–305, doi: http://dx.doi.org/10.1016/j.irfa.2005.09.003. brockman, p., chung, d.y., perignon, c. (2009), commonality in liquidity: a global perspective, journal of financial and quantitative analysis, 44(4), 851–882, doi: http://dx.doi.org/10.1017/s0022109009990123. campbell, j.y., lo, a.w., mackinlay, a.c. (1997), the econometrics of financial markets, princeton university press, new jersey. chordia, t., roll, r., subrahmanyam, a. (2000), commonality in liquidity, journal of financial economics, 56, 3–28, doi: http://dx.doi.org/10.1016/s0304-405x(99)00057-4. dickey, d.a., fuller, w.a. (1981), likelihood ratio statistics for autoregressive time series with a unit root, econometrica, 49(4), 1057–1072, doi: http://dx.doi.org/10.2307/1912517. dimson, e. (1979), risk measurement when shares are subject to infrequent trading, journal of financial economics, 7, 197–226, doi: http://dx.doi.org/10.1016/0304-405x(79)90013-8. elliott, g., rothenberg, t.j., stock, j.h. (1996), efficient tests for an autoregressive unit root, econometrica, 64(4), 813–836, doi: http://dx.doi.org/10.2307/2171846. engle, r.f. (1982), autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflations, econometrica, 50, 987–1007, doi: http://dx.doi.org/10.2307/1912773. fabre, j., frino, a. (2004), commonality in liquidity: evidence from the australian stock exchange, accounting and finance, 44, 357–368, doi: http://dx.doi.org/10.1111/j.1467-629x.2004.00117.x. fong, k.y.l., holden, c.w., trzcinka, c. (2017), what are the best liquidity proxies for global research?, review of finance, 21, 1355–1401, doi: http://dx.doi.org/10.1093/rof/rfx003. foran, j., hutchinson, m.c., o’sullivan, n. (2015), liquidity commonality and pricing in uk equities, research in international business and finance, 34, 281–293, doi: http://dx.doi.org/10.1016/j.ribaf.2015.02.006. joanna olbryś dynamic econometric models 18 (2018) 67–79 78 goyenko, r.y., holden, c.w., trzcinka, c.a. (2009), do liquidity measures measure liquidity?, journal of financial economics, 92, 153–181, doi: http://dx.doi.org/10.1016/j.jfineco.2008.06.002. hamilton, j.d. (2008), macroeconomics and arch. working paper 14151 nber working paper series, cambridge. kamara, a., lou, x., sadka, r. (2008), the divergence of liquidity commonality in the crosssection of stocks, journal of financial economics, 89(3), 444–466, doi: http://dx.doi.org/10.1016/j.jfineco.2007.10.004. kang, w., zhang, h. (2013), limit order book and commonality in liquidity, financial review, 48(1), 97–122, doi: http://dx.doi.org/10.1111/j.1540-6288.2012.00348.x. karolyi, g.a., lee, k.-h., van dijk, m.a. (2012), understanding commonality in liquidity around the world, journal of financial economics, 105(1), 82–112, doi: http://dx.doi.org/10.1016/j.jfineco.2011.12.008. kempf, a., mayston, d. (2008), liquidity commonality beyond best prices, journal of financial research, 31(1), 25–40, doi: http://dx.doi.org/10.1111/j.1475-6803.2008.00230.x. korajczyk, r., sadka, r. (2008), pricing the commonality across alternative measures of liquidity, journal of financial economics, 87(1), 45–72, doi: http://dx.doi.org/10.1016/j.jfineco.2006.12.003. miralles marcelo, j.l., miralles quirós, m., oliveira, c. (2015), systematic liquidity: commonality and intertemporal variation in the portuguese stock market, cuadernos de gestion, 15(2), 39–64. doi: http://dx.doi.org/10.5295/cdg.140472mm. narayan, p.k., zhang, z., zheng, x. (2015), some hypotheses on commonality in liquidity: new evidence from the chinese stock market, emerging markets finance & trade, 51, 915–944, doi: http://dx.doi.org/10.1080/1540496x.2015.1061799. newey, w.k., west, k.d. (1987), a simple, positive semi-define, heteroskesticity and autocorrelation consistent covariance matrix, econometrica, 55(3), 703–708, doi: http://dx.doi.org/10.2307/1913610. nowak, s., olbryś, j. (2016), direct evidence of non-trading on the warsaw stock exchange, research papers of wroclaw university of economics, 428, 184–194. olbryś, j. (2014), is illiquidity risk priced? the case of the polish medium-size emerging stock market, bank i kredyt, 45(6), 513–536. olbryś, j. (2018), intra-market commonality in liquidity. new evidence from the polish emerging stock market, ssem euroconference 2018: emerging market economies, june 7–8, 2018, lodz, poland. olbryś, j., mursztyn, m. (2018), on some characteristics of liquidity proxy time series. evidence from the polish stock market, in: tsounis n., vlachvei a. (eds) advances in time series data methods in applied economic research. springer proceedings in business and economics, springer, cham, 177–189, doi: http://dx.doi.org/10.1007/978-3-030-02194-8_13. pukthuanthong-le, k., visaltanachoti, n. (2009), commonality in liquidity: evidence from the stock exchange of thailand, pacific-basin finance journal, 17(1), 80–99, doi: http://dx.doi.org/10.1016/j.pacfin.2007.12.004. sensoy, a. (2016), commonality in liquidity: effects of monetary policy and macroeconomic announcements., finance research letters, 16, 125–131, doi: http://dx.doi.org/10.1016/j.frl.2015.10.021. syamala, s.r., wadhwa, k., goyal, a. (2017), determinants of commonality in liquidity: evidence from an order-driven emerging market, north american journal of economics and finance, 42, 38–52, doi: http://dx.doi.org/10.1016/j.najef.2017.07.003. tsay, r.s. (2010), analysis of financial time series, john wiley, new york. the non-trading problem in assessing commonality in liquidity… dynamic econometric models 18 (2018) 67–79 79 wang, j. (2013), liquidity commonality among asian equity markets, pacific-basin finance journal, 21(1), 1209–1231, doi: http://dx.doi.org/10.1016/j.pacfin.2012.06.003. wpływ problem braku transakcji na badanie wspólności w płynności na małych rynkach giełdowych z a r y s t r e ś c i. artykuł przedstawia badanie wspólności w płynności na siedmiu małych giełdach europy środkowo-wschodniej w kontekście problemu braku transakcji, czyli bardzo dużej liczby dni z zerowym wolumenem. przeprowadzono konieczną redukcję niepłynnych spółek na badanych rynkach, co spowodowało znaczny spadek liczby firm uczestniczących w badaniu. jako estymator dziennej płynności zastosowano zmodyfikowaną miarę amihuda. w celu identyfikacji wzorców w płynności wykorzystano modele ols-hac oraz garch. nie stwierdzono efektu wspólności w płynności na żadnym z badanych rynków giełdowych. s ł o w a k l u c z o w e: europa środkowo-wschodnia; wspólność w płynności; garch; hac; problem braku transakcji. microsoft word 00_tresc.docx © 2013 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2013.004 vol. 13 (2013) 69−85 submitted october 10, 2013 issn accepted december 30, 2013 1234-3862 milda maria burzała* determination of the time of contagion in capital markets based on the switching model a b s t r a c t. this article attempts to compare conclusions made about market contagion based on the periods indicated by using the markov-switching model and based on a range for unconditional correlations as well as on arbitrary arrangements. dcc-model was used to control for correlation change over time. determination of extremely high correlations by using a range for unconditional correlations and the ms(3) switching model yields similar results regarding conclusions about the occurrence of the process of contagion in a market. conclusions about contagion are, however, made at a higher significance level in the case of the switching model. k e y w o r d s: switching model, dcc-garch model, contagion. j e l classification: g01, g15, c24. introduction current economical and financial crises in general have international – character. propagation mechanisms across countries and markets are called the transmissions for fundamental linkages. in literature contagion term is applied only to the financial markets, however it should not be identified only with the financial linkages – it can also concern the markets which are not significantly financially connected. many authors claim that increase in financial integration intensifies contagion effects. on the subject of interde * correspondence to: milda burzala, department of econometrics, faculty of informatics and electronic economy, poznan university of economics, towarowa 53, 61-896 poznań, poland, e-mail: m.burzala@ue.poznan.pl. milda maria burzała dynamic econometric models 13 (2013) 69–85 70 pendence between markets, contagion effects and transmission channels treat, among others, the works of: eichengreen et al. (1995, 1996), goldstein (1998), masson (1998), kaminsky and reinhart (2000, 2002), forbes and rigobon (2002), pericoli and sbracia (2003), pesaran and pick (2004), dungey et al. (2005). the most restrictive definition provided by the world bank assumes that market contagion occurs when the correlation between markets in a time of crisis is significantly higher than during the period of tranquillity1. it is possible to control for correlation change over time by using, for example, a dynamic conditional correlation model. researchers often also adopt an additional definition of contagion that would “suit the purpose of a given research method”. if volatility models are used, then contagion is identified with the spread of uncertainty across financial markets. the assessment of the significance of a contagion process requires dividing a sample into observations from the time of tranquillity and from the time of crisis in financial markets. a period of tranquillity is a benchmark period for determining connections between markets, which is a point of reference for changes observed during a crisis. the transition from a tranquillity period to the period of crisis is usually established based on events which may change the behaviour of certain indicators. the results of research studies depend on the division which has been made and the time of crisis often covers both high and low correlations between researched markets. establishing a potential time of market contagion by using the markovswitching model makes it possible to make an assumption about the differences in a stochastic process that determines correlations in particular regimes. the main hypothesis refers to the possibility of using the onedimensional markov switching model to determine the time of contagion in capital markets. results were compared with conclusions made about market contagion based on a range for unconditional correlations as well as on arbitrary arrangements. the consequences of adopting particular divisions are, in fact, important information for researchers. research results presented in this paper concern the assessment of the significance of contagion in certain capital markets in the years 2007–2009. selected stock exchange indices represent the situation in securities markets2. in empirical studies that are described later, the concept of market 1 contagion of financial crises, world bank, http://www.worldbank.org/economicpolicy/managing%20volatility/contagion/definitions.htm (14 may 2012). this definition is cited based on forbes and rigobon’s paper (2002). 2 capital market crisis is identified with sharp decline in stock prices, maintaining for an extended period of time. role of stock market indexes is broadly described by jajuga (2006). determination of the time of contagion in capital markets... dynamic econometric models 13 (2013) 69–85 71 contagion means a contagion spreading from an index representing the u.s. market to an index representing the studied market3. section 1 presents the dcc-garch model and section 2 describes the markov-switching model which has been used in the research. section 3 contains information on the tested stock exchange indices as well as the criteria for an arbitrary division of the set of observations into those relating to the time of crisis and those relating to the period of tranquillity in securities markets. the obtained research results are presented in section 4. 1. a dynamic conditional correlation model let us assume that an n-dimensional vector of rates of return ts (t = 1,…,t) can be decomposed into the following form: ,ttt εμs += (1) ,2/1 tt ξhε = (2) where tμ is the vector of conditional expected values of vector st based on model var(p). in empirical research it is usually assumed that p =14. the student’s t-distribution was used because of an increased kurtosis for process tξ . the dynamic conditional correlation (dcc) model can be formulated as (engle, 2002): ,tttt drdh = (3) 11, ,( ,..., ),t t nn tdiag h h=d (4) 3 causality tests are sometimes used for establishing the direction of contagion (cheung, ng, 1996; coporale, pittis and spagnolo, 2002). their usefulness, however, is limited. this is because these tests are based on granger’s concept related to analysing correlations between studied processes and the consequences of events. it is often a researcher who decides whether to test a particular causal relationship exists based on his or her knowledge and experience (osińska, 2006; fiszeder, 2009). 4 a vector autoregression model also controls for the mutual interdependence between markets through connections between the delayed values of endogenous variables. empirical studies described in literature have found that linear relationship between stock returns are low significant. some researchers suggest that it is better to resign from expected value model than include incorrectly specified model, especially in the case of total model for expected values and variances (doman, doman, 2009). milda maria burzała dynamic econometric models 13 (2013) 69–85 72 2 2 , , , , , 1 1 ( ( 0) ) , 1,... , q p i t i ij i t j ij i t j i t j ij i t j j j h c i h i n α ε γ ε ε β− − − − = = = + + < + = ∑ ∑ (5) 1/ 2 1/ 2( ( )) ( ( )) ,t t tdiag diag − −= tr q q q (6) ' 1 1 1 1 1 , k l k l t k l t k t k t k l t l k l k l α β α β− − − = = = = ⎛ ⎞ = − − + +⎜ ⎟ ⎝ ⎠ ∑ ∑ ∑ ∑q q ξ ξ q (7) matrix rt is a positively defined symmetric matrix with ones along the main diagonal; vector 1t t t −=ξ d ε in this case denotes the vector of standardised residuals from model var(1). matrix dt was estimated based on the onedimensional gjr-garch(1,1) model (glosen, jagannathan, runkle, 1993). in equation (5) )(⋅i is an indicator function and it assumes the value of 1 for 0, <− jtiε and the value of 0 for 0, ≥− jtiε ( ni ,...,1= ). positive values of parameter ijγ which are significantly different from zero prove that the leverage effect occurs5. covariance stationarity and thus a finite variance in equation (5) is ensured by satisfying these conditions: 1 1 0, , 0 and ( / 2) 1. q p i ij ij ij ij ij j j c α β α γ β = = > ≥ + + <∑ ∑ (8) in equation (6) tq denotes a square matrix of unconditional covariances of the vector tξ variables. in addition, it is required that , 0,k lα β ≥ 1 1 1 k l k l k l α β = = + <∑ ∑ . the model’s parameters are estimated in two stages. the logarithmic likelihood function is the sum of likelihood functions for a volatility model and likelihood functions for the parameters of dynamic correlations (engle, 2002). 5 the leverage effect results from an asymmetric response of rates of return to positive and negative information reaching the financial market. it is a consequence of a negative correlation between securities prices and the volatility of rates of return. the higher the value of parametr ijγ , the stronger the leverage effect (the additional impact of negative information). determination of the time of contagion in capital markets... dynamic econometric models 13 (2013) 69–85 73 2. the markov-switching model in markov-switching models it is assumed that a switch between the behaviours of rates of return in regimes (periods), and thus the process of contagion in a market, depend on certain hidden factors which are not directly observable. one can only observe the external symptoms of regime change by observing, for example, the mutual correlations between rates of return. theoretically, for a time series of dynamic conditional correlations tρ , a one-dimensional switching model can be used, in which switches occur as a result of changes in the expected value μ , variance σ2 or the expected value and variance of the studied correlations (hamilton, 1989; davidson, 2013). if a switching model is only constructed for the purpose of classifying the already obtained theoretical values of correlations, it can be assumed that, in each regime, values are generated by independent processes with a different constant expected value and constant variance: ,2,1,0),(,0(~)( 2 ==+== iirnir titittit σεεμρ (9) where )( irt =μ , )( 2 irt =σ denote the expected value and variance of conditional correlations, respectively, in the i-regime. such an approach allows one to use a one-dimensional model, in which it is assumed that three regimes will be analysed, i.e. tr = i (i = 0, 1, 2), which are related to an extremely low, average and extremely high correlation between rates of return. the proposed sequential procedure entails estimating the model of dynamic conditional correlations and the switching model separately, which makes it possible to avoid many problems related to estimating multidimensional models6. the series of random variables rt in the subsequent moments in time t (t=1,..., t) has the markov property, i.e. its value at the time moment t+1, i.e. rt+1, depends only on the regime at the t moment, rather than on all the preceding regimes, which is formally formulated as: 6 in practice, the use of multidimensional switching models is associated with many problems because of the number of estimated parameters which grows exponentially, as in the multidimensional vechgarch model (billio, lo duca, pelizzon, 2005). if one assumes that only two states (of high and low volatility) can occur in each of two studied markets, than one already allows for the occurrence of four regimes, and the matrix of conditional probabilities has dimensions [4 x 4]. the final number of parameters depends on the assumptions regarding the differences between processes determining the behaviour of rates of return in particular regimes. milda maria burzała dynamic econometric models 13 (2013) 69–85 74 1 1 1( , ,...) ( ) , , 0,1, 2. t t t t t ijp r j r i r k p r j r i p i j + − += = = = = = = = (10) probabilities pij denote the probability of transition of the dependence between rates of return from regime i to regime j. if at the t – 1 moment the process was under the rt-1 = i regime, then the conditional density function of the explained variable tρ can be represented as 1( , )t t tf s r i i −= , where 1−ti denotes the history of the process until the t – 1 moment. any suppositions on the i regime may be made by means of a conditional probability: , )(),( )(),( )( 2 0 11 11 ∑ = −− −− =⋅= =⋅= == j ttttt ttttt tt ijrpijrsf iirpiirsf iirp (11) where ∑ = −−− === 2 0 111 ).()( j ttijtt ijrppiirp (12) the model’s parameters are estimated by using the maximum likelihood method (davidson, 2013)7. 3. the statistical material and an arbitrary division of the sample in the empirical research, daily continuously compounded rates of return on six indices representing the situation on stock exchanges during the period from august 17, 2005 to july 31, 2009 were used (1022 observations for each stock exchange): )).ln()(ln(100 1,, −−⋅= titiit pps two indices from strong eu economies – dax and cac, representing the situation on the german and french stock exchanges, as well as two indices from weaker economies from the “old” european union – the spanish ibex and the greek atex, and two indices from the countries of central and eastern europe – the hungarian bux and the polish wig20 were selected for the purpose of the analysis (source: the stooq database). the dow jones 7 the relevant likelihood function is presented as part of the description of the tsm (time series modelling) program. it is not easy to estimate the model’s parameters. numerical problems result from the occurrence of local extrema of the logarithmic likelihood function. this is why normally two, up to three, regimes under which a process may be are distinguished. determination of the time of contagion in capital markets... dynamic econometric models 13 (2013) 69–85 75 industrial average (djia) index represented the situation on the u.s. stock exchanges. gaps in the data were filled by using the linear interpolation method. due to the different quotation times, data were smoothed by using a two-period moving average (dungey et al., 2007). figure 1. the dow jones industrial average index during the period from august 17, 2005 to july 31, 2009 (lower figure – daily rates of returns from djia index) the behaviour of the dow jones industrial index during the studied period is shown in figure 1. an arbitrary division of the observation set into two subsets related to the period of crisis (high volatility of rates of return) and the period of tranquillity (low volatility) in securities markets should be made in such a way that the time of tranquillity immediately precedes the time of crisis. this is because it constitutes a benchmark for comparisons. quotations preceding, for example, the collapse of lehman brothers, hardly represented a time of tranquillity as the dow jones industrial average had already been declining for some time and the rate of return on the index had been characterised by increased volatility. therefore, it was decided that the information about the difficulties related to evaluating assets which was announced by the french bnp paribas would be used when dividing the set of observations. on 9 august 2007 this bank suspended payments from three funds investing in the market of bonds secured by subprime mortgages. the period of crisis was extended beyond the time when securities were trading at the lowest level because of the increased volatility of rates of return which persisted until the end of july 2009. there were 511 observations for the time of crisis determined in this way. in order to ensure comparability of results, 511 former observations were analysed for events contributing to 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000 20 05 -0 817 20 05 -1 117 20 06 -0 217 20 06 -0 517 20 06 -0 817 20 06 -1 117 20 07 -0 217 20 07 -0 517 20 07 -0 817 20 07 -1 117 20 08 -0 217 20 08 -0 517 20 08 -0 817 20 08 -1 117 20 09 -0 217 20 09 -0 517 09-08-2007 bnp paribas cannot determine security values march-2009 min on the stock 31-07-2009 end of the sample 15-09-2008 bankruptcy of the bank lehman brothers 17-08-2005 the beginning of the sample -6 -4 -2 0 2 4 6 05 06 07 08 09 milda maria burzała dynamic econometric models 13 (2013) 69–85 76 financial turmoil. it was a time of rising asset prices, with minor adjustments, which is why it has been assumed to be a period of tranquillity on the stock exchange. 4. research results the estimation of a model of dynamic conditional correlations should be preceded by a test justifying their use. the results of such tests depend on the adopted specification of volatility models. therefore, two tests were used in the research, i.e. the tse test (2000) as well as engle and sheppard test (2001) in two versions with delays p=5 and p=10. for all indices at least one test indicated the reasonableness of constructing a dynamic conditional correlation model8. a significant increase of correlations in the time of crisis confirms the occurrence of the process of market contagion. forbes and rigobon (2002) propose using fisher’s transformation of correlation coefficients while testing the significance of change in correlation between rates of return. after fisher’s transformation, the sample correlation coefficient can be treated as the realization of a random variable with a normal distribution with the expected value of ˆ1 1 ( ) ln ˆ2 1 e ρ ρ ρ + = − , where ρ̂ is the estimated correlation coefficient. the variance of this variable is 1 ( ) 3 var t ρ = − . the null hypothesis 0 : k sh ρ ρ≤ is tested against an alternative hypothesis, i.e. 1 : k sh ρ ρ> (index s means the tranquility period, k the time of potential market contagion). the empirical statistic in the test for two expected values is as follows: ˆ ˆ1 1 0, 5ln 0, 5 ln ˆ ˆ1 1 . 1 1 3 3 k s k s k s fr t t ρ ρ ρ ρ ⎛ ⎞ ⎛ ⎞+ + −⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠= + − − (13) the fr-statistic have a normal distribution n(0,1) and even if the sample is small it allows one to use the critical values of a standard normal distribution. 8 calculations were carried out by using the oxmetrics 6.10 program. determination of the time of contagion in capital markets... dynamic econometric models 13 (2013) 69–85 77 the parameters of the gjr-garch(1,1) model are presented in the upper part of table 1. let us remember that the models were estimated on the basis of residuals from model var(1). therefore, a slightly different model for the djia index was connected with each index. in all the volatility models for the djia index, the alfa parameter which describes the impact of positive residual impulses was insignificant. in the models for the european indices, the alfa parameter was only significant for the greek (ath), the hungarian (bux) and the polish (wig20) indices. the parameters that were significant were beta and gamma which describe the impact of past variance as well as the leverage effect (an additional impact of negative information reaching the market). the model’s assumptions require that the alfa parameter be significantly greater than zero. thus, in order to standardise the residuals from model var(1), it was finally the garch(1,1) models that were used9: ,,...1, 1 1 , 2 ,, nihch q j p j jtiiijjtiijitij =++= ∑ ∑ = = −− βεα (14) where 1 1 1 q p ij ij j j α β = = + <∑ ∑ , .0,0,0 ≥≥> ijijic βα this time the obtained parameter estimates meet the models’ assumptions; the arch effect is significant in all of the models (the lower part of table 1). this means that the impact of negative information on many markets was considerably stronger than the impact of positive information. such markets could be identified by estimating the gjr-garch model at the first stage of the research. estimates of parameter β exceed the value of 0.8, which confirms the volatility clustering phenomenon. both in the garch(1,1) model and in the conditional correlation model dcc-garch(1,1) the requirement of covariance stationarity is satisfied. also the conditions for variance (nonnegative, significant model parameters) are met. the sum of parameters ( )α β+ in the garch model is close to one, which means the occurrence of persistence (long-term dependencies) and suggest test of integrated model (igarch, figarch) in further studies. the highest unconditional correla 9 the same volatility model was adopted for two time series due to program limitations. in the research garch (1,2), garch (2,1) and garch (2,2) models were also tested – only in garch (1,1) model all parameters were significant and was the lowest information criterion value (aic). conditional correlations from two models (dcc-gjr-garch (1,1) and dcc-garch) were slightly different. milda maria burzała dynamic econometric models 13 (2013) 69–85 78 tion with the dji index was recorded for the cac and the dax indices and the lowest for the bux and the wig20 indices. table 1. the dcc-garch model’s parameters gjr–garch(1,1) index for index i for index djia const(i) alfa(i) beta(i) gamma(i) const alfa beta gamma ath 0.017*** 0.069** 0.842*** 0.146*** 0.006*** –0.003 0.894*** 0.211*** cac 0.010*** 0.022 0.900*** 0.123*** 0.006*** 0.002 0.891*** 0.201*** dax 0.013*** 0.032 0.868*** 0.162*** 0.006*** –0.005 0.892*** 0.214*** ibex 0.015** 0.043 0.853*** 0.164*** 0.007*** 0.006 0.880*** 0.212*** bux 0.037** 0.113*** 0.798*** 0.135*** 0.006*** 0.005 0.881*** 0.209*** wig20 0.027** 0.059** 0.889*** 0.064** 0.006*** –0.003 0.889*** 0.221*** garch(1,1) index for index i for index djia const(i) alfa(i) beta(i) const alfa beta ath 0.016** 0.165*** 0.823*** 0.005** 0.111*** 0.882*** cac 0.009** 0.107*** 0.883*** 0.005** 0.116*** 0.877*** dax 0.011** 0.129*** 0.861*** 0.005** 0.112*** 0.881*** ibex 0.015** 0.159*** 0.824*** 0.006** 0.118*** 0.875*** bux 0.034** 0.172*** 0.806*** 0.004** 0.116*** 0.880*** wig20 0.026* 0.098*** 0.883*** 0.005** 0.114*** 0.879*** dcc–garch(1,1) index alfa beta df unconditional correlations log–likelihood ath 0.022*** 0.958*** 16.960*** 0.384*** –2060.58 cac 0.016* 0.962*** 12.636*** 0.641*** –1820.90 dax 0.027** 0.951*** 10.960*** 0.649*** –1801.13 ibex 0.016** 0.971*** 11.360*** 0.590*** –1845.05 bux 0.075 0.829*** 16.020*** 0.319*** –2292.27 wig20 0.018** 0.967*** 16.971*** 0.378*** –2346.02 note: a parameter’s significance for α = 0.01 is marked with three asterisks, for α = 0.05 with two asterisks and for α = 0.1 with one asterisk. the parameters of the ms(3) switching model are provided in table 2. regime 2 is related to an extremely high correlation (the shaded area in figure 2). as for the two indices – cac and dax, regime 2 covered both significantly high and significantly low correlation between markets. this probably resulted from a very high volatility of extreme correlations. a correct classification was obtained by simplifying the process to a model in which only the expected value would change. for the remaining four indices, the variance of extremely high and extremely low correlations was significantly higher than the variance of condidetermination of the time of contagion in capital markets... dynamic econometric models 13 (2013) 69–85 79 tional correlations in the time of tranquillity, and the model made it possible to make a correct classification10. table 2. the estimates of the ms(3) switching model’s parameters switches occur as a result of changes → the expected value and variance the expected value regime ath ibex bux wig20 cac dax number of observations 0 283 423 178 312 223 222 1 379 387 405 336 614 461 2 360 212 439 374 185 339 expected value 0 0.264 0.511 0.103 0.255 0.567 0,520 1 0.372 0.598 0.271 0.372 0.638 0,632 2 0.482 0.672 0.440 0.468 0.691 0,710 fr-statistic 1,820** 1.449* 2.811*** 1.545* 1.132 1.989** variance 0 0.060 0.031 0.085 0.047 x x 1 0.027 0.017 0.052 0.029 x x 2 0.049 0.029 0.078 0.038 x x h-statistic 601.6*** 589.2*** 854.1*** 596.2*** 492.4*** 650.4*** d-statistic 0 15.89 *** 12.90*** 16.69*** 13.77*** 10.83*** 11.61*** 1 24.53*** 23.10*** 29.23*** 23.78*** 19.97*** 22.72*** probability of transition p_{0/0} 0.977 0.983 0.880 0.987 0.978 0,961 p_{0/1} 0.023 0.017 0.120 0.013 0.000 0,039 p_{0/2} 0.000 0.000 0.000 0.000 0.022 0,000 p_{1/0} 0.017 0.018 0.052 0.012 0.008 0,018 p_{1/1} 0.962 0.970 0.880 0.964 0.984 0,965 p_{1/2} 0.021 0.012 0.068 0.024 0.008 0,017 p_{2/0} 0.000 0.000 0.061 0.019 0.000 0,000 p_{2/1} 0.022 0.021 0.000 0.000 0.046 0,021 p_{2/2} 0.978 0.979 0.939 0.981 0.954 0,979 log-likelihood 1708,9 2257.9 1063.5 1824.5 2307.3 1797.5 aic –3,325 –4.399 –2.062 –3.551 –4.499 –3.502 note: the fr-statistic refers to the difference in correlation between regimes 2 and 1; the d-statistic refers to the difference distributions between regime 2 and regime 0 or 1. the jarque-berra test rejects at conventional significance level the normality of correlation in three regimes (not reported). it is the reason of the use of nonparametric variance analysis (kruskal and wallis-test) to evaluate the quality of classification (division of the sample into observations from the time of tranquillity and from the time of potential market contagion). in the first research stage, h-statistic indicates the diversification of distribution at least in two regimes. in the second stage, d-statistics indicates the 10 high variance in 0 regime (extremely low correlations) indicates, that differentiation of the sign of returns on two markets causes the increase of uncertainty among investors. milda maria burzała dynamic econometric models 13 (2013) 69–85 80 diversification of distribution in all regimes for all studied indices. relevant statistics are provided in table 2. results confirms the legitimacy of the use of one-dimensional switching model. simple model can give satisfactory results. corr_ath_djia corr_ibex_djia corr_bux_djia corr_wig20_djia corr_cac_djia corr_dax_djia figure 2. potential periods of market contagion as determined based on conditional correlations from the dcc-garch(1.1) model note: the shaded area: the time of extremely high correlations (regime 2); ── the average value of conditional correlations in the time of crisis (august 9, 2007 to july 31, 2009); the upper limit of the range: unconditional correlation + 2 error. determination of the time of contagion in capital markets... dynamic econometric models 13 (2013) 69–85 81 the occurrence of extreme correlations under regime 2 only means the time of potential market contagion. only the rejection of the null hypothesis in the test for two expected values means that the process of market contagion has occurred. the expected value of correlation in regime 2 is significantly higher than the expected value of correlation in regime 1 (the time of tranquillity in the market) in all securities markets except for the french market (cac). the significance level that allows one to reject the null hypothesis is, however, varied, which is highlighted in the table. the frstatistics assumes the highest value for the hungarian market. it can be assumed that the value of statistics reflects the effects of contagion. the higher the value of statistics, the more severe the effects of market contagion. the probability of remaining under each of the regimes, which is provided in table 2, is high, which means that the highlighted regimes are persistent. the analysis of charts in figure 2 allows one to compare the frequency of an index being under regime 2 and the time of remaining there. the additional, horizontal and dashed line makes it possible to relate indicated periods to the time of potential market contagion as identified based on a range for unconditional correlations. in this case, those correlations which fall outside the upper and the lower limits determined by a double estimation error are assumed to be extremely high and extremely low correlations, respectively. the time of tranquillity in a market is represented by correlations from a range determined in this way. as for arbitrary arrangements, it should be remembered that the sample was only divided into two subsets. total duration times of the potential market contagion period are provided in table 3. the longest time is for an arbitrary division and the shortest for the range for unconditional correlations. the switching model indicated that the period of potential contagion in the hungarian market was the longest. table 3. the number of observations during the potential period of market contagion metod ath ibex bux wig20 cac dax arbitrary arrangements 511 511 511 511 511 511 switching model – regime 2 360 212 439 374 185 339 range for unconditional correlations 161 89 243 101 57 81 a comparison of the results of tests of the significance of occurrence of contagion spreading from the u.s. market to a given market is presented in table 4. milda maria burzała dynamic econometric models 13 (2013) 69–85 82 table 4. a comparison of the results of testing the significance of contagion in a market method index ath ibex bux wig20 cac dax arbitrary arrangements contagion 0.416 0.594 0.360 0.427 0.650 0.654 tranquility 0.349 0.562 0.267 0.315 0.614 0.613 fr-statistic 1.309* 0.770 1.641* 2.072** 0.966 1.096 switching model contagion 0.482 0.672 0.440 0.468 0.691 0.710 tranquility 0.372 0.598 0.271 0.372 0.638 0.632 fr-statistic 1.820** 1.449* 2.811*** 1.545* 1.132 1.989** range for unconditional correlations contagion 0.535 0.702 0.492 0.574 0.731 0.771 tranquility 0.384 0.584 0.321 0.405 0.639 0.653 fr-statistic 2.019** 1.788** 2.647*** 1.572* 1.237 2.030** note: “contagion” means the time of potential market contagion. for five indices the values of the fr-statistics recorded in the case of the switching model are lower than the corresponding statistics in the analysis of unconditional correlations, which has an effect on conclusions about contagion. the occurrence of the contagion process is registered more often (for lower significance levels) in the analysis of a range for unconditional correlation. the opposite is true only for the bux index, which probably results from small differences between the duration times of market contagion that are determined by using different methods. the lowest values of the frstatistics are usually recorded for an arbitrary division. detailed results and significance levels are provided in table 4. conclusions it is relatively difficult to date a crisis in financial markets. during periods determined based on events that change the behaviour of rates of return, both high and low correlation between markets can be observed. this paper proposes indicating the periods of potential market contagion on the basis of a one-dimensional switching model. tests made confirm the legitimacy of the use of simple switching model to determine potential market contagion periods. further studies should be conducted – it is important to compare obtained results with the results from multidimensional model, where switches are determined based on the changes of expected value, variance, and covariance. in the paper such comparisons were not made because of the lack of appropriate software. occurrence of persistence suggest, that further studies should also include inference based on the integrated model. results confirm the conclusions made by the author on the subject of contagion on the basis of logit model for panel data (burzała, 2012). significant contagion effects were observed on german market, less significant – determination of the time of contagion in capital markets... dynamic econometric models 13 (2013) 69–85 83 on polish, and the lack of significant contagion effects were observed on french market. determination of extremely high correlations by using a range for unconditional correlations and the ms(3) switching model yields similar results regarding conclusions about the occurrence of the process of contagion in a market. conclusions about contagion are, however, made at a higher significance level in the case of the switching model. it is worth emphasising that it is necessary that the appropriate tests be conducted which would confirm the significance of the increase of correlation between markets. also, the time of potential market contagion determined on the basis of a regime with an extremely high correlation (the switching model) is longer. in further studies more attention should be paid to the issue of determining the direction of contagion as well as extremely low correlations which may be a harbinger of a crisis. references aczel, a. d. (2000), statystyka w zarządzaniu (statistic in business), pwn, warszawa. billio, m., lo duca, m., pelizzon, l. (2005), contagion detection with switching regime models: a short and long run analysis, working paper 05.01, university ca’foscari of venice, italy, doi: http://dx.doi.org/10.2139%2fssrn.676956. burzała, m. m. (2012), efekty zarażania giełd europejskich w czasie kryzysu finansowego 2008-2009 model logitowy dla danych panelowych (contagion effects on european stock exchanges during 2008–2009 financial crisis – logit model for panel data), in: appenzeller d. (ed.), matematyka i informatyka na usługach ekonomii: metody, analizy, prognozy (mathematics and computer science at the service of economics: methods, analysis, forecasts), 31–43, wydawnictwo naukowe uep, poznań. cieciura, m., zacharski, j. (2007), metody probabilistyczne w ujęciu praktycznym (probabilistic methods: a practical approach), vizja press&it, warszawa. davidson, j. (2013), time series modelling, version 4.38, university of exeter, http://www.timeseriesmodelling.com/tsmod4doc.pdf, (13.06.2013), doi: http://dx.doi.org/10.2307%2f2231972. doman, m., doman, r. (2009), modelowanie zmienności i ryzyka. metody ekonometrii finansowej (volatility and risk modeling. methods of financial econometrics), oficyna, kraków. dungey, m., fry, r. a., gonzalez-hermosillo, b., martin, v. l. (2005), empirical modeling of contagion: a review of methodologies, quantitative finance, 5(1), 9–24, doi: http://dx.doi.org/10.1080%2f14697680500142045. dungey, m., fry, r., a., gonzález-hermosillo, b., martin, v., l. (2007), contagion in global equity markets in 1998: the effects of the russian and ltcm crises, north american journal of economics and finance, 18(2), 155–174, doi: http://dx.doi.org/10.1016%2fj.najef.2007.05.003. eichengreen, b., rose, a., wyplosz, c., dumas, b., weber, a. (1995), exchange market mayhem: the antecendents and aftermath of speculative attacks, economic policy, 21, 249–312, doi: http://dx.doi.org/10.2307%2f1344591. milda maria burzała dynamic econometric models 13 (2013) 69–85 84 eichengreen, b., rose, a., wyplosz, c. (1996), contagious currency crises: first tests, the scandinavian journal of economics, 98(4), 463–484, doi: http://dx.doi.org/10.2307%2f3440879. engle, r. f. (2002), dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models, journal of business and economic statistics, 20(3), 339–350, doi: http://dx.doi.org/10.1198%2f073500102288618487. fiszeder, p. (2009), modele klasy garch w empirycznych badaniach finansowych (the garch-class models in empirical financial research), wydawnictwo naukowe umk, toruń. forbes, k., rigobon, r. (2002), no contagion, only interdependence: measuring stock market comovements, the journal of finance, 57(5), 2223–2261. doi: http://dx.doi.org/10.1111%2f0022-1082.00494. glosten, l., jagannathan, r., runkle, d. (1993), on the relation between the expected value and the volatility of the nominal excess return on stocks, journal of finance, 48, 1179–1801, doi: http://dx.doi.org/10.2307%2f2329067. goldstein, m. (1998), the asian financial crisis: causes, cures and systemic implications, institute for international economics, peterson institute. hamilton, j. d. (1989), a new approach to the economic analysis of nonstationary time series and the business cycle, econometrica, 57(2), 357–384, doi: http://dx.doi.org/10.2307%2f1912559. jajuga, k. (2006), rynek wtórny papierów wartościowych (secondary market securities), fundacja edukacji rynku kapitałowego, warszawa. kaminsky, g. l., reinhart, c. m. (2000), on crises, contagion and confusion, journal of international economics, 51(1), 145–168, doi: http://dx.doi.org/10.1016%2fs0022-1996%2899%2900040-9. kaminsky, g. l., reinhart, c. m. (2002), the center and the periphery: tales of financial turmoil, mimeo, george washington university. masson, p. (1998), contagion: monsoonal effects, spillovers and jumps between multiple equilibria, imf working paper wp/98/142, doi: http://dx.doi.org/10.1017%2fcbo9780511559587.017. osińska, m. (2006), ekonometria finansowa (financial econometrics), pwe, warszawa. pericoli, m., sbracia, m. (2003), a primer on financial contagion, journal of economic surveys, 17(4), 571–608, doi: http://dx.doi.org/10.1111%2f1467-6419.00205. pesaran, m. h., pick, a. (2004), econometric issues in the analysis of contagion, university of cambridge, working paper in economics 0402, cambridge. world bank, http://www.worldbank.org/economicpolicy/managing%20volatility/contagion/ definitions.htm (14.05.2012). determination of the time of contagion in capital markets... dynamic econometric models 13 (2013) 69–85 85 wyznaczanie czasu zarażania rynków kapitałowych na podstawie modelu przełącznikowego z a r y s t r e ś c i. w artykule podjęto próbę porównania wnioskowania o zarażaniu rynków na podstawie okresów wskazanych przez model przełącznikowy markowa z wnioskowaniem opartym na przedziale dla korelacji bezwarunkowych i ustaleniach arbitralnych. w celu kontrolowania zmieniających się w czasie korelacji wykorzystano model dcc. ustalenie ekstremalnie wysokich korelacji przy wykorzystaniu przedziału dla korelacji bezwarunkowych lub modelu przełącznikowego ms(3) prowadzi do podobnych rezultatów w zakresie wnioskowania o wystąpieniu procesu zarażania rynku. wnioski o zarażaniu są jednak stawiane przy wyższym poziomie istotności w przypadku modelu przełącznikowego. s ł o w a k l u c z o w e: model przełącznikowy, model dcc, zarażanie. microsoft word 00_tresc.docx © 2013 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2013.008 vol. 13 (2013) 145−162 submitted october 17, 2013 issn accepted december 30, 2013 1234-3862 anna czapkiewicz, artur machno* empirical verification of world’s regions profitability in dynamic international investment strategy a b s t r a c t. the main goal of the work is to present the empirical verification of the investment attractiveness in a given world financial region. the attractiveness of a region is represented by the share of assets from this region in the optimal portfolio. the multivariate garch model has been used to describe international dependencies. optimal portfolios based on value at risk and expected shortfall minimization have been compared to the markowitz portfolio. indications, which should be taken into account by investors willing to invest in different world regions, have been presented as the result. k e y w o r d s: optimal portfolio, value at risk, expected shortfall, international dependency. j e l classification: c52, g11, g15, g32. introduction the trading digitization of the last two decades has allowed investors to invest their money into financial markets of a given region of the world with no impediments. the easy access to the knowledge about the economic situation of a given region helps to decide if it is worthy to invest in it. moreover, the capital flow between markets from different parts of the word leads to evaluation of the attractiveness of an investing on a global scale. there are questions to be answered: how to define the attractiveness of a region and * correspondence to: artur machno , agh university of science and technology, faculty of management, 10 gramatyka street, 30-067 kraków, poland, e-mail: artur.machno@gmail.com. anna czapkiewicz, artur machno dynamic econometric models 13 (2013) 145–162 146 how to measure it. the attractiveness of a region could be understood as an amount of invested assets of this region in profitable investments. however, accepting this definition of the attractiveness of a region makes its empirical verification difficult because of numerous possibilities of asset investing in this region. thus, we suggest building a global index characterizing financial conditions of a region and then the attractiveness of this region could be understood as a share of the corresponding global index in the optimal portfolio of all global indices. the optimal portfolio is assumed to be the least risky one in the class of portfolios with the assumed return. the first attempt to build the optimal portfolio was undertaken by markowitz (1959). this method is based on a historical data and it assumes the multivariate normal distribution of variables. the risk of a financial position is described by the variance. despite of the serious simplification, the model is commonly in use. some approaches to a construction of optimal portfolios discussed in the literature are based on other measures of risk. the optimal portfolios based on value-at-risk (var) or expected shortfall (es) values were discussed by rockafellar and uryasev (2000, 2002). the var is the most popular measure of risk and it achieved the high status of being written into industry regulations (see, for instance, jorion (1996)). however, var is unstable and it is difficult to determine even numerically when variables have non-normal distributions. moreover, var fails to be coherent in the sense of artzner et al. (1999). the expected shortfall (es) is another risk measure, with an economic interpretation similar to var, which avoids most of the var’s drawbacks. it seems that the construction of optimal portfolios based on var or es values is more reasonable than markowitz portfolio as it takes into consideration time series distributions which usually differ from normal distributions. that is why in the case of constructing optimal portfolio based on var or es values it is necessary to know the distribution of time series being a multivariate financial data. researchers have proposed numerous models in order to describe a multivariate financial data. the empirical research suggests using a dynamic multidimensional model to describe the relationship among financial time series. the multivariate garch model was proposed by bollerslev (1990), where the conditional correlation was assumed to be constant (ccc model). in the literature there are proposed models where the conditional correlation is dynamic, such as the dcc model of engle (2002) and tse and tsui (2002) where the correlation matrix changes at every point of the time. pelletier (2006) proposed the regime switching dynamic correlation (rsdc) model where the covariance was decomposed into correlations and standard deviations and both the correlations and the standard deviations were dynamic. empirical verification of world’s regions profitability… dynamic econometric models 13 (2013) 145–162 147 capiello et al. (2006) described the model with asymmetric dynamics of dependences among considered financial time series. the multivariate garch model has been used in the construction of optimal portfolios by e.g. billio et al. (2006) and palomba (2008). the main goal of this article is to carry out empirical verification of the attractiveness of regions understood as a share of the corresponding global index in the optimal portfolio of all global indices. the following five regions were taken into consideration: northern america, pacific asia, japan, western and eastern europe. the index characterizing the financial condition of each region has been constructed as the capitalization weighted rate of regional stock market indices. to build the optimal portfolio three approaches have been implemented: the minimization of var, the minimization of es and the standard markowitz procedure, all of them under the fixed expected return assumption. in the empirical study, the short selling has been permitted. the multivariate garch (dcc) model with a vector autoregressive mean has been chosen to model the multidimensional time series. as an additional result of this empirical work the comparison of the properties of the three optimal portfolios has been obtained. according to our knowledge, this is the first analysis of this type. the paper is organized as follows: in section 1, the model is presented (section 1.1), the risk measure theory is briefly depicted (section 1.2) and the portfolio optimization methods used in further analysis are described in section 1.3. the empirical study is presented in section 2, the data set is described in section 2.1, the estimated parameters of the model are shown in section 2.2 and the outcome of the optimal investment strategy is discussed in section 2.3. the obtained results are summarized and interpreted in conclusions. 1. methodology 1.1. multivariate model consider the dimensional stochastic vector process satisfying the following vector autoregressive formula: , (1) where and denote the constant vector and the autoregression matrix, respectively, and denotes the error term. let ω denotes the information set generated by the observed series up to the time 1 . we assume that the process is conditionally heteroscedastic, represented by: anna czapkiewicz, artur machno dynamic econometric models 13 (2013) 145–162 148 / , (2) where is the dynamic covariance matrix at the time , and is a sequence of dimensional i.i.d. random vector, such that and . therefore |ω and |ω . furthermore, let us assume that ~ 0, , where is a continuous density function. there are various parametric formulations to specify the covariance matrix introduced. in this paper, the dynamic conditional correlation (dcc) specification, introduced by engle (2002) and tse, tsui (2002), is considered. hence, the covariance matrix is decomposed as follows: , (3) where is the time varying conditional correlation matrix of the vector and / is the diagonal matrix whose i-th diagonal entry is given by the conditional standard deviation , of , . conditional variances , can be estimated separately and written in the following vector form based on garch 1,1 model: , , , , 1,2, … , . (4) in order to model the joint distribution, the most popular of dcc models has been used, due to engle (2002), where the correlation matrix is presented as follows: / / , (5) where the proxy process is defined by: 1 , , 0, 1, (6) where , and denotes the unconditional matrix of the standardized errors . in this paper, the density function is assumed to be the multidimensional student’s-t distribution. the model parameters are estimated by maximum likelihood method according to the procedure described by ghalanos (2013) and implemented in the r environment. the estimation of the realized correlation is conducted by the recursive procedure. 1.2. risk measures in this paper two measures of risk are discussed: value at risk (var) and expected shortfall (es). we follow the notation presented by föllmer empirical verification of world’s regions profitability… dynamic econometric models 13 (2013) 145–162 149 and schied (2011). for the financial position and 0,1 , value at risk at level λ is defined as: inf |p 0 . (7) value at risk is the smallest amount of capital which, if added to and invested in the risk-free asset, keeps the probability of a negative outcome below some fixed level. generally, value at risk is not a convex risk measure excluding the case when the set of all attainable positions consists of normally distributed financial positions. the absence of the convexity in the case of non-normal distributions is a substantial objection. it appears that, so called expected shortfall (es), is a convex risk measure. this measure has similar interpretation to var. for the financial position and 0,1 , es at level is defined as: : e | . (8) for a portfolio , … , of financial n dimensional vector , representations of var and es at the time are defines as follows: inf |p · 0|ω , (9) and e | , (10) where ω denotes the information set up to the time . since the conditional distribution in the var(1)-dcc-garch(1,1) model described in the section 1.1 is assumed to be t-student, and are estimated using monte carlo method. 1.3. portfolio optimization the portfolio optimization problem is the procedure providing us with the portfolio with the minimal risk in the class of portfolios with a given expected return. in this article, we construct three different optimal portfolios: the portfolio with the minimal var, the portfolio with the minimal es, both under the var(1)-dcc-garch(1,1) model assumption, and classical markowitz portfolio. let the n-dimensional stochastic vector process be represented by the model presented in the section 2.1. the portfolio is obtained as the result of the following optimization problem: ∑ , 1, e ∑ , · , | , (11) anna czapkiewicz, artur machno dynamic econometric models 13 (2013) 145–162 150 ∑ , · , | min!, where , are coordinates of the vector , and , | denotes the conditional distribution of the i-th asset return, for 1, … , . the vector is the n-dimensional time series, which represents the investment with the expected return at the time 1 and the minimal var. analogously, the portfolio is built as the result of the following optimization problem: ∑ , 1, e ∑ , · , | , (12) ∑ , · , | min!, where , are coordinates of the vector . the vector is the n-dimensional time series, which represents the investment at the time with the expected return at the time 1, and the minimal es. the third constructed optimal portfolio – is obtained using the standard markowitz’s mean-variance model: ∑ , 1, e ∑ , · ̂ , | , (13) var ∑ , · ̂ , | min!, where , are coordinates of the vector . the vector is the n-dimensional time series, which represents the investment at the time with the expected return at the time 1, and the minimal variance. it is assumed that the vector | with coordinates ̂ , | is normally distributed with the mean vector and the covariance matrix obtained from the preceding observations. 2. empirical study 2.1. data the investigation covers five global indices constructed using market indices from all over the world. a random sample, diversified enough to capture all specific properties in terms of both geographical and economical dimensions are selected to study. these markets are combined into five regions: (i) north america, (ii) japan (iii) pacific asia, (iv) western europe and (v) eastern europe. the global index of northern american includes the empirical verification of world’s regions profitability… dynamic econometric models 13 (2013) 145–162 151 usa (djia, nasdaq) and canadian (tse300) indices. the global asian index includes indices of india (bse), hong kong (hsi), indonesia (jci), malaysia (klci), south korea (kospi), china (shbs) and singapore (sti). the global index of western europe includes indices of germany (dax), the netherlands (aex), austria (atx), france (cac40), the uk (ftse), switzerland (smi) and spain (ibex).the index of eastern europe includes indices of poland (wig), czech republic (px), hungary (bux) and turkey (xu100). the japanese market is represented by the nikkei index. the data set used in the study has been taken from the stooq database (http://www.stooq.com). all considered indices have been denominated in the us dollar. daily returns come from the period from october 2002 to april 2012. to deal with the missing data in the sample, the linear approximation has been performed. daily returns are computed as the difference between the logarithm of the closing price on the day and the logarithm of the closing price on the day – 1. the global indices’ returns are constructed as the mean of component indices’ returns weighted by the corresponding market capitalization. therefore, the return of a global index at the time is given by: ∑ , , , (14) where , is the return of the j-th component index at the time , and the weight factor , , for 1, … , , is defined by: , , ∑ , , (15) where , is the capitalization of the market corresponding to the j-th index at the time . in this article we have used the annual data of capitalization. therefore, every weight factor is constant in the period of a particular year. table 1 presents summary statistic for the five discussed percent (multiplied by 100) logarithmic returns of the global indices. table 1. descriptive statistics of the data (october 2002 to april 2012) average median std dev. skewness kurtosis p-value of engle test p-value of ljung–box test northern american 0.0629 0.1774 1.8628 –0.5225 7.8354 0.0000 0.0000 western europe 0.0163 0.0463 1.6036 –0.0715 9.9381 0.0000 0.8772 eastern europe 0.0279 0.0776 1.2675 –0.1971 11.2966 0.0000 0.0012 asian 0.0565 0.1082 1.2586 –0.3713 7.4732 0.0000 0.0256 japan 0.0207 0.0842 1.5479 –0.7544 10.0213 0.0000 0.0000 note: presented values are calculated for corresponding global indices defined in (14) anna czapkiewicz, artur machno dynamic econometric models 13 (2013) 145–162 152 averages of returns are close to zero. an asymmetry is suggested by the higher value of the median than the average, for all global indices. this is confirmed by the negative skewness for all series under study. however, this skewness seems to be relatively small so it is not included in the model. the kurtosis is high, taking values from 7.47 to 11.30, which suggests the fattailedness of analyzed time series. in order to examine the properties of the time series, especially, autocorrelation and heteroscedasticity, the ljung-box and engle tests were performed. the test results indicated existence of autocorrelation for all indices, except for western europe, and the garch effect for all considered returns. the presented properties of the 5-dimensional time series justify the use of the multivariate dcc-garch model with the vector autoregressive mean described in section 1.1 by formulas (1)–(6). 2.2. estimation results for model estimating purposes, the time zone difference between studied regions has been taken into account. the non-synchronous trading can cause a bias in the estimation. several transformations of the indices returns were considered to omit this problem. lagging american index return or accelerating the asian seems adequate. however, there is the period during a day, when american and european stock exchanges are working at the same time. therefore, in the analysis we have used only the accelerated asian index return. in the preliminary step, the structure of the multivariate model has been investigated. the five dimensional var(1)-dcc-garch(1,1) model with the student’s-t conditional distribution and dynamic conditional correlations (dcc) has been considered to describe the returns’ process. in order to determine whether the chosen lags in the model are proper, ljung-box and engle tests on the obtained residual were conducted. the results of the tests showed clearly that there is no autocorrelation, nor garch effect in the residual series. table 2. vector autoregressive parameters eastern europe western europe northern america asia japan eastern europe –0.0621** 0.0193 0.3062*** 0.5204*** 0.1173*** western europe –0.0031 –0.2267*** 0.2442*** 0.4951*** 0.0857*** northern america –0.0165 0.0294 –0.1637*** 0.3344*** –0.0304 asia 0.0188 –0.0287 0.1103*** 0.0679** –0.1026*** japan 0.0059 –0.0225 0.1102** 0.2430*** –0.2338*** note: the significance levels are used: *** for 0.001, ** for 0.01 and * for 0.05. empirical verification of world’s regions profitability… dynamic econometric models 13 (2013) 145–162 153 table 2 presents the estimation result of the var(1) part of the considered model. table 2 presents elements of the matrix from equation (1) calculated for the entire sample. estimates presented in table 2 show that the current returns of the index data are strongly affected by lagged returns of themselves. the eastern european region is significantly dependent on the others, apart from the western europe region. on the other hand, the western european region is also dependent on the others and it is not dependent only on the preceding state of markets of eastern europe. northern american markets are affected only by the asian region. pacific asian and japanese regions are dependent on northern america and on each other. it can be noticed that asian and northern american markets affect all the other regions strongly and positively. the japanese market impacts european markets also positively but weaker than american or asian markets. the only negative impact is observed in the case of japan market influencing the asian one. it indicates that a growth on japanese markets results in decreases on asian markets. note that, described relations are irrespective to correlations among the series itself. in particular, both european regions are strongly correlated, however, ones current value does not depend on the preceding value of the other. furthermore, considered regions (all except the pacific asian) tend to correct daily returns, because of significant negative values on the diagonal of the autoregressive matrix. table 3. the dcc model parameters eastern europe western europe northern america asia japan joint 0.0796** 0.0101 0.0122 0.0098 0.0577* 0.0096* 0.0981** 0.0923*** 0.0795** 0.0664** 0.0989*** 0.9853*** 0.8710*** 0.9063*** 0.9103*** 0.9280*** 0.8748*** df 11.786 note: the significance levels are used: *** for 0.001, ** for 0.01 and * for 0.05. table 3 summarize results of the dcc model estimation. the and the parameters are similarly framed in each series, about 90% of the standard deviation value comes from the previous value and about 10% comes from the realization of the previous error term. the conditional correlation matrix is more persistent. figure 1 presents the estimated correlations between considered global indices. there are presented the correlation between eastern europe to other discussed regions (1a), the correlation between western europe and the others (1b), the correlations between northern america and the others (1c), the correlation between asian index and the others (1d) and the correlations of japan (1e). anna czapkiewicz, artur machno dynamic econometric models 13 (2013) 145–162 154 figure 1a. the estimated realized correlations between eastern europe and the others figure 1b. the estimated realized correlations between western europe and the others figure 1 shows that the global index of the eastern europe region is the most correlated with the global index of western europe region, similar to western europe, it is the least correlated with the global index of pacific asian region. on the other hand, the western europe region is the most correlated with northern american, except for the beginning of the financial crisis (october 2007 to july 2008), when it was the most correlated with eastern europe. the northern american region is the most correlated with empirical verification of world’s regions profitability… dynamic econometric models 13 (2013) 145–162 155 western europe markets, partially with the japanese market and it is the least correlated with eastern europe (up to the middle of the year 2005) and the pacific asian region (after the year 2006). the noted region (pacific asian) is the most correlated with japan. the intensity of the relationships between region global indices are time varying. for example, until september 2006, eastern european was correlated with northern america, pacific figure 1c. the estimated realized correlations between northern america and the others figure 1d. the estimated realized correlations between pacific asia and the others anna czapkiewicz, artur machno dynamic econometric models 13 (2013) 145–162 156 figure 1e. the estimated realized correlations between japan and the others asia and japan at similar levels, after that it was correlated more strongly with japan and northern america. the american region appeared to be correlated at the same level with japan and western europe until march 2009, after that it was correlated more strongly to western europe. until the turn of the years 2009/2010 america was the least correlated to eastern europe, from the beginning of 2010 to march 2011 it was the least correlated to pacific asia, after that it was correlated to pacific asia and eastern europe at similar levels. 2.3. properties of optimal portfolios dynamic dependences which derive from the analysis presented in the paragraph 2.2 should be taken into consideration in the process of building optimal portfolios. that is why two dynamic portfolios are constructed: the portfolio of the minimal var and portfolio of the minimal es. to show properties of these procedures, results of the markowitz portfolio optimization is presented. for this purpose, at every time, starting from january 2009, parameters of var(1)-dcc-garch(1,1) model, described in section 2.1 have been estimated using preceding 7 years (1601 observations). the estimation has been conducted using the data of the same length at every time point and reestimated every 50 observations. it provides us with the 27-month series (863 moments) in the ex post analysis. for each , the distribution of the random vector of returns | , with coordinates , | , for 1, … ,5, is approximated using monte carlo method. we simulate 100,000 realizaempirical verification of world’s regions profitability… dynamic econometric models 13 (2013) 145–162 157 tions at each time point. technically, the analysis has been performed as if we have not known the future and been doing it for 27 months. the estimates in tables 2 and 3 have been presented in order to visualize additional properties of the series. these results differ from the estimates obtained during the procedures. in the analysis, we reestimate the model repeatedly, therefore we do not omit parameters which occurred to be insignificant in tables 2 and 3. it would be difficult to control significance of parameters at every step and would not change results of the analysis substantially. figure 2. the portfolio shares obtained by the es (dynamic) and the markowitz (static) models, of: 2a) eastern europe 2b) western europe 2c) northern america index 2d) pacific 2e) japan the results are discussed for 0.1, the similar results are obtained by taking 0.05, which are standard levels for var. in the regulations such as the basel regulations, var levels are 0.01 or even 0.005. the presented procedures have no relation to these regulations. according to these regulations, an investor is obliged to keep the var at the level 0.01 (or 0.005) below some value. it does not mean, that one do not fulfill this condition using presented method. moreover, conducting the same procedure for 0.01 does not assure investor that he fulfills the regulations. for lower levels, such as 0.01, the analysis is not stable. we fixed the expected interest anna czapkiewicz, artur machno dynamic econometric models 13 (2013) 145–162 158 rate on the level 1%, which is much higher than the average of any analyzed variable (see table 1). the realized interest rates of the constructed portfolios are computed and analyzed. the realizations in the 1 time point of portfolios constructed in the time are considered. the compositions of both dynamic portfolios (var and es) are similar, therefore, corresponding portfolios have similar interest rate. thus, we present only returns of the portfolio obtained by minimizing es under the dynamic var(1)-dcc-garch(1,1) model assumption (dynamic portfolio) and portfolio under the markowitz’s mean-variance model (static portfolio) in figure 1. figure 2 presents shares of the static and the dynamic portfolios. firstly, we have noticed the differences in the scale of the portfolio shares. large absolute values of shares in the markowitz portfolios are caused by the height of the assumed expected return. if the expected return had been assumed to be lower, the absolute values of shares in the markowitz portfolio would have decreased, however, it would not have changed the preferences essentially. despite the assumption about height of the expected return, the shares of the es and var optimal portfolios remain relatively small. additionally, we have noticed differences of biases of estimation of the expected interest rate. it turns out that the average of the realized returns of the markowitz portfolios is equal to 0.605%, which is far from the assumed 1%, whereas the return average of the es and the var optimal portfolios are very close to 1% ( 0.98% and 1.14%, respectively). secondly, vivid differences in shares of the static and dynamic portfolios can be observed. comparing the two portfolios we notice that markowitz portfolio gives clear verification of the attractiveness of regions while shares obtained by minimizing the var or the es portfolios under the dynamic model assumption are highly variable and it is hard to notice any tendencies. in the case of markowitz model, it can easily be noticed that the pacific asian region is the most attractive for investors during the period under study. at the beginning of the period the eastern europe region was also attractive but in the last year its attractiveness decreased radically. the regions of western europe and japan are unattractive for investors during the whole period, while the american region after an initial phase of selling tendency proved to be quite attractive for investors. figure 3 illustrates that returns of the static portfolio are much more volatile than returns of the dynamic portfolio. without a doubt, markowitz procedure provides us with the much more risky portfolio than the two other proposed. empirical verification of world’s regions profitability… dynamic econometric models 13 (2013) 145–162 159 figure 3. the realized optimal portfolio returns of the es (dynamic) and the markowitz (static) portfolios in order to illustrate investment regions preferred by the dynamic portfolio, the figure 4 shows the monthly moving average shares for the es strategy. figure 4. the moving average for the dynamic strategy the optimal investment strategy strongly suggests buying pacific asian assets most of the time. the eastern europe region is also regarded to be attractive for investors but it is less preferred than the pacific asian region. anna czapkiewicz, artur machno dynamic econometric models 13 (2013) 145–162 160 optimal shares of northern american, western europe and japan assets are mostly negative. the meaning of these findings is not apparent. the attractiveness read from the figure 4 should be interpreted correctly. it does not mean that optimal portfolios had been containing a positive value of pacific asian assets during the analyzed period, it means that it had been containing a positive value of the assets in average. however, the change of the composition of the optimal portfolio is intense, an investor has to rebuild the portfolio completely every day in order to keep it optimal. conclusions we have constructed, for the region attractiveness study purposes, global indices which represent five regions: northern america, pacific asia, japan, eastern europe and western europe. the study covers the period of january 2009 to april 2012. the period of october 2002 to january 2009 has been used only for the model estimation. the five-dimensional time series’ mean has been modeled by the vector autoregressive term and the variance’s dynamics have been described by the generalized autoregressive conditional heteroscedastic model with dynamic conditional correlation. the estimated correlations between considered regions have been computed due to the var(1)-dcc-garch(1,1) model, they illustrate relations among the investigated regions. the analysis confirms the claim that dependences between financial markets are higher in a period of crisis than during a prosperity time. dynamic dependences were included in the construction of the optimal portfolio. that is why two dynamic optimal portfolios have been built: the portfolio of the minimal var and the portfolio of the minimal es. in order to show the properties of these portfolios, the result of the implementation of the markowitz model, still very commonly used in practice, has been presented. the share of the index corresponding with a given region in the optimal portfolio have determined the region’s daily attractiveness. the markowitz portfolio composition gives clear results of the attractiveness of a region, however, properties of this portfolio have been interior to properties of the dynamic portfolio. the markowitz procedure estimates the mean of a portfolio incorrectly. moreover, the risk of the portfolio obtained using this method is much higher than the risk of the portfolio obtained under the dynamic model assumption. the optimal portfolio obtained using the dynamic model have attained the assumed return with relatively small risk. the diversification of the optimal dynamic portfolio is highly volatile and it has hardly any connection to a global financial situation. the cost of empirical verification of world’s regions profitability… dynamic econometric models 13 (2013) 145–162 161 such an investment strategy is a daily rebuilding of the portfolio. it requires almost complete decomposition of the portfolio every day. therefore it is not possible to summarize the profitability of the regions in general. the daily attractiveness is dynamical and it should be taken into account by international investors. as a illustration of the result, monthly moving averages of the daily attractiveness (measured by the share of assets from the region in the optimal portfolio) have shown pacific asian region as the most attractive during the period under study. eastern european markets also have appeared to be profitable. western european, american and especially japanese markets have appeared to be unattractive. references aas, k., czado, c., frigessi, a., bakken, h. (2009), pair-copula constructions of multiple dependence. insurance, mathematics and economics, 44(2), 182–198, doi: http://dx.doi.org/10.1016/j.insmatheco.2007.02.001. artzner, p., delbaen, f., eber, j., heath, d. (1999), coherent measures of risk. mathematical finance, 9, 203–228, doi: http://dx.doi.org/10.1111/1467-9965.00068. billio, m., caporin, m, gobbo, m., flexible dynamic conditional correlation multivariate garch models for asset allocation, applied financial economics letters, 2(2), 123–130, doi: http://dx.doi.org/10.1080/17446540500428843. bollerslev, t. (1990), modeling the coherence in short-run nominal exchange rates: a multivariate generalized arch model, review of economics and statistics, 72, 498–505, doi: http://dx.doi.org/10.2307/2109358. cappiello, l., engle, r., sheppard, k. (2006), asymmetric dynamics in the correlations of global equity and bond returns, journal of financial econometrics, 4, 537–572, doi: http://dx.doi.org/10.1093/jjfinec/nbl005. engle, r. f. (2002), dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models, journal of business and economic statistics, 20, 339–350, doi: http://dx.doi.org/10.1198/073500102288618487. föllmer, h., schied, a. (2011), stochastic finance: an introduction in discrete time, walter de gruyter berlin, new york, doi: http://dx.doi.org/10.1515/9783110212075. ghalanos, a. (2013), the rmgarch models: background and properties, cran, http://cran.r-project.org/web/packages/rmgarch/vignettes/the_rmgarch_models.pdf (9.04.2013). markowitz, h. m. (1959), portfolio selection: efficient diversification of investments, new york: john wiley & sons. palomba, g. (2008), multivariate garch models and the black-litterman approach for tracking error constrained portfolios: an empirical analysis, global business and economics review, 10(4), 379–413, doi: http://dx.doi.org/10.1504/gber.2008.020592. pelletier, d. (2006), regime-switching for dynamic correlation, journal of econometrics, 131, 445-473, doi: http://dx.doi.org/10.1016/j.jeconom.2005.01.013. rockafellar, r. t., uryasev, s. (2000), optimization of conditional value-at-risk, journal of risk, 2, 21–41. anna czapkiewicz, artur machno dynamic econometric models 13 (2013) 145–162 162 rockafellar, r. t., uryasev, s. (2002), conditional value-at-risk for general loss distributions, journal of banking and finance, 26, 1443–1471, doi: http://dx.doi.org/10.1016/s0378-4266(02)00271-6. tse, y. k., tsui, a. k. c. (2002), a multivariate generalized autoregressive conditional heteroscedasticity model with time-varying correlations, journal of business and economic statistics, 20, 351–362, doi: http://dx.doi.org/10.1198/073500102288618496. badanie zyskowności wybranych regionów świata w międzynarodowej dynamicznej strategii inwestycyjnej z a r y s t r e ś c i. artykuł przedstawia empiryczną weryfikacji przydatności wybranych regionów świata do inwestowania na ich rynkach finansowych. do badania wybrano pięć regionów świata: region ameryki północnej, region europy zachodniej, region europy wschodniej, region azji i pacyfiku oraz japonię. jako reprezentant nastrojów finansowych danego regionu zdefiniowano globalne wskaźniki będące ważoną kapitalizacją stopą zwrotu głównych indeksów rynków wchodzących w skład rozważanych regionów. do modelowania zależności pomiędzy tak utworzonymi globalnymi wskaźnikami zastosowano wielowymiarowy model var(1)-dcc-garch(1,1) z warunkowym wielowymiarowym rozkładem t-studenta. atrakcyjność regionu jest definiowana jako udział globalnego wskaźnika związanego z danym regionem w optymalnym portfelu. rozważono trzy typy konstrukcji optymalnych portfeli. własności optymalnych portfeli opartych na minimalizacji value at risk oraz na minimalizacji expected shortfall zostały porównane do własności standardowego portfela markowitza. s ł o w a k l u c z o w e: portfel optymalny, wielowymiarowe modele dynamiczne, miary ryzyka. acknowledgements we would like to thank two anonymous referees for the valuable comments on an earlier version of the paper. microsoft word dem_2018_115to127.docx © 2018 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2018.007 vol. 18 (2018) 115−127 submitted october 29, 2018 issn (online) 2450-7067 accepted december 28, 2018 issn (print) 1234-3862 ewa majerowska, magdalena gostkowska-drzewicka* impact of the sector and of internal factors on profitability of the companies listed on the warsaw stock exchange a b s t r a c t. the aim of the article is to assess the impact of the sector environment and of selected internal factors on the profitability level of the companies listed on the warsaw stock exchange in 1998–2016. an increase in the financial leverage, financial liquidity, non-debt tax shield and enterprise size cause a drop in the roa. an increase in the ratio of fixed assets to the total assets results in an increase in the roa. similar results were obtained for the models estimated for the roe. it means, that profitability of the examined companies results from the decisions made by the managers and from the impact of the sector environment. k e y w o r d s: return on assets (roa), return on equity (roe), industry effect, panel estimation. j e l classification: g30. introduction identification of the factors shaping enterprise profitability is an important research trend that has been considered on many levels of economic sciences. in the traditional approach that is based on the s-c-p concept (structure-conduct-performance), industrial economics focuses on the sectoral factors shaping the competitive advantage. these determinants entail the concentration, the scale effect as well as the entry and exit barriers (porter 1992; slater and * correspondence to: ewa majerowska, faculty of management, university of gdańsk, ul. armii krajowej 101, 81-824 sopot, poland, e-mail: ewa.majerowska@ug.edu.pl; magdalena gostkowska-drzewicka, faculty of management, university of gdańsk, ul. armii krajowej 101, 81-824 sopot, poland, e-mail: mgostkowska@wzr.ug.edu.pl. ewa majerowska, magdalena gostkowska-drzewicka dynamic econometric models 18 (2018) 115–127 116 olson, 2002). by contrast, in accordance with the rbv concept (resourcebased view), the role of company-specific internal factors is emphasized. according to this approach, the organizational structure and the managers’ managerial skills are the source of the differences in the level of profitability of individual companies. discussion has been continuing in the literature on the subject about the impact of both groups of factors on the financial results of enterprises. the attempts to confirm the validity of both approaches have been supported by many empirical studies, the results of which, however, are ambiguous. the hypothesis formulated assumes, therefore, that both the sector environment and the internal factors exert specific impact on the level of a given entity’s profitability. moreover, profitability is the main pillar for any company to survive in the long run. for this reasons, the aim of the article is to assess the impact of the sector environment and of selected internal factors on the profitability level of the companies listed on the warsaw stock exchange in the years 1998–2016. implementation of a goal formulated as such and verification of the research hypothesis required estimation of panel data models. to group the examined enterprises according to their sectoral affiliation and to estimate the impact of the sector on their financial results, the sectoral classification used for the needs of the warsaw stock exchange was applied. the profitability (roa) and the capital (roe) ratios of the examined entities were adopted as the dependent variables. the sector environment (industry) is one of the independent variables. other exogenous variables are the internal factors. the value of debt (leverage) has been expressed as the total debt and total assets ratio, thus it describes the structure of the capital, which is identified with the structure of financing. the growth rate (growth) has been described by the percentage change in the sales revenue, relative to the previous year. financial liquidity (liq) has been designated as the ratio of current assets to current liabilities. the non-debt tax shield (ndts) was calculated as the ratio of the depreciation to the total assets. the natural logarithm of the total assets’ value has been assumed as the volume (size). the structure of assets (tang), also referred to as asset flexibility (tangibility), has been described by the ratio of the tangible fixed assets to the total assets. calculations were carried out using the gretl package. the article consists of an introduction, four parts and an ending. the first part concerns the shaping of the enterprise performance results, in the context of the positional and the resource-related concept of the competitive advantage. in the second part, an overview of the studies based on these theories was made. the third part is methodical. the following part presents the results of the research, which have been summarized in the final part of the paper. impact of the sector and of internal factors on profitability… dynamic econometric models 18 (2018) 115–127 117 1. enterprise profitability in the light of the positional and the resource-related concept of the competitive advantage the factors shaping enterprise profitability can be classified as: internal factors (specific for a given entity), those related to the sector environment and to the macro-environment (pierścionek, 1997, p. 105–107). impact of the internal factors and of the sector environment on the company's financial results can be considered in the light of two opposing concepts of the competitive advantage: the positional and the resource-related school of thought. supporters of the positional school of thought emphasize the role of the sector environment in shaping the results of enterprise performance. the classic research approach consistent with this theory is the aforementioned s-c-p concept (structure-conduct-performance), initiated by bain (1951). it should be emphasized that the sector structure determines the activities of enterprises, in terms of price formation, research and development or investments, which in turn affect their financial results. due to the superior role of the sector, impact of managerial decisions on the company's profitability is thus limited. in his pioneering work, bain (1951) proved that the average level of profitability obtained by the enterprises belonging to the sectors with a high degree of concentration is higher than in dispersed industries. the greater degree of concentration provides these companies with a bargaining advantage and the ability to put pressure on the partners. in contrast to the positional school of thought, the rbv (resource-based view) approach exposes the importance of internal factors in achieving a competitive advantage. this means that the key role in shaping a company's financial results is the way of managing the assets it owns (wernerfelt, 1984; barney, 1991). therefore, market success of a company is determined by the ability to use the available resources, owing to which it is able to develop a highly competitive position. porter (1992, p. 21–23; 2008), emphasizes that relating a company to its environment constitutes the essence of a competition strategy formulation. the concept of five competitive forces developed by this researcher refers to the structural features of a sector, which determine the strength of the competitive forces, and therefore the level of profitability. this means that different sectors vary, in terms of the ultimate potential for generating profit. this author subjected this thesis to verification in his later works (porter, 1991; mcgahan and porter, 1997). the research carried out confirmed, to a certain extent, previous observations. it turned out that both the sector-related and the internal factors had impact on the profitability of the examined enterprises. porter (1991) emphasizes that a company is partly under the influence of the sector environment, however, to some extent, it can also affect it. therefore, it is possible to determine to what extent a company's financial results are ewa majerowska, magdalena gostkowska-drzewicka dynamic econometric models 18 (2018) 115–127 118 shaped by the sector environment and to which by the internal factors associated with the decisions made by management. 2. overview of the studies on the factors shaping the results of enterprise performance impact of the sector environment and internal factors on the profitability of enterprises has been the subject of numerous studies. their results are diversified, ambiguous and even contradictory. schmalensee (1985) analyzed the impact of the sector on the financial results of enterprises, on a sample of 456 us manufacturing companies, for the year 1975. the author stated that the sectoral effect is the key factor shaping the profitability of the examined entities and explains about 19% of their financial results. the same research shows that the internal factors, affecting profitability from the level of an enterprise, have negligible impact on it. rumelt (1991) conducted analogous analyzes for the period of 1974–1977, in which he included 432 to 471 production enterprises for each year. this author obtained opposite results. according to him, the sectoral effect was not significant, it explained only 4% of the volatility of the profitability ratios in the examined entities. on the other hand, internal factors were shown to have significant impact on the profitability of the examined entities (44%). however, in the studies carried out by both cited authors, a very short period of time was analyzed, which could significantly distort the results obtained by them. in response to the studies carried out by rumelt (1991) and schmalensee (1985), mcgahan and porter (1997) analyzed a much longer period, covering all phases of the business cycle, i.e. 1981–1994 on a sample of 5196 us companies from all sectors, except the finance sector. the research conducted showed that the sectoral effect explained 19% of the volatility of the profitability ratios in the examined entities. what is more, the strength of this effect varied, depending on the sector. and so, the sectoral effect was of less significance in the case of manufacturing enterprises, and of larger importance in entertainment, commercial and transportation industries. in turn, internal factors explained the financial results of the surveyed entities in as many as 32%. the studies conducted by bamiatzi and hall (2009), on a sample of as many as 71750 entities operating in great britain in the years 2002–2004, show that the financial results of micro-, smalland mediumas well as largesized enterprises during the analyzed period were affected by both the sectoral factors, and the internal ones. however, the strength of their impact varied. the reason for this lies in the fact, that entities of various sizes operate in different segments of the market, which can be distinguished within individual sectors. the company size turned out to be an important factor impacting the impact of the sector and of internal factors on profitability… dynamic econometric models 18 (2018) 115–127 119 financial results of us public companies in the years 1987–2006. in turn, impact of the sector environment on the profitability of the examined entities was small (lee, 2009). also, ruefli and wiggins (2003) believe that the sector environment does not have any significant impact on enterprise performance. the authors emphasize that managers play a key role in formation of the financial results and that it is primarily the financial situation of a company that determines its performance. similar results were obtained by stierwald (2010), who analyzed 961 large australian enterprises in the years 1995– 2005. the author concluded that, in contrast to the internal factors, impact of the sectoral effect on the profitability of the analyzed entities was scant. the authors of the above-cited studies used the return on assets, and thus considered enterprise performance in accounting terms. in contrast, hawawini, subramanian and verdin (2003), applied measures based on enterprise value, i.e. the economic value added (eva) and the market value added (mva), in addition to the return on assets (roa). the research carried out by these authors confirms that a given sector has significant impact on the financial results of the enterprises within it, while the strength of its impact is greater than that of the internal factors specific for individual entities. only in the case of the enterprises operating as market leaders and those least competitive within a given sector, a reverse relation was observed. for both groups of entities, the strength of the impact of internal factors was greater than that of the sectoral ones. dragonić (2014) conducted the study on croatian fast-growing small and medium businesses. the research showed that an impact of internal factors and sector on company’s profitability depends on the period, i.e. life cycle stage of a company and general state of the economy. margaretha and supartika (2016) examined internal factors affecting profitability and industry affiliation of smes firm listed in indonesia stock exchange. the results confirmed that firm size, growth, lagged profitability, productivity and industry affiliation significantly effect on profitability. the industry affiliation has a positive impact on profitability. the authors emphasize that for further improve company’s performance the manager should define a strategy to increasing profitability with focusing on productivity and industry affiliation. many other studies confirmed that industry affiliation influences company’s profitability ratio. for example, vijayakumar (2011) concluded that vertical integration is significantly associated with profitability. the studies conducted by salman and yazdanfar (2012) and yazdanfar (2013) indicate that firm industry affiliation has and impact on its profitability. the influence of internal factors on company’s performance was also analyzed by many other authors. tailab (2014) analyzed an impact of leverage, ewa majerowska, magdalena gostkowska-drzewicka dynamic econometric models 18 (2018) 115–127 120 liquidity, inventory, growth, size and firm’s age on financial performance of 100 top non-financial american firms listed on fortune 500 in 2009–2013. alarussi and alhaderi (2018) examined the internal factors affecting profitability in malaysian-listed companies. this study applies the resource-based theory. research is based on five independent variables that were empirically examined for their relationship with profitability. the findings show a strong positive relationship between firm size, working capital, company efficiency and profitability. the results also show a negative relationship between leverage and profitability. in the available literature on the subject, few studies can be found regarding the impact of the sector environment and the internal factors on the profitability of polish enterprises (matyjas, 2012, 2016). the author analyzed companies listed on the warsaw stock exchange. the research results vary, depending on the number of the subjects admitted to the sample and the period covered by the analysis. thus, during the period of 2008–2011, 389 companies were examined. the author proved that internal factors played an important role in shaping those companies’ profitability. sector environment, in turn, did not affect it (matyjas, 2012). in further studies, the same author obtained different results (matyjas, 2016). this time, the subject of the analysis entailed 221 companies listed on the warsaw stock exchange in the years 2007–2012. it turned out that their profitability was impacted by both the internal factors and the sector environment. it should be noted, however, that the strength of the company-specific factors was, in this case, much higher than that of the sector-specific ones. 3. the research sample and description of the research method the subject of the analysis entails the companies listed on the main market of the warsaw stock exchange in the years 1998–2016, as of 15 december 2017. to select the entities, the sector classification used for the needs of the warsaw stock exchange was applied to the sample. out of 477 companies, companies from the following sectors and subsectors were accepted for the research: fuels and energy (14 companies); chemistry and raw materials (37 companies); construction (44 companies); electromechanical industry (24 companies); transport and logistics, business supplies and enterprise services (20 companies in total); consumer goods (39 companies); wholesale trade, retail chains and e-commerce (13 companies in total); recreation and leisure, media and games (26 companies in total); health care (15 companies) and technologies (28 companies). companies from the financial sector were impact of the sector and of internal factors on profitability… dynamic econometric models 18 (2018) 115–127 121 excluded from the research. therefore, 64 entities were rejected, with the exception of the enterprises included in the real-estate subsector (20 companies), which carry out property development activity, consisting in the construction of real estate and then its sale or rental. the sector, and more generally – the real estate market can be considered from two perspectives: the financial and the material one. in the first case, it should be perceived through the prism of the links with the financial capital market, while in the second – in terms of the real estate supply and demand (wiśniewska, 2004, p.79). due to the presumptions resulting from the second conceptualization, real estate development companies were accepted for research. the sample excluded the entities that did not submit complete financial statements during the period under consideration, i.e. 52 enterprises. the companies in bankruptcy or under restructuring, i.e. 15 enterprises, were also rejected. moreover, only the entities that were continuously listed on the warsaw stock exchange for a period of at least 5 years were admitted to the study. therefore, 66 companies were excluded from the sample. ultimately, 280 enterprises were qualified for the research, i.e. almost 59% of the pre-selected entities. to identify the impact of selected internal factors on the profitability of the examined companies and the dependence of profitability on the sectoral affiliation, estimation of an econometric model, in a panel approach, was proposed. due to the fact that not all companies were listed on the warsaw stock exchange or operated since 1998, it was an unbalanced panel. a linear model of the dependence of the profitability level on the affiliation to the sector and on the internal factors has the following form: 𝑦"# = 𝛽& + (𝛽)𝑆) ++ ),+ + 𝛼+𝐿𝐴𝑉𝐸𝑅𝐴𝐺𝐸"# + 𝛼4𝐺𝑅𝑂𝑊𝑇𝐻"# + 𝛼9𝐿𝐼𝑄"# + 𝛼<𝑁𝐷𝑇𝑆"# + 𝛼?𝑆𝐼𝑍𝐸"# + 𝛼a𝑇𝐴𝑁𝐺"# + 𝜉"# where is an endogenous variable representing the profitability of the i-th company in the period t. profitability is defined by the relation of the profit to the total assets (roa) or the ratio of the net profit to the total equity (roe). the variables sj are centered dummy variables, taking the value of one if it belongs to a given sector, zero in other cases. while identifying individual sectors, the following markings were adopted: the real estate sector (res); the fuels and energy (fe); chemistry and raw materials (crm); construction (c); electromechanical industry (ei); transport and logistics, business supplies and services for enterprises (tls); consumer goods (cg); wholesale trade, retail chains and e-commerce (trec); recreation and leisure, media and games (rm); health care (hc) and technologies (te). ity ewa majerowska, magdalena gostkowska-drzewicka dynamic econometric models 18 (2018) 115–127 122 additionally, the remaining exogenous variables that represent the internal factors of companies are: leverage – debt, the total debt and total assets ratio; growth – increase, a percentage change in sales revenues in relation to the previous year; liq – financial liquidity, the relation of current assets to current liabilities; ndts – investment tax shield, the depreciation and total assets ratio; size – size, natural logarithm from the value of total assets; tang – the structure of assets, the ratio of fixed assets to the total assets. symbols are the structural parameters, while signifies the model’s random component. the proposed model was estimated using the least squares panel method (pooled model). then, using the wald test, it was checked whether estimation of a fixed effects would be correct. 4. research results as already mentioned, the empirical analysis aims to indicate whether internal factors and sectoral affiliation determine the profitability level of an enterprise. the above-proposed model was estimated in a panel approach. while modeling the shaping of the roa variable, the results of the estimations of the 4 versions of the model have been presented in table 1. model (1) includes the statistically significant variables. this means that an increase in the financial leverage, the financial liquidity, the non-debt tax shield and the size of an enterprise cause a decrease in the roa, while an increase in the share of the fixed assets in the total assets, expressing the structure of assets, results in an increase in the roa. the negative relationship between the financial leverage and the asset profitability is explained by the pecking order theory. the most profitable enterprises are in low debt due to the fact that they prefer internal financing first (they use their earned profits for this purpose), not because they ultimately set a low level of the debt ratio. in contrast, enterprises with low profitability are more willing to use debt, because they do not have sufficient resources from their internal sources. the tendencies described are supported by the studies on the capital structure of polish enterprises, carried out by other authors (marzec, 2010; janus, 2006; lisińska 2012; barburski, 2014, wrońskabukalska, 2014; jaworski and czerwonka, 2017). the negative relationship between asset profitability of and financial liquidity is justified by the fact that purchase of inventories exceeding current needs generates additional costs, which do not bring additional revenues – it provides a hedge, in the event of a sudden increase in the demand or in the 611110 ,...,,,...,, aabbb itx impact of the sector and of internal factors on profitability… dynamic econometric models 18 (2018) 115–127 123 prices of supplies and raw materials. on the other hand, current assets and therefore liquidity increase, while profitability does not. asset profitability may even drop, because the value of assets increases, while profits may decrease, since excessive stocks create additional costs. table 1. estimates of the linear regression model for roa variable model (1) (2) (3) (4) pooled fe pooled fe const 2,4175*** 4,4642*** 2,5358*** 4,5556*** res –0,3417 fe 0,3121** crm –0,0195 c –0,0858 ei –0,0646 tls –0,3982*** cg 0,0352 trec –0,0365 rm 0,2938*** 0,4461** hc –0,1665 –0,7783*** te 0,4717*** –1,0186*** leverage –0,1599*** –0,1531*** –0,1588*** –0,1536*** growth 0,0000 liq –0,0064** –0,0099*** –0,0059** –0,0105*** ndts –37,7074*** –47,928*** –38,8988*** –47,6416*** size –0,1476*** –0,2990*** –0,1574*** –0,3086*** tang 2,9309*** 3,4679*** 3,0362*** 3,6041*** joint significance test 3,7365# na 3,6960# na breusch-pagan test 376,6740# na 300,6340# na hausman test 388,53538# na 429,5220# na note: **) ***) statistically significant at the significance levels 0.05 and 0.01 respectively; #) statistically significant at the significance level of 0.05. the non-debt tax shield – expressed as the ratio of the depreciation to the total assets, is a substitute for an interest tax shield. enterprises collecting funds through depreciation do not have to involve debt in their investment financing. therefore, entities that have the option of financing investments from internal sources, use a non-debt tax shield. however, making high depreciation charges applies only to modern components of the fixed assets, with a high initial value. hence, if high-value modern fixed assets dominate in the structure of the company's assets, the value of the return on assets may be lower than in the entities that have efficient but used-up fixed assets. the negative relationship between the size of an enterprise and the profitability of the assets can be justified by the fact that there are components within the asset structure that do not bring additional revenues over a short period of time, e.g. some long-term investments or stocks of raw materials and ewa majerowska, magdalena gostkowska-drzewicka dynamic econometric models 18 (2018) 115–127 124 supplies, providing security in the event of a sudden increase in the demand or in the prices of supplies and raw materials. in enterprises with a high share of such assets in the total assets, profitability may be low, because their value is high, while profits may be reduced, because a high level of non-production assets creates additional costs. the positive relationship between the structure of assets and the value of roa should be explained by the significant share of the investments yielding a high rate of return in the asset structure of the companies under examination. it causes an increase in the profitability level of these entities. model (1) indicates predominance of the fixed effects model, hence the version (2), which is an estimation of the model describing the impact of internal factors, which is the fixed effects one. version (3) contains estimations of the model with sectoral effects and factors, using the pooled model. the final version (4) indicates the sectoral effects, statistically significant, and the internal factors of the fixed effects model. the results indicate in which sectors statistically significant differences in the roa level were noted, while impact of individual factors on the roa level was indicated (table 1). table 2. estimates of the linear regression model for roe variable model (1) (2) (3) (4) pooled fe pooled fe const 0,2354 0,6781 2,5358*** 4,5556*** res –0,3417 fe 0,3121** crm –0,0195 c –0,0858 ei –0,0646 tls –0,3982*** cg 0,0352 trec –0,0365 rm 0,2938*** 0,4461** hc –0,1665 –0,7783*** te 0,4717*** –1,0186*** leverage –0,1588*** –0,1536*** growth 0,0000 liq –0,0059** –0,0105*** ndts 16,0422** 17,0020* –38,8988*** –47,6416*** size –0,1574*** –0,3086*** tang 3,8527** –5,8666** 3,0362*** 3,6041*** joint significance test 1,1625# na 3,6960# na breusch-pagan test 0,1408 na 300,6340# na hausman test 0,4527 na 429,5220# na note: **) ***) statistically significant at the significance levels 0.05 and 0.01 respectively; #) statistically significant at the significance level of 0.05. impact of the sector and of internal factors on profitability… dynamic econometric models 18 (2018) 115–127 125 next, a model was estimated by adopting the value of roe as a determinant of the profitability level. the results are presented in table 2. similarly, as in the case of the roa, it was estimated in four versions. generally, the results justify using the panel ols method (pooled model), only the model in version (4) contains estimates with fixed effects (within group estimator). it can be noticed that the value of roe statistically significantly differs in the consumer goods and services sectors, in the case of a model containing only sectoral effects. the significance of internal factors confirmed the results obtained for the roa index. conclusions the empirical analysis carried out enabled indication of the dependence of the profitability level of companies on their sectoral affiliation and on selected internal factors characterizing those companies. the model estimated for the roa indicates a statistically significant level of this index, higher than the average in the service sector, and higher in the sectors of health care and technology. in addition, an increase in the financial leverage, financial liquidity, non-debt tax shield and enterprise size cause a drop in the roa. on the other hand, an increase in the value of the ratio of fixed assets to the total assets results in an increase in the roa. similar results were obtained for the models estimated for the roe. therefore, it can be concluded that there are no grounds to reject the hypothesis assumed in the introduction, about the profitability of companies listed on the warsaw stock exchange being dependent on their sectoral affiliation and on selected factors. in other words, profitability of the examined enterprises results from the decisions made by the managers and from the impact of the sector environment. references alarussi, a. s., alhaderi, s. m., (2018), factors affecting profitability in malaysia, journal of economic studies, 45(3), 442–458, doi: http://dx.doi.org/10.1108/jes-05-2017-0124. bamiatzi, v., hall, g. (2009), firm versus sector effects on profitability and growth: the importance of size and interaction, international journal of the economics of business, 16(2), 205–220, doi: http://dx.doi.org/10.1080/13571510902917517. bain, j. (1951), relation of profit rate to industry concentration: american manufacturing, 1936–1940, quarterly journal of economics, 65, 293–324, doi: http://dx.doi.org/10.2307/1882217. barburski, j. (2014), kapitały własne jako podstawa bezpieczeństwa działalności gospodarczej na przykładzie przedsiębiorstw wig 20 (equity as a basis for business security, on the example of wig 20 enterprises), finanse, rynki finansowe, ubezpieczenia (finance, financial markets, insurance), 67, 177–136. barney, j. b. (1991), firm resources and sustained competitive advantage, journal of management, 17(1), 99–120, doi: http://dx.doi.org/10.1177/014920639101700108. ewa majerowska, magdalena gostkowska-drzewicka dynamic econometric models 18 (2018) 115–127 126 dragonić, d. (2014), impact of internal and external factors on the performance of fast–growing small and medium businesses, management, 19(1), 119–159. hawawini, g., subramanian, v., verdin, p. (2003), is performance driven by industry – or firm–specific factors? a new look at the evidence, strategic management journal, 24(1), 1–16, doi: http://dx.doi.org/10.1002/smj.278. janus, a. (2006), kapitał własny jako źródło finasowania działalności małych i średnich przedsiębiorstw (equity as a source of financing of small and medium enterprise activity), folia oeconomica, 200, 69–78. jaworski, j., czerwonka, l. (2017), determinanty struktury kapitału przedsiębiorstw notowanych na gpw w warszawie. sektor usług (determinants of the capital structure of enterprises listed on the warsaw stock exchange. the service sector), annales h – oeconomia, 51(4), 133–142. lee, j. (2009), does size matter in firm performance? evidence from us public firms, international journal of the economics of business, 16(2), 189–203, doi: http://dx.doi.org/10.1080/13571510902917400. lisińska, k. (2012), struktura kapitałowa przedsiębiorstw produkcyjnych w polsce, niemczech i portugalii (capital structure of production enterprises in poland, germany and portugal), prace naukowe uniwersytetu ekonomicznego we wrocławiu (scientific papers of the university of economics in wroclaw), 27, 449–458. margaretha, f., supartika, n. (2016), factors affecting profitability of small medium enterprises (smes) firm listed in indonesia stock exchange, journal of economics, business and management, 4(2), 132–137, doi: http://dx.doi.org/10.7763/joebm.2016.v4.379. marzec, j. (2010), złote reguły finansowania w praktyce małych i średnich przedsiębiorstw w polsce (golden rules of financing in practice for small and medium enterprises in poland), ekonomiczne problemy usług (issues of economic service), 51, 143–152. matyjas, z. (2012), wpływ czynników oddziałujących na poziomie firmy oraz czynników sektorowych na wyniki finansowe spółek w świetle badań empirycznych (impact of the company-level and sectoral factors affecting the financial results of companies, in the light of empirical research), zarządzanie i finanse (management and finance), 4(2), 23–33. matyjas, z. (2016), wpływ poziomu sektora oraz firmy na wyniki przedsiębiorstw (impact of the sector and the company level on enterprise results), prace naukowe uniwersytetu ekonomicznego we wrocławiu (scientific works of the university of economics in wroclaw), 444, 307–316, doi: http://dx.doi.org/10.15611/pn.2016.444.28. mcgahan, a. m., porter, m. e. (1997), how much does industry matter, really?, strategic management journal, 18, 15–30, doi: http://dx.doi.org/10.1002/(sici)1097-0266(199707). pierścionek, z. (1997), strategie rozwoju firmy (company development strategies), scientific publishing pwn, warsaw. porter, m. e. (2008), the five competitive forces that shape strategy, harvard business review, 86(1), 79–93. porter, m. (1991), towards a dynamic theory of strategy, strategic management journal, winter special issue 12, 95–117. porter, m. e. (1992), strategia konkurencji. metody analizy sektorów i konkurentów (competition strategy. methods for analyzing sectors and competitors), publishing: polskie wydawnictwo ekonomiczne, warsaw. ruefli, t. w., wiggins, r. r. (2003), industry, corporate, and segment effects and business performance: a non-parametric approach, strategic management journal, 24, 861–879, doi: http://dx.doi.org/10.1002/smj.350. rumelt, r. p. (1991), how much does industry matter?, strategic management journal, 12, 167–185, doi: http://dx.doi.org/10.1002/smj.4250120302. impact of the sector and of internal factors on profitability… dynamic econometric models 18 (2018) 115–127 127 salman, a. k., yazdanfar, d. (2012) profitability in swedish sme firms: a quantile regression approach, international business research, 5(8), 94–106. schmalensee, r. (1985), do markets differ much?, american economic review, 75, 341–351. slater, s., olson, e. (2002), a fresh look at industry and market analysis, business horizons, 45(1), 15–22. stierwald, a. (2010), determinants of profitability: an analysis of large australian firms, melbourne institute working paper, 3/10, 1–23, doi: http://dx.doi.org/10.2139/ssrn.1632749. tailab, m., (2014), analyzing factors effecting profitability of non-financial u.s. firms, research journal of finance and accounting, 5(22), 17–26. vijayakumar, a. (2011), an empirical study of firm structure and profitability relationship: the case of indian automobile firms, international journal of research in commerce and management, 1(2), 100–108. wernerfelt, b. (1984), a resource-based view of the firm, strategic management journal, 5(1), 171–80. wiśniewska, e. (2004), rynek nieruchomości a gospodarka (real estate vs. and economy), (in:) zachodnie rynki nieruchomości (western real estate markets), collective work, ed. by: e. kucharska-stasiak, twigger, warsaw. wrońska-bukalska, e. (2014), zróżnicowanie branżowe poziomu i struktury kapitału własnego (industry diversification of the level and the structure of equity), annales universitatis mariae curie-skłodowska, 48(3), 393–402. yazdanfar, d., (2013) profitability determinants among micro firms: evidence from swedish data, the international journal of managerial finance, 9(2), 150–160. wpływ sektora i czynników wewnętrznych na rentowność spółek notowanych na gpw w warszawie z a r y s t r e ś c i. celem artykułu jest ocena wpływu otoczenia sektorowego oraz wybranych czynników wewnętrznych na poziom rentowności spółek notowanych na gpw w warszawie w latach 1998–2016. wzrost dźwigni finansowej, płynności finansowej, nieodsetkowej tarczy podatkowej oraz wielkości przedsiębiorstwa powodują spadek roa. z kolei wzrost wskaźnika rzeczowych aktywów trwałych do aktywów ogółem powoduje wzrost roa. analogiczne wyniki otrzymano w przypadku modeli oszacowanych dla roe. oznacza to, że rentowność badanych przedsiębiorstw jest wypadkową decyzji podejmowanych przez menedżerów i wpływu otoczenia sektorowego. s ł o w a k l u c z o w e: rentowność aktywów roa, rentowność kapitału roe, efekt sektorowy, estymacja panelowa. dem_2019_41to56 © 2019 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2019.003 vol. 19 (2019) 41−56 submitted july 3, 2019 issn (online) 2450-7067 accepted december 20, 2019 issn (print) 1234-3862 sebastian a. roy * demonetisation as an economic policy tool: macroeconomic implications of a monetary market shock. the example of the indian monetary reform a b s t r a c t. this paper discusses the implementation of the 2016 india demonetisation, and analyses its macroeconomic consequences. the pivotal issue here is a regional heterogeneity of cpi inflation caused by demonetisation. a dynamic panel cpi model has been estimated in order to find out whether unequal accessibility of banking services determines the inflation heterogeneity. the findings suggest that financial services accessibility is not a significant inflation-driving factor. hence a hypothesis about a redistribution of wealth between rural and urban areas with different access to banking might be rejected. k e y w o r d s: developing economies; inflation; macroeconomic policy; monetary policy; panel econometrics. j e l classification: e02, e26, e31, e52; c01, c23 introduction demonetisation in india was announced on 8 november 2016 and included pulling from circulation banknotes of the two highest denominations (500 inr and 1,000 inr). subsequently, new series of 500 inr and 2,000 inr notes were issued, but in order to exchange the old notes it was required to make a bank account deposit (killawala, 2016b). according to oecd, the * correspondence to: sebastian a. roy, sgh warsaw school of economics, kazimierza promyka 5/80, 01-604 warszawa, e-mail: sebastian.amit.roy@gmail.com sebastian a. roy dynamic econometric models 19 (2019) 41–56 42 demonetised notes composed 86% of cash in circulation (beyes and bhattacharya, 2017). the demonetisation was designed to achieve four goals: curbing black economy, tackling bribery, targeting counterfeit notes, and cutting resources of terrorist organisations (modi, 2017). such a selection of aims implies that the demonetisation needs to be analysed not only in its economic dimension, but also in a social one. whatever its political and institutional motivations, indian monetary reform remains a macroeconomic experiment of substantial interest, perhaps the most sweeping change in currency policy that has occurred anywhere in the world in decades, according to the former us treasury secretary lawrence summers (sharma, 2016). it is worth noting that experiment is scarcely used in macroeconomic research – this is due to the fact that hardly would anybody allow economists to wreak havoc in lives of millions for sake of their curiosity (czarny, 2017). moreover, the demonetisation may be considered as a ceteris paribus negative shock in the m1 money supply, since it was introduced in times of solid economic growth and in a stable political environment. this allows usage of the post-demonetisation macroeconomic data in empirical studies of price flexibility in emerging markets and inflation-driving factors. demonetisation has received limited interest from the researchers. first analyses, most of them penned by economic journalists, were published in december 2016 by media such as the economist (2016), forbes (worstall, 2016), and al jazeera (sharma, 2016). they were followed by a number of speculative papers published by mostly indian think-tanks and universities. their common characteristic was, however, lack of empirical evidence for discussed hypotheses – none of them employed formal statistical inference in order to support authors’ reasoning, except for the paper by chodorow-reich, gopinath, mishra et al. (2018). in polish literature demonetisation has attracted even weaker attention. in december 2016 the issue was examined in monitoring makroekonomiczny (roy, 2016). this paper aims at discussing implementation of the 2016 indian demonetisation and its macroeconomic consequences. in the first section it presents a detailed analysis of the reform implementation process, based on the authorities’ announcements. the second section aims to analyse possible macroeconomic consequences of driving 86% of cash out of the circulation. it comprises an empirical study alongside with a broad discussion of the literature. the key hypothesis here is presence of a regional heterogeneity of cpi inflation caused by demonetisation. a dynamic, individual effects cpi model has been estimated in order to find out whether unequal accessibility of banking services determines the inflation heterogeneity. moreover, the analysis also demonetisation as an economic policy tool… dynamic econometric models 19 (2019) 41–56 43 discusses the reform’s possible impacts on gdp and the role of elasticities in shaping the real sector response to a nominal shock. the cpi heterogeneity research is based on a panel dataset of monthly observations for 35 indian states with distinction between rural and urban areas in the post-demonetisation period. estimates show that a hypothesis of a positive short-run correlation between banking accessibility and cpi growth rates can be rejected under all circumstances. hence unequal accessibility of banking services couldn’t have driven cross-sectional cpi inflation variability following demonetisation. 1. implementation of the demonetisation 1.1. primary demonetisation mechanism the earliest source of information about the technical details of the process of withdrawing and exchanging the old notes was an address by the pm narendra modi broadcasted on 8 november 2016. more formally, demonetization was scheduled by two reserve bank of india (rbi) statements (vijaya kumar, 2016a; dave, 2016). later some elements of the reform were corrected and adjusted in response to certain difficulties arising. the first demonetization schedule comprised the following points (modi, 2017): 1. owners of the old 500 inr and 1,000 inr notes may deposit them at a bank or post office from 10 november to 30 december 2016 with no limitations. 2. cash withdrawals from bank accounts shall not exceed 10,000 inr per day and 20,000 inr per week. the limit is to be relaxed in a short time. 3. over-the-counter notes exchange for immediate needs is possible at every bank or post office. such an exchange is limited to 4,000 inr. after 24 december the limit shall be increased1. 4. any person unable to exchange their old notes before 30 december 2016 may do the exchange at certain rbi branches before 31 march 2017. 5. on 9 december 2016 (in particular cases, on 10 december as well) atms shall be shut due to recalibration. after recalibration cash withdrawals are limited to 2,000 inr per card. the limit shall be increased to 4,000 inr in a short time. 6. cashless transactions via cheques or electronic transfers may be conducted with no restrictions. 1 eventually the limit was not changed. on 24 november 2016 rbi decided to end the otc exchanges (beyes and bhattacharya, 2017). sebastian a. roy dynamic econometric models 19 (2019) 41–56 44 7. due to convenience and humanitarian reasons the following units will accept payments made in the old notes within 72 hours from the announcement of the demonetization: a. public hospitals and drug stores located therein (only for medicines prescribed by a physician) b. bus, train and flight ticket booking points c. petrol stations authorised by the national oil companies d. consumer cooperative stores authorized by the state or central government e. milk booths authorized by the state authorities f. burial grounds and crematoria g. currency exchange offices at international airports the vital part of the demonetisation policy, designed to tackle bribery and black economy, was obligatory fiscal investigation of every newly created bank deposit of 250,000 inr or more (roy, 2016). failure in proving that the due tax had been payed could end up with being fined 200% of the overdue fiscal liabilities (rowlatt, 2016). 1.2. modifications of the demonetisation mechanism the primary organization of the reform was subject to multiple alterations after 8 november 2016. due to a significant number of modifications oecd experts peter beyes and reema bhattacharya described indian demonetisation as evolutionary (beyes and bhattacharya, 2017). according to their research, in the central period of the reform (that is from 8 november to 30 december 2016) rbi issued 50 announcements concerning demonetisation, majority of whose significantly modified its implementation. on the other hand, ministry of finance issued 19 such announcements, part of which actually mirrored those produced by the rbi. the most important modifications of the reform concerned means of depositing and exchanging old notes as well as limits of cash withdrawals. those were issues of outstanding significance for the rural areas inhabitants. cash shortages posed a serious threat to farmers who could not pay for seeds for winter sowing. on 14 november 2016 rbi decided to withdraw the right to exchange2 notes from two most popular rural financial institutions, district central cooperative bank (dccb) and primary agricultural credit society (pacs) (vijaya kumar, 2016c). rbi decision was founded on a suspicion that dccb and 2 those institutions were no more allowed to deposit old notes. demonetisation as an economic policy tool… dynamic econometric models 19 (2019) 41–56 45 pacs were used for money laundering, as between 8 and 14 november 2016 they noted outstandingly high inflow of cash (beyes and bhattacharya, 2017). as a result, accessibility to the banking services was seriously restrained in the rural areas, whose inhabitants had to commute to towns or cities in order to exchange their notes. demonetisation timing further exacerbated circumstances for the agricultural sector. in november indian farmers harvest their summer crops and sow winter seeds. free access to financial services is of utmost importance as it allows for depositing resources obtained from summer crops sales and provides funding for sowing seeds. therefore, rbi decision concerning dccb and pacs met strong resistance. in response to the social distress, on 20 november 2016 farmers were allowed to purchase seed with old banknotes (dhoot, 2016). moreover, on 21 november rbi raised the cash withdrawal limit to 25,000 inr for farmers (vijaya kumar, 2016b). 2. macroeconomic consequences of the reform 2.1. literature review the literature discussing demonetisation in india comprises two groups of articles. first set contains articles published in various economic magazines shortly after the reform had been announced. among them three deserve special interest. the first, authored by white and rajagopalan (2016) from foundation for economic education (fee), discusses selected issues of remonetisation, including seigniorage; second, by forbes’ worstall (2016), provides one of the earliest discussions of the impact demonetisation might have on gdp growth; the third, by the economist (2016), analyses social impact of the reform. the second set comprises research papers covering multivariate issues of the demonetisation. in a keynesian-inspired paper impact of demonetisation on india: a macro-theoretic analysis a. ghosh (2017) develops a disaggregated demand model with two sectors: organised and unorganised. the unorganised sector uses organised sector output as the only input and produces goods that have perfectly flexible prices. due to illiteracy and restricted access to collateral, the unorganised sector settles all transactions in cash, hence it must posess certain cash holdings. on the other hand, the organised sector is an oligopoly that uses two inputs: capital and labour. there are markups on wages and capital rental rates, with short-run wages constant. the organised sector may be considered cashless except for transactions with the unorganised sector. sebastian a. roy dynamic econometric models 19 (2019) 41–56 46 ghosh uses his model to determine possible impact of demonetisation on the indian economic growth, but his findings are ambiguous. cash supply slump may lower production of the unorganised sector, which in turn should make the organised sector output decrease – all in all, total production shall fall. on the other hand, lower cash supply could increase demand for the goods produced in the organised sector, thus boosting gdp. however, ghosh argues that such a demand shift shall be limited due to low accessibility of the organised sector output in less affluent areas and reluctance towards cashless payments. one of the earliest macroeconomic papers on the monetary reform is the one by the national institute of public finance and policy (nipfp) from 14 november 2016 (rao, mukherjee, kumar et al., 2016). it provides an analysis of shortand medium-run effects of the negative money supply shock based on four types of transaction demand for cash. those include accounted transactions, unaccounted transactions, informal sector transactions, and illegal transactions. furthermore, it includes credit supply and public finance issues. in the short-run the report predicts a sharp fall in households’ income and consumption due to a plunge in money supply. similarly, to ghosh (2017), the nipfp experts suggest that the informal sector might be affected stronger than the formal one. interestingly, they oppose a common notion that the reform will induce cpi deflation. contrarily – if cash shortage makes firms produce less, prices may even rise. finally, demand of those who do not have access to digital payment methods shall decrease, thus lowering the aggregate demand. in the medium-run rao, mukherjee, kumar, et al. stress that impact of the reform will depend on the remonetisation scale. if the authorities manage to conduct remonetisation swiftly3, effects should be mild and scarce. however, there are several sectors with high probability of being affected, including agriculture and construction. the report notes also a possible positive effect that growing deposits may have on the credit supply and, eventually, on the whole economy. das and rawat (2017) from institute for studies in industrial development (isid) evaluate demonetisation’s performance against its intended targets alongside discussing it from a gdp growth perspective. their conclusions are pessimistic. they find that the reform’s efficiency in fighting black economy and counterfeit currency is too low to outweigh macroeconomic threats it 3 that is, if remonetisation does not exceed two months (rao, mukherjee, kumar et al., 2016). demonetisation as an economic policy tool… dynamic econometric models 19 (2019) 41–56 47 poses. moreover, they interpret early post-demonetisation macroeconomic indicators as disappointing. siddiqui, mishra and tiwari (2017) present an analysis including fiscal consequences of the demonetisation. they suggest that allowing for paying taxes in demonetised notes might be a strong incentive for quicker and more comprehensive filling for tax return. after examining ministry of finance fiscal data, they argue that the fiscal authorities noted a 260% rise in municipal and local tax payments within 14 days from demonetisation. 2.2. regional heterogeneity of inflation. price arbitrage hypothesis interestingly, issue of post-demonetisation regional diversity of prices is not discussed in the literature except for the paper by white and rajagopalan (2016). arguably, unequal access to financial services could contribute to cross-sectional deflation heterogeneity. in the process of exchanging old notes commercial banks played a crucial role. this could lead to regional differentiation in remonetisation dynamics, conditional on local accessibility to financial services. arguably, m1 money supply in the rural areas with poorer access to banks might have been lower than in urban localities, where financial institutions were better accessible, particularly given restrictions on dccb and pacs (vijaya kumar, 2016c). according to the keynesian aggregated demand framework, prices are sticky in the short-run (czarny, 2017). this assumption has a robust justification; indeed, there is a wide consensus among economists that flexible price assumptions of the quantity theory of money are not satisfied but in a longrun analysis (sławiński, 2011). however, indian economy is heavily cash-dependent as 90% of all transactions there are settled in cash (beyes and bhattacharya, 2017). hence in the indian case of an abrupt demonetisation of 86% of cash in circulation prices can be flexible even in the short-run. it is worth noting that ghosh (2017) reaches a similar conclusion while specifying his macro-theoretic model. summing up, it is justified to expect that in the areas with high banking accessibility slump in the m1 money supply was milder, thus dampening change of prices. specifically, cpi deflation could be deeper in poorer states, but also there could occur within-state price index heterogeneity between rural and urban areas. unlike international price differentiation, neither withinnor betweenstate cpi heterogeneities could not be neutralized through the exchange rate as inr is a legal tender in the whole indian territory. hence temporary or persistent purchasing power diversity could be employed in a following arbitrage scheme: sebastian a. roy dynamic econometric models 19 (2019) 41–56 48 1. capital transfer to the locality with higher purchasing power (equivalently, with deeper deflation) 2. purchase of goods at lower prices 3. shipping the goods back to the area with lower purchasing power (equivalently, with milder deflation) 4. selling the goods with profit arbitrage results with a wealth transfer from regions suffering from deeper deflation to those with higher prices. since cpi dynamics depends on financial services accessibility, which is higher in more affluent, often urbanised regions, post-demonetisation arbitrage could lead to deepening economic inequalities in india. this effect could be further amplified by the fact that business entities from the wealthy regions were able to take advantage of their well-developed distribution networks. nonetheless; conducting such an arbitrage scheme would pose multiple difficulties, e.g. in specifying goods whose prices differ regionally. tackling it in detail would require scrutinizing trade flows in india, which falls beyond scope of this paper. therefore the empirical study presented in the final chapter discusses regional price heterogeneity only, leaving aside question of whether arbitrage actually took place. 2.3. real economy response to the money supply shock. nominal rigidities and shift of the long-run as curve (las) demonetisation impact on the gdp has been widely discussed since the very first days of the reform, with the earliest paper tackling this issue being that of rao, mukherjee, kumar et al. (2016) published five days from the reform commencement. while there is a broad consensus that in the short run consumption (and consequently aggregated demand) shrank, opinions about demonetisation’s long-run impact on the real economy vary significantly. this chapter studies long-run response of the real sector to the money supply shock, paying particular attention to the possibility of a long-run as curve shift caused by a substantial shrink in the money supply. demonetisation as an economic policy tool… dynamic econometric models 19 (2019) 41–56 49 figure 1. ad-as model real economy analysis will be founded on the ad-as framework (compare czarny, 2017). in ad-as model economy reaches equilibrium at the intersection of negative aggregated demand (ad) curve, flat and positive shortrun aggregated supply (sas) curve and vertical long-run aggregated supply (sas) curve. in the fig. 1 equilibrium is denoted as e. under ad-as approach a money supply slump is a negative demand shock, which shifts the demand from ad to ad’. hence temporary equilibrium is reached at the point e’, where ad’ and sas intersect. the temporary equilibrium is characterized with slightly lower prices and substantially decreased output. in the long run, however, sas curve shifts downward and a new equilibrium e’’ is reached with the pre-shock level of output, but lower prices. this illustrates that in the long run money the real sector is independent from nominal shocks. the ad-as analysis above could be summarized in a following way: a demonetisation-caused negative demand shock in the short run lowers the output due to nominal rigidities. after certain adjustments, the economy reaches new equilibrium with potential output and lower prices. that means that in the long run a negative demand shock affects the nominal economy only. there are two assumptions in the ad-as approach that deserve special scrutiny. the first is: were prices in india actually sticky in the post-demonetisation period? price stickiness is a crucial assumption for the ad-as analysebastian a. roy dynamic econometric models 19 (2019) 41–56 50 sis. there is massive literature covering the nominal rigidity issue, both theoretical, such as burda and wyplosz (2012), begg, fischer, vernasca et al. (2014), czarny (2017), and empirical, e.g. banerjee and bhattacharya (2017), chong, zhu and rafiq (2013); it is also in line with assumptions of the benchmark demonetisation research made by chodorow-reich, gopinath, mishra et al. (2018). however, is it justified to assume that after such a dramatic demand shock as demonetisation of 86% of cash in circulation, prices remained sticky? on the one hand, it is reasonable to suspect that about 7 times higher decrease of cash supply in an economy with approx. 100% of all transactions volume and 70% of their value settled in cash should cause a fall in cpi. on the other hand, remonetisation started immediately after the reform was announced, which could limit its effect on inflation. price stickiness assumption may be verified empirically with a time series analysis. figure 2 shows the reserve bank of india (rbi) data about year-onyear cpi inflation and year-on-year m1 money growth rate. according to the data there is no ground for rejection of the short-run price stickiness hypothesis. within two quarters from the beginning of the reform there occurred a slight cpi deflation; nevertheless, negative price dynamics not exceeding 5 percentage points do not differ significantly from its pre-demonetisation levels. figure 2. inflation and m1 money supply dynamics -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 0 1 2 3 4 5 6 7 8 9 ja n -2 01 6 fe b20 16 m ar -1 6 a pr -2 01 6 m a y20 16 ju n -2 01 6 ju l20 16 a u g -2 01 6 se p20 16 o ct -2 01 6 n o v20 16 d ec -2 01 6 ja n -2 01 7 fe b20 17 m ar -1 7 a pr -2 01 7 m a y20 17 ju n -2 01 7 ju l20 17 a u g -2 01 7 se p20 17 o ct -2 01 7 n o v20 17 d ec -2 01 7 ja n -2 01 8 fe b20 18 inflation m1 money supply dynamics (right axis) demonetisation as an economic policy tool… dynamic econometric models 19 (2019) 41–56 51 the second ad-as model assumption that needs to be re-considered, is constant potential output or, in other words, issue whether las curve is fixed. standard justification for this assumption says that the potential output depends solely supply-driven, depending only on technology and means of production. constant potential output assumption may be true only if hardships caused by demonetisation did not result with substantial number of insolvencies between indian business entities. among them small enterprises which generally settle in cash and have limited access to credit4, could be affected in a stronger way. hence long run demonetisation impact on the aggregated supply should depend on general market flexibility. if entrepreneurs failed to display enough perseverance to operate under cash shortages and became insolvent, decrease in potential output should be expected. such a wave of insolvencies would directly result with persistent decrease of different types of capital: physical (machines in the case of craftsmen, vehicles in the case of rickshaw drivers), human (know-how) and institutional (cutting employer-employee relations). there is empirical, macroeconomic evidence that in the months following demonetisation the number of bankruptcies among small entrepreneurs actually rose (shirley, 2017). rbi negatively corrected the mpc5 forecasts of gdp growth for the 2016-17 accounting year. they were adjusted from 7.6% on 4 october 2016 (killawala, 2016a) to 6.9% on 8 february 2017 (killawala, 2017). multiple international organisations lowered their forecasts as well. the data presented below imply that the demonetisation impact on the gdp growth was only temporary, which supports the thesis that there was no shift of the las curve. however, post-demonetisation estimation of the potential product of the indian economy remains a promising field of research. table 1. gdp growth forecasts institution wb imf cmie icra fitch pre-demonetisation growth forecast [%] 7.6 7.6 7,.5 7.7 7.4 post-demonetisation forecast [%] 7.0 6.6 6.0 6.8 6.9 forecast change [p. points] –0.6 –1.0 –1.5 –0.9 –0.5 note: wb – world bank, imf – international monetary fund, cmie – centre for monitoring of indian economy, icra – investment information and credit rating agency of india ltd. 4 this is partially justified by ghosh, who says that producers and workers in the unorganized sector cannot access institutional financial facilities because of illiteracy and lack of collateral (2017). 5 monetary policy committee of india. sebastian a. roy dynamic econometric models 19 (2019) 41–56 52 3. dynamic panel cpi model 3.1. research hypotheses a dynamic panel cpi model has been developed in order to verify hypothesis of regional heterogeneity of inflation driven by unequal access to financial services. formally, methods of statistical inference have been used to study the following hypotheses: h1: banking accessibility is a positive, statistically significant inflation-driving factor h2: banking accessibility has a stronger impact on inflation in rural territories estimation was computed in r software using plm package (croissant and millo, 2008). 3.2. the data the model employs a panel dataset of monthly observations for 35 indian states with distinction between rural and urban areas in the post-demonetisation period. the data has been downloaded from database on indian economy (dbie) governed by the rbi and from the census 20116 database. observations for the state of telangana have been dropped due to missing demographic data. all the variables are discussed in detail in table 2. 3.3. model specification. discussion of the method. hypotheses presented in 3.1 have been verified with three dynamic panel cpi models with fixed effects, estimated separately for rural, urban and aggregate inflation. such a specification is able to account for cross-sectional heterogeneity and inflation time-persistence. detailed specification is given below: 𝑖𝑛𝑓!" = 𝛽#𝑖𝑛𝑓!,"%# + 𝛽&𝑑𝑦𝑛𝑀1" + 𝛽'𝑏𝑎𝑛𝑘𝑖𝑛𝑔𝑃𝐶!" + 𝜇! + 𝜀!" 𝑅𝑖𝑛𝑓!" = 𝛽#𝑅𝑖𝑛𝑓!,"%# + 𝛽&𝑑𝑦𝑛𝑀1" + 𝛽'𝑅𝑏𝑎𝑛𝑘𝑖𝑛𝑔𝑃𝐶!" + 𝜇! + 𝜀!" 𝑈𝑖𝑛𝑓!" = 𝛽#𝑈𝑖𝑛𝑓!,"%# + 𝛽&𝑑𝑦𝑛𝑀1" + 𝛽'𝑈𝑏𝑎𝑛𝑘𝑖𝑛𝑔𝑃𝐶!" + 𝜇! + 𝜀!" coefficients have been estimated with arellano and bond (1991) estimator. 6 between 2011 and 2016 no census survey had been conducted. demonetisation as an economic policy tool… dynamic econometric models 19 (2019) 41–56 53 table 2. dataset description variable description type source comments rinf cpi yoy inflation rate at rural areas [%] panel, monthly dbie rbi uinf cpi yoy inflation rate at urban areas [%] panel, monthly dbie rbi inf cpi yoy inflation rate [%] panel, monthly dbie rbi dynm1 m1 money supply yoy growth rate [%] time series, monthly dbie rbi own calculations rbranches number of commercial bank branches at rural areas panel, quarterly dbie rbi ubranches number of commercial bank branches at urban areas panel, quarterly dbie rbi mbranches number of commercial bank branches at metropolitan areas panel, quarterly dbie rbi branches total number of commercial bank branches panel, quarterly dbie rbi own calculations rpopulation rural population cross-section census upopulation urban population cross-section census population total population cross-section census rbankingpc banking accessibility coefficient at rural ar-eas7 panel, quarterly own calculations ubankingpc banking accessibility coefficient at urban areas8 panel, quarterly own calculations bankingpc total banking accessibility coefficient panel, quarterly own calcula-tions 3.4. regression analysis table 3 presents the estimates with selected post-estimation statistics9. due to strong second-order autocorrelation of the residuals, robust variancecovariance matrix is used. all the tests assume standard 5% significance level. each panel is balanced. aggregate inflation panel is of width n=35 and length t=12 with n=420 observations altogether, out of which 338 are used in estimation. rural inflation panel is of width n=35 and length t=12 with n=420 observations altogether, out of which 348 are used in estimation. urban inflation panel is of width n=35 and length t=12 with n=420 observations 7 coefficient calculated as a fraction of number of bank branches at particular area over its population. 8 metropolitan areas will be treated as a subset of urban territories, hence 𝑈𝑏𝑎𝑛𝑘𝑖𝑛𝑔𝑃𝐶 = !"#$%&'()*+"#$%&'() !,-,./$01-% . 9 p-values for statistical tests are given in brackets. sebastian a. roy dynamic econometric models 19 (2019) 41–56 54 altogether, out of which 340 is used in estimation. differences in sample sizes are due to the missing values structure. table 3. estimation summary dependent variable 𝜷𝟏 𝜷𝟐 𝜷𝟑 wald chi-squared statistic arellano-bond 2nd order correlation test 𝒊𝒏𝒇 0.86 (0) 2.47 (0) 39,700 (0.35) 416.793 (0) -2.5 (0.01) 𝑹𝒊𝒏𝒇 0.89 (0) 2.72 (0) 46,496 (0.58) 153.673 (0) -2.55 (0.01) 𝑼𝒊𝒏𝒇 0.74 (0) 3.28 (0) 117,360 (0.18) 420.943 (0) -1.24 (0.22) each model tested with wald chi-squared statistic is statistically significant. autoregressive part of each model is also significant. rural inflation proves to be the most persistent with adjacent coefficient taking value of 0.89. money supply dynamics is also a significant, positive factor shaping inflation. its impact is strongest in the urban areas, where money supply growth by 1 percentage points rises inflation by 3.28 percentage points. banking accessibility, however, turns to be uncorrelated with cpi inflation. its coefficients are statistically insignificant with p-values of 0.18 for the urban, 0.58 for the rural and 0.35 for the aggregate panel. it is noteworthy that the urban panel is the only one in which no autocorrelation null hypothesis in the second-order arellano-bond test cannot be rejected. this suggests that standard variance-covariance matrix can be used instead of the robust one. under standard variance-covariance matrix banking accessibility becomes significant with p-value close to zero. however, since robust standard errors account not only for autocorrelation, but also other types of non-spherical variance-covariance matrix, conclusions shall be drawn in accordance to the robust error estimates. conclusions empirical study shows that both h1 and h2 hypotheses stated in subsection 3.1 can be rejected. banking accessibility proves not to be a significant inflation-shaping factor; therefore, one cannot say that its impact is stronger in the rural areas. this is a strong argument in support of a thesis that citizens of india did not face difficulties connected to unequal access to the financial institutions. lack of correlation between banking accessibility and inflation cannot verify whether restrictions imposed by the rbi on dccb and pacs did not influence prices significantly. this is so because dccb and pacs counted demonetisation as an economic policy tool… dynamic econometric models 19 (2019) 41–56 55 into the overall number of bank branches in india regardless of their ability to exchange notes. references arellano, m. & bond, s. (1991). some tests of specification for panel data: monte carlo evidence and an application to employment equations. review of economic studies, 58(2), 277–297. banerjee, sh. & bhattacharya, r. (2017). micro-level price setting behaviour in india: evidence from group and sub-group level cpi-iw data. nipfp working paper series, 17, new delhi. begg, d., fisher, s., vernasca, g. et al. (2014). makroekonomia. warszawa: pwn. beyes, p., & bhattacharya, r. (2017). india's 2016 demonetisation drive: a case study on innovation in anti-corruption policies, government communications and political integrity. 2017 oecd global anti-corruption & integrity forum. burda, m. & wyplosz, c. (2012). makroekonomia. podręcznik europejski. oxford university press. chodorow-reich, g., gopinath, g., mishra, p. et al. (2018). cash and the economy: evidence from india’s demonetisation. nber, 25370 chong, t. t. l., zhu, t., rafiq, m.s. (2013) are prices sticky in large developing economies? an empirical comparison of china and india. mpra paper, 60985. croissant, y. & millo, g. (2008). panel data econometrics in r: the plm package. journal of statistical software, 27(2), 1–43 czarny, b. (2017). podstawy ekonomii. warszawa: oficyna wydawnicza sgh. das, s. k. & rawat, p. s. (2017). demonetisation: macroeconomic implications for indian economy. institute for studies in industrial development, working paper, 197 new delhi. dave, n. s. (2016). closure of atm operations notification. rbi notifications (rbi/201617/111). https://www.rbi.org.in/scripts/notificationuser.aspx?id=10683&mode=0 dhoot, v. (2016). farmers can purchase seeds with banned rs. 500 notes. the hindu, november 21 ghosh, a. (2017). impact of demonetization on india: a macro-theoretic analysis. trade and development review, 9(1–2), 57–73. killawala, a. (2016a). fourth bi-monthly monetary policy statement, 2016–17 resolution of the monetary policy committee (mpc), reserve bank of india. rbi press releases. killawala, a. (2016b). rbi press release 2016–2017/1142. https://rbi.org.in/scripts/bs_pressreleasedisplay.aspx?prid=38520 killawala, a. (2016b). rbi press release 2016–2017/1142. https://rbi.org.in/scripts/bs_pressreleasedisplay.aspx?prid=38520 killawala, a. (2017). sixth bi-monthly monetary policy statement, 2016–17 resolution of the monetary policy committee (mpc), reserve bank of india. rbi press releases. modi, n. (2017). pm modi's 2016 demonetisation speech that shocked india. business standard, 8. rao, k., mukherjee, s., kumar, s. et al. (2016). demonetisation: impact on the economy. new delhi: nipfp working paper series, 182. rowlatt, j. (2016). why india wiped out 86% of its cash overnight. bbc news, november 14. sebastian a. roy dynamic econometric models 19 (2019) 41–56 56 roy, s. a. (2016). czy 14% tygrysa to wciąż tygrys? reforma demonetyzacyjna w indiach. monitoring makroekonomiczny. sharma, v. (2016, 12 6). india's demonetisation: 'modi didn't think of the poor'. al jazeera. https://www.aljazeera.com/indepth/features/2016/11/india-demonetisation-modi-didnpoor-161123073645259.html shirley, m. a. (2017). impact of demonetization in india. international journal of trend in research and development, special issue, 20–23. http://www.ijtrd.com/papers/ijtrd7787.pdf siddiqui, m. a., mishra, m. k., tiwari, p. (2017). impact of demonetisation in india. international journal of advance scientific research and engineering trends, 2(1), 14–18. sławiński, a. (2011). polityka pieniężna. warszawa: c. h. beck. the economist. (2016). the dire consequences of india’s demonetisation initiative. the economist, december 3. https://www.economist.com/news/finance-and-economics/21711035-withdrawing-86-value-cash-circulation-india-was-bad-idea-badly vijaya kumar, p. (2016a). rbi instructions to banks notification. rbi notifications https://rbi.org.in/scripts/notificationuser.aspx?id=10684&mode=0 vijaya kumar, p. (2016b). revisions for farmers/traders registered with apmc/mandis. rbi notifications. https://www.rbi.org.in/scripts/notificationuser.aspx?id=10730&mode=0 vijaya kumar, p. (2016c). withdrawal of legal tender character of existing ₹ 500/and ₹ 1000/bank notes – applicability of the scheme to dccbs. rbi notifications https://rbidocs.rbi.org.in/rdocs/notification/pdfs/130nta11a67017ce54b49a788afdc1599277d.pdf white, l., & rajagopalan, s. (2016). the indian government's $100 billion heist. foundation for economic education, december 3. https://fee.org/articles/the-indian-governments-100-billion-heist/ worstall, t. (2016). absolutely true – demonetization will boost india's gdp – but how true is it? forbes, december 4. https://www.forbes.com/forbes/welcome/?tourl=https://www.forbes.com/sites/timworstall/2016/12/04/absolutely-true-demonetisation-will-boost-indias-gdp-but-howtrue-is-it/&refurl=https://www.google.com/&referrer=https://www.google.com/ microsoft word 00_tresc.docx © 2013 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2013.006 vol. 13 (2013) 107−125 submitted june 30, 2013 issn accepted december 14, 2013 1234-3862 agnieszka kapecka* fractal analysis of financial time series using fractal dimension and pointwise hölder exponents∗∗ a b s t r a c t. this paper presents a fractal analysis application to the verification of assumptions of fractal market hypothesis and the presence of fractal properties in financial time series. in this research, the box-counting dimension and pointwise hölder exponents are used. achieved results lead to interesting observations related to nonrandomness of price series and occurrence of relationships binding fractal properties and variability measures with the presence of trends and influence of the economic situation on financial instruments’ prices. k e y w o r d s: fractal analysis, fractal dimension, box-counting dimension, pointwise hölder exponents, hurst exponent. j e l classification: g14, g15, g17. introduction ever since mandelbrot had published his works on the application of r/s analysis to long-memory dependencies in time series (mandelbrot, wallis, 1969; mandelbrot, 1972) and since peters had presented his fractal market hypothesis (peters, 1991) as an alternative to commonly acknowledged efficient market hypothesis, this approach is being explored with regard to financial time series. mulligan examined the use of lo’s modified rescaled range analysis on foreign exchange markets (mulligan, 2000), proving the * correspondence to: agnieszka kapecka, warsaw school of economics, the collegium of economic analysis, al. niepodległości 162, 02-554 warsaw, poland, e-mail: a.kapecka@gmail.com. ∗∗ this work was financed by the author. agnieszka kapecka dynamic econometric models 13 (2013) 107–125 108 presence of long memory dependencies and fractal structure of analysed price series. another published research assumed hurst exponent estimation using geometrical interpretation (granero, segovia, pérez, 2008) applied to stock market indices or the search for periodic and nonperiodic components in s&p 500 time series (bohdalová, greguš, 2010). another group of studies focused on the analysis of variation of hurst exponent over time, showing the possible impact of capital flow and trading volume on the decrease of hurst exponent values (cajueiro, tabak, 2004), the influence that the end of bretton woods system had on efficiency of us stock markets (alvarezramirez, alvarez, rodriguez, fernandez-anaya, 2008), or the relationship between local hurst exponent and stock market crashes with example of the warsaw stock exchange index (grech, pamuła, 2008). due to certain limitations of classical r/s analysis approach and hurst exponent itself, some of the authors explored hölderian pointwise regularity of some major stock market indices (bianchi, pantanella, 2010) and usage of multifractal spectra analysis in order to discover patterns of change in price series before the 1987 market crash and other significant market drawdowns (los, yalamova, 2004). in the presented paper, the initial assumption is that the markets are not efficient, but are fractal in their nature. despite the fact that efficient market hypothesis (emh) has been commonly accepted as a default theory explaining the fundamentals of financial markets’ behavior, plenty of criticism and doubts have been adressed towards it. the critical remarks are mainly related to too strong assumptions underlying this hypothesis, vastly mangling the real world behavior of the markets. in terms of real market behavior, a chaos theory based fractal market hypothesis (fmh) seems to be much more appriopriate. it assumes – on the contrary to emh, which uses linear differential equation – that the market is a nonlinear dynamic system, which allows to suppose that „real feedback systems involve long-term correlations and trends, because memories of long-past events can still affect the decisions made in the present” (peters, 1997, p. 6). actions of market participants usually generate nonlinear behavior of financial processes. functions describing investors’ attitute towards risk, their expectations towards stock market returns or financial instruments pricing formulae are nonlinear as well (osińska, 2006, p. 118). the most characteristic property of fmh is acknowledgement that stock market returns time series have fractal (selfsimilar) structure. fmh also allows chaotic behavior of the market during particular periods and under certain conditions (peters, 1994, pp. 46–48). fmh was described by peters based on the results of long studies conducted by hurst in the first half of the 20th century (peters, 1991). the main fractal analysis of financial time series using fractal dimension… dynamic econometric models 13 (2013) 107–125 109 conclusion was that most of natural systems do not follow the random walk model, but are subject to fractional brownian motion. such theory is in line with peters’ assumption about the markets, which says that globally (in the long-term) the market is deterministic, while locally (in the short-term), due to randomly occuring information and emotional reactions of market participants, the market is random (peters, 1997, pp. 45, 64). empirical confirmation of this hypothesis is presented in the section 4 of this paper. 1. fractal dimension one of the most substantial characteristics of geometric object is its dimension. however, an attempt to analyse this matter is nontrivial due to the fact that so far the scientists provided many different definitions of dimension, namely: the topological dimension, the hausdorff dimension, the fractional dimension, the box-counting dimension, the self-similarity dimension etc. the reasoning behind using particular types of dimension depends on certain conditions and while sometimes using different types of dimensions can lead to similar results, it might as well show varying results for the same object (peitgen, jürgens, saupe, 2002, p. 274). in brief, a dimension describes the way in which a geometric object (or time series) fills the space. a common characteristic of all fractal objects is presence of self-similarity, which means that there is a relation between the reduction coefficient (a scale of similarity) and the amount of reduced fragments similar to the original object. while analyzing zigzag-shaped financial instrument time series chart in terms of dimension, it is easy to conclude that its dimension falls into range ( )1, 2 . as zigzag is not a straight line, it has dimension which is definitely distinct from 1, but it is not two-dimensional as it does not fill the entire plane. there are several kinds of fractal dimensions. one of them is the boxcounting dimension, which – in view of its use as a research tool – is worth presenting. below description of the box-counting dimension was written based on: peitgen, jürgens, saupe (2002), mastalerz-kodzis (2003), kudrewicz (2007) and borys (2011). let f be a certain geometric object embedded in n -dimensional euclidean space ,nr covered with a set of small cubes (boxes called hypercubes) which sides are equal to ε (e.g., for 1=n it will be segments, for 2=n squares, for 3=n cubes). theoretically these cubes can be discretionary oriented with respect to the axes of a coordinate system, but they can be as well spheres or other convex solids with a diameter .ε let ( )ε,fn be agnieszka kapecka dynamic econometric models 13 (2013) 107–125 110 a minimum amount of cubes that can completely cover the entire object .f having satisfied this, below relations are true: − ( )ε,fn ~ ε/1 – for f being a segment of a smooth line (amount of cubes is approximately inversely proportional to the length of cube side), − ( )ε,fn ~ ( )2/1 ε – for f being a piece of smooth plane, − ( )ε,fn ~ ( )3/1 ε – for f being an area contained in .3r with respect to the above, a generalization can be made that there are such geometric objects, for which – assuming small values of ε (which is a scale of similarity, so that the smaller the values of ,ε the better the approximation) – below commensurateness is approximately met: ( )ε,fn ~ ( ) ,/1 dε (1) whereas – while in majority of cases there is only an approximation of commensurateness between the two sides of the equation – in case of strictly self-similar objects an equality sign can be placed so that: ( )( , ) 1 / ,dn f ε ε= (2) where d (not necessarily being an integer number) can be treated as a dimension of the object .f then the limit: ( ) ( ) ( ) , /1log ,log lim 0 ε ε ε fn fd f → = (3) if it exists, is called a fractal dimension (in this case, a box-counting dimension)1. it is easy to conclude from (3), which is applicable to exact fractals, that a dimension of such objects is defined by: ( ) ( ) ( ) . /1log ,log ε εfn fd f = (4) a possibility to apply the box-counting dimension led to creation of algorithms capable of estimating the dimension of time series. in this method, the analyzed structure is placed on a regular grid with a size of ,ε followed by 1 if a given unit gets reduced ε times, the measured line will approximately contain ε times more units than in the previous iteration step (approximation is in this case a result of a possible presence of small curves disturbing the existing commensurateness). given commensurateness is exact only in the limit, because after reducing the length of the sunit it is possible to achieve a infinitely small unit only in the limit (tempczyk, 1995, p. 135). fractal analysis of financial time series using fractal dimension… dynamic econometric models 13 (2013) 107–125 111 counting all the cubes (boxes), which contain the fragments of analyzed structure. the amount of fragments is obviously dependent on the value of .ε in the next stage it is necessary to repeat the calculations for smaller values of ε and plot the obtained values on the logarithmic chart (marking ( )log ,n f ε with respect to ( )log 1 / ε ). the slope of the straight line fitted to the points marked on the plot determines the box-counting dimension of the time series (daros, 2010, p. 13). 2. measures of variability of the graph of the function one of the most popular and useful measures of variability (irregularity) are hurst exponent and hölder exponents. the hurst exponent is a numerical characteristic of the entire price series, whereas hölder exponents can be used to analyze the complexity of the function and the trajectory of some stochastic processes in the vicinity of any point of the graph of the function (mastalerz-kodzis, 2003, p. 37). the most important turning point in the subject of long-term dependency analysis of time series was without any doubt the creation of rescaled range analysis n)s/r( method (hurst, 1951). a starting point of his analysis was einstein’s work on brownian motion (einstein, 1908), which presented an equation for the distance r that a particle travels in time ,t which is defined by ,tcr = where c is a nonnegative constant. this equation was applicable when the series of increments of the distance travelled by the particle in time was a random walk, characterised by the independency of normally distributed random variables (weron, weron, 1998, p. 323). however, during almost fourty years of research, hurst has reached a conclusion that the majority of natural phenomena is not subject to gaussian random walk, but rather to processes with „long memory”, later called by the name of fractional brownian motion, which is a combination of a trend and noise (peters, 1997, p. 64; mastalerz-kodzis, 2003, pp. 37–38). derivation of a formula for rescaled range allowed the comparison of different types of time series. creation of this dimensionless indicator, which should increase over time, allowed to formulate the following equation being an extension of brownian motion model proposed by einstein. ,nc)s/r( hn ⋅= (5) where )s/r( – rescaled range, n – number of observations, c – positive constant, h – hurst exponent. agnieszka kapecka dynamic econometric models 13 (2013) 107–125 112 in order to calculate hurst exponent, one has to calculate the average value of n)s/r( for different n and then solve the following equation using linear regression: log ( / ) log( ) log( ),ne r s h n c= + (6) where n)s/r(e – expected value of rescaled range. in above equation, the hurst exponent can be treated as the regression coefficient and estimated using the least squares method (jajuga, papla, 1997). the hurst exponent is strictly linked to the fractal dimension of time series, therefore the search for the hurst exponent is in fact a search for the fractal properties of the series. this relation is described by the following equation (grech, 2012, p. 10): .2 hd f −= (7) this equation has a huge practical importance as it can be used to classify the type of a time series depending on the fractal dimension of a given object. following cases can be distinguished based on hurst exponent values (peters, 1997, p. 76–77): − if ,5.00 << h then 25.1 << d (antipersistent time series), − if ,5.0=h then 5.1=d (random walk), − if ,15.0 << h then 5.11 << d (persistent time series). first case ( 5.00 <≤ h ) applies to antipersistent (ergodic) time series. such a series has a mean reversion tendency. if in a given period the value of the series increased, then in the following period it will most probably decrease and vice versa. the closer the value of h to ,0 the more ergodic the behavior of the system and the time series graph has more jagged line, which is a result of a frequent trend reversion. in such case the fractal dimension of the series 2→fd as the series fills the plane more and more. the lower the h value, the more noise can be observed in the system. speaking in the probability language, if e.g. ,2.0=h then there is 80% probability that in the future the market will change the direction, which will be equal to trend reversion (stawicki, janiak, müller-frączek, 1997, p. 37). despite the fact that mean reversion plays a dominant role in economic and financial literature, so far only a few antipersistent time series have been observed. the second case applies to a situation, when with ∞→n ,5.0=h which corresponds to a random walk (the consecutive elements of the series fractal analysis of financial time series using fractal dimension… dynamic econometric models 13 (2013) 107–125 113 are independent). the fractal dimension of the series equals ,5.1=d the series itself is unpredictable, the present does not influence the future, and the past did not influence the present. the probability distribution function can be gaussian, but not necessarily. both in natural and economic phenomena the h exponent’s value usually differs from ,5.0 and the natural processes most often have a long-term data dependency. when ,15.0 ≤< h the time series are persistent, which means that they bolster the trend. it is caused by the presence of long-term data dependency. when ,1→h the trend gets stronger. as hurst exponent defines the probability of consecutive rises or drops of the prices, with ,1→h there are more consecutive rises or drops and the level of noise becomes smaller. for example, if ,8.0=h then there is 80% probability that a given trend will be sustained in the future. the fractal dimension ( )5.1,1∈d as the more persistent the time series, the less it fills the plane and the smoother are the curves created by a given system. fractal time series is obviously not purely deterministic, it is rather an intermediate form between a completely random time series and a deterministic system. persistent time series are fractional brownian motions, which means that their important feature is a biased random walk, and the strength of bias increases when ,1→h so when the hurst exponent value recedes from .5.0 despite n)s/r( analysis is a very useful tool, it poses a major disadvantage – it does not take into consideration the changes in particular subperiods. for example, if a particular stock is subject to rapid price changes, whereas the prices of other stocks show only minor price changes, it is not possible to detect this periodical changes using hurst exponent. therefore, a good measure of variability of the graph of a function over time is a hölder function2, which values in particular points are equal to pointwise hölder exponents (mastalerz-kodzis, 2003, p. 115). in order to further discuss pointwise hölder exponents, let us first define the hölder function3. let )d,x( x and )d,y( y be metric spaces, then function yx:f → is called a hölder function with exponent ,α where ,0>α if 2 although pointwise hölder exponents are considered to be the best measure of function regularity in the vicinity of a certain point, other measures used include: local box-counting dimension, local hausdorff dimension, the degree of fractional differentiability. 3 the definition of hölder function and pointwise hölder exponents was written based on: mastalerz-kodzis 2003, pp. 49–51; kuperin, schastlivtsev, 2008, pp. 4–6.  agnieszka kapecka dynamic econometric models 13 (2013) 107–125 114 for each xy,x ∈ such that 1)y,x(d x < the function satisfies the following inequality: ,),()](),([ αyxcdyfxfd xy ≤ (8) where c – a positive constant. assuming that function rd:f → and that parameter ),1,0(∈α function f is a αc class ( α∈cf ) hölder function, if such 0c > and 0h0 > constants exist that for every x and every ( )0h,0h∈ the following inequality is satisfied: .ch)x(f)hx(f α≤−+ (9) assuming that 0x is an arbitrary point from the domain of function ,f so that ,rdx0 ⊂∈ function rd:f → is a α 0x c class ( α∈ 0x cf ) hölder function in ,x0 if such 0c > and 0>ε constant exist that for every )x,x(x 00 ε+ε−∈ the following inequality is satisfied: .xxc)x(f)x(f 00 α −≤− (10) by definition hölder function is continous in its entire domain and when this assumption is satisfied, the graph of the function has fractal nature (gabryś, 2005, p. 24). if the hölder function is not continuous, it is called generalized hölder function. it is worth noting that thanks to its time-varying values, hölder function can take different types of random walks in different ranges (kutner, 2009, p. 36). after providing a hölder function definition it is possible to define a pointwise hölder exponent of function f in .x0 by definition it is a number )x( 0fα given by the following equation: { }. cf:sup)x( 0x0f α∈α=α (11) approximated pointwise hölder exponent is not a flawless measure, however its major advantage is the ability to accomodate the stationarity of the series. interpretation of pointwise hölder exponent is the same as for hurst exponent with a difference that pointwise hölder exponent estimates local, not global value. it is also worth mentioning that hölder functions are not constant as they are time-varying (mastalerz-kodzis, 2003, p. 121). fractal analysis of financial time series using fractal dimension… dynamic econometric models 13 (2013) 107–125 115 3. tools and analysis methodology 3.1. data all monthly and daily price series were downloaded from http://stooq.pl. table 1 presents the list of stock market indices and forex currency pairs used in this research. table 1. financial instruments chosen for the analysis symbol available data period market type djia 1896.05–2012.12 mature market s&p 500 1923.01–2012.12 mature market dax 1959.10–2012.12 mature market nikkei225 1949.05–2012.12 mature market hang seng 1969.11–2012.12 mature market wig20 1991.04–2012.12 emerging market bovespa 1992.01–2012.12 emerging market rts 1995.09–2012.12 emerging market sensex30 1979.04–2012.12 emerging market sci 1990.12–2012.12 emerging market xu100 1990.01–2012.12 emerging market eur/usd 1980.01–2012.12 major currency pair gbp/usd 1971.01–2012.12 major currency pair usd/jpy 1971.01–2012.12 major currency pair chf/pln 1990.01–2012.12 exotic currency pair eur/pln 1990.01–2012.12 exotic currency pair usd/brl 1995.01–2012.12 exotic currency pair usd/rub 1995.10–2012.12 exotic currency pair usd/inr 1973.01–2012.12 exotic currency pair usd/cny 1984.01–2012.12 exotic currency pair usd/try 1984.01–2012.12 exotic currency pair the above selection is reasoned by analysis of both mature and emerging markets and the associated currency pairs in order to check if any relationships interesting from the fractal analysis point of view are present. 3.2. tools during the analysis, microsoft excel was used to calculate the common logarithms of price values and plot the charts of logarithmic price series and pointwise hölder exponents. for the purposes of box-counting dimension estimation and pointwise hölder exponents calculation, fraclab 2.0 was used. as quoted from fraclab homepage (http://fraclab.saclay.inria.fr/): „fraclab is a general purpose signal and image processing toolbox based on fractal and multifractal methods. (…) a large number of procedures allow to agnieszka kapecka dynamic econometric models 13 (2013) 107–125 116 compute various fractal quantities associated with 1d or 2d signals, such as dimensions, hölder exponents or multifractal spectra. (…) fraclab is a free software developed in the regularity team at inria saclay/ecole centrale de paris.” 3.3. analysis methodology of the fractal analysis methods described in sections 1 and 2, in this research the box-counting dimension, hurst exponent and pointwise hölder exponents were used for long-term dependency analysis of chosen financial time series. table 2. values of fractal dimension d and hurst exponent h calculated on the entire data range of monthly and daily price series of chosen financial instruments symbol period monthly data daily data d h d h djia 1896.05–2012.12 1.44 0.56 1.43 0.57 s&p 500 1923.01–2012.12 1.40 0.60 1.40 0.60 dax 1959.10–2012.12 1.50 0.50 1.47 0.53 nikkei225 1949.05–2012.12 1.36 0.64 1.44 0.56 hang seng 1969.11–2012.12 1.47 0.53 1.50 0.50 wig20 1991.04–2012.12 1.43 0.53 1.48 0.52 bovespa 1992.01–2012.12 1.22 0.78 1.48 0.52 rts 1995.09–2012.12 1.47 0.53 1.44 0.56 sensex30 1979.04–2012.12 1.37 0.63 1.46 0.54 sci 1990.12–2012.12 1.50 0.50 1.46 0.54 xu100 1990.01–2012.12 1.31 0.69 1.46 0.54 eur/usd 1980.01–2012.12 1.49 0.51 1.47 0.53 gbp/usd 1971.01–2012.12 1.56 0.44 1.44 0.56 usd/jpy 1971.01–2012.12 1.49 0.51 1.42 0.58 chf/pln 1990.01–2012.12 1.32 0.68 1.44 0.56 eur/pln 1990.01–2012.12 1.35 0.65 1.43 0.57 usd/brl 1995.01–2012.12 1.34 0.66 1.40 0.60 usd/rub 1995.10–2012.12 1.20 0.80 1.36 0.64 usd/inr 1973.01–2012.12 1.23 0.77 1.32 0.68 usd/cny 1984.01–2012.12 1.16 0.84 1.36 0.64 usd/try 1984.01–2012.12 1.08 0.92 1.39 0.61 fractal analysis of financial time series using fractal dimension… dynamic econometric models 13 (2013) 107–125 117 4. fractal analysis using fractal dimension and pointwise hölder exponents 4.1. presentation of results the research was divided into two parts. in the first part, the dimensions of chosen price series were estimated and equation (7) was used to derive hurst exponent from the estimated dimension. first, the dimensions of price series were calculated for the entire historical data range (see table 2). table 3. values of fractal dimension d and hurst exponent h calculated on the oct 1995 to dec 2012 data range of monthly and daily price series of chosen financial instruments symbol period monthly data daily data d h d h djia 1995.10–2012.12 1.41 0.59 1.48 0.52 s&p 500 1995.10–2012.12 1.36 0.64 1.45 0.55 dax 1995.10–2012.12 1.39 0.61 1.44 0.56 nikkei225 1995.10–2012.12 1.50 0.50 1.48 0.52 hang seng 1995.10–2012.12 1.60 0.40 1.51 0.49 wig20 1995.10–2012.12 1.52 0.48 1.48 0.52 bovespa 1995.10–2012.12 1.46 0.54 1.49 0.51 rts 1995.10–2012.12 1.46 0.54 1.44 0.56 sensex30 1995.10–2012.12 1.51 0.49 1.47 0.53 sci 1995.10–2012.12 1.53 0.47 1.44 0.56 xu100 1995.10–2012.12 1.37 0.63 1.45 0.55 eur/usd 1995.10–2012.12 1.53 0.47 1.49 0.51 gbp/usd 1995.10–2012.12 1.52 0.48 1.52 0.48 usd/jpy 1995.10–2012.12 1.50 0.50 1.49 0.51 chf/pln 1995.10–2012.12 1.51 0.49 1.50 0.50 eur/pln 1995.10–2012.12 1.57 0.43 1.49 0.51 usd/brl 1995.10–2012.12 1.36 0.64 1.40 0.60 usd/rub 1995.10–2012.12 1.20 0.80 1.36 0.64 usd/inr 1995.10–2012.12 1..40 0.60 1.39 0.61 usd/cny 1995.10–2012.12 1.26 0.74 1.56 0.44 usd/try 1995.10–2012.12 1.21 0.79 1.40 0.60 in the second part, pointwise hölder exponents were calculated for six chosen markets. figures 1–6 present pointwise hölder exponents plotted together with common logarithms of time series values. the range of values taken by pointwise hölder exponents’ values is presented on the right of each graph. dynamic 118 4.2. an the especial the evidenc majority long-ter out of 6.0≥h nature, h series w remark econom not bein a fractio figure 1 ano fraction wherea variety case of exponen monthly econometric nalysis of re e interpretatio lly in the ligh e part of res ce supporting y of the inve rm trends du 21 markets .6 only 4 out having 5.0 ≤ was detected about rare my. such resu ng a random onal brownia . s&p500 pri other interest nal dimension as the researc of values ( 0 daily data fo nt values are y data and fa ag c models 13 (2 esults on of the res ht of fractal a search based g the fractal estigated mar uring entire a s having sig t of 21 mark ,51.0≤≤ h a (gbp/usd w occurrence ults seem to m walk, but an motion. ce series and p ting evidence n and hurst ch carried ou 044.0 ≤≤ h or 16 out of 2 e usually co all into a narr gnieszka kapec 2013) 107–12 sults leads to analysis and d on fractal nature of fi rkets (16 out available his gnificantly n kets displaye and just in a with 4.0=h of antipersi o prove peter rather a co pointwise höl e was gained exponent va ut for monthl 92. for the e 21 examined onsiderably l row range ( 0 cka 25 o some inter fractal mark dimension inancial time t of 21) reve story of the p nonrandom ed a random single case t 44 ), which i istent time s rs’ concept mbination o lder exponent d based on th alues for mon ly price serie entire history d markets it t lower than r 6.05.0 ≤≤ h resting obser ket hypothesi provided int e series. firs ealed the pre price series, nature, den m or close to the antiperist is in line with series in nat of financial of trend and ts he compariso nthly and da es resulted in y of price se turned out th respective re 6 for the entir rvations, is. teresting st of all, esence of with 12 noted by random tent time h peters’ ture and markets noise – on of the aily data. n a wide eries), in hat hurst sults for re histo figure 2 figure 3 ry of pr data ran fractal m is determ tively v ularity o tomated fractal ana . dax price s . nikkei225 p rice series a nge). this m market hypo ministic, wh erified. mor of electronic d algorithmic lysis of financi d series and poin price series an and 48.0 ≤ h might mean othesis, sayin hile locally (i eover, it seem c trading, lar c trading and ial time series dynamic econ ntwise hölder nd pointwise h 57.0≤h for that anothe ng that globa in the shortms that durin rger trading d high freque using fractal nometric mod r exponents hölder expone r october 19 er peters’ as ally (in the lo -term) it is ra ng recent yea volumes and ency trading dimension… els 13 (2013) ents 995–decemb ssumption re ong-term) the andom, can ars the grow d introductio systems, mig 107–125 119 ber 2012 elated to e market be posiwing popon of aught have dynamic 120 increase biggest us and ingly, o the exot (usd/b of signi suggests for this is out o only fra figure 4 the range an type lea correspo of hurs od refle 1990s a respect seng is during l and 198 econometric ed the efficie increase of japan stock one group of tic currency brl, usd/c ificant trends s that the un group of ma of scope of t actal analysis . xu100 price e observation nd october ad to a concl ond to the u t exponent v ects the dyna and most of tively, the de indicative f last two deca 80s. another ag c models 13 (2 ency of the m efficiency is k market indi f investigated pairs linked cny, usd/in s on the dail nderlying eco arkets. neve this paper an s, but also ec e series and po ns made duri 1995–decem lusion that f underlying ec values for dj amic growth 2000s in the ecrease of hu for the econ ades after the interesting o gnieszka kapec 2013) 107–12 markets in th s visible for ices or majo d markets is to emerging nr, usd/ru ly data interv onomic phen ertheless, furt nd might req onomic facto ointwise höld ing the analy mber 2012 d fractal dimen conomic situ jia, s&p500 h and propiti e economies urst exponen nomic slowd e preceeding observations cka 25 he short-term the world’s or forex curre an exceptio g markets ex ub, usd/tr val indicated omena are su ther research quire more s ors, e.g. capi der exponents ysis of result ata range wi nsion and hu uation. for ex 0 and dax d ious econom s of united s nt values for n down in japa g very aggres include the m (especially biggest mark ency pairs). n to this rule xamined in th ry) show a p d by 6.0≥h ubstantially h on such hy studies inclu ital flows. ts for the en ith respect to urst exponen xample, the during the lat mic condition states and g nikkei225 a an and sout sive growth i growth of ef y that the kets like intereste. all of his paper presence 6 , which different ypothesis uding not ntire data o market nt values increase tter perins in the germany. and hang th korea in 1970s fficiency of majo reflects econom figure 5 figure 6 the hölder random fractal ana or forex curr the huge in my. . eur/usd p . usd/jpy pr e second part exponents in nature of th lysis of financi d rency pairs nfluence that price series an rice series and t of the resea n fractal ana he markets. ial time series dynamic econ or the rando t chinese go d pointwise h d pointwise h arch, focused alysis, provid for all six e using fractal nometric mod omness of s overnment h hölder expone ölder exponen d on the app ded addition examined ma dimension… els 13 (2013) sci, which p has on the c ents nts lication of p nal evidence arkets, the v 107–125 121 probably country’s pointwise on nonvalues of agnieszka kapecka dynamic econometric models 13 (2013) 107–125 122 the pointwise hölder exponents are significantly higher than 5.0 during most of the entire data range, which proves the presence of long-term dependencies in the investigated time series. moreover, the graphs expose the relationship between the trend strength and hölder exponents’ values as they tend to considerably decrease and converge to 5.0 during recent years sideways price movements (which is especially visible for nikkei225, dax or eur/usd). one of the pros of using the box-counting dimension method in this article is that due to its nature, it does not require assumption on the selfsimilarity of the analysed object, which allows to avoid making a possibly false statement even before the beginning of the research. another considerable advantage of the methods used in this article was their algorithmic accessibility, thus making it affordable from the calculation complexity point of view and available even for the users which do not have a high-end processor cluster at their disposal. nevertheless, as the estimation method assumes that the hölder function is constant over each of the intervals, it might introduce some error of method to the results. conclusions the analysis conducted in this research provides solid empirical evidence in favour of fractal market hypothesis, with the presence of longterm dependencies in the financial time series and the confirmation of the global determinism and local randomness of the markets being the most important ones. an important observation supporting the nonrandomness of the markets is a relationship between fractal properties of the investigated time series and the underlying economic situation. one of the most interesting researches that could be made in the future based on this would be a repetition of the calculations at some fixed intervals of time – after ten, twenty or thirty years. such approach could possibly verify if the opinions stated by economists a posteriori major changes in economic conditions are in line with the conclusions drawn from the fractal characteristics of analysed markets. another study that could be made as a continuation of research conducted in this article could involve inclusion of additional test data, this time not limited to market time series. inclusion of some regularly published economic indicators like gross domestic product, consumer price index or money supply could potentially reveal some additional relationships and regularities and shed more light on the macroeconomic processes. fractal analysis of financial time series using fractal dimension… dynamic econometric models 13 (2013) 107–125 123 moreover, the analysis of results suggests that fractal analysis can be a valuable tool for the evaluation of market trends, which might be of practical use for institutional and individual investors. references alvarez-ramirez, j., alvarez, j., rodriguez, e., fernandez-anaya, g. (2008), time-varying hurst exponent for us stock markets, physica a, 387, 6159–6169, doi: http://dx.doi.org/10.1257/002205103765762743. bianchi, s., pantanella, a. (2010), stock returns declustering under time dependent hölder exponent, international conference on e-business, management and economics, 14–21 (ipedr, 3 (2011) © (2011) iacsit press, hong kong). bohdalová, m., greguš, m. (2010), markets, information and their fractal analysis, eleader, new york: casa, 2, 1–8. borys, p. (2011), sztuczki karciane, wylewy nilu i wykładnik hursta (card tricks, nile effusions and hurst exponent), foton (photon), 113, 4–22. cajueiro, d. o., tabak b. m. (2004), the hurst exponent over time: testing the assertion that emerging markets are becoming more efficient. physica a, 336, 521–537, doi: http://dx.doi.org/10.1257/002205103765762743. daros, ł. (2010), analiza porównawcza fraktalnych własności polskich i obcojęzycznych tekstów pozaliterackich (comparative analysis of fractal properties of polish and foreign illiterary texts), master thesis, akademia górniczo-hutnicza, kraków. einstein, a. (1908), elementare theorie der brownschen bewegung, zeitschrift für elektrochemie und angewandte physikalische chemie, 14, 50, 496–502, doi: http://dx.doi.org/10.1257/002205103765762743. gabryś, a. (2005), rynek kapitałowy w ujęciu fraktalnym (fractal approach to capital market), webdoc, http://www.aureamediocritas.pl/30-lista-publikacji (11.08.2009). grech, d. k. (2012), nowatorskie metody badania szeregow czasowych i układow złożonych z zastosowaniami w ekonofizyce i fizyce (innovative methods of research on time series and complex systems with applications in econophysics and physics), instytut fizyki teoretycznej, uniwersytet wrocławski, wrocław. grech, d., pamuła g. (2008), the local hurst exponent of the financial time series in the vicinity of crashes on the polish stock exchange market, physica a, 387, 4299–4308, doi: http://dx.doi.org/10.1257/002205103765762743. hurst, h. e. (1951), long term storage capacity of reservoirs, transactions of the american society of civil engineers, 116, 770–799. jajuga, k., papla, d. (1997), teoria chaosu w analizie finansowych szeregów czasowych – aspekty teoretyczne i badania empiryczne (chaos theory in financial time series analysis – theoretical aspects and empirical research), uniwersytet mikołaja kopernika w toruniu, toruń, http://www.dem.umk.pl/dme/1997.htm (12.11.2009). kudrewicz, j. (2007), fraktale i chaos (fractals and chaos), wydawnictwa naukowotechniczne, warszawa. kutner, r. (2009), symulacje komputerowe procesów syngularnych i osobliwych w finansach – wybrane algorytmy (computer-aided simulations of singular and peculiar processes in finance – selected algorithms), working paper, http://www.fuw.edu.pl/tl_files/ studia/materialy/ef /hurst_finance.pdf (12.02.2010). agnieszka kapecka dynamic econometric models 13 (2013) 107–125 124 kuperin, yu. a, schastlivtsev, r. r. (2008), modified hölder exponents approach to prediction of the usa stock market critical points and crashes, arxiv: 0802, 4460, physics and society, 2008, 20, http://xxx.lanl.gov. los, c.,yalamova r. (2004), multifractal spectral analysis of the 1987 stock market crash, kent state university working paper, finance 0409050, econwpa. mandelbrot, b. (1972), statistical methodology for nonperiodic cycles from covariance to r/s analysis, annals of economic and social measurement, 1, 259–290. mandelbrot, b., wallis, j. r. (1969), robustness of the rescaled range r/s in the measurement of noncyclic long-run statistical dependence, water resources research, 5, 967–988, doi: http://dx.doi.org/10.1257/002205103765762743. mastalerz-kodzis, a. (2003), modelowanie procesów na rynku kapitałowym za pomocą multifraktali, (process modelling on the capital market using multifractals), wydawnictwo akademii ekonomicznej w katowicach, katowice. mulligan, r. f. (2000), a fractal analysis of foreign exchange markets, international advances in economic research, 1, 33–49, doi: http://dx.doi.org/10.1257/002205103765762743. osińska, m. (2006), ekonometria finansowa (financial econometrics), polskie wydawnictwo ekonomiczne, warszawa. peitgen, h-o., jürgens, h., saupe, d. (2002), granice chaosu: fraktale 1 (borders of chaos: fractals 1), polskie wydawnictwo naukowe, warszawa. peters, e. e. (1991), chaos and order in the capital markets: a new view of cycles, prices, and market volatility, wiley, new york. peters, e. e. (1994), fractal market analysis: applying chaos theory to investment and economics, wiley, new york. peters, e. e., (1997), teoria chaosu a rynki kapitałowe (chaos theory in the capital markets), wig-press, warszawa. sánchez granero, m. a., trinidad segovia, j. e., garcía pérez, j. (2008), some comments on hurst exponent and the long memory processes on capital markets, physica a: statistical mechanics and its applications, 387, 5543–5551, doi: http://dx.doi.org/10.1257/002205103765762743. stawicki, j., janiak, e. a., müller-frączek, i. (1997), różnicowanie fraktalne szeregów czasowych – wykładnik hursta i wymiar fraktalny (fractal differentiation of time series – hurst exponent and fractal dimension), uniwersytet mikołaja kopernika w toruniu, toruń, http://www.dem.umk.pl/dme/1997.htm (12.11.2009). tempczyk, m. (1995), świat harmonii i chaosu (world of harmony and chaos), państwowy instytut wydawniczy, warszawa. weron, a., weron, r. (1998), inżynieria finansowa (financial engineering), wydawnictwo naukowo-techniczne, warszawa. fractal analysis of financial time series using fractal dimension… dynamic econometric models 13 (2013) 107–125 125 fraktalna analiza finansowych szeregów czasowych z wykorzystaniem wymiaru fraktalnego oraz punktowych wykładników höldera z a r y s t r e ś c i. artykuł przedstawia propozycję zastosowania analizy fraktalnej w celu weryfikacji niektórych założeń hipotezy rynku fraktalnego oraz występowania fraktalnych właściwości w finansowych szeregach czasowych. w celu przeprowadzenia badań wykorzystany został wymiar pudełkowy oraz punktowe wykładniki höldera. rezultaty osiągnięte dla badanych rynków pozwoliły dokonać interesujących obserwacji dotyczących nielosowości szeregów cenowych oraz występowania relacji między fraktalnymi właściwościami i miarami zmienności a obecnością trendów i wpływem sytuacji ekonomicznej na ceny instrumentów finansowych. s ł o w a k l u c z o w e: analiza fraktalna, wymiar fraktalny, wymiar pudełkowy, punktowe wykładniki höldera, wykładnik hursta. microsoft word 00_tresc.docx © 2013 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2013.002 vol. 13 (2013) 33−50 submitted july 27, 2012 issn accepted april 3, 2013 1234-3862 joanna olbryś* asymmetric impact of innovations on volatility in the case of the us and ceec–3 markets: egarch based approach a b s t r a c t. the main goal of this study is to investigate the asymmetric impact of innovations on volatility in the case of the us and three biggest emerging ceec–3 markets, using univariate egarch approach. we compare empirical results for both the whole sample from jan 3, 2007 to dec 30, 2011, and two equal subsamples: the ‘down market’ period, and the ‘up market’ period. pronounced negative asymmetry effects are presented in the case of all markets, and are especially strong in the ‘down market’ period, which is closely connected with the 2007 us subprime crisis period. k e y w o r d s: volatility, asymmetry effect, down and up market, overlapping information set, univariate egarch model. j e l classification: c32, c58, g15. introduction the us stock market is found to be the most influential market in the world. the results of many studies support the evidence for us dominance in the international stock markets, and therefore the s&p 500, the main index of the new york stock exchange, is universally accepted as a benchmark index, both in the case of developed and emerging markets research (e.g. eun, shim, 1989; hamao et al., 1990; koutmos, booth, 1995; tse et al., 2003; syriopoulos, 2007; lee, stewart, 2010; baumöhl, výrost, 2010; * correspondence to: joanna olbryś, faculty of computer science, bialystok university of technology, wiejska 45a, 15-351 bialystok, poland, e-mail: j.olbrys@pb.edu.pl. joanna olbryś dynamic econometric models 13 (2013) 33–50 34 olbryś, 2013). the recent research evaluates the transmission of the us subprime crisis to both developed and emerging markets. however, the emerging markets responded very strongly to the deteriorating situation in the us financial system and real economy. for example, dooley’s and hutchson’s (2009) regression ‘event study’ focusing on 15 types of news, indicates that a range of financial and real economic news emanating from the us had statistically and economically large impacts on 14 emerging markets and several news events uniformly moved markets. as a matter of course, there are extremely more ‘bad’ than ‘good’ news during the ‘crisis’ period. nelson (1991) points out that researchers beginning with black (1976) found evidence that stock returns are negatively correlated with changes in returns volatility, i.e. volatility tends to rise in response to ‘bad news’ (excess returns lower than expected) and to fall in response to ‘good news’ (excess returns higher than expected). the purpose of this paper is to investigate the asymmetric impact of innovations on volatility in the case of three biggest emerging ceec–31 markets, and (for comparison) for the us stock market, using univariate egarch approach (nelson, 1991). we try to deal with the ‘nonsynchronous trading effect ii’ by using a ‘common trading window’ procedure and estimating suitable egarch models based on daily open–to–close logarithmic returns for the four major stock market indexes: s&p 500 (new york), wig (warsaw), px (prague), and bux (budapest). the main goal is to obtain an overlapping information set in the case of the ceec–3 markets, as we test the impact of common ‘bad’ and ‘good’ news. we compare empirical results for both the whole sample from jan 3, 2007 to dec 30, 2011 and two equal subsamples: feb 12, 2007 to mar 9, 2009 as the ‘down market’ period and mar 10, 2009 to mar 10, 2011 as the ‘up market’ period. we observe pronounced negative asymmetry effects in the case of all markets, especially in the ‘down market’ period, which is closely connected with the 2007 us subprime crisis period. to the best of author’s knowledge, no such comparative investigation has been undertaken for the us and ceec–3 stock markets. as mentioned above, the impact of ‘bad’ and ‘good’ news is described in terms of univariate egarch models while e.g. büttner and hayo (2012) advocate to take into consideration ‘actual news’ in economic sense (e.g. emu–related news, news from the ecb, and the like)2. they analyze the 1 three biggest emerging central and eastern european countries (ceec-3), in order of largest population size are: poland, the czech republic, and hungary (büttner, hayo, 2012). 2 emu – european economic and monetary union; ecb – european central bank. asymmetric impact of innovations on volatility… dynamic econometric models 13 (2013) 33–50 35 impact of news on three financial markets in poland, the czech republic, and hungary. the remainder of the paper is organized as follows. section 1 specifies a methodological background and a brief literature review. first, we stress the validity of the nonsynchronous trading problem. next, we present the motivation for the choice of ‘down market’ and ‘up market’ subperiods. a brief theoretical framework concerning the egarch(p, q) models is also presented. in section 2, we present the data and an empirical analysis of the asymmetric impact of innovations on volatility in the case of the us developed stock market (as a benchmark market) and the three biggest emerging ceec–3 markets. then we discuss the results obtained. conclusion recalls the main findings and sums them up. 1. methodological background 1.1. the non-trading problem some studies distinguish between two nonsynchronous trading effect problems. the first problem, called ‘nonsynchronous trading effect i’, occurs when we analyze one selected domestic stock market. stock tradings do not occur in a synchronous manner. different stocks have different trading frequencies, and even for a single stock the trading intensity varies from hour to hour and from day to day. the actual time of last transaction of the stock varies from day to day. as such we incorrectly assume daily returns as an equally spaced time series with a 24-hour interval (tsay, 2010, p. 232). the non–trading effect induces potentially serious biases in the moments and comoments of asset returns such as their means, variances, covariances, betas, and autocorrelation and cross-autocorrelation coefficients (e.g. campbell et al., 1997; doman, 2011). the second and potentially serious problem, called ‘nonsynchronous trading effect ii’, occurs when we examine the relations between stock markets in various countries. the national stock markets are operating in diverse time zones with different opening and closing times, thereby making return observations nonsynchronous (eun, shim, 1989). these differences arise naturally from the fact that trading days in different countries are subject to different national and religious holidays, unexpected events, and so forth (baumöhl, výrost, 2010). this paper investigates the asymmetric impact of innovations on volatility in the case of the three biggest emerging ceec–3 stock markets, and (for comparison) for the us market. for this reason, we have to deal with the ‘nonsynchronous trading effect ii’. many studies attempted various methods joanna olbryś dynamic econometric models 13 (2013) 33–50 36 to deal with the ‘nonsynchronous trading effect ii’. some researchers use weekly or monthly data to avoid the non–trading problem. such solutions, however, may lead to small sample sizes and cannot capture the information transmission in shorter (daily) timeframes (baumöhl, výrost, 2010). other papers present various daily data–matching procedures. for example, hamao et al. (1990) divide daily close–to–close returns into their close–to–open and open–to–close components. to examine how a nonsynchronous problem would affect the relationships between selected markets, some researchers estimate suitable garch–type models (e.g. multivariate egarch models) based on the open–to–close returns (cf. koutmos, booth, 1995; tse et al., 2003; olbrys, 2013). syriopoulos (2007, p. 46) says: ‘(…) it would have been ideal to use both the open and close stock market prices, in order to reduce potential non–synchronous trading bias between the us and the european stock markets.’ in many studies the following approach, also called a ‘common trading window’, is very popular: the data are collected for the same dates across the stock markets, removing the data for those dates when any series has a missing value due to no trading (e.g. eun, shim, 1989; booth et al., 1997; olbrys, 2013). 1.2. motivation for the choice of subperiods in our research, we compare empirical results of asymmetric effects of innovations on volatility in the case of the us and ceec–3 markets for both the whole sample from jan 3, 2007 to dec 30, 2011 and two equal subsamples: feb 27, 2007 to mar 9, 2009 as the ‘down market’ period and mar 10, 2009 to mar 10, 2011 as the ‘up market’ period (each consists of 476 observations). syczewska (2010) proposed somewhat different subsamples as ‘crisis’ (‘down market’) and ‘post-crisis’ (‘up market’) periods, but we advocate feb 27, 2007 as the beginning of the ‘down market’ period following dooley and hutchison (2009), and march 9, 2009 as the end of the ‘down market’ period because of the global minimum of the s&p 500 index value in the whole sample achieved on this day. the overall s&p 500 index fell from 1399.04 (feb 27, 2007) to 676.53 (march 9, 2009). it lost 51.64% of previous value during the ‘down market’ period, which is closely connected with the 2007 us subprime crisis period. dooley and hutchison (2009) focus their analysis on the links between the us and a broad range of emerging equity markets over a subprime crisis sample period from feb 2007 to march 2009, including poland, hungary, and the czech republic amongst others. mun and brooks (2012) extend the dooley’s and hutchison’s analysis to a broader set of individual developed and emerging markets, and also asymmetric impact of innovations on volatility… dynamic econometric models 13 (2013) 33–50 37 extend the whole sample period to feb 2010 (full 3 years). frank and hesse (2009) find that end-february 2007 was a period when early signs of stress began to emerge in global markets prior to the time when the subprime crisis was revealed in mid–2007. 1.3. the exponential garch model many researchers documented that stock return volatility tends to rise following ‘good’ and ‘bad’ news. this phenomenon was noted both for individual stocks and for market indexes (braun et al., 1995). since nelson (1991) introduced the univariate exponential generalized autoregressive conditionally heteroskedastic (egarch) model, some papers employ this model to capture the asymmetric effect of innovations on volatility. several studies present various applications of univariate and multivariate egarch models. in (koutmos, booth, 1995) the transmission mechanism of price and volatility spillovers across the new york, tokyo and london stock markets from three different time zones is investigated, using the egarch approach. jane and ding (2009) propose the multivariate extension of nelson’s univariate egarch model and compare their model with the existing one given by koutmos and booth (1995). booth et al. (1997) provide the evidence on price and volatility spillovers among four scandinavian (nordic) stock markets. bhar (2001) applies an extended bivariate egarch model to provide evidence of linkages between the equity market and the index futures market in australia. reyes (2001) examines volatility transfers between size–based indexes from the tokyo stock exchange, using a bivariate egarch model. tse et al. (2003) employ a bivariate egarch model that allows for both mean and variance spillovers between the us and polish stock markets. balaban and bayar (2005) test the relationship between stock market returns and their forecast volatility derived from the symmetric and asymmetric garch–type models in 14 countries. lee and stewart (2010) examine asymmetric effects on volatility in the case of the baltic and nordic major stock indexes, using both univariate and multivariate egarch models. olbrys (2013) investigates the interdependence of price volatility across the us developed stock market and two emerging central and eastern european (cee) markets in warsaw and budapest using a multivariate modified egarch model. as a matter of fact, the asymmetric effects of innovations on volatility for one selected domestic stock market could be well described by the univariate egarch model, although it is now widely accepted that a multivariate modeling framework (in the case of the group of markets) leads to joanna olbryś dynamic econometric models 13 (2013) 33–50 38 more relevant empirical models than working with separate univariate models (bauwens et al., 2006). but it is worth stressing that the multivariate egarch model estimation is particularly difficult due to the large number of estimated parameters. the univariate time series }{ tr can be expressed as: ,ttt εμr += (1) where: )( 1−= ttt freμ is the conditional expectation of tr given the past information 1−tf , tε is the innovation of the series at time t . nelson’s univariate egarch(p, q) model can be represented as follows (tsay, 2010): 1 1 12 0 1 1 1 1 ln , 1 ~ (0, ), ~ (0,1), q q t tp p t t t t t t t β b β b (σ ) α g(z ) α b α b ε z , f n z n σ ε σ − − − − + + + = + ⋅ − − − = ⋅ (2) ,][ )ze(zγzθ)g(z tttt −⋅+⋅= (3) where: )( 1−= ttt frvarσ is the conditional variance of tr given the past information 1−tf , 0α is a constant, b is the back-shift (or lag) operator such that )g(z)bg(z tt 1−= , 1 111 − −+++ q q bβbβ and p p bαbα −−− 11 are polynomials with zeros outside the unit circle and have no common factors. the value of )g(zt depends on several elements. nelson (1991) points out that to accommodate the asymmetric relation between stock returns and volatility changes, the value of )g(zt must be a function of both the magnitude and the sign of tz . in eq. (3), )g(zt is a linear combination of tz and )][ tt ze(z − with coefficients θ and γ . the term in the bracket measures the magnitude effects and the coefficient γ relates lagged standardized innovations to volatility in a symmetric way. the term tz⋅θ measures the sign asymmetric impact of innovations on volatility… dynamic econometric models 13 (2013) 33–50 39 effects and the coefficient θ relates standardized shocks to volatility in an asymmetric style. ∞−∞= ,}{ tt )g(z is an i.i.d. random sequence with mean zero (jane, ding, 2009). for 0<θ the future conditional variances will increase proportionally more as a result of a negative shock than for a positive shock of the same absolute magnitude (bollerslev, mikkelsen, 1996). both tz and )][ tt ze(z − are zero mean i.i.d. random sequences with continuous distributions. the asymmetry of )g(zt can be easily seen by rewriting it as: ⎩ ⎨ ⎧ <⋅−⋅− ≥⋅−⋅+ = .0)( ,0)( ttt ttt t zif)ze(γzθ zif)ze(γzθ )g(z γ γ (4) since egarch(p, q) = egarch(1, 1) is a simple case, eq. (2) becomes: ,1ln1 101 2 1 )g(zαb)α()(σb)α( tt −+⋅−=⋅− (5) eq. (5) can be rewritten (subscript of 1α is omitted) and then: ,lnln 1 2 10 2 )g(z)(σαα)(σ tt * t −− +⋅+= (6) where .0 constα * = the parameter α in eq. (6) determines the influence of the past conditional volatility on the current conditional volatility. for the conditional volatility process to be stationary, 1<α is required. the persistence of volatility may be also quantified by examination of the half–life ( hl ) defined by: α ).( hl ln 50ln = (7) which measures the time period required for the innovations to be reduced to one–half of their original size. an additional advantage of the egarch model is that no parameter restrictions are required to insure positive variances at all times (fiszeder, 2009). let ti ti ti o c r , , , ln100 ⋅= be the open–to–close percentage logarithmic return at time t for market i ( 4,3,2,1=i , where 1 = new york, 2 = warsaw, joanna olbryś dynamic econometric models 13 (2013) 33–50 40 3 = prague, and 4 = budapest). then, the univariate ar(1)–egarch(1, 1) model for market i may be written as follows: .)ze(zγzθ)(σαα)(σ ,εrr ttitii,ti * i,i,t i,ti,tii,i,t ][lnln 2 10 2 10 −⋅+⋅+⋅+= ++= − −ϕϕ (8) 2. empirical results 2.1. data description and preliminary statistics the raw data consists of daily opening and closing prices of major stock market indexes for new york (s&p 500 index), warsaw (wig index), prague (px index), and budapest (bux index). as mentioned in introduction, the main goal was to obtain the overlapping information set in the case of the ceec–3 markets, as we tested the impact of common ‘bad’ and ‘good’ news. we used the ‘common trading window’ procedure and removed the data for those dates when any series has a missing value due to no trading. thus all the data are collected for the same dates across the four markets and finally there are 1181 observations for each series for the period beginning jan 3, 2007 and ending dec 30, 2011. since ceec–3 countries are geographically close, the trading hours for the markets are about the same. trading at the wse (wig index) starts at 9:00 a.m. and finishes at 5:40 p.m. cet (central european time). prague (px index) trades from 9:00 a.m. to 4:30 p.m., budapest (bux index) trades from 9:00 a.m. to 5:00 p.m. while the nyse (s&p 500 index) trades from 3:30 p.m. to 10:00 p.m. cet3. the trading overlap between the ceec–3 and new york markets is approximately equal to one and a half hours, i.e. late trading in warsaw, prague or budapest corresponds to early trading in new york. we advocate to use daily open–to–close logarithmic returns, as these returns inform about the situation on a given stock market between the opening and closing time. we compute daily close–to–close, close–to–open, and open–to–close logarithmic returns for the four stock indexes. following hamao et al. (1990), we divide daily close-to-close (c–c) logarithmic returns into their close–to–open (c–o) and open–to–close (o–c) components: ,ln 1− =− t t c c cc ,ln 1− =− t t c o oc ,ln t t o c co =− (9) 3 sources: http://www.standardandpoors.com/ ; http://www.gpw.pl/ ; http://www.pse.cz/ ; http://bse.hu/. asymmetric impact of innovations on volatility… dynamic econometric models 13 (2013) 33–50 41 we obtain consequently that a daily close–to–close logarithmic return can be expressed as: ,lnlnlnln 111 −−− =⋅=+ t t t t t t t t t t c c o c c o o c c o (10) where tc and 1−tc are the closing prices of days t and )1( −t , respectively, and to is the opening price of day t . note that on a given day t , because the ceec–3 markets open before the us market, daytime information set from the us market would have an influence on the ceec–3 markets on the next day. an information set can be seen in broad terms as the set of all information relevant for pricing an asset at a given time (baumöhl, výrost, 2010). therefore, the information on the opening and closing values of the ceec–3 and us stock markets indexes does not belong to the same information set, however, the information set for the ceec–3 markets is overlapping. table 1 reports summarized statistics for the close–to–close, close–to– open, and open–to–close logarithmic returns for four stock indexes: s&p 500, wig, px, and bux, as well as statistics testing for normality and interdependence. the sample means are not statistically different from zero. the measures for skewness and excess kurtosis show that all return series are negatively skewed and highly leptokurtic with respect to the normal distribution. likewise, the doornik–hansen (2008) test rejects normality for each of the return series at the 5 per cent level of significance. the ljung– –box (1978) statistic at the lag tq ln≈ , where t is the number of data points (tsay, 2010, p. 33), calculated for both the return and the squared return series, indicates the presence of significant linear and non-linear dependencies, respectively, except the wig o–c and px o–c series. the linear dependences may be due to the ‘nonsynchronous trading effect i’ of the stocks that make up each index (e.g. campbell et al., 1997). the non–linear dependences may be due to the autoregressive conditional heteroskedasticity (e.g. nelson, 1991; koutmos, booth, 1995; booth et al., 1997). all calculations were done using gretl 1.9.11 (adkins, 2012). joanna olbryś dynamic econometric models 13 (2013) 33–50 42 table 1. summarized statistics for the close–to–close, close–to–open, and open–to– close logarithmic returns for four stock indexes: s&p 500, wig, px, and bux number of obs. mean standard deviation skewness excess kurtosis doornik– hansen test lb(7) lb 2(7) s&p 500 c–c 1260 –9⋅10 –5 0.017 –0.25* 6.53* 729.71* [0.0] 34.42* 742.66* s&p 500 c–o 1260 –9⋅10 –5 0.002 –0.28* 7.29* 836.62* [0.0] 28.13* 173.47* s&p 500 o–c 1260 4⋅10 –7 0.016 –0.31* 6.62* 724.33* [0.0] 33.84* 740.99* wig c–c 1256 –2⋅10 –4 0.015 –0.37* 2.63* 174.69* [0.0] 17.57* 268.82* wig c–o 1256 6⋅10 –4 0.009 –0.76* 5.38* 370.39* [0.0] 27.44* 552.66* wig o–c 1256 –8⋅10 –4 0.013 –0.29* 2.83* 209.38* [0.0] 7.27 255.28* px c–c 1258 –4⋅10 –4 0.018 –0.57* 12.66* 1541.46* [0.0] 31.08* 778.58* px c–o 1258 4⋅10 –4 0.013 –0.71* 13.35* 1545.51* [0.0] 28.21* 609.05* px o–c 1258 –8⋅10 –4 0.013 –1.34* 12.00* 731.53* [0.0] 9.44 199.15* bux c–c 1254 –3⋅10 –4 0.020 –0.02* 5.67* 624.96* [0.0] 52.16* 584.76* bux c–o 1254 7⋅10 –4 0.011 –0.29* 9.61* 1190.26* [0.0] 49.11* 869.96* bux o–c 1254 –1⋅10 –3 0.017 –0.47* 3.81* 288.20* [0.0] 19.77* 320.37* note: the table is based on all sample observations during the period jan 2, 2007–dec 31, 2011. c-c, c-o, and o-c stand for close-to-close, close-to-open, and open-to-close logarithmic returns for four stock indexes (s&p 500, wig, px, bux), respectively. * denotes significance at the 5 per cent level. the test statistic for skewness and excess kurtosis is the conventional t-statistic. the doornik-hansen test (2008) has a χ2 distribution if the null hypothesis of normality is true. numbers in brackets are p-values. lb(q) and lb2(q) are the ljung-box (1978) statistics for returns and squared returns, respectively, distributed as χ2 (q), q≈lnt, where t is the number of data points (tsay, 2010). the χ2 (7) critical value is 14.07 (5%). 2.2. asymmetric impact of innovations on volatility to examine asymmetric effects between positive and negative index return innovations, we first estimate the univariate ar(1)–egarch(1,1) models of the four stock indexes: s&p 500, wig, px, and bux, in the whole sample period from jan 3, 2007 to dec 30, 2011. the robust qml asymmetric impact of innovations on volatility… dynamic econometric models 13 (2013) 33–50 43 (bollerslev, wooldridge, 1992) estimates of the parameters of the model (8) are presented in table 24. table 2. results from the ar(1)–egarch(1, 1) models of the four stock indexes: s&p 500, wig, px, and bux. full sample period from jan 3, 2007 to dec 30, 2011 (1181 daily open–to–close percentage logarithmic returns) new york (i = 1) warsaw (i = 2) prague (i = 3) budapest (i = 4) conditional mean equation ,0iφ 0.024 (0.033) –0.060* (0.029) –0.052* (0.026) –0.092* (0.039) iφ –0.084* (0.027) –0.022 (0.026) –0.055 (0.030) –0.056 (0.031) conditional variance equation * ,0iα –0.101* (0.020) –0.108* (0.028) –0.166* (0.034) –0.117* (0.030) iα 0.978* (0.007) 0.983* (0.007) 0.977* (0.011) 0.979* (0.010) iθ –0.141* (0.020) –0.083* (0.019) –0.047* (0.020) –0.034 (0.020) iγ 0.142* (0.026) 0.144* (0.037) 0.227* (0.049) 0.179* (0.044) conditional density parameters iν 7.515* (1.713) 10.923* (2.989) 5.743* (0.846) 6.538* (1.178) iλ –0.227* (0.041) –0.019 (0.033) –0.078* (0.039) –0.007 (0.043) asymmetry effect for market i /i i iδ θ γ= –0.99 –0.58 –0.21 –0.19 half–life (hl) 31.16 40.43 29.79 32.66 log–likelihood –1872.80 –1833.79 –1735.00 –2145.68 bic 3802.19 3724.16 3526.59 4347.92 aic 3761.60 3683.58 3486.00 4307.64 lb(20) 11.42 [0.93] 14.23 [0.82] 7.97 [0.99] 16.80 [0.67] lb2(20) 25.76 [0.17] 14.39 [0.81] 16.60 [0.68] 13.89 [0.84] note: the table is based on all sample observations during the period jan 3, 2007–dec 30, 2011; * denotes significance at the 5 per cent level; the heteroskedastic consistent standard errors are in parentheses; the variance-covariance matrix of the estimated parameters is based on the qml algorithm; the distribution for the innovations is supposed to be skewed t; ν and λ are conditional density parameters (lucchetti, balietti s, 2011, p. 3); the asymmetry coefficient is defined in the text; the half-life is defined in the text and represents the time it takes for the shock to reduce its impact by one-half; bic and aic are the information criterions; lb(20) and lb2(20) denotes the ljung-box (1978) statistics for standardized innovations and squared standardized innovations, respectively (baillie, bollerslev, 1990); numbers in brackets are p-values. 4 in the case of all periods analyzed, the choice of an appropriate version of the egarch model was conducted based on the bic and aic information criterions, and distributions for the innovations were supposed to be normal, t-student, or skewed t. as it turned out, the univariate ar(1)–egarch(1, 1) models with skewed t as the distribution for the innovations are the most adequate. due to the space restrictions, details and calculations are available upon request. joanna olbryś dynamic econometric models 13 (2013) 33–50 44 for model checking, the ljung–box statistics lb(20) for the standardized innovation process, and lb2(20) for the squared standardized innovations were applied (baillie, bollerslev, 1990). the evidence is that there is no serial correlation or conditional heteroskedasticity in the standardized innovations of the fitted models. the estimated ar(1)–egarch(1, 1) models are adequate (tsay, 2010, p. 146). several results presented in table 2 are worth special notice. the autoregressive coefficients iϕ are negative, and this coefficient is statistically significant only for the new york market. the conditional variance is a function of past conditional variances and past innovations. the relevant coefficients iα , iθ , and iγ are statistically significant at the 5 per cent level in the case of all models (except 4θ ). in addition, all of the iγ coefficients are positive. for positive iγ , if 0/ <= iii γθδ , then negative innovations have a higher impact on volatility than positive innovations; if 0=iδ ( 0=iθ and 0>iγ ), then the magnitude terms raises (lowers) volatility when the magnitude of market movements is large (small); if 10 << iδ , then positive innovations would increase volatility but negative innovations decrease volatility. these pronounced negative asymmetry effects are present in table 2. for new york, warsaw, prague, and budapest, negative innovations increase volatility considerably more than positive innovations. our findings suggest that the four stock markets are more sensitive to ‘bad’ than ‘good’ news. the persistence of volatility may be interpreted by using the half–life concept (7), which measures the time it takes for an innovation to reduce its impact by one half. numerically, the hl coefficients for the new york, warsaw, prague, and budapest indexes are equal to: 31.16, 40.43, 29.79, and 32.66 days, respectively. it is worth stressing that the half–life coefficients are surprisingly high, however scheicher (2001, p. 37) documents half–life coefficients for the ctx, htx, and ptx indexes5, which are equal to: 16.39, 1.95, and ∞ (!) days. for example, bhar (2001) documents half–life coefficients equal to 2.63 and 3.86 days for two australian spot and futures markets, respectively. figure 1. presents time plots of conditional variances from the univariate ar(1)–egarch(1, 1) models for the s&p 500, wig, px, and bux index 5 ctx, htx, and ptx are the czech, hungarian, and polish traded indexes, which are computed by the central european clearing houses and exchange (cece) in vienna (scheicher (2001, p. 28). asymmetric impact of innovations on volatility… dynamic econometric models 13 (2013) 33–50 45 es, in the whole sample period from jan 3, 2007 to dec 30, 2011. the global financial crisis was reflected evidently in all stock exchanges (cf. figure 1). figure 1. conditional variances from the univariate ar(1)–egarch(1, 1) models for the s&p 500, wig, px, and bux indexes, in the whole sample period from jan 3, 2007 to dec 30, 2011 (table 2). tables 3a–3b present further analysis, including details about results from the ar(1)–egarch(1, 1) models of the four stock indexes in: − the ‘down market’ period from feb 27, 2007 to mar 9, 2009 (table 3a), − the ‘up market’ period from march 10, 2009 to mar 10, 2011 (table 3b). 0 5 10 15 20 25 30 2007 2008 2009 2010 2011 c o n d it io n a l v a ri a n ce conditional variances from egarch(1, 1) (s&p 500) 0 2 4 6 8 10 12 2007 2008 2009 2010 2011 c o n d it io n a l v a ri a n ce conditional variances from egarch(1, 1) (wig) 0 5 10 15 20 25 30 2007 2008 2009 2010 2011 c o n d it io n a l v a ri a n ce conditional variances from egarch(1, 1) (px) 0 5 10 15 20 25 2007 2008 2009 2010 2011 c o n d it io n a l v a ri a n ce conditional variances from egarch(1, 1) (bux) joanna olbryś dynamic econometric models 13 (2013) 33–50 46 table 3a. results from the ar(1)–egarch(1, 1) models of the four stock indexes: s&p 500, wig, px, and bux. the ‘down market’ period from feb 27, 2007 to mar 9, 2009 (476 daily open–to–close percentage logarithmic returns) new york (i = 1) warsaw (i = 2) prague (i = 3) budapest (i = 4) conditional mean equation ,0iφ –0.086 (0.060) –0.139* (0.056) –0.056 (0.047) –0.119 (0.061) iφ –0.141* (0.041) –0.058 (0.043) –0.063 (0.048) 0.030 (0.047) conditional variance equation * ,0iα –0.069* (0.028) –0.061 (0.034) –0.131* (0.052) –0.162* (0.058) iα 0.965* (0.009) 0.970* (0.011) 0.960* (0.013) 0.972* (0.016) iθ –0.182* (0.035) –0.127* (0.029) –0.136* (0.043) –0.048* (0.038) iγ 0.117* (0.034) 0.098* (0.040) 0.188* (0.068) 0.250* (0.082) conditional density parameters iν 17.061 (14.100) 15.610 (10.273) 6.484* (1.710) 6.026* (1.651) iλ –0.253* (0.059) 0.017 (0.062) –0.126 (0.080) –0.025 (0.059) asymmetry effect for market i /i i iδ θ γ= –1.56 –1.30 –0.72 –0.19 half–life (hl) 19.46 22.76 16.98 24.41 log–likelihood –841.76 –806.98 –763.88 –858.78 bic 1732.82 1663.26 1577.06 1766.86 aic 1699.52 1629.96 1543.75 1733.55 lb(20) 15.35 [0.76] 23.40 [0.27] 11.69 [0.93] 23.47 [0.27] lb2(20) 27.21 [0.13] 13.60 [0.85] 17.09 [0.65] 12.19 [0.91] note: the table is based on observations during the ‘down market’ period february 27, 2007–march 9, 2009; * denotes significance at the 5 per cent level; the heteroskedastic consistent standard errors are in parentheses; the variance-covariance matrix of the estimated parameters is based on the qml algorithm; the distribution for the innovations is supposed to be skewed t; ; ν and λ are conditional density parameters (lucchetti, balietti s, 2011, p. 3); the asymmetry coefficient is defined in the text; the half-life is defined in the text and represents the time it takes for the shock to reduce its impact by one-half; bic and aic are the information criterions; lb(20) and lb2(20) denotes the ljung-box (1978) statistics for standardized innovations and squared standardized innovations, respectively (baillie, bollerslev, 1990); numbers in brackets are p-values. the results in tables 3a–3b clearly show that the asymmetric effects between positive and negative index return innovations are especially strong in the ‘down market’ period (cf. table 3a). all of the iγ coefficients are significantly positive, all of the iθ coefficients are significantly negative, and then suitable iii γθδ /= coefficients are negative, therefore we conclude that negative innovations have a higher impact on the volatility than positive innovations. it is worthwhile to note that the asymmetry effect is extremely strong asymmetric impact of innovations on volatility… dynamic econometric models 13 (2013) 33–50 47 in the case of the new york ( 56.11 −=δ ) and warsaw ( 30.12 −=δ ) markets. suitable half-life coefficients for the new york, warsaw, prague, and budapest indexes are equal to: 19.46, 22.76, 16.98, and 24.41 days, and are numerically comparable. table 3b. results from the ar(1)–egarch(1, 1) models of the four stock indexes: s&p 500, wig, px, and bux. the ‘up market’ period from mar 10, 2009 to mar 10, 2011 (476 daily open–to–close percentage logarithmic returns) new york (i = 1) warsaw (i = 2) prague (i = 3) budapest (i = 4) conditional mean equation ,0iφ 0.101* (0.000) –0.0003 (0.042) –0.065 (0.043) –0.028 (0.067) iφ –0.065* (0.000) 0.016 (0.049) –0.010 (0.039) –0.071 (0.052) conditional variance equation * ,0iα –0.159* (0.034) –0.129* (0.054) –0.146* (0.068) –0.074 (0.049) iα 0.971* (0.016) 0.984* (0.013) 0.962* (0.032) 0.989* (0.015) iθ –0.112* (0.041) –0.037 (0.034) 0.026 (0.036) –0.006 (0.025) iγ 0.216* (0.048) 0.162* (0.071) 0.193* (0.096) 0.105 (0.075) conditional density parameters iν 5.439* (1.183) 9.942* (3.997) 5.412* (1.215) 10.093* (3.981) iλ –0.148* (0.051) 0.057 (0.070) –0.021 (0.064) 0.045 (0.066) asymmetry effect for market i /i i iδ θ γ= –0.52 –0.23 0.13 –0.06 half–life (hl) 23.55 42.97 17.89 62.67 log–likelihood –674.60 –678.30 –649.60 –892.64 bic 1398.51 1405.90 1348.51 1834.60 aic 1365.21 1372.59 1315.20 1801.29 lb(20) 19.77 [0.47] 21.10 [0.39] 11.03 [0.95] 26.96 [0.14] lb2(20) 23.61 [0.26] 10.76 [0.95] 19.01 [0.52] 11.27 [0.94] note: the table is based on observations during the ‘up market’ period march 10, 2009–march 10, 2011; * denotes significance at the 5 per cent level; the heteroskedastic consistent standard errors are in parentheses; the variance-covariance matrix of the estimated parameters is based on the qml algorithm; the distribution for the innovations is supposed to be skewed t; ; ν and λ are conditional density parameters (lucchetti, balietti s, 2011, p. 3); the asymmetry coefficient is defined in the text; the half-life is defined in the text and represents the time it takes for the shock to reduce its impact by one-half; bic and aic are the information criterions; lb(20) and lb2(20) denotes the ljung-box (1978) statistics for standardized innovations and squared standardized innovations, respectively (baillie, bollerslev, 1990); numbers in brackets are p-values. as for the ‘up market’ period the evidence is that the estimated univariate egarch models are qualitatively rather poor. most of the parameters are not statistically significant at the 5 per cent level. essentially, the research provides evidence that the four markets are not homogeneous joanna olbryś dynamic econometric models 13 (2013) 33–50 48 regarding the asymmetric impact of innovations on volatility in the ‘up market’ period, as well as the half-life coefficient size. the asymmetric effects between positive and negative index return innovations are present in the case of three markets (i.e. new york, warsaw, and budapest). suitable half– life coefficients for the new york, warsaw, prague, and budapest indexes are equal to: 23.55, 42.97, 17.89, and 62.67 days, and are substantially higher compared those in the ‘down market’ period. this evidence confirms that the four stock markets are more sensitive to ‘bad’ than ‘good’ news. conclusions our research provides evidence for pronounced asymmetric impact of innovations on volatility in the case of the us and ceec–3 markets, especially in the ‘down market’ period (feb 27, 2007–march 9, 2009). we conclude that negative innovations have a higher impact on volatility than positive innovations. our findings suggest that the four stock markets are more sensitive to ‘bad’ than ‘good’ news. a possible and interesting direction for further investigation would be an asymmetry effects investigation in the case of the us and ceec–3 markets, in terms of other asymmetric garch–type models (cf. engle, 2000; bauwens et al., 2006). references adkins, l. c. (2012), using gretl for principles of econometrics, 4th edition, version 1.03. balaban, e., bayar, a. (2005), stock returns and volatility: empirical evidence from fourteen countries, applied economics letters, 12, 603–611, doi: http://dx.doi.org/10.1080/13504850500120607. baillie, r. t., bollerslev, t. (1990), a multivariate generalized arch approach to modeling risk premia in forward foreign exchange rate markets, journal of international money and finance, 9, 309–324, doi: http://dx.doi.org/10.1016/0261-5606(90)90012-o. baumöhl, e., výrost, t. (2010), stock market integration: granger causality testing with respect to nonsynchronous trading effects, finance a uver: czech journal of economics and finance, 60(5), 414–425. bauwens, l, laurent, s., rombouts, j.v.k. (2006), multivariate garch models: a survey, journal of applied econometrics, 21, 79–109, doi: http://dx.doi.org/10.1002/jae.842. bhar, r. (2001), return and volatility dynamics in the spot and futures markets in australia: an intervention analysis in a bivariate egarch–x framework, journal of futures markets, 21, 833–850, doi: http://dx.doi.org/10.1002/fut.1903. black, f. (1976), studies of stock market volatility changes, 1976 proceedings of the american statistical association, business and economic statistics section, 177–181. asymmetric impact of innovations on volatility… dynamic econometric models 13 (2013) 33–50 49 bollerslev, t., mikkelsen, h. o. (1996), modeling and pricing long memory in stock market volatility, journal of econometrics, 73, 151–184, doi: http://dx.doi.org/10.1016/0304-4076(95)01736-4. bollerslev, t., wooldridge, j. m. (1992), quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances, econometric reviews, 11, 143–179, doi: http://dx.doi.org/10.1080/07474939208800229. booth, g. g., martikainen t., tse y. (1997), price and volatility spillovers in scandinavian stock markets, journal of banking & finance, 21, 811–823, doi: http://dx.doi.org/10.1016/s0378-4266(97)00006-x. braun, p. a., nelson, d. b., sunier, a. m. (1995), good news, bad news, volatility, and betas, the journal of finance, 50(5), 1575–1603, doi: http://dx.doi.org/10.2307/2329327. büttner, d., hayo, b. (2012), emu-related news and financial markets in the czech republic, hungary and poland, applied economics, 44(31), 4037–4053, doi: http://dx.doi.org/10.1080/00036846.2011.587775. campbell j.y., lo a.w., mackinlay a.c. (1997), the econometrics of financial markets, princeton university press, new jersey. doman, m. (2011), mikrostruktura giełd papierów wartościowych (stock exchange microstructure), poznan university of economics press. dooley, m., hutchison, m. (2009), transmission of the u.s. subprime crisis to emerging markets: evidence on the decoupling – recoupling hypothesis, journal of international money and finance, 28, 1331–1349, doi: http://dx.doi.org/10.1016/j.jimonfin.2009.08.004. doornik, j. a., hansen, h. (2008), an omnibus test for univariate and multivariate normality, oxford bulletin of economics and statistics, 70, 927–939, doi: http://dx.doi.org/10.1111/j.1468-0084.2008.00537.x. engle, r. f. (ed.) (2000), arch. selected readings, oxford university press. eun, c. s., shim, s. (1989), international transmission of stock market movements, the journal of financial and quantitative analysis, 24(2), 241–256, doi: http://dx.doi.org/10.2307/2330774. fiszeder, p. (2009), modele klasy garch w empirycznych badaniach finansowych (the class of garch models in empirical finance research), torun, nicolaus copernicus university press. frank, n., hesse, h. (2009), financial spillovers to emerging markets during the global financial crisis, finance a uver: czech journal of economics and finance, 59(6), 507–521. hamao, y., masulis, r. w., ng, v. (1990), correlations in price changes and volatility across international stock markets, review of financial studies, 3(2), 281–307, doi: http://dx.doi.org/10.1093/rfs/3.2.281. jane, t-d, ding, c. g. (2009), on the multivariate egarch model, applied economics letters, 16, 1757–1761, doi: http://dx.doi.org/10.1080/13504850701604383. koutmos, g., booth, g. g. (1995), asymmetric volatility transmission in international stock markets, journal of international money and finance, 14(6), 747–762, doi: http://dx.doi.org/10.1016/0261-5606(95)00031-3. lee, j., stewart, g. (2010), asymmetric volatility and volatility spillovers in baltic and nordic stock markets, european journal of economics, finance and administrative sciences, 25, 136–143. ljung, g., box, g. e. p. (1978), on a measure of lack of fit in time series models, biometrika, 66, 67–72. joanna olbryś dynamic econometric models 13 (2013) 33–50 50 lucchetti, k., balietti, s., (2011), the gig package, version 2.2. mun, m., brooks, r. (2012), the roles of news and volatility in stock market correlations during the global financial crisis, emerging markets review, 13, 1–7, doi: http://dx.doi.org/10.1016/j.ememar.2011.09.001. nelson, d. b. (1991), conditional heteroskedasticity in asset returns: a new approach, econometrica, 59, 347–370, doi: http://dx.doi.org/10.2307/2938260. olbryś, j. (2013), price and volatility spillovers in the case of stock markets located in different time zones, emerging markets finance & trade, 49(2), 145–157, doi: http://dx.doi.org/10.2753/ree1540-496x4902s208. reyes, m .g. (2001), asymmetric volatility spillover in the tokyo stock exchange, journal of economics and finance, 25(2), 206–213, doi: http://dx.doi.org/10.1007/bf02744523. scheicher, m. (2001), the comovements of stock markets in hungary, poland and the czech republic, international journal of finance and economics, 6, 27–39, doi: http://dx.doi.org/10.1002/ijfe.141. syczewska, e. m. (2010), changes of exchange rate behavior during and after crisis, quantitative methods in economics, wuls press, 11(1), 145–157. syriopoulos, t. (2007), dynamic linkages between emerging european and developed stock markets: has the emu any impact?, international review of financial analysis, 16, 41–60, doi: http://dx.doi.org/10.1016/j.irfa.2005.02.003. tsay, r. s. (2010), analysis of financial time series, john wiley, new york. tse, y., wu, c., young, a. (2003), asymmetric information transmission between a transition economy and the u.s. market: evidence from the warsaw stock exchange, global finance journal, 14, 319–332, doi: http://dx.doi.org/10.1016/j.gfj.2003.09.001. asymetryczny wpływ dodatnich i ujemnych stóp zwrotu na zmienność w przypadku rynków stanów zjednoczonych, polski, czech i węgier: podejście oparte na modelu egarch z a r y s t r e ś c i. artykuł przedstawia badania dokumentujące asymetryczny wpływ dodatnich i ujemnych stóp zwrotu na zmienność w przypadku rynków stanów zjednoczonych, polski, czech i węgier, z wykorzystaniem jednorównaniowych wykładniczych modeli egarch (nelson, 1991). porównawcze analizy empiryczne dotyczą okresu styczeń 2007–grudzień 2011 oraz dwóch jednakowo licznych podokresów: spadków (27.02.2007– –9.03.2009) i wzrostów (10.03.2009–10.03.2011). stwierdzono wyraźny efekt asymetrii na wszystkich badanych rynkach, szczególnie silny w wyróżnionym okresie spadkowym, wyznaczonym w oparciu o zmiany wartości indeksu s&p500 i ściśle związanym z okresem kryzysu finansowego w stanach zjednoczonych. s ł o w a k l u c z o w e: zmienność, efekt asymetrii, okresy spadków i wzrostów, wspólny zbiór informacji, model egarch. acknowledgements i would like to thank prof. evzen kocenda and prof. jan hanousek for providing me with the data on prague stock exchange index. gacr grant no. 403/11/0020 as a source of the data is acknowledged. i am especially indebted to anonymous referees for their valuable comments and suggestions which greatly improved the paper. microsoft word 00_tresc.docx dynamic econometric models vol. 10 – nicolaus copernicus university – toruń – 2010 piotr płuciennik adam mickiewicz university, national bank of poland forecasting financial processes by using diffusion models a b s t r a c t. time series forecasting is one of the most important issues in the financial econometrics. in the face of growing interest in models with continuous time, as well as rapid development of methods of their estimation, we try to use the diffusion models to modeling and forecasting time series from various financial markets. we use monte-carlo-based method, introduced by cziraky and kucherenko (2008). received forecasts are confronted with those determined with the commonly applied parametrical time series models. k e y w o r d s: diffusion models, ex-post forecasts, monte-carlo simulation, the garch model, the arima model, unit-root. 1. introduction models with continuous time and its particular case – diffusion models are exceptionally important class of models. on the developed financial markets there are available quotations containing full information about transaction prices, so called tick-by-tick data. it provides natural motivation to applying diffusion models or jump diffusion models to examination of financial instruments price series. diffusion models were initially used to short-term rate modeling (merton, 1973; vašiček, 1977; cox, at al. 1985). they gained in importance in the early 70s, when black and scholes (1973) introduced european call and put option pricing model in which the underlier price was modeled with simple diffusion model called geometric brownian motion. in the following years many modification of black-scholes models were introduced. merton (1974) supposed additionally that the risk-free rate is modeled by diffusion model, and heston (1993) assumed that the volatility process is described by mean reverting diffusion models. forms of blackscholes formula were introduced for american options, term options and even volatility index options, which had just appeared on the market. piotr płuciennik 52 other application areas of models with continuous time are issues concerning modeling and forecasting interest rates (term structure) and valuation of very complicated derivatives based on debt instruments. this subject was brought up among others by jagannathan at al. (2004), which applied multidimensional cir models to caps and swaptions pricing, tamba (2006), which used hull-white diffusion model to bermudian swaption pricing, and mannolini, mari, renò (2008), which priced caps and floors by extended cir models. the aforementioned derivatives are of outstanding significance. they played the crucial role in risk management and in aggressive investment strategies. the first hedging strategies were proposed by black and scholes (1973). nowadays there are strategies which allow to hedge the positions in swaption (javaheri at al., 2004; howison at al., 2004) and vix options (psychoyios, skiadopoulos 2006; sepp 2008; broadie, jain 2008). in the following article we use diffusion models to forecast the logarithmic levels of dax, cac40, nasdaq and wig20 indexes. the parameter estimates were obtained by modern phillips and yu (2009) method and more classical, introduced by hansen (1982), generalized method of moments (gmm) with covariance matrix, estimated by using bartlett kernels, as a weight matrix (newey, west, 1987). we determine the forecasts by using monte-carlo methods and compare its quality with the forecasts which we obtained by using popular parametrical time series models. 2. models we use popular diffusion models to describe logarithmic prices of financial instruments. the diffusion models were originally used to describe the evolution of short-term rates. their significant feature is the mean reverting property. the most simple diffusion model – vašíček (1977) model – assume that the price process is modeled by the following stochastic differential equation ttt dbdtxdx σμκ +−= )( , with initial condition .00 xx t = parameters ,κ μ and σ are strictly positive. parameter μ can be interpreted as a long term mean level, κ as a speed of reversion, and σ as a instantaneous volatility. cox, ingersoll and ross (1985) introduced model called cir, which is an extension of vašíček model. the tx evolution is described by the formula tttt dbxdtxdx σμκ +−= )( , with initial condition .00 xx t = parameters ,κ μ and σ are strictly positive, and have the same interpretation as in vašíček model. the square root in the forecasting financial processes by using diffusion models 53 diffusion function allows to avoid the possibility of nonpositive values of tx , provided that the condition 22 σκμ > is met. chan at al. (1992) introduced model called ckls. authors assume that tx evolution is described by the following diffusion models tttt dbxdtxdx βσμκ +−= )( , with initial condition .00 xx t = similarly as in vašíček and cir model, ,κ μ and σ are strictly positive. an additional β parameter is called the elasticity of variance parameter, and ]1,0[∈β . by simply placing the appropriate restrictions on the four parameters ,κ ,μ σ and β we can obtain 7 other diffusion models – among others the vašíček and the cir model. (see chan at al., 1992). 3. determination of one-day ex-post forecast from diffusion models denote the l-step forecast of ltx + as )(ˆ lx t . assuming that the minimum squared error is the loss function, the forecast )(ˆ lx t is the random variable chosen such that ,)],...,([min)](ˆ[ 21 2 xxgxlxx ttgtlt −≤−+  where ),...,( 1xxg t is measurable function towards σ-algebra generated by the information available up to time t inclusive. we can show that )|()(ˆ thtt xhx += ( . therefore, if we assume that process is described by diffusion models and length of one step equals ,h then ,)()ˆ,,()ˆ,,( )()ˆ,,(+)ˆ,,( )()ˆ,,(+)ˆ,,()(ˆ 21 21 0 2 0 10 ∫ ∫ ∫∫ ∫∫ + + ++ ++ ++= ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ += ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ += δ δ δδ δδ θσθμ θσθμ θσθμ ht t ht t sst ht t s ht t st t ht s ht st sdbsxdssxx sdbsxdssxx sdbsxdssxxhx  + + where 1̂θ and 2̂θ are parameter estimates vectors of drift and diffusion respectively, obtained on the quotations up to time s . as we know, piotr płuciennik 54 the quotations are available only in discrete intervals. consequently, we approximate the forecast )(ˆ lx t by the euler scheme of the form .)ˆ,,( ,)ˆ,,()(ˆ 1 0 1 0 )( 1 )( |1 )( | )()( ∑ ∑ − = − = +++ ++= h k h k ktnktnkttt txtxxhx δδδδδ εδθσδθμ (1) moreover, for all t holds .]/[ )( δ δ tt xx = cziraky i kucherenko (2008) obtain estimates of )(ˆ )( lx t δ by repeating the above recursion using n independent realizations of innovations vectors ),,....,( )()( 1 δδ εε knn ++ and for any realization they determine the trajectory (1) with initial condition .)(δtx the mc estimator of )(ˆ lx t is then given by the average of the last elements of every trajectory. one-step ex-post forecasts are obtained by fitting the model using data up to time t, and then computing the usual fitted equation and residuals for periods 1+t to ,ft + with additional assumption that the quotation which precede the forecast is known. 4. the data we take into account daily levels of german dax, french cac40, american nasdaq and polish wig20 indexes from the period 2. january 2001 to 29. december 2006. in both series we observe the logarithmic trend. during the mentioned period the trend is growing. therefore, we decide to model logarithmic levels of the indexes. the significance of the trend is then marginalized. the descriptive statistics are given in the following table 1. table 1. descriptive statistics of the logarithmic levels of indexes cac40, dax, nasdaq and wig20 from the period 2. january 2001 to 29. december 2006 time series obs. number mean std. dev. skewness kurtosis min. max. cac40 dax nasdaq wig20 1534 1528 1554 1546 8.3138 8.40231 7.55651 7.4555 0.19922 0.24511 0.1818 0.34958 -0.1759 -0.3745 -0.8033 0.4318 2.0605 2.3597 3.0256 2.0328 7.7845 7.6976 7.0158 6.8979 8.6993 8.824 7.9583 8.1745 5. empirical research in the following section we present results of one day ex-post forecasts quality testing of examined time series, which we obtain from diffusion model. the model is estimated by using the modern two-stage phillips and yu (2009) method and the generalized method of moments. we obtain the parameter estimation of diffusion models by using our own procedures in matlab (phillipsyu method) and by using the matlab libraries by cliff (2003) (gmm). the values of parameters estimations are given in table 2. forecasting financial processes by using diffusion models 55 table 2. parameter estimations of diffusion models obtained by using the phillips and yu method and the generalized method of moments for logarithmic levels of cac40, dax, nasdaq and wig20 indexes estimation method: phillips and yu model parameter cac40 dax nasdaq wig20 vašíček κ μ σ -0.10636 8.1772 0.16926 0.29613 8.4126 0.21296 0.0000258 7.5829 0.12692 0.0012536 7.4681 0.16780 cir κ μ σ -0.10702 8.1667 0.058477 0.29529 8.4223 0.072836 0.081467 7.5632 0.045667 0.00776 7.3712 0.052088 ckls κ μ σ β 0.41333 8.3644 5.5273 -4.8234 0.51308 8.4174 5.3016 -4.2147 0.081748 7.5632 0.72314 -0.85398 -0.85398 6.557 0.4757 -0.51843 estimation method: gmm model parameter cac40 dax nasdaq wig20 vašíček κ μ σ 0.034582 8.3644 0.11064 0.042929 8.4174 0.12291 0.91407 7.5811 0.26122 -0.12749 6.5984 0.23064 cir κ μ σ 0.03683 8.3589 0.03826 0.044431 8.4169 0.042108 0.91274 7.5827 0.094545 -0.13399 6.6412 0.084197 ckls κ μ σ β 0.033606 8.3669 0.17815 -0.22647 0.041253 8.4177 0.43424 -0.59029 0.97411 7.5632 4.5291 -3.7814 -0.11478 6.5059 1.7382 -1.007 starting values: ,0=κ μ equales the mean of the sample, σ determines the starting value for σ by using yoshida (1992) estimator for vašíček and cir model. as the starting value for σ in ckls model we take earlier obtained estimation from cir model. as the ,β we take 0.5. for examined time series we determine 100 ex-post forecasts, by using 10000 monte-carlo simulations, and to assess the quality of the forecasts we use common error measures. small values of error measures are indicative of good quality of the forecasts, and consequently of good model fitting. the good quality also implies using diffusion models as the alternative for parametric models of time series. forecast errors were compared with errors obtained from popular time series models forecasts. the grade of time series models had been selected by using the schwarz information criterion. the occurrence of unit root was verified by using dickey-fuller (said, dickey, 1984) and phillips-perron (1988) tests. in the case of failure to reject the h0 hypothesis we modeled the conditional mean by arima(p,1,q) model. independently we used two innovation distributions – the student and the generalized error distribution developed by nelson (1991). moreover, we verified the existence of the arch effect by using engle (1982) and mcleod-li (1983) tests. the latter consists in applying ljung-box (1978) test to squared residuals of linear model. piotr płuciennik 56 as we can observe in tables 3-6, the forecast errors for wig20 index are bigger than for other indexes. polish financial market is still a raising market, and wig20 volatility is bigger than volatility of indexes traded in mature markets. the type of applied model do not have big influence on the forecast quality. for indexes cac40, dax and wig20 the forecast was a little bit better when we modeled the logarithmic prices by using diffusion models, but for nasdaq index the forecast errors were smaller for parametric time series models. from among diffusion models, the best forecasts we obtained using cir models. moreover, we can notice that for all examined time series, phillips and yu method of parameters estimation leads to smaller forecast errors than gmm method. table 3. values of the forecast errors obtained by using diffusion models. logarithmic levels of cac40 index error phillips and yu method generalized method of moments vašiček cir ckls vašiček cir ckls mse mede me mae rmse mape amape ll 6.9069e-5 2.5575e-5 0.00077931 0.0062754 0.0083108 0.00072612 0.00036308 9.2627e-7 6.8895e-5 2.6669e-5 0.00077539 0.0062588 0.0083003 0.0007242 0.00036212 9.2395e-7 6.902e-5 2.6053e-5 0.00078897 0.0062567 0.0083078 7.2396e-6 3.62e-6 9.2561e-7 6.8355e-5 2.6771e-5 0.00032694 0.0062394 0.0082677 0.00072199 0.000361 9.1675e-7 6.817e-5 2.6696e-5 0.00032105 0.0062339 0.0082565 0.00072135 0.00036067 9.1422e-7 6.9836e-5 2.7293e-5 0.0012119 0.0062989 0.0083568 0.00072881 0.00036444 9.3652e-7 error time series parametric models arima(0,1,2) (student) arima(0,1,2) (ged) arima(0,1,2) garch(1,1) (student) arima(1,1,1) garch(1,1) (ged) mse mede me mae rmse mape amape ll 7.0223e-5 2.6884e-5 0.00092873 0.0063371 0.0083799 0.00073332 0.00036668 9.4187e-7 7.0214e-5 2.6873e-5 0.00092695 0.0063367 0.0083794 0.00073327 0.00036665 9.4175e-7 7.0232e-5 2.6895e-5 0.00093054 0.006338 0.0083805 0.00073342 0.00036673 9.42e-7 table 4. values of the forecast errors obtained by using diffusion models. logarithmic levels of dax index error phillips and yu method generalized method of moments vašiček cir ckls vašiček cir ckls mse mede me mae rmse mape amape ll 7.7296e-5 2.6917e-5 0.0014902 0.0066737 0.0087918 0.00075446 0.00037726 9.8999e-7 7.7358e-5 2.7475e-5 0.0014927 0.0066797 0.0087953 0.00075514 0.0003776 9.9078e-7 7.7691e-5 2.742e-5 0.0014958 0.0066842 0.0088142 7.5564e-6 3.7785e-6 9.9505e-7 7.8574e-5 2.706e-5 0.0017948 0.0067379 0.0088642 0.00076169 0.00038089 1.0063e-6 7.8449e-5 2.9142e-5 0.0017836 0.0067449 0.0088571 0.00076248 0.00038128 1.0047e-6 8.0539e-5 3.0036e-5 0.0023101 0.0068758 0.0089744 0.00077721 0.00038867 1.0312e-6 error time series parametric models arima(0,1,2) (student) arima(0,1,2) (ged) arima(0,1,2) garch(1,1) (student) arima(1,1,1) garch(1,1) (ged) mse mede me mae rmse mape amape ll 7.8051e-5 2.7454e-5 0.0015387 0.0067188 0.0088346 0.00075963 0.00037985 9.9984e-7 7.8615e-5 2.7438e-5 0.0016234 0.0067622 0.0088665 0.00076455 0.0003823 1.007e-6 7.8058e-5 2.7546e-5 0.0015409 0.0067197 0.0088351 0.00075973 0.0003799 9.9994e-7 7.8058e-5 2.7546e-5 0.0015409 0.0067197 0.0088351 0.00075973 0.0003799 9.9994e-7 table 5. values of the forecast errors obtained by using diffusion models. logarithmic levels of nasdaq index error phillips and yu method generalized method of moments vašiček cir ckls vašiček cir ckls mse mede me mae rmse mape amape ll 7.3033e-5 1.8082e-5 0.055699 0.0062114 0.0085459 0.00079535 0.00039763 1.1983e-6 7.3078e-5 1.7335e-5 0.00064038 0.0062146 0.0085486 0.00079575 0.00039783 1.199e-6 7.3089e-5 1.8443e-5 0.055435 0.0062015 0.0085492 7.9409e-6 3.9699e-6 1.1992e-6 7.5079e-5 2.1237e-5 0.0013926 0.0064573 0.0086648 0.00082676 0.00041338 1.2319e-6 7.4249e-5 1.9876e-5 0.0013442 0.0064043 0.0086168 0.00081998 0.00040998 1.2183e-6 7.5569e-5 2.1606e-5 0.0015984 0.0065172 0.008693 0.00083443 0.00041722 1.24e-6 error time series parametric models arima(0,1,2) (student) arima(0,1,2) (ged) arima(0,1,2) garch(1,1) (student) arima(1,1,1) garch(1,1) (ged) mse mede me mae rmse mape amape ll 7.2567e-5 1.7764e-5 0.056472 0.0062011 0.0085187 0.00079404 0.00039697 1.1907e-6 7.2567e-5 1.7769e-5 0.056466 0.006201 0.0085186 0.00079404 0.00039697 1.1907e-6 7.282e-5 1.8811e-5 0.055536 0.0062328 0.0085335 0.00079811 0.00039901 1.1949e-6 7.282e-5 1.8819e-5 0.055532 0.0062328 0.0085334 0.00079811 0.00039901 1.1948e-6 piotr płuciennik 58 table 6. values of the forecast errors obtained by using diffusion models. logarithmic levels of wig20 index error phillips and yu method generalized method of moments vašiček cir ckls vašiček cir ckls mse mede me mae rmse mape amape ll 0.00018948 6.1233e-5 0.00063017 0.010608 0.013765 0.0013032 0.00065164 2.8646e-6 0.0001892 6.3602e-5 0.00063866 0.010619 0.013755 0.0013046 0.00065232 2.8603e-6 0.00018936 6.5722e-5 0.058766 0.010609 0.013761 1.3034e-5 6.5172e-6 2.8627e-6 0.00018897 6.4217e-5 0.0001788 0.010582 0.013747 0.0013001 0.00065007 2.8568e-6 0.00018938 6.4506e-5 0.00032633 0.010595 0.013762 0.0013016 0.00065083 2.8629e-6 0.00018973 6.7774e-5 -0.0001169 0.010586 0.013774 0.0013007 0.00065032 2.8682e-6 error time series parametric models arma(0,2) (ged) arma(1,1) (student) mse mede me mae rmse mape amape ll 0.01904131 6.6946e-05 0.00035112 1.06889157 0.13799026 0.131329 0.00065666 2.8789e-06 0.019061629 6.87936e-5 0.000398688 1.071935667 0.138063858 0.1317012 0.000658529 2.88198e-6 note: mse – the mean squared error, mede – the mean median error, me – the mean error, mae – the mean average error, rmse – the root of the mean squared error, mape – the mean average percentage error, amape – corrected average percentage error, and ll – logarithmic loss function (cf. welfe 1998; doman, doman, 2004). 6. conclusions the high quality of the forecast obtained from the diffusion models is indicative of good fitting of the diffusion models to the studied time series. the values of the forecast errors are often smaller when diffusion models were used. it is notable that in diffusion models the volatility depends only on white noise and optionally on current value of the process. in arima-garch models the volatility is described by the second parametric equation. the most surprising fact is that the ckls model leads to worse quality of the forecast than the vašíček and cir model. after all, both models are special cases of the ckls model. the reason for that situation lies in very bad fitting of the ckls model to the examined time series. the estimates of β are negative, while the model assumed that ].1,0[∈β it is notable that determining the forecasts by using diffusion models is not laborious. the matlab procedures used in the conducted research need only a fraction of a second to estimate parameters by using two-stage phillips-yu method and a few seconds if we decide to use the gmm method. the most laborious part of the calculation is determining 10000 sample paths, but it takes up to two minutes to do this operation. forecasting financial processes by using diffusion models 59 references black, f., scholes, m. (1973), the pricing of option and corporate liabilities, journal of political economy, 81, 637–659. broadie, m., jain, a. (2008), pricing and hedging volatility derivatives, journal of derivatives, 15, issue 3, 7–24. chan, k. c., karolyi, g. a., longstaff, f. a., sanders, a. b. (1992), an empirical comparison of alternative models of short term interest rates, journal of finance, 47, 1209–1227. cliff, m. t. (2003) gmm and minz program libraries for matlab, krannert graduate school of management purdue university. cox, j. c., ingersoll, j., ross, s. (1985), a theory of the term structure of interest rates, econometrica, 53, 385–407. cziraky, d., kucherenko, s. (2008), monte carlo forecasting from cir square root diffusion models, broda ltd. http://www.broda.co.uk (8.01.2009). detemple, j., osakwe, s. (1999), the valuation of volatility options, cirano paper, scientific series. doman, m., doman, r. (2004), econometric modeling of polish financial market dynamic (in polish), poznań university of economics, poznań. engle, r. f. (1982), autoregressive conditional heteroscedacticity with estimates of the variance of united kingsdom inflation, econometrica, 50, 987–1007. hansen, l. p. (1982), large sample properties of generalized method of moments estimators, econometrica, 50, 1029–1054. heston, s. l. (1993), a closed-form solution for options with stochastic volatility with applications to bond and currency options, the review of financial studies, 6, issue 2, 327–343. howison, s., rafailidis, a., rasmussen, h. (2004), on the pricing and hedging of volatility derivatives, applied mathematical finance, 11, issue 2, 317–346. javaheri, a., wilmott, p., haug, e. (2002), garch and volatility swaps, wilmott magazine, january, 1–17. jagannathan r., kaplin a., sun s. g. (2004), an evaluation of multi-factor cir models using libor, swap rates, and cap and swaption prices, journal of econometrics, 116, 113–146. ljung, g. m., box, g. e. p. (1978), on a measure of lack of fit in time series models, biometrika, 65, 297–303. mannolini, a., mari, c., renò, r. (2008), pricing caps and floors with the extended cir model, international journal of finance & economics, 13 (4) 386–400. mcleod, a. i., li, w. k. (1983), diagnostic checking arma time series models using squared residual autocorrelations, journal of time series analysis, 4, 269–273. merton, r. c. (1973), theory of rational option pricing, bell journal of economics and management science, 4 (1) 141–183. merton, r. c. (1974), on the pricing of corporate debt: the risk structure of interest rates, journal of finance, 29, 449–470. nelson, d. (1991), conditional heteroskedasticity in asset returns: a new approach, econometrica, 59, 347–370. newey, w. k., west, k. d. (1987), a simple, positive semidefinite, heteroskedasticity and autocorrelation consistant covariance matrix, economerica, 59, 347–370. phillips, p. c. b., perron, p. (1988), testing for a unit root in time series regressions, biometrika, 75, 335–346. phillips, p. c. b., yu, j. (2009), a two-stage realized volatility approach to estimation of diffusion processes with discrete data, journal of econometrics, 150, issue 2, 139–150. psychoyios, d., skiadopoulos, g. (2006), volatility options: hedging effectiveness, pricing, and model error, journal of futures markets, 26, 1–31. said, e., dickey, d.a. (1984), testing for unit roots in autoregressive moving average models of unknown order, biometrika, 71, 599–607. sepp, a. (2008), vix option pricing in a jump-diffusion model, risk, april, 84–89. piotr płuciennik 60 vašiček, o. (1977), an equilibrium characterization of the term structure, journal of financial economics, 5, 177–188. tamba, y. (2006), pricing the bermudan swaption with the efficient calibration, nucb journal of economics and information science, 51, issue 1, 17–31. welfe, a. (1998), econometrics (in polish), polskie wydawnictwo ekonomiczne, warszawa. yoshida, n. (1992), estimation for diffusion processes from discrete observation, journal of multivariate analysis, 41, 220–242. prognozowanie procesów finansowych za pomocą modeli dyfuzji z a r y s t r e ś c i. prognozowanie szeregów czasowych jest jednym z najważniejszych zagadnień współczesnej ekonometrii finansowej. w obliczu rosnącego zainteresowania modelami z czasem ciągłym i szybkiego rozwoju metod ich estymacji, podejmujemy w pracy próbę modelowania i prognozowania szeregów czasowych z różnych rynków finansowych za pomocą modeli dyfuzji. stosujemy w tym celu bazującą na symulacjach monte-carlo metodę wprowadzoną przez cziraky i kucherenko (2008). jakość otrzymanych prognoz zostaje skonfrontowana z jakością prognoz otrzymanych za pomocą powszechnie stosowanych parametrycznych modeli szeregów czasowych. s ł o w a k l u c z o w e: model arima, model garch, modele dyfuzji, pierwiastek jednostkowy, prognozy ex-post, symulacja monte-carlo. microsoft word 00_tresc.docx © 2013 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2013.003 vol. 13 (2013) 51−68 submitted october 15, 2013 issn accepted december 30, 2013 1234-3862 monika papież, sławomir śmiech* economic growth and energy consumption in post-communist countries: a bootstrap panel granger causality analysis∗∗ a b s t r a c t. the aim of this paper is to identify granger causality between energy consumption and economic growth in post-communist countries in the period 1993 to 2011. bootstrap panel granger causality test was used as a research tool in order to accommodate for countryspecific heterogeneity and to avoid the problem of cross-sectional dependence. the analysis allowed for the verification of the hypothesis regarding the links between economic growth and energy consumption in nine countries. the hypotheses were confirmed: the growth hypothesis in three countries and the feedback hypothesis in one country. k e y w o r d s: energy consumption, economic growth, bootstrap panel granger causality test, energy efficiency. j e l classification: c33, q43. introduction improving energy security is the priority of the eu policy. growing dependence of the eu on energy import (its volume is predicted to increase from 50% at present to 80% in 2035) makes it imperative to limit energy * correspondence to: monika papież, cracow university of economics, department of statistics, rakowicka 27, 31-510 krakow, poland e-mail: papiezm@uek.krakow.pl; sławomir śmiech, cracow university of economics, department of statistics, rakowicka 27, 31-510 krakow, poland e-mail: smiechs@uek.krakow.pl. ∗∗ we are grateful for the financial support of the polish national science centre (the project dec-2011/03/b/hs4/01134). monika papież, sławomir śmiech dynamic econometric models 13 (2013) 51–68 52 consumption (measured by co2 emission) and replace non-renewable energy sources with renewable ones. on the other hand, energy consumption is inextricably linked with economic growth (the environmental kuznets curve). dynamically developing countries consume more and more energy, and curbing energy consumption can lead to stagnation or the drop in economic growth rate. the analysis of links between economic growth and energy consumption was addressed by numerous research studies beginning with a pioneering study by kraft and kraft (1978). four hypotheses regarding causal relations between energy consumption and economic growth can be found in the literature dealing with this topic: the growth hypothesis, the conservation hypothesis, the feedback hypothesis and the neutrality hypothesis. the growth hypothesis assumes that there are countries in which the growth of energy consumption is an important element of their economic development. in this case, the energy conservation policy advocating the introduction of limits in energy consumption will have a negative impact on economic growth. the growth hypothesis is based on unidirectional granger causality running from energy consumption to economic growth. the conservation hypothesis claims that the changes in energy consumption stem from the changes in economic activity, and energy conservation policy does not negatively affect economic growth. this hypothesis is confirmed if unidirectional granger causality running from gdp growth to energy consumption can be observed. the feedback hypothesis assumes that there are countries with bi-directional granger causality between energy consumption and economic growth. the neutrality hypothesis states that there are countries in which gdp does not depend on energy consumption and vice versa. as karanfil (2009) demonstrates in his survey of empirical literature devoted to this issue, the relations between economic growth and energy consumption are not unambiguous. the differences can be attributed to different econometric approaches, differently specified time frames and different sets of variables used in each of those studies. the occurrence of the relationship between economic growth and energy consumption is related to the changes in energy efficiency. energy efficiency aims at reducing the amount of energy required to provide products and services in a given country. most studies underline a positive influence of energy efficiency on economy and the environment. for example, sarkar and singh (2010) show that energy efficiency programmes can: conserve natural resources, reduce the environmental pollution and carbon footprint of the energy sector, reduce a country’s dependence on fossil fuels, thus enhancing its energy security, ease infrastructure bottlenecks and impacts of temporary economic growth and energy consumption in post-communist countries... dynamic econometric models 13 (2013) 51–68 53 power shortfalls, as well as improve industrial and commercial competitiveness through reduced operating costs. however, rebound effects (see turner, 2009; turner and hanley, 2011 for a recent review) can appear. when energy becomes more productive, and its price falls, the increase of energy use through the substitution effect can be observed. this paper examines the relationship between energy consumption and economic growth in nine eastern european countries and the baltic states. it focuses on the production side model of the energy consumption-growth nexus, with labour and capital included in it (stern, 1993). the analysis covers the period from 1993 to 2011. such a choice was dictated by the need to cover the relations in the analysed countries, all of which witnessed rapid political and economic changes in the 1990s. the aim of the analysis was to investigate the relations between energy consumption and economic growth in selected eastern european countries and the baltic states on the basis of the relations between overall energy efficiency gains (industry, transport, households) in 2000–20101 and countries' economic growth in the same period. the following research hypotheses were formulated: a) countries which increased energy efficiency the most will confirm the growth hypothesis, feedback hypothesis or conservation hypothesis (bidirectional unidirectional causality between energy consumption and economic growth). b) countries in which energy efficiency was not considerably increased will confirm the neutrality hypothesis. the hypotheses result from the following reasoning. the former group of countries had to bear the costs of the increase in energy efficiency on the one hand, and, on the other hand, modernisation allows for reducing the amount of energy used and, consequently, its costs, which should result in the appearance of causal relations between energy production and economic growth. in the latter group of countries the effects mentioned are nonexistent, which rules out any relations between them. we applied a bootstrap panel causality approach proposed by kónya (2006), which allows for simultaneous examination of cross-sectional dependence. the paper consists of the following sections. section 1 presents the most important findings from studies dealing with energy-economy nexus in central and eastern european countries. in section 2 the relations between over 1 the choice of the period for the analysis was dictated by the availability of the data provided by http://www.odyssee-indicators.org. monika papież, sławomir śmiech dynamic econometric models 13 (2013) 51–68 54 all energy efficiency gains and economy growth are described. section 3 introduces methodology used in the study, and section 4 presents the data and results obtained. the paper ends with the conclusion and the interpretation of the results. the paper contributes to the existing literature because the analysis focuses on countries from eastern europe and the baltic states. most of them have not been analysed from this angle before. they have similar gdp and a similar level of energy consumption. as member states of the eu, they are obliged to follow a common energy policy. thanks to these similarities, the data used were characterised by cross-sectional dependence. the application of the methodology suggested by kónya (2006) made correct inference on causalities in this situation possible. two additional variables, labour and capital, were used to compare the models, which makes this study more general in scope than other studies. our findings may provide valuable information for developing more effective energy policies with respect to both energy consumption and environmental protection. 1. review of literature existing literature offers a wide range of perspectives and insights into the issue of energy consumption-growth nexus, which sometimes report contradicting results. it can be divided into country-specific case studies and multi-country studies (karanfil, 2009). in both types various econometric methods, the choice of the period analysed, and the choice of control variables can be found. taking into consideration the methodological perspective, four generations of contributions were identified (belke et al., 2011; costantini and martini, 2010). the first-generation studies were based on var methodology (kraft and kraft, 1978) and assumed that the time series were stationary. the second-generation studies accounted for non-stationarity and applied engle-granger two-step procedure to test pairs of variables for cointegrating relationships. the third-generation studies used multivariate estimators (johansen, 1991). this approach allowed for more than two variables in cointegration relationship and for analysing causality both in the shortand long-run simultaneously. the fourth-generation studies were based on panel methods testing for unit roots, cointegration and granger causality. using panel cointegration has several advantages. it allows for higher degrees of freedom, reduces multicollinearity between regressors, and improves the power of the cointegration test, especially in case of annual data. the main economic growth and energy consumption in post-communist countries... dynamic econometric models 13 (2013) 51–68 55 disadvantage of this approach is the need to assume cross-sectional independence, which is difficult to satisfy in a panel data. what is more, different countries are treated as an entity. as a result, it is impossible to identify the difference in the dynamic relationship between energy consumption and economy (slope homogeneity). as a result, it is impossible to identify the difference in the dynamic relationship between energy consumption and economy (slope homogeneity). what is more, in most studies based on panel models (except panel var, see surveys by canova and ciccarelli (2013), which has not been used in such analyses so far) different countries are treated as an entity. in spite of a substantial number of studies concerning relations between energy consumption and economic growth, not all eastern european countries were analysed, for example, the baltic states were not included in any of them, and among the countries from central and eastern europe only the following ones were studied: poland (gurgul and lach, 2011a, 2011b, 2012), romania (apergis and danuletiu, 2012), albania, bulgaria, hungary, and romania (ozturk and acaravci, 2010). gurgul and lach (2011b) found that energy consumption granger caused gdp in poland during the last decade. they (gurgul and lach, 2011a) also investigated causal relations between coal consumption and economic growth. in another paper gurgul and lach (2012) investigated causal interdependences between electricity consumption and gdp in poland. apergis and danuletiu (2012) showed that energy consumption granger caused gdp in romania in the period 20002011. however, ozturk and acaravci (2010) did not find any relationship between energy consumption and real gdp in romania and bulgaria, while found bidirectional strong granger causality between these variables in hungary in the period 1980–2006. using a two-way fixed effects model, menegaki and ozturk (2013) confirmed bidirectional causality between growth and political stability, capital and political stability, and capital and fossil energy consumption for 26 european countries in a multivariate panel framework over the period 1975–2009. 2. energy efficiency in eastern european countries and the baltic states energy efficiency is considered to be one of the most cost effective ways of meeting the demands of sustainable development and lower fossil fuel dependence. so, the efficient use of energy is an important topic in public policy debates. unfortunately, in literature dealing with energy issues there is no consensus on the appropriate method for defining and measuring en monika papież, sławomir śmiech dynamic econometric models 13 (2013) 51–68 56 ergy efficiency. the authors of this paper adopted the definition of energy efficiency given by the odyssee project2. the “odex” energy efficiency indicator provides an overall perspective of energy efficiency trends by sector and combines the trends of indicators by end-use or sub-sector. it represents a better proxy to evaluate energy efficiency trends at an aggregate level (overall economy, industry, households, transport, services). the odex indicators by sector (industry, transport, households) are calculated from unit consumption trends by sub –sector (or end-use or mode of transport) by aggregation of unit consumption indices by sub-sector in one index for the sector on the basis of the current weight of each sub-sector in the sector’s energy consumption. the odex can be defined as the ratio between the actual energy consumption of the sector in year t and the sum of fictive energy consumption of each underlying sub-sector/end-use that would have been observed in year t had the unit consumption of the subsector been that of a reference year. the energy efficiency gains are calculated from odex and reflect efficiency gains since 2000. figure 1 presents relations between overall energy efficiency gains (industry, transport, households) in the period 2000–2010 and real growth of gdp per capita in the same period for eastern europe and the baltic states. this diagram identifies two groups of countries with similar levels of energy efficiency and a similar level of economic development. the first group comprises countries with high overall energy efficiency gains (above 18 percent) and high (and medium) growth of gdp per capita in the period 2000–2010. the second group includes countries with low overall energy efficiency gains (below 14 percent) and low (and medium) growth of gdp per capita in the period 2000–2010. in the analysed eastern european countries and the baltic states in the period 2000–2010 the mean energy efficiency index for the whole economy (odex) decreased by 15.3 percent. countries with the highest improvement in energy efficiency in the period analysed include: poland (25.7 percent), bulgaria (23 percent), latvia (21 percent), lithuania (19.8 percent), and romania (19.4 percent). countries with the lowest improvement in energy efficiency in the same period include: slovakia (3.7 percent), the czech republic (5.2 percent), estonia (7.3 percent), and hungary (13 percent). it should be noticed that countries which did not improve their energy efficiency substantially also had a lower increase growth in gdp per capita 2 odyssee is a project between ademe, the eie programme of the european commission/dgtren and energy efficiency agencies, or their representative, in the 27 countries in europe plus norway and croatia (http://www.odyssee-indicators. org). economic growth and energy consumption in post-communist countries... dynamic econometric models 13 (2013) 51–68 57 (except for slovakia3) than countries with a considerable increase in energy efficiency. figure 1. relations between overall energy efficiency gains (industry, transport, households) in the period 2000-2010 and growth of gdp per capita in the period 2000–2010 3. methodology the choice of a suitable method allowing for the analysis of causality for panel data requires the assessment of cross-sectional dependence. if crosssectional dependence exists, the seemingly unrelated regressions (sur) are more efficient then the ordinary least-squares (ols) (zellner, 1962). kónya (2006) proposed a method which takes into account the characteristics of cross-sectional dependence. therefore, before considering causality, we investigated the characteristics of panel data. the tools used for bootstrap panel causality test are presented below. 3 a specific situation of slovenia is a result of numerous phenomena which are described in detail in the study: energy efficiency policies and measures in slovakia in 2012, odysseemure 2010, http://www.odyssee-indicators.org/publications/pdf/slovakia_nr.pdf [30.12.2013]. monika papież, sławomir śmiech dynamic econometric models 13 (2013) 51–68 58 3.1. tests of cross-sectional dependence an important issue to be considered in a panel data analysis is testing for cross-sectional dependence across countries, because a shock that affects one country may spillover on other countries. let us consider the standard panel data model: ititiiit uy ++= x'βα , (1) where ni ,,2,1 …= represents the cross section dimension, tt ,,2,1 …= refers to the time series dimension, itx is a (k × 1) vector of observed regressors (individual-specific as well as common regressors). the individual intercepts iα and the slope coefficients iβ are defined on a compact set and allowed to vary across i. for each i, ( )2,0~ iuit iidu σ for all t, although they may exhibit crosssectional dependence. the null hypothesis of no-cross-sectional dependence – ( ) 0:0 =jtituucovh for all t and ji ≠ – is tested against the alternative hypothesis of cross-sectional dependence – ( ) 0:1 ≠jtituucovh , for at least one pair of ji ≠ . in literature several tests for error cross-sectional dependence have been proposed. breusch and pagan (1980) proposed a lagrange multiplier (lm) statistic for testing the null hypothesis of no-cross-sectional dependence, which is defined as: ∑ ∑ − = += = 1 1 1 2ˆ n i n ij ijtlm ρ , (2) where ijρ̂ is the sample estimate of the pair wise pearson correlation coefficient of the residuals from the ordinary least-squares (ols) estimation of eq. (1) for each i. lm is asymptotically distributed as chi-squared with 2/)1( −nn degrees of freedom under the null hypothesis, as ∞→t , with n fixed. it is important to note that the lm test is valid for relatively small n and sufficiently large t. so, where ∞→t and ∞→n , pesaran (2004) proposed the following lm statistic for the cross-sectional dependence test (the so-called cd test): ( ) ( )∑ ∑ − = += − − = 1 1 1 2 1ˆ 1 1 n i n ij ijlm tnn cd ρ . (3) economic growth and energy consumption in post-communist countries... dynamic econometric models 13 (2013) 51–68 59 under the null hypothesis, the lmcd test converges to the standard normal distribution. however, this test is likely to exhibit substantial size distortions for n large and t small, a situation that can frequently arise in empirical applications. to overcome this problem, pesaran (2004) proposed the following simple alternative test, which is based on the pair-wise correlation coefficients rather than their squares used in the lm test: ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = ∑ ∑ − = += 1 1 1 ˆ 1 2 n i n ij ijnn t cd ρ , (4) and showed that under the null hypothesis of no cross-sectional dependence ( )1,0ncd d→ for ∞→n and t sufficiently large. pesaran (2004) concluded that it is also clear that, since the mean of cd is exactly equal to zero for all fixed 1+> kt and n, the test is likely to have good small sample properties (for both n and t small). in pesaran et al. (2008) the authors concluded that the cd test has an important drawback, namely it will lack power in certain situations where the population average pairwise correlations are zero, although the underlying individual population pairwise correlations are non-zero. that is why pesaran et al. (2008) proposed a bias-adjusted test, which is a modified version of the lm test, by using the exact mean and variance of the lm statistic. the bias-adjusted lm test is as follows: ( ) ( ) ∑ ∑ − = += −− − = 1 1 1 2ˆ 1 2 n i n ij tij tijij adj v kt nn lm μρ , (5) where tijμ and 2 tijv are respectively the exact mean and variance of ( ) 2ˆijkt ρ− provided in pesaran et al. (2008 p.108). pesaran et al. (2008) showed that under the null hypothesis of no cross-sectional dependence with ∞→t first followed by ∞→n , the statistics adjlm follow an asymptotic standard normal distribution. 3.2. bootstrap panel granger causality test taking into account cross-sectional dependence and heterogeneity across country groups requires a method of analysis which would be able to capture both these features. the bootstrap panel causality approach proposed by kónya (2006) seemed to be a suitable method. this approach uses seemingly monika papież, sławomir śmiech dynamic econometric models 13 (2013) 51–68 60 unrelated regression (sur) and, therefore, is able to deal with crosssectional dependence across the members of the panel. the test for the direction of causality is based on wald tests with country-specific bootstrap critical values. that is why it does not impose a joint hypothesis across all members of the panel. using kónya (2006) approach allows for the identification of specific countries in which a granger causal relationship exists. what is more, kónya (2006) claimed that 'this approach does not require pretesting for unit roots and cointegration', which is important 'since the unit-root and cointegration tests in general suffer from low power'. on the other hand, ignoring potential (common) stochastic trends results in a situation in which the results of the suggested procedure can be used only for the evaluation of short-term causality (one-period-ahead forecast). kónya's (2006) panel causality approach models the data as a system of two sets of the following equations4: 1 1 1 1 1, 1,1 1,1, 1, 1,1, 1, 1,1, 1, 1 1 1 1,1, 1, 1,1, 1 , mly mlx mlz t l t l l t l l t l l l l ml l t l t l y y x z v ν α β δ γ ϑ ε − − − = = = − = = + + + + + + ∑ ∑ ∑ ∑ …. (6) 1 1 1 1 , 1, 1, , , 1, , , 1, , , 1 1 1 1, , , 1, , 1 , mly mlx mlz n t n n l n t l n l n t l n l n t l l l l mlv n l n t l n t l y y x z v α β δ γ ϑ ε − − − = = = − = = + + + + + + ∑ ∑ ∑ ∑ and 2 2 2 2 1, 2,1 2,1, 1, 2,1, 1, 2,1, 1, 1 1 1 2,1, 1, 2,1, 1 , mly mlx mlz t l t l l t l l t l l l l mlv l t l t l x y x z v α β δ γ ϑ ε − − − = = = − = = + + + + + + ∑ ∑ ∑ ∑ …. (7) 4 it is possible to include a deterministic component into the system of equations. economic growth and energy consumption in post-communist countries... dynamic econometric models 13 (2013) 51–68 61 2 2 2 2 , 2, 2, , , 2, , , 2, , , 1 1 1 2, , , 2, , 1 , mly mlx mlz n t n n l n t l n l n t l n l n t l l l l mlv n l n t l n t l x y x z v α β δ γ ϑ ε − − − = = = − = = + + + + + + ∑ ∑ ∑ ∑ where tiy , denotes economic growth (in country i and t period), tix , refers to energy consumption, tiz , is the capital formation, tiv , is the labour participation rate5, n denotes the number of countries in the panel ( ni ,,2,1 …= ), t is time period ( tt ,,2,1 …= ), and l is the number of lags in equations. titi ,,2,,1 ,εε are supposed to be correlated contemporaneously across equations (due to common random shocks). the system of equations allows for testing unidirectional and bidirectional granger causality for each country separately. there is unidirectional granger causality running from economic growth to energy consumption (the equivalent of the conservation hypothesis) if in (7) not all i,2β 's are zero, but in (6) all i,1δ 's are zero. there is unidirectional granger causality running from energy consumption to economic growth in country i (the equivalent of the growth hypothesis) if not all i,1δ 's are zero, but all i,2β 's are zero in (7). there is bi-directional granger causality between energy consumption and economic growth if neither all i,1δ 's nor all i,2β 's are zero. finally, there is no granger causality between energy consumption and economic growth if all i,1δ 's and all i,2β 's are zero. the country-specific bootstrap6 critical values are obtained as follows7: 1. a system of equations (6) is estimated under the null hypothesis of noncausality running from energy consumption to economic growth (that is imposing the 0,,1 =liδ restriction for all i and l) and the residuals are obtained: 5 z and ν are treated as an auxiliary variable, and they will not be directly involved in the granger causality analysis. 6 on bootstrapping in general see e.g. horowitz (2003). on bootstrapping in sur models see atkinson et. al (1992), and rilstone and veall (1996). 7 we present a procedure for testing granger causality running from x to y. similar steps are required for testing causality running from y to x. monika papież, sławomir śmiech dynamic econometric models 13 (2013) 51–68 62 1 1 1 0 , , , 1, 1, , 1, 1, , 1, 1, , 1, 1 1 1 ˆ ˆˆ ˆ , mly mlz ml h i t i t i i l t l i l t l i l t l l l l e y y z v ν α β γ ϑ− − − = = = = − − − −∑ ∑ ∑ (8) for 1, ,i n= … and tt ,,1 …= . from these residuals n×t matrix [ ]tihe ,,0 is developed. 2. these residuals are re-sampled by randomly selecting a full column form the matrix [ ]tihe ,,0 , and denote the selected bootstrap residuals as [ ]* ,,0 tihe where *1, 2,3,..., .t t= 3. the bootstrap sample of y is generated under the assumption of no causality running from energy consumption to economic growth, i.e.: 1 1 1 0 * * * , 1, 1, , , 1, , 1, 1, , 1, , , 1 1 1 ˆ ˆˆ ˆ . mly mlz ml i t i i l i t l i l t l i l t l h i t l l l y y z v e ν α β γ ϑ− − − = = = = + + + +∑ ∑ ∑ (9) 4. *,tiy is substituted for tiy , and a system of equations is re-estimated (without any restrictions). the wald test for each country is implied by the no-causality null hypothesis. 5. the empirical distributions of the wald test statistics are developed by repeating steps 2 – 4. the bootstrap critical values are specified by selecting appropriate percentiles of these sampling distributions. eventually, wald test statistics obtained from original series are compared with the bootstrap critical values. specifying the number of lags in all equations is a crucial step in kónya's approach. following kónya (2006), we decided to allow for different lags in each system but did not allow for different lags across countries. assuming that the number of lags ranges from 1 to 4, we estimated all equations and used the akaike information criterion (aic) to determine the optimal solution. the akaike information criterion8 (aic) was evaluated as: t qn aick 22 ||ln += w , (10) where w stands for estimated residual covariance matrix, n is the number of equations, q is the number of coefficients per equation, t is the sample size. 8 kónya (2006) presented also schwartz information criterion. economic growth and energy consumption in post-communist countries... dynamic econometric models 13 (2013) 51–68 63 4. data and empirical results the analysis of causal relationship between energy consumption and economic growth based on the annual panel data was carried out over the period 1993–2011 for nine european countries: bulgaria, the czech republic, estonia, hungary, latvia, lithuania, poland, romania, and slovakia. two variables from the world bank development indicators were chosen for the analysis: real gross domestic product per capita (gdp) in constant 2000 u.s. dollars and energy consumption (ec), represented by energy use in kg of oil equivalent per capita. taking into consideration rapid economic changes experienced by the countries analysed, a set of variables was extended to include real gross fixed capital formation per capita (k) in constant 2000 us dollars as a proxy of capital and labour participation rate (l)9. all variables were in natural logarithms. till 1989 eastern european countries and the baltic states were under the communist rule with centrally planned economies. in 1989 communism fell in bulgaria, czechoslovakia, hungary, poland, and romania. after the dissolution of the soviet union in 1991, estonia, latvia, and lithuania reappeared on the map, and in 1993 czechoslovakia was divided into two countries: the czech republic and slovakia. that is why year 1993 was chosen as an initial period of the analysis of causality between economic growth and energy consumption. table 1. cross-sectional dependence tests variable cross-sectional dependence test lm cdlm cd lmadj gdp 150,62*** 13,51*** 6,23*** 21,654*** ec 324,84*** 30,50*** 17,84*** 36,375*** note: ***, **, and * indicate significance at the 1, 5, and 10% levels, respectively. the first step in analysing panel data granger causality is testing for cross-sectional dependence. table 1 shows the results obtained for four different cross-sectional dependence test statistics: lm (breusch and pagan, 1980), cdlm (pesaran, 2004), cd (pesaran, 2004), and lmadj (pesaran et al., 2008). the results indicate that for all countries with significance level p = 0.05 we reject the null hypothesis of no cross-sectional dependence among the four variable examined. these findings show that a shock which 9 the use of real gross fixed capital as a proxy of capital follows the works by sari and soytas (2007) in assuming that under the perpetual inventory method with a constant depreciation rate, the variance in capital is closely related to the change in investment. monika papież, sławomir śmiech dynamic econometric models 13 (2013) 51–68 64 occurred in one post-communist country will be transmitted to other countries. the existence of cross-sectional dependence in these countries means that it is justified to use the bootstrap panel granger causality testing method. for each system of equations the number of lags was chosen according to the aic criterion10. additionally, specifications incorporating deterministic trend were taken into account. the results from the bootstrap11 panel granger causality analysis are reported in table 2 and table 3. table 2. the bootstrap panel granger causality analysis countries h0: energy consumption does not granger cause gdp (h1: ec → gdp) wald statistics bootstrap critical value 10% 5% 1% bulgaria 12.915** 7.638 11.186 21.785 czech republic 0.686 7.715 11.176 19.899 estonia 0.574 10.351 16.084 35.753 hungary 0.317 7.751 11.784 20.686 latvia 9.265* 8.842 13.331 24.626 lithuania 3.378 8.856 14.397 29.298 poland 18.917** 6.597 10.029 20.283 romania 8.372* 8.245 11.548 25.975 slovakia 2.235 7.584 12.652 23.370 note: ***, **, and * indicate significance at the 1, 5, and 10% levels, respectively. bootstrap critical values are obtained from 10,000 replications. table 2 and table 3 present the results obtained for nine transition countries in eastern europe and the baltic states. the results confirm the growth hypothesis for bulgaria, poland (both at the significance level 5%) and romania (at the significance level 10%). this means that economies in those three countries can be called ‘energy dependent’, and that energy consumption plays an important role in their economic growth, both directly and indirectly in the production process as a complementary factor to labour and capital. consequently, we may conclude that energy is a limiting factor to economic growth and, hence, shocks to energy supply will have an impact 10 we used the aic criterion to compare the specifications with and without a linear trend. finally, we constructed sur with one lag and without a linear trend. 11 following the original paper of kónya (2006) and several others, e.g. nazlioglu et. al (2011), we used 10000 replications in the procedure. andrews and buchinsky (2000) provide an exact method of evaluating the adequacy of the chosen number of replications. economic growth and energy consumption in post-communist countries... dynamic econometric models 13 (2013) 51–68 65 on economic growth. additionally, it can be claimed that excessive energy protection and a reduction in energy consumption may lead to stagnation. table 3. the bootstrap panel granger causality analysis countries h0: gdp does not granger cause energy consumption (h1: gdp → ec) wald statistics bootstrap critical value 10% 5% 1% bulgaria 0.425 13.234 20.697 41.039 czech republic 0.330 12.162 18.667 37.114 estonia 0.696 8.989 13.623 29.087 hungary 1.244 12.947 20.043 42.007 latvia 23.841** 13.824 22.365 61.726 lithuania 4.246 9.716 14.063 26.881 poland 12.580 17.470 25.420 51.422 romania 2.910 11.334 17.488 34.534 slovakia 0.632 10.430 15.556 31.457 note: ***, **, and * indicate significance at the 1, 5, and 10% levels, respectively. bootstrap critical values are obtained from 10,000 replications. the feedback hypothesis was confirmed only for latvia. this means that energy consumption and economic growth are jointly determined and affected at the same time. the results support the neutrality hypothesis for 5 countries: the czech republic, hungary, estonia, lithuania, and slovakia. the neutrality hypothesis states that energy consumption and economic growth are not sensitive to one another. therefore, any policy with respect to the consumption of energy, conservative or expansive, is expected to have a negligible effect on economic growth. conclusions and discussion we investigated the relations between energy consumption and economic growth. labour and real gross fixed capital formations were added to the analysis in order to avoid the problem of impact of omitted-variables bias. the methodology used in the study, kónya's procedure (2006), firstly, allowed for the assessment of causality in countries with cross-sectional dependence, and, secondly, avoided the problem of incorrect specification connected with unit root and cointegration. empirical results confirm the linkages between energy consumption and economic growth in four of nine countries. the growth hypothesis was confirmed in 3 countries: bulgaria, poland, and romania. energy consumption seems to be the bottleneck in their economic growth, and, hence, shocks to monika papież, sławomir śmiech dynamic econometric models 13 (2013) 51–68 66 energy supply will have an impact on this growth. latvia is the only country for which the feedback hypothesis was confirmed. in such countries energy consumption and economic growth are complementary to each other. the results obtained in the study indicate that causality between energy consumption and economic growth can be equated with energy efficiency. the group of countries with the best energy efficiency indicators consisted of almost the same countries as the group in which the growth hypothesis, the conservation hypothesis or the feedback hypothesis were confirmed (the only exception was latvia). thus, it can be assumed that the hypotheses formulated in the introduction were confirmed. unfortunately, the empirical strategy used in this study did not allow for showing the reactions of economic growth to the changes in energy consumption and vice versa. so, it is not possible to conclude whether modernisation of economies, which can be equated to energy efficiency, exerts a positive or negative impact on economy (the rebound effect). in conclusion, a special situation of poland and bulgaria, countries confirming the growth hypothesis, should be mentioned. they rely on coal as the most important source of energy. in 2011 in poland it accounted for 56.2 percent and in bulgaria for 42.3 percent of the country’s total primary energy consumption. the pressure on those countries is especially heavy because of the amount of their emission of carbon dioxide, which are the highest for coal. the necessity of limiting the use of this energy source without access to alternative energy sources can be a serious threat for their economies. references apergis, n., danuletiu, d. (2012), energy consumption and growth in romania: evidence from a panel error correction model, international journal of energy economics and policy, 2 (4), 348-356, http://www.econjournals.com/index.php/ijeep/article/view/316 andrews, d. w. k., buchinsky, m. (2001), evaluation of a three-step method for choosing the number of bootstrap repetitions, journal of econometrics, 103, 345–386, doi: http://dx.doi.org/10.1016/s0304-4076(01)00047-1. atkinson, s. e., wilson p.w. (1992), the bias of bootstrapped versus conventional standard errors in the general linear and sur models, econometric theory, 8, 258-275, doi: http://dx.doi.org/10.1017/s0266466600012792. belke, a., dobnik, f., dreger, c. (2011), energy consumption and economic growth: new insights into the cointegration relationship, energy economics, 33 (5), 782–789, doi: http://dx.doi.org/10.1016/j.eneco.2011.02.005. breusch, t., pagan, a. (1980), the lagrange multiplier test and its application to model specifications in econometrics, reviews of economics studies, 47, 239–253. canova, f., ciccarelli, m. (2013), panel vector autoregressive models: a survey, working paper series 1507, european central bank, http://ssrn.com/abstract=2201610. economic growth and energy consumption in post-communist countries... dynamic econometric models 13 (2013) 51–68 67 costantini, v., martini, c. (2010), the causality between energy consumption and economic growth: a multi-sectoral analysis using non-stationary cointegrated panel data, energy economics, 32 (3), 591–603, doi: http://dx.doi.org/10.1016/j.eneco.2009.09.013. gurgul, h., lach, ł. (2011a), the role of coal consumption in the economic growth of the polish economy in transition, energy policy, 39, 2088–2099, doi: http://dx.doi.org/10.1016/j.enpol.2011.01.052. gurgul, h., lach, ł. (2011b), the interdependence between energy consumption and economic growth in the polish economy in the last decade. managerial economics, 9, 25–48, http://www.managerial.zarz.agh.edu.pl/em_on_line/managerial%20economics%20%209%20%282011%29.pdf. gurgul, h., lach, ł. (2012), the electricity consumption versus economic growth of the polish economy, energy economics, 34(2), 500–510, doi: http://dx.doi.org/10.1016/j.eneco.2011.10.017. horowitz, j. l. (2003), the bootstrap in econometrics, statistical science, 18, 211–218, doi: http://dx.doi.org/10.1214/ss/1063994976. johansen, s. (1991), estimation and hypothesis testing of cointegration vectors in gaussian vector autoregressive models, econometrica, 59 (6), 1551–1580, doi: http://dx.doi.org/10.2307/2938278. karanfil, f. (2009), how many times again will we examine the energy–income nexus using a limited range of traditional econometric tools?, energy policy, 36, 1191–1194, doi: http://dx.doi.org/10.1016/j.enpol.2008.11.029. kónya, l. (2006), exports and growth: granger causality analysis on oecd countries with a panel data approach, economic modelling, 23, 978–992, doi: http://dx.doi.org/10.1016/j.econmod.2006.04.008. kraft, j., kraft, a. (1978), on the relationship between energy and gnp, journal of energy development, 3, 401–403. menegaki, a. n., ozturk, i. (2013), growth and energy nexus in europe revisited: evidence from fixed effects political economy model, energy policy, 61, 881–887, doi: http://dx.doi.org/10.1016/j.enpol.2013.06.076. nazlioglu, s., lebe, f., kayhan, s. (2011), nuclear energy consumption and economic growth in oecd countries: cross-sectionally dependent heterogeneous panel causality analysis, energy policy, 39, 6615–6621, doi: http://dx.doi.org/10.1016/j.enpol.2011.08.007. odysseemure 2010, (2012), energy efficiency policies and measures in slovakia in 2012, monitoring of eu and national energy efficiency targets, slovak innovation and energy agency, bratislava, http://www.odyssee-indicators.org/publications/pdf/slovakia_nr.pdf [30.12.2013] ozturk, i., acaravci, a. (2010), the causal relationship between energy consumption and gdp in albania, bulgaria, hungary and romania: evidence from ardl bound testing approach, applied energy, 87, 1938–43, doi: http://dx.doi.org/10.1016/j.apenergy.2009.10.010. pesaran, m. h. (2004), general diagnostic tests for cross section dependence in panels. cambridge working papers in economics no. 0435. faculty of economics, university of cambridge, doi: http://www.dspace.cam.ac.uk/handle/1810/446. pesaran, m. h., ullah, a., yamagata, t. (2008), a bias-adjusted lm test of error crosssection independence, the econometrics journal, 11, 105–127, doi: http://dx.doi.org/10.1111/j.1368-423x.2007.00227.x. monika papież, sławomir śmiech dynamic econometric models 13 (2013) 51–68 68 rilstone, p., veall, m. r. (1996), using bootstrapped confidence intervals for improved inferences with seemingly unrelated regression equations, econometric theory, 12, 569–580, doi: http://dx.doi.org/10.1017/s026646660000685x. sarkar, a., singh, j. (2010), financing energy efficiency in developing countries—lessons learned and remaining challenges, energy policy, 38, 5560–5571, doi: http://dx.doi.org/10.1016/j.enpol.2010.05.001. soytas, u., sari, r. (2007), the relationship between energy and production: evidence from turkish manufacturing industry, energy economics, 29, 1151–1165, doi: http://dx.doi.org/10.1016/j.eneco.2006.05.019. stern, d. i. (1993), energy and economic growth in the usa, energy economics, 15, 137–150, doi: http://dx.doi.org/10.1016/0140-9883(93)90033-n. turner, k. (2009), negative rebound and disinvestment effects in response to an improvement in energy efficiency in the uk economy, energy economics, 31, 648-666, doi: http://dx.doi.org/10.1016/j.eneco.2009.01.008. turner, k., hanley, n. (2011), energy efficiency, rebound effects and the environmental kuznets curve, energy economics, 33, 709–720, doi: http://dx.doi.org/10.1016/j.eneco.2010.12.002. zellner, a. (1962), an efficient method of estimating seemingly unrelated regressions and tests for aggregation bias, journal of the american statistical association, 57, 348–368, doi: http://dx.doi.org/10.1080/01621459.1962.10480664. wzrost gospodarczy i zużycie energii w krajach postkomunistycznych bootstrapowa panelowa analiza przyczynowości z a r y s t r e ś c i. celem artykułu jest identyfikacja zależności przyczynowych (w sensie grangera) pomiędzy zużyciem energii i wzrostem gospodarczym w krajach europy środkowo wschodniej oraz w krajach bałtyckich w okresie 1993–2011. jako narzędzie badawcze wykorzystano procedurę bootstrapowej panelowej analizy przyczynowości zaproponowaną przez kónya (2006). procedura ta pozwala na wnioskowanie w przypadku występowania zależności przestrzennych w badanej próbie i nie wymaga wstępnej analizy stacjonarności oraz umożliwia opis relacji dla poszczególnych analizowanych obiektów. przeprowadzone badanie wskazuje na prawdziwość hipotezy wzrostu w przypadku trzech krajów oraz hipotezy sprzężenia zwrotnego w przypadku jednego kraju. otrzymane wyniki pozwalają podejrzewać, że zależność przyczynowa pomiędzy wzrostem gospodarczym i konsumpcją energii jest związana z zmianami efektywności energetycznej. s ł o w a k l u c z o w e: zużycie energii, wzrost gospodarczy, bootstrapowa panelowa analiza przyczynowości, efektywność energetyczna. acknowledgements we thank the anonymous referee for providing constructive comments and suggestions and help in significantly improving the contents of this paper. microsoft word 00_tresc.docx dynamic econometric models vol. 10 – nicolaus copernicus university – toruń – 2010 witold orzeszko nicolaus copernicus university in toruń measuring nonlinear serial dependencies using the mutual information coefficient† a b s t r a c t: construction, estimation and application of the mutual information measure have been presented in this paper. the simulations have been carried out to verify its usefulness to detect nonlinear serial dependencies. moreover, the mutual information measure has been applied to the indices and the sector sub-indices of the warsaw stock exchange. k e y w o r d s: nonlinearity, mutual information coefficient, mutual information, serial dependencies. 1. introduction measuring relationships between variables is an extremely important area of research in econometrics. to this end the pearson correlation coefficient is commonly used. however, the pearson coefficient is not a proper tool for measuring nonlinear dependencies. therefore, in the case of nonlinearity other methods must be used. the mutual information coefficient is one of the most important tools to detect nonlinear relationships. it comes from the information theory and is based on a concept of entropy. the mutual information coefficient may be applied to measure dependencies between two time series or serial dependencies in a single time series. 2. measuring nonlinear dependencies in time series there are various methods to measure nonlinear dependencies in time series (cf. granger, terasvirta, 1993; maasoumi, racine, 2002; bruzda, 2004). one of the most important is the mutual information measure (mi hereafter), given by the formula: † financial support of nicolaus copernicus univerity in toruń for the project umk 397-e is gratefully acknowledged. witold orzeszko 98 , )()( ),( log),(),( 21 ∫∫ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = dxdy ypxp yxp yxpyxi (1) where ),( yxp is a joint probability density function and )(1 xp and )(2 yp are marginal densities for random variables x and y. it can be shown that for all x and y the measure ),( yxi takes non-negative values and 0),( =yxi only if x and y are independent. it is convenient to define the mutual information coefficient, given by the expression: .1),( ),(2 yxieyxr −−= (2) it can be shown that the mutual information coefficient has the following properties (cf. granger, terasvirta, 1993; granger, lin, 1994): 1. 1),(0 ≤≤ yxr , 2. 0),( =yxr ⇔ x and y are independent, 3. 1),( =yxr ⇔ )( xfy = , where f is some invertible function, 4. r is unaltered if x, y are replaced by instantaneous transformations )(),( 21 yhxh , i.e. ( ) ( ))(),(, 21 yhxhryxr = , 5. if ( )yx , (or ( ))(),( 21 yhxh , where 1h and 2h are instantaneous) has a joint gaussian distribution with correlation ),( yxρ , then ),(),( yxyxr ρ= . in the literature one can find several methods for estimating a value of ),( yxi . essentially, due to the technique of estimating the probability density functions in equation 1, they can be divided into three main groups (cf. dionisio, menezes, mendes, 2003): − histogram-based estimators, − kernel-based estimators, − parametric methods. the kernel-based estimators have many adjustable parameters such as the optimal kernel width and the optimal kernel form, and a non-optimal choice of those parameters may cause a large bias in the results. for the application of parametric methods one needs to know the specific form of the generating process (dionisio, menezes, mendes, 2003)). therefore a standard way is to estimate the densities by means of histograms (cf. darbellay, wuertz, 2000). one can also define auto mutual information at lag k for a stationary discrete-valued stochastic process nxxx ,...,, 21 as the mutual information between random variables tx and ktx + : measuring nonlinear serial dependencies using the mutual information coefficient 99 . )()( ),( log),(),( ∑∑ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = + + ++ t ktx x ktt ktt kttktt xpxp xxp xxpxxi (3) since the process is stationary, ),( ktt xxi + is independent of t and so we can refer to the mutual information at lag k, as )(ki (fonseca, crovella, salamatian, 2008). this means that, the mutual information measure may be used to measure serial dependencies in a single time series as well. to this end, the past realizations of the investigated data x should be taken as the variable y. it should be emphasized that mi measures both linear and nonlinear dependencies, so to identify serial nonlinear relationships, analyzed data must be prefiltered by an estimated arma-type model. 3. application of the mutual information measure to detect serial dependencies 3.1. simulated data the aim of the simulations was to verify, if the mutual information measure may be effectively applied to detect nonlinear serial dependencies. the time series produced from five different generating models and two different sample sizes (with each of those models) were used in the simulations. this data was generated by barnett et al. (1998) to compare the power of some popular tests for nonlinearity and chaos1. specifically, these were: five time series of 2000 observations – m1, m2, m3, m4, m5 and five time series of their first 380 observations – m1s, m2s, m3s, m4s, m5s. the investigated series were generated from the following models2: i) m1 – logistic map3: ),1(57.3 11 −− −= ttt xxx (4) ii) m2 – garch(1,1) process: ,ttt uhx = (5a) ,8.01.01 1 2 1 −− ++= ttt hxh (5b) where 10 =h and 00 =x . 1 the data was downloaded from the homepage of w.a. barnett: http://econ.tepper.cmu.edu/ /barnett/papers.html. 2 in all cases, the white-noise disturbances – ut were sampled independently from the standard normal distribution. 3 the logistic map with the parameter equaled to 3.57 generates chaotic dynamics. witold orzeszko 100 iii) m3 – nonlinear moving average process (nlma): ,8.0 21 −−+= tttt uuux (6) iv) m4 – arch(1) process: ,5.01 2 1 ttt uxx −+= (7) v) m5 – arma(2,1) process: ,3.015.08.0 121 −−− +++= ttttt uuxxx (8) where 10 =x and 7.01 =x . in each case the mutual information measure was calculated for the raw series and for its residuals from the fitted arma model. first, stationarity was verified using the augmented dickey-fuller test. the null hypothesis of a unit root was strongly rejected for the all investigated data, except m5s. thus, instead of m5s, the series of its first differences – m5s_diff was chosen for further research. in table 1 the arma models fitted to analyzed series are presented4. table 1. arma models for the generated series series arma model series arma model m1 white noise (ex=0.648) m1s white noise (ex=0.649) m2 white noise (ex=0.034) m2s white noise (ex=0.067) m3 white noise (ex= 0.007) m3s white noise (ex= 0.033) m4 white noise (ex= 0.011) m4s white noise (ex= 0.018) m5 arma(1,1) m5s_diff ma(1) next, the ljung-box test was applied to test if the residual series are white noise. the test confirmed that no investigated residuals contain linear dependencies. to estimate the mutual information measure the method proposed by fraser and swinney (1986) was used5. this method is based on an analysis of the twodimensional histogram. briefly speaking, it consists in covering the twodimensional plane containing pairs ( )tt yx , with rectangular partitions and calculating frequencies of points in each partition. next, equation 1 is used, i.e. the calculated frequencies are estimators of the probability density functions and the integration is carried out numerically. let ki denotes an estimated value of the mutual information measure between variables tx and ktx − . due to a purpose of the research, the key task is to verify the hypothesis of mutual information measure’s insignificance (i.e the hypothesis of independence). to this end, for each investigated series and for 4 the models were selected based on the schwarz criterion. 5 in the calculations the m-file created by a. leontitsis was used. measuring nonlinear serial dependencies using the mutual information coefficient 101 each 10...,,2,1=k , the p-value was evaluated through bootstraping6 with 00010 repetitions7. in tables 2-6 the calculated values of ki and the corresponding p-values (at the bottom) are summarized. the p-values not larger than 0.005 are bolded8. table 2. values of ki for m1s and m1 k series 1 2 3 4 5 6 7 8 9 10 m1s 1.6927 0.0000 1.6963 0.0000 1.6123 0.0000 1.7148 0.0000 1.5919 0.0000 1.6849 0.0000 1.5412 0.0000 1.6381 0.0000 1.5379 0.0000 1.6560 0.0000 m1 2.0139 0.0000 2.0090 0.0000 2.0064 0.0000 2.2520 0.0000 1.9981 0.0000 1.9991 0.0000 1.9940 0.0000 2.2737 0.0000 1.9891 0.0000 1.9891 0.0000 table 3. values of ki for m2s and m2 k series 1 2 3 4 5 6 7 8 9 10 m2s 0.0848 0.9616 0.1538 0.0201 0.1191 0.3802 0.1308 0.1786 0.1231 0.3052 0.1616 0.0081 0.1701 0.0029 0.1162 0.4412 0.1281 0.2187 0.1228 0.3090 m2 0.0541 0.0053 0.0562 0.0025 0.0477 0.0808 0.0488 0.0536 0.0492 0.0451 0.0509 0.0227 0.0541 0.0052 0.0461 0.1303 0.0449 0.1868 0.0334 0.9315 table 4. values of ki for m3s and m3 k series 1 2 3 4 5 6 7 8 9 10 m3s 0.1857 0.0492 0.1586 0.3316 0.1425 0.6241 0.1469 0.5429 0.1323 0.8032 0.1028 0.9927 0.1897 0.0353 0.1600 0.3096 0.1525 0.4389 0.1606 0.2987 m3 0.0725 0.0000 0.0658 0.0001 0.0307 0.9634 0.0429 0.2065 0.0309 0.9599 0.0383 0.5426 0.0372 0.6274 0.0404 0.3724 0.0389 0.4868 0.0456 0.0976 table 5. values of ki for m4s and m4 k series 1 2 3 4 5 6 7 8 9 10 m4s 0.1365 0.2663 0.1667 0.0205 0.1442 0.1562 0.1349 0.2940 0.1198 0.6104 0.1367 0.2613 0.1347 0.2959 0.1327 0.3361 0.1435 0.1641 0.1464 0.1303 m4 0.1053 0.0000 0.0472 0.0051 0.0363 0.3383 0.0379 0.2324 0.0286 0.9261 0.0344 0.5058 0.0370 0.2866 0.0475 0.0039 0.0368 0.3074 0.0344 0.5059 6 bootstrap without replacement (i.e. permutation) was performed. bootstrapped p-values correspond to a one-sided test. 7 in this way, for each of the filtered series an expected distribution of mi(1) (i.e. the mi measure with k=1) was determined. next, this distribution has led to evaluation of the p-value for each k=1,2,...,10. 8 note that the rejection of the null of ki insignificance for at least one k=1,2,...,10 implies the rejection of the hypothesis of serial independence. therefore, adopting the value 0.005 for each k implies that the probability for a type i error (in the test of serial independence) is approximately 5%. witold orzeszko 102 table 6. values of ki for m5s and m5 k series 1 2 3 4 5 6 7 8 9 10 m5s 1.4787 0.0000 1.1206 0.0000 0.9817 0.0000 0.8640 0.0000 0.7505 0.0000 0.6895 0.0000 0.6344 0.0000 0.6310 0.0000 0.6173 0.0000 0.6070 0.0000 m5s_ diff 0.1390 0.5519 0.1658 0.1199 0.1288 0.7509 0.1438 0.4542 0.1496 0.3452 0.2012 0.0039 0.1642 0.1351 0.1297 0.7340 0.1161 0.9125 0.1387 0.5560 m5s_ diffma 0.1224 0.7971 0.1584 0.1595 0.1225 0.7942 0.1242 0.7668 0.1444 0.3745 0.1391 0.4816 0.1624 0.1193 0.1510 0.2584 0.1495 0.2821 0.1474 0.3179 m5 1.7145 0.0000 1.3154 0.0000 1.0949 0.0000 0.9504 0.0000 0.8414 0.0000 0.7597 0.0000 0.6958 0.0000 0.6449 0.0000 0.5917 0.0000 0.5584 0.0000 m5arma 0.0422 0.2714 0.0375 0.6530 0.0417 0.3103 0.0412 0.3438 0.0355 0.8012 0.0396 0.4685 0.0419 0.2963 0.0486 0.0398 0.0434 0.2030 0.0397 0.4640 in tables 7-8 the results of nonlinearity detection carried out by the mi measure are summarized. table 7. results of nonlinearity detection for the long series series serial dependencies nonlinearity m1 yes yes m2 yes yes m3 yes yes m4 yes yes m5 yes no table 8. results of nonlinearity detection for the short series series serial dependencies nonlinearity m1s yes yes m2s yes yes m3s no no m4s no no m5s_ diff yes no as it is clearly seen, the mi measure correctly identified each of the investigated long series. in an application to the short series it led to erroneous conclusions in the case of m3s and m4s. the obtained result is consistent with studies by other authors, i.e. it indicates that histogram-based estimators may be unreliable in a case of a small number of observations (e.g. dionisio, menezes, mendes, 2003). 3.2. stock market indices in this section the indices and the sector sub-indices of the warsaw stock exchange from 2.01.2001–15.04.2009 (2078 observations) were analyzed. for the each index, the three time series were investigated: daily log returns, residuals from their arma and arma-garch models. investigation of the residuals from the arma model gives information, if dependencies are nonlinear. if so, the standardized residuals from the arma-garch model were anameasuring nonlinear serial dependencies using the mutual information coefficient 103 lyzed, to verify if this class of processes can capture nonlinear dynamics found in the investigated data9. the results of this analysis are presented in tables 9-20. table 9. values of ki for the wig index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0458 0.0000 0.0444 0.0003 0.0605 0.0000 0.0612 0.0000 0.0486 0.0000 0.0518 0.0000 0.0350 0.0338 0.0365 0.0153 0.0522 0.0000 0.0559 0.0000 ma(1) 0.0412 0.0010 0.0455 0.0000 0.0549 0.0000 0.0632 0.0000 0.0427 0.0002 0.0500 0.0000 0.0379 0.0074 0.0313 0.1530 0.0552 0.0000 0.0566 0.0000 ma(1)garch(3,1) 0.0458 0.0225 0.0498 0.0033 0.0336 0.7074 0.0395 0.2254 0.0359 0.5142 0.0306 0.8964 0.0302 0.9124 0.0352 0.5738 0.0328 0.7700 0.0309 0.8823 table 10. values of ki for the wig20 index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0514 0.0000 0.0415 0.0106 0.0577 0.0000 0.0690 0.0000 0.0489 0.0002 0.0509 0.0000 0.0388 0.0381 0.0388 0.0373 0.0438 0.0029 0.0537 0.0000 ma(1) 0.0456 0.0011 0.0471 0.0006 0.0579 0.0000 0.0687 0.0000 0.0506 0.0001 0.0510 0.0001 0.0402 0.0187 0.0458 0.0009 0.0439 0.0028 0.0545 0.0000 ma(1)garch(3,1) 0.0441 0.0410 0.0457 0.0222 0.0337 0.6723 0.0382 0.2962 0.0311 0.8499 0.0333 0.6978 0.0307 0.8683 0.0384 0.2806 0.0303 0.8867 0.0272 0.9756 table 11. values of ki for the mwig40 index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0728 0.0000 0.0508 0.0000 0.0630 0.0000 0.0603 0.0000 0.0545 0.0000 0.0660 0.0000 0.0508 0.0000 0.0343 0.0058 0.0397 0.0002 0.0428 0.0000 ar(3) 0.0511 0.0000 0.0458 0.0000 0.0539 0.0000 0.0569 0.0000 0.0508 0.0000 0.0462 0.0000 0.0465 0.0000 0.0376 0.0002 0.0379 0.0002 0.0460 0.0000 ar(3)garch(1,2) 0.0340 0.0964 0.0301 0.3434 0.0278 0.5657 0.0404 0.0039 0.0264 0.6980 0.0377 0.0182 0.0250 0.8188 0.0309 0.2750 0.0283 0.5131 0.0295 0.3955 table 12. values of ki for the swig80 index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0911 0.0000 0.0551 0.0000 0.0680 0.0000 0.0579 0.0000 0.0597 0.0000 0.0546 0.0000 0.0498 0.0000 0.0416 0.0014 0.0440 0.0003 0.0426 0.0006 arma(1.2) 0.0478 0.0000 0.0386 0.0031 0.0538 0.0000 0.0502 0.0000 0.0397 0.0020 0.0479 0.0000 0.0371 0.0074 0.0340 0.0349 0.0369 0.0083 0.0376 0.0056 arma(1,2) garch(1,1) 0.0268 0.7878 0.0300 0.5014 0.0367 0.0616 0.0282 0.6646 0.0258 0.8545 0.0345 0.1451 0.0255 0.8722 0.0295 0.5471 0.0278 0.7014 0.0309 0.4072 9 the fit of all estimated models was positively verified using the box-ljung and the engle tests. table 13. values of ki for the wig-banking index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0429 0.0000 0.0439 0.0000 0.0628 0.0000 0.0556 0.0000 0.0485 0.0000 0.0518 0.0000 0.0476 0.0000 0.0516 0.0000 0.0602 0.0000 0.0443 0.0000 ma(1) 0.0421 0.0000 0.0469 0.0000 0.0566 0.0000 0.0542 0.0000 0.0609 0.0000 0.0530 0.0000 0.0421 0.0000 0.0544 0.0000 0.0565 0.0000 0.0496 0.0000 ma(1)garch(1,2) 0.0387 0.0534 0.0346 0.2279 0.0347 0.2242 0.0278 0.8131 0.0308 0.5525 0.0320 0.4328 0.0276 0.8250 0.0302 0.6136 0.0306 0.5702 0.0354 0.1790 table 14. values of ki for the wig-construction index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0525 0.0000 0.0301 0.2070 0.0415 0.0009 0.0400 0.0016 0.0365 0.0119 0.0451 0.0001 0.0460 0.0001 0.0326 0.0823 0.0321 0.1004 0.0451 0.0001 arma(2,1) 0.0336 0.0145 0.0386 0.0003 0.0428 0.0000 0.0387 0.0003 0.0350 0.0064 0.0365 0.0022 0.0391 0.0002 0.0221 0.7270 0.0320 0.0311 0.0481 0.0000 arma(2,1) garch(1,1) 0.0286 0.5966 0.0289 0.5661 0.0321 0.2637 0.0293 0.5281 0.0239 0.9422 0.0305 0.4099 0.0263 0.8061 0.0251 0.8875 0.0301 0.4452 0.0338 0.1569 table 15. values of ki for the wig-developers index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.1392 0.0063 0.1477 0.0013 0.1290 0.0246 0.1154 0.1188 0.1859 0.0000 0.1292 0.0240 0.1255 0.0392 0.1370 0.0091 0.1699 0.0000 0.1353 0.0118 arma(1,1) 0.1479 0.0022 0.1562 0.0006 0.1466 0.0028 0.1144 0.1664 0.1506 0.0017 0.1515 0.0014 0.1488 0.0021 0.1258 0.0531 0.1412 0.0079 0.1484 0.0022 arma(1,1) garch(1,2) 0.0928 0.9250 0.1147 0.5124 0.0999 0.8324 0.0929 0.9245 0.1195 0.3976 0.1153 0.4980 0.1189 0.4135 0.1199 0.3861 0.0869 0.9708 0.1241 0.3014 table 16. values of ki for the wig-food index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0544 0.0000 0.0361 0.0000 0.0449 0.0000 0.0473 0.0000 0.0446 0.0000 0.0311 0.0037 0.0314 0.0028 0.0337 0.0003 0.0305 0.0052 0.0342 0.0002 arma(1,1) 0.0371 0.0000 0.0365 0.0000 0.0418 0.0000 0.0433 0.0000 0.0358 0.0000 0.0270 0.0055 0.0310 0.0003 0.0409 0.0000 0.0230 0.0601 0.0298 0.0007 arma(1,1)garch(1,1) 0.0311 0.5233 0.0340 0.2766 0.0338 0.2873 0.0347 0.2198 0.0239 0.9738 0.0309 0.5448 0.0281 0.7944 0.0281 0.7938 0.0311 0.5252 0.0362 0.1404 measuring nonlinear serial dependencies using the mutual information coefficient 105 table 17. values of ki for the wig-it index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0486 0.0000 0.0388 0.0038 0.0466 0.0000 0.0573 0.0000 0.0454 0.0000 0.0443 0.0001 0.0335 0.0580 0.0513 0.0000 0.0469 0.0000 0.0434 0.0002 ar(1) 0.0585 0.0000 0.0359 0.0449 0.0476 0.0000 0.0619 0.0000 0.0553 0.0000 0.0488 0.0000 0.0314 0.2556 0.0499 0.0000 0.0543 0.0000 0.0409 0.0032 ar(1)garch(1,1) 0.0362 0.0778 0.0251 0.8876 0.0270 0.7622 0.0339 0.1810 0.0222 0.9796 0.0260 0.8343 0.0282 0.6611 0.0244 0.9176 0.0303 0.4641 0.0293 0.5646 table 18. values of ki for the wig-media index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0481 0.0555 0.0560 0.0049 0.0448 0.1229 0.0562 0.0047 0.0475 0.0642 0.0456 0.1020 0.0350 0.6139 0.0422 0.2144 0.0304 0.8539 0.0393 0.3531 ma(1) 0.0484 0.1063 0.0571 0.0097 0.0516 0.0473 0.0529 0.0333 0.0519 0.0432 0.0398 0.4922 0.0464 0.1644 0.0450 0.2159 0.0446 0.2322 0.0426 0.3298 ma(1)garch(1,1) 0.0484 0.1380 0.0414 0.4673 0.0427 0.3925 0.0481 0.1451 0.0510 0.0735 0.0352 0.8260 0.0397 0.5785 0.0363 0.7691 0.0465 0.2020 0.0370 0.7320 table 19. values of ki for the wig-oil&gas index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0825 0.0185 0.0780 0.0503 0.0761 0.0733 0.0711 0.1660 0.0862 0.0076 0.0658 0.3335 0.0562 0.7510 0.0685 0.2422 0.0829 0.0166 0.0667 0.3009 ar(2) 0.0816 0.0203 0.0652 0.3279 0.0619 0.4671 0.0824 0.0183 0.0820 0.0195 0.0863 0.0062 0.0573 0.6711 0.0823 0.0184 0.0878 0.0043 0.0771 0.0493 ar(2)garch(1,1) 0.0451 0.8837 0.0611 0.2362 0.0493 0.7406 0.0573 0.3809 0.0716 0.0368 0.0573 0.3787 0.0524 0.6043 0.0652 0.1240 0.0592 0.3055 0.0472 0.8178 table 20. values of ki for the wig-telecom index k series 1 2 3 4 5 6 7 8 9 10 log returns 0.0467 0.0072 0.0395 0.1307 0.0429 0.0369 0.0687 0.0000 0.0440 0.0234 0.0393 0.1405 0.0417 0.0579 0.0469 0.0062 0.0419 0.0518 0.0514 0.0007 garch(1.3) 0.0311 0.5186 0.0340 0.2752 0.0338 0.2834 0.0347 0.2234 0.0239 0.9693 0.0309 0.5411 0.0281 0.7826 0.0281 0.7821 0.0311 0.5207 0.0362 0.1448 the results summarized in tables 9-20 indicate that evidence of serial dependencies was found for the most investigated indices10. the same conclusion may be drawn for the residuals from the arma models, which means that the detected dependencies are nonlinear. in most cases the estimated armagarch models were able to capture these nonlinearities. only in the case 10 the exception is the wig-oil&gas index. in this case the obtained result is rather unusual, i.e. filtering data by the arma model caused the appearance of significance of the mi measure. witold orzeszko 106 of wig and mwig40 indices there are reasons to believe that identified nonlinearity is not caused by an arch effect. references barnett w. a., gallant a. r., hinich m. j., jungeilges j. a., kaplan d., jensen m. j. (1998), a single-blind controlled competition among tests for nonlinearity and chaos, journal of econometrics, 82.1, 157–192. bruzda j. (2004), miary zależności nieliniowej w identyfikacji nieliniowych procesów ekonomicznych (measures of nonlinear relationship in identification of nonlinear economic processes), acta universitatis nicolai copernici, 34, 183–203. darbellay g.a, wuertz d. (2000), the entropy as a tool for analysing statistical dependencies in financial time series, physica a, 287, 429–439. dionisio a., menezes r., mendes d.a. (2003), mutual information: a dependence measure for nonlinear time series, working paper, http://129.3.20.41/eps/em/papers/0311/ /0311003.pdf (10.02.2010). fonseca n., crovella m., salamatian k. (2008), long range mutual information, proceedings of the first workshop on hot topics in measurement and modeling of computer systems (hotmetrics ’08), annapolis. fraser a.m., swinney h.l. (1986), independent coordinates for strange attractors from mutual information, physical review a, 33.2, 1134–1140. granger c. w. j., terasvirta t. (1993), modelling nonlinear economic relationship, oxford university press, oxford. granger c. w. j., lin j-l. (1994), using the mutual information coefficient to identify lags in nonlinear models, journal of time series analysis, 15, 371–384. maasoumi e., racine j. (2002), entropy and predictability of stock market returns, journal of econometric”, 107, 291–312. orzeszko w. (2010), detection of nonlinear autodependencies using hiemstra-jones test, financial markets. principles of modeling, forecasting and decision-making, eds. milo w., szafrański g., wdowiński p., 157–170. współczynnik informacji wzajemnej jako miara zależności nieliniowych w szeregach czasowych z a r y s t r e ś c i. w artykule scharakteryzowano konstrukcję, estymację oraz możliwości zastosowania współczynnika informacji wzajemnej. przedstawiono wyniki symulacji, prowadzących do weryfikacji jego przydatności w procesie identyfikacji zależności nieliniowych w szeregach czasowych. ponadto zaprezentowano wyniki zastosowania tego współczynnika do analizy indeksów giełdy papierów wartościowych w warszawie. s ł o w a k l u c z o w e: nieliniowość, współczynnik informacji wzajemnej, mutual information, identyfikacja zależności. microsoft word 00_tresc.docx dynamic econometric models vol. 10 – nicolaus copernicus university – toruń – 2010 joanna bruzda† nicolaus copernicus university in toruń european equity market integration and optimal investment horizons – evidence from wavelet analysis a b s t r a c t. in the paper the process of equity market integration in europe is examined from the wavelet perspective. the method applied is the continuous discrete wavelet transform that enables to perform global and local wavelet variance and correlation decompositions. in particular, questions about changes of the investment risk and the possibility of international portfolio diversification under different investment horizons are addressed. the study documents both convergence of the central and eastern european equity markets as well as their segmentation on the european market. the latter enables reduction of portfolio returns variability by an international portfolio diversification, especially for long investment horizons. k e y w o r d s: equity market integration, time-scale analysis, wavelet variance, wavelet correlations 1. introduction one consequence of financial globalization are comovements of prices on different stock markets. investigation of these processes is important due to both investors allocation decisions and policy-makers actions. from the point of view of investors the ongoing integration of capital markets increases the importance of a sectoral portfolio diversification at the expense of an international diversification. from a global perspective integration of financial markets is fundamentally related to economic growth via improvement of allocative efficiency, risk sharing and reduction of macroeconomic variability (see kim et al., 2005). convergence of stock markets, through the income channel, influences also the effectiveness of monetary policy and – as such – should be of considerable interest to policy-makers. † the author acknowledges the financial support from the polish ministry of science and higher education under the grant no. n n111 285135. joanna bruzda 16 studies by longin and solnik (1995) point out increasing integration of major world stock exchanges over the period 1960–1990 and find also an additional rise of correlation in the time of high volatility. also more recent studies document convergence of stock markets, although the process is non-uniform both in time and market segments (kim et al., 2005), asymmetric, i.e. negative shocks are more strongly transmitted via borders (fratzscher, 2002) and leaves place for partial segmentation of certain stock markets (bessler, yang, 2003). behind capital market integration stands real and nominal macroeconomic convergence, and therein reduction of currency risk and convergence of monetary policy with respect to interest rates and inflation (fratzscher, 2002; phengpis et al., 2004). integration of major european equity markets and the rise in their world significance first of all results from formation of the common currency area (fratzscher, 2002; kim et al., 2005; hardouvelis et al., 2006). only a couple of papers undertakes the task of examining the convergence of central and eastern european (cee) capital markets and their integration with the world capital market (scheicher, 2001; voronkova, 2004; chelleysteeley, 2005; gilmore et al., 2008; harrison, moore, 2009). chelley-steeley (2005) documents that for the period 1994–1999 correlations of stock index returns between cee and developed european markets were usually below 0.3, while at the same time correlations between indices on developed european markets are often above 0.5. the majority of empirical investigations points out a certain kind of segmentation of central-eastern european markets and the lack of uniformly increasing integration with the rest of europe, although cee markets remain under a significant influence of the world capital market. recent theoretical studies underline the importance of agents heterogeneity in asset pricing models (see, e.g., the fractal market hypothesis of peters, 1994). also survey studies confirm that investors acting on financial markets have different investment strategies and different investment horizons – from one day to several years (see the discussion in demary, 2009). agents with long investment horizons concentrate on economic fundamentals driving trends, while speculators want to beat the market in the short run and often resort to methods of technical analysis. heterogeneity of agents in a natural way gives rise to analyze stock prices according to different time scales (investment horizons). a method which enables investigation of stochastic processes decomposed according to scales is time-scale (wavelet) analysis. the aim of this study is to apply wavelet analysis to investigate the convergence process between central european capital markets and developed european markets. as by design the subject of this investigation are not the directly observed results of the globalization process, and therein the so-called contagion effects on financial markets, but rather their time-scale consequences for stock investors, stock indices have been denominated in one currency (polish zloty) in order to compensate for foreign currency exposure. european equity market integration and optimal investment horizons… 17 the empirical investigation spans indices from three developed european stock markets – frankfurt (dax), london (ftse1000) and paris (cac40) – as well as six emerging markets from the central and eastern europe – prague (px50), bratislava (sax), budapest (bux), sofia (sofix), bucharest (bet) and warsaw (wig). the study concentrates on the following questions: − do european stock markets converge (emerging markets alone as well as both developing and developed european markets), and if so, is the process uniform in time and across investment horizons (scales)? − for which investment horizons is international portfolio diversification most efficient? − is the process of the fast development of cee stock markets accompanied by the rise of investment risk? − is there an eu’s effect – an increase in stock exchange comovements caused by policy coordination, the rise in trade and investment and the opening of labor markets that foster the process of real macroeconomic convergence? the rest of the paper is organized as follows. section 2 describes briefly the tools of wavelet analysis that are used in the study, section 3 presents main empirical results, while the last section shortly concludes. 2. methodology wavelet analysis consists in decomposing a signal into shifted and scaled versions of a basis function, )(⋅ψ , called the mother wavelet. the decomposition can be continuous or discrete depending on the kind of the wavelet transform applied. the discrete wavelet transform (dwt) provides a parsimonious representation of the data and is particularly useful in noise reduction and information compression, while the continuous wavelet transform (cwt) is more helpful in recognizing local features of signals, especially those that are defined over the entire real axis, although this results in excessive redundancy of information. the continuous wavelet transform of a function )(⋅f is defined as follows: ∫ ∞ ∞− = dxxfxtw t )()(),( ,λψλ , (1) where: ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = λ ψ λ ψ λ tx xt 1 )(, , 0>λ . (2) by applying the cwt we obtain a set of wavelet coefficients, )(, xtλψ , depending on scale λ and time t. let us consider a vector of length jn 2= in the joanna bruzda 18 form ),,,( 110 ′= −nxxx …x . for jj ,,2,1 …= and 12,,1,0 −= − jjt … we define the discrete wavelet transform of vector x: ( )∑ − = = 1 0 ,, n n tjntj nxw ψ , (3) where )(, ⋅tjψ are shifted and scaled versions of the mother wavelet with dyadic shifts and scales, i.e.: ( )txx jjtj −= −− 22)( 2/, ψψ . (4) for a given j the coefficients tjw , correspond to scale 12 −= jjλ . the dwt results from a critical sampling of the cwt, which means that it contains the minimal amount of wavelet coefficients for complete reconstruction of the signal. here in the paper, following, e.g., percival and walden (2002), we concentrate exclusively on the dwt considering it as a more natural way of analyzing discrete time series. among wavelet tools based on the discrete transform are the wavelet variances and the wavelet correlations (also known as the wavelet coherences – see sanderson et al., 2009).1 for a stochastic process ty the timedependent wavelet variance is defined as: )var( 2 1 )( , 2 tj j jt wλ λσ = . (5) assuming that (5) does not depend on time2, the following variance decomposition according to time scales is obtained (see percival, walden, pp. 296–298): ∑ ∑ ∞ = ∞ = == 1 1 2 , )()var( 1 2 1 )var( j j jtj j t wy λσλ . (6) the wavelet variance at level j corresponding to scale 12 −= jjλ , )( 2 jλσ , informs about variation of oscillations with period lengths approximately in the 1 other tools based on the dwt are: multiresolution analysis taking advantage of both the dwt and its inversion, wavelet cross-correlations computed via the continuous discrete (maximal overlap) wavelet transform operating on discrete scale and continuous time as well as methods from complex-valued wavelet analysis, especially the discrete wavelet phase angle taking advantage of the maximal overlap discrete hilbert wavelet transform (modhwt) – see gençay et al. (2002); whitcher, craigmile (2004). the measures mentioned above can also be applied to only a portion of wavelet coefficients, what results in local (short-time) versions of the tools – see, e.g., sanderson et al. (2009). 2 such an assumption is fulfilled also for nonstationary processes provided that they are integrated of order d and the width of the daubechies wavelet filter, l, is sufficient to eliminate nonstationarity (i.e. l ≥ 2d) – see percival, walden (2000), p. 304. in what follows we concentrate exclusively on daubechies filters, although they are not the only ones that are interpretable in terms of generalized differences of weighted averages. european equity market integration and optimal investment horizons… 19 interval j2 – 12 +j . similarly, the wavelet covariance and wavelet correlation are introduced. for stochastic processes ty1 and ty2 the wavelet covariance for scale jλ is defined as: ),cov( 2 1 )( ,,2,,1 tjtj j j wwλ λγ = . (7) as in the case of variance decomposition (6), wavelet covariances are obtained by decomposing the covariance between ty1 and ty2 according to different scales jλ . next, let us define the wavelet correlation coefficient for scale jλ via: )()( )( )( 21 jj j j λσλσ λγ λρ = . (8) the quantity (8) is normalized in the interval [-1, 1] and indicates the strength and direction of a relationship between two processes for a given resolution level (i.e. for a given time scale jλ ). in practice, when estimating the wavelet variance, covariance and correlation, instead of the dwt their modification in the form of the so-called maximal overlap (continuous discrete) dwt is used. the modwt can be thought of as a subsampling of the cwt at dyadic scales, but leaving all times t instead of only those that are multiples of powers of 2. for this reason it is called also the non-decimated wavelet transform. this enables to eliminate certain artifacts produced by the dwt resulting from the lack of time-invariance, does not require data sets of length j2 and – what is important from the point of view of this study – provides more statistically efficient estimators of the wavelet variance (gençay et al., 2002, p. 135). an unbiased estimator of the wavelet variance is then defined as: ∑ − −= = 1 1 2 , 2 ~ ~ 1 )(~ n lt tj j j j w n λσ , (9) where tjw , ~ are the modwt coefficients, 1)1)(12( +−−= ll jj is the width of the wavelet filter for scale jλ (l is the width of the basic wavelet filter at the first stage of the pyramid algorithm that computes the dwt) and 1 ~ +−= jj lnn is the number of wavelet coefficients not affected by the boundary. by using (9) we assume that the wavelet filter applied eliminates all deterministic components of the process under scrutiny. if the process has constant, nonzero mean value, the formula (9) is modified appropriately. an approximate (1–α)% confidence interval for )(2 jλσ is computed as follows: joanna bruzda 20 5,0 , ~ 2 ~ )0(ˆ )(~ 2 ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ ± j jw j n f αςλσ , (10) where 2 ας is the (1–α/2) quantile of the standard normal distribution and )0(ˆ , ~ jw f is an estimator of the spectral density of scale jλ squared wavelet coefficients at frequency 0. estimates of wavelet covariances and wavelet correlations are computed via the following formulas: tj n lt tj j j ww n j ,,2 1 1 ,,1 ~~ ~ 1 )(~ ∑ − −= =λγ , (11) )(~)(~ )(~ )(~ 21 jj j j λσλσ λγ λρ = . (12) an approximate (1–α)% ci for )( jλγ is obtained as previously with )0(ˆ ,~ jwf being an estimate of the cross-spectrum at frequency 0. in the case of wavelet correlations gençay et al. (2002), p. 259–260, suggest an approach taking advantage of the fact that the dwt approximately decorrelates even long memory processes. in such a case, in order to obtain confidence limits that are placed in the interval ±1 the fisher z-transformation can be applied, which under the gaussian assumption leads to the following (1–α)% ci for scale jλ : [ ] ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ±− 5,0 1 3ˆ 1 )(~tanhtanh 2 j j n αςλρ , (13) where jn̂ is the number of (conventional) dwt coefficients associated with scale jλ that is treated here as a measure of the scale-dependent sample size. 3. empirical results in the empirical examination daily closing prices of nine indices denominated in polish zloty at the national bank of poland (nbp) exchange rates have been used. we decided on examining prices instead of returns due to the fact that, firstly, using wavelets we do not need to transform data to stationarity prior to the analysis, secondly, examining indices provides better interpretation in terms of synchronization of stock market cycles and, thirdly, wavelet coefficients obtained with the haar wavelet filter can be thought of being local estimates of multi-period returns and – as such – our results of wavelet variance and correlation analysis will have practical implications for portfolio construceuropean equity market integration and optimal investment horizons… 21 tion. the main advantage of using wavelets over more traditional approaches is their efficiency in data exploration resulting from optimal time-frequency resolution and the simplicity in analyzing scale-dependent phenomena. the indices examined here are from both developed european capital markets (dax, ftse, cac) and cee markets (px, sax, bux, sofix, bet and wig) and span the period 2.01.2002–30.04.2009 (1912 observations), except for sax, for which the sample starts on 2.01.2003, as well as sofix and bet, where the samples begin on 3.04.2007. the reason for the differences in data length is the lack of daily quotations of slovak koruna, bulgarian lev and romanian leu to the beginnings of our samples. after linear interpolation of missing observations the quotations have been transformed into logarithms. the computations were performed on the entire sample and in three subsamples: 01.2002–04.2004 (608 observations), 05.2004–03.2007 (761 observations), 04.2007–04.2009 (543 observations). the first subsample spans the period prior to the enlargement of the european union in 2004, the second comprises first three years after the enlargement, while the last covers the most recent period that includes also the data on the latest financial crisis and is the only one spanning all nine indices. the results of group unit root tests on an unbalanced panel comprising all indices and all observations performed with lag length chosen on the base of the schwartz criterion, spectrum estimation at frequency 0 with the bartlett kernel, newey-west bandwidth selection, individual intercepts for the levels and no intercept for first differences are given in table 1. the results (except for the t* statistic) allow to treat our multivariate process as integrated of order 1. table 1. results of group unit root tests test statistic p-value statistic p-value level first difference h0: common unit root levin, lin, chu t* -3.029 0.001 -92.054 0.000 breitung t 3.199 0.999 87.593 0.000 h0: individual unit root im, pesaran, shin w 0.596 0.725 adf-fisher χ2 15.915 0.599 1153.20 0.000 pp-fisher χ2 16.155 0.582 529.606 0.000 h0: no common unit root hadri z 66.89 0.000 depending on data length a 6or 8-level maximal overlap discrete wavelet transforms were executed. one of the daubechies least asymmetric wavelet filters – la(8) – was applied, which is nearly linear in phase and has the width 8. besides, in the case of wavelet correlations the examination was also performed with the haar wavelet filter3. the decomposition levels 1–6 corres 3 as they confirm results obtained with the la(8) filter, we do not present them here. joanna bruzda 22 pond to oscillations with period lengths approximately in the following intervals: 2–4, 4–8 (up to one and a half weeks), 8–16 (up to 3 weeks), 16–32 (up to 6 weeks), 32–64 (up to one quarter) and 64–128 (up to 2 quarters). when the whole samples were analyzed, two further decomposition levels were added that correspond to fluctuations with periods 128–256 (up to one year) and 256–512 (up to 2 years), respectively. the computations ware executed in matlab after modifying codes developed by b. whitcher (the wavecov package at www2.imperial.ac.uk/~bwhitche/software) as well as d. b. percival and a. t. walden (the wmtsa toolkit at www.atmos.washington.edu/~wmtsa) and supplementing them with own programs. figures 1–3 present decompositions of the wavelet variances with the help of la(8) wavelet. the results point out that for the three mature markets the investment risk in the second period was significantly below its level in the first and third part of the sample. in the third period it rose for all investment horizons (for the london stock exchange even above its level in the first subsample). in the case of the three examined emerging markets – prague, budapest and warsaw – the situation looks differently: in the preand post-accession periods the wavelet variances do not differ at all scales, while in the last period they rose significantly above their previous levels, except for the longest investment horizons (above 6 weeks in the case of bux and wig and one quarter in the case of px). this lack of a significant increase in the investment risk for the longest horizons can be thought of as a sign that central and eastern european capital markets were more robust to the financial crisis at the beginning of it than the developed markets. a more detailed examination, of which we present here only the comparison between dax and wig (see figure 3), performed with the help of the local wavelet variance4, shows, however, a quite similar level of volatility for medium and long horizon investments on the polish market as compared to frankfurt, with periods of a rapid increase of risk, except for the highest level of examination. a similar situation was observed for the other developing and developed markets. figures 4–10 show decompositions of the wavelet correlations. there are very strong dependences present for the developed capital markets at all investment horizons. the three markets can be thought of as a one investment possibility. analyzing correlations between the developed and developing markets it is seen that the mature markets significantly influence the cee stock indices, except for bratislava and for longer horizons also sofia. one important finding is the lack of systematic convergence of the mature and the cee mar 4 the local wavelet variances were computed for windows of width 100 after aligning them with the original data and exclusion of all wavelet coefficients affected by the boundary (the circular filtering), what cut results at the beginning and the end of the period and is especially pronounced for higher resolution levels. the number of affected coefficients at the beginning (end) of the sample for the eight decomposition levels is, respectively, 3 (4), 10 (11), 24 (25), 52 (53), 108 (109), 220 (221), 444 (445), 892 (893). european equity market integration and optimal investment horizons… 23 kets: in the majority of cases the highest correlations occur in the last period, what can be attributed to the contagion effects on the financial markets, while the lowest are usually in the middle subsample, although they do not differ significantly from those computed for the first period. generally, our conclusion is that we do not observe a uniformly increasing dependence between these two types of markets. this finding has been also confirmed with a more detailed examination with the help of the local wavelet correlations analysis – see figure 9 – that was executed for data windows of length 200. figure 1. results of 6-level wavelet variance decomposition using the la(8) wavelet filter in subsamples: 01.2002–04.2004 (–ο–), 05.2004–03.2007 (– – ), 04.2007–04.2009 (–∗– ) on the other hand, a systematic rise in the degree of dependence is found for three of the emerging markets: prague, budapest and warsaw, what is especially pronounced for the lowest decomposition levels (see figure 10). besides there are also significant relationships between these indices and bet, while sax seem to be uncorrelated with all the other indices and sofix is best correlated with dax and bet. figure 2. comparison of wavelet variance in subsamples together with the 95% confidence intervals using the la(8) wavelet filter: 01.2002–04.2004 (–ο–), 05.2004–03.2007 (– –), 04.2007–04.2009(–∗–); thick lines correspond to the later periods european equity market integration and optimal investment horizons… 25 figure 3. local wavelet variance at 8 decomposition levels for dax (thick solid line) and wig (dashed line) using the la(8) wavelet filter figure 4. results of wavelet correlation decompositions together with the 95% confidence intervals using the la(8) wavelet filter as for international portfolio diversification very promising seem to be the relationships between ftse and wig as well as cac and wig, especially for long investment horizons. besides, we notice insignificant wavelet correlations of px, bux and wig with the indices from sofia and bratislava. for the majority of pairs of indices the wavelet correlations seem to be homogenous across scales, except for the longest investment horizons. 500 1000 1500 0 1 2 3 4 x 10 -4 scale 1 500 1000 1500 0 1 2 3 4 x 10 -4 scale 2 500 1000 1500 0 2 4 6 x 10 -4 scale 4 500 1000 1500 0 0.5 1 x 10 -3 scale 8 500 1000 1500 0 0.5 1 1.5 2 x 10 -3 scale 16 500 1000 1500 0 1 2 3 x 10 -3 scale 32 500 1000 1500 0 1 2 3 x 10 -3 scale 64 500 1000 1500 0 0.005 0.01 0.015 scale 128 figure 5. results of wavelet correlation decompositions (continued) figure 6. results of wavelet correlation decompositions in subsamples using the la(8) wavelet filter: 01.2002–04.2004 (–ο–), 05.2004–03.2007 (– – ), 04.2007–04.2009 (–∗– ) figure 7. comparison of wavelet correlations in subsamples together with the 95% confidence intervals using the la(8) wavelet filter: 01.2002–04.2004 (–ο–), 05.2004–03.2007 (– –), 04.2007–04.2009(–∗–); thick lines correspond to the later periods figure 8. comparison of wavelet correlations in subsamples (continued) joanna bruzda 28 figure 9. local wavelet correlations at 6 decomposition levels for the dax-wig (thick solid line) and ftse-wig (dashed line) relationships using the la(8) wavelet filter figure 10. local wavelet correlations at 6 decomposition levels for the bux-wig (thick solid line) and px-wig (dashed line) relationships using the la(8) wavelet filter 4. conclusions of the countries investigated an approximately uniform rise in integration takes place for the ‘big three’ emerging cee equity markets: the czech republic, hungary and poland. we do not observe systematic convergence among emerging capital markets and the three mature stock exchanges (frankfurt, london and paris), as the rise in the wavelet correlations documented for the most recent data investigated can be explained by contagion effects on financial 500 1000 1500 -1 -0.5 0 0.5 1 scale 1 500 1000 1500 -1 -0.5 0 0.5 1 scale 2 500 1000 1500 -1 -0.5 0 0.5 1 scale 4 500 1000 1500 -1 -0.5 0 0.5 1 scale 8 500 1000 1500 -1 -0.5 0 0.5 1 scale 16 500 1000 1500 -1 -0.5 0 0.5 1 scale 32 600 800 1000 1200 -1 -0.5 0 0.5 1 scale 64 500 1000 1500 -1 -0.5 0 0.5 1 scale 1 500 1000 1500 -1 -0.5 0 0.5 1 scale 2 500 1000 1500 -1 -0.5 0 0.5 1 scale 4 500 1000 1500 -1 -0.5 0 0.5 1 scale 8 500 1000 1500 -1 -0.5 0 0.5 1 scale 16 500 1000 1500 -1 -0.5 0 0.5 1 scale 32 600 800 1000 1200 -1 -0.5 0 0.5 1 scale 64 european equity market integration and optimal investment horizons… 29 markets. our empirical results point out a certain kind of segmentation of cee capital markets, although these markets remain under a significant influence of the developed stock exchanges. so, it seems that in the rather short postaccession period the developing capital markets do not face the eu’s effect yet. the distribution of wavelet correlations across scales is relatively homogenous. departures from homogeneity take usually place for longer horizons and have the form of bidirectional deviations. for certain pairs of indices we found significantly negative wavelet correlations for the longest investment horizons investigated. this makes it possible substantially to reduce portfolio returns variability by international portfolio diversification. for shorter investments zero or even negative wavelet correlations have been found with the sax and sofix indices. generally, the fast development of central and eastern european equity markets is accompanied by a relatively moderate risk for mediumterm investments, with rapid changes in volatility. references bessler, d. a., yang, j. (2003), the structure of interdependence in international stock markets, journal of international money and finance, 22, 261–287. chelley-steeley, p. l. (2005), modelling equity market integration using smooth transition analysis: a study of eastern european stock markets, journal of international money and finance, 24, 818–831. connor, j., rossiter, r. (2005), wavelet transforms and commodity prices, studies in nonlinear dynamic and econometrics, 9 (1), article 6. demary, m. (2009), transaction taxes and traders with heterogeneous investment horizons in an agent-based financial market model, economics. the open-access, open-assessment e-journal, discussion paper no. 2009–47. fratzscher, m. (2002), financial market integration in europe: on the effects of emu on stock markets, international journal of finance and economics, 7, 165–193. gençay, r. f., selçuk, f., whitcher, b. (2002), an introduction to wavelets and other filtering methods in finance and economics, academic press, san diego. gilmore, c. g., lucey, b. m., mcmanus, g. m. (2008), the dynamics of the central european equity market comovements, quarterly review of economics and finance, 48, 605–622. hardouvelis, g. a., malliaropulos, d., priestley, r. (2006), emu and european stock market integration, journal of business, 79, 365–392. harrison, b., moore, w. (2009), spillover effects from london and frankfurt to central and eastern european stock markets, applied financial economics, 19, 1509–1521. kim, s. j., moshirian, f., wu, e. (2005), dynamic stock market integration driven by the european monetary union: an empirical analysis, journal of banking and finance, 29, 2475– 2502. longin, f., solnik, b. (1995), is the correlation in international equity returns constant:1960– 1990?, journal of international money and finance, 14, 3–26. percival, d. b., walden, a. t. (2000), wavelet methods for time series analysis, cambridge university press, cambridge. peters, e. e. (1994), fractal market analysis. applying chaos theory to investment and economics, john wiley & sons, new york. phengpis, c., apilado, v. p., swanson, p. e. (2004), effects of economic convergence on stock market returns in major emu member countries, review of quantitative finance and accounting, 23, 207–227. joanna bruzda 30 sanderson, j., fryzlewicz, p., jones, m. (2009), measuring the dependence between nonstationary time series using the locally stationary wavelet model, to appear in biometrika. scheicher, m. (2001), the comovements of stock markets in hungary, poland and the czech republic, international journal of finance and economics, 6, 27–39. voronkova, s. (2004), equity market integration in central european emerging markets: a cointegration analysis with shifting regimes, international review of financial analysis, 13, 633–647. whitcher, b., craigmile, p. f. (2004), multivariate spectral analysis using hilbert wavelet pairs, international journal of wavelets, multiresolution and information processing, 2, 567–587. integracja giełd europejskich i optymalne horyzonty inwestycyjne w świetle analizy falkowej z a r y s t r e ś c i. w artykule prezentuje się wyniki badania procesu integracji giełd europejskich przeprowadzonego z użyciem narzędzi ciągło-dyskretnej transformaty falkowej, a dokładniej globalnych i lokalnych (krótkookresowych) wariancji i korelacji falkowych. w szczególności odpowiada się na pytania o zmiany ryzyka inwestycyjnego oraz możliwość międzynarodowej dywersyfikacji portfeli przy uwzględnieniu różnych horyzontów inwestycyjnych. badanie pokazuje, że ma miejsce proces konwergencji giełd środkowoeuropejskich, ale rynki te jako całość wykazują pewną segmentację. daje to możliwość międzynarodowej dywersyfikacji portfeli, przede wszystkim dla dłuższych horyzontów inwestycyjnych. s ł o w a k l u c z o w e: integracja giełd, analiza czasowo-skalowa, wariancje falkowe, korelacje falkowe. microsoft word dem_2018_49to65.docx © 2018 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2018.003 vol. 18 (2018) 49−65 submitted october 30, 2018 issn (online) 2450-7067 accepted december 4, 2018 issn (print) 1234-3862 mitra lal devkota and humnath panta * an inquiry into the effect of the interest rate, gold price, and the exchange rate on stock exchange index: evidence from nepal a b s t r a c t. this study examines the causal relationship between the nepalese stock exchange (nepse) index, the interest rate, gold price, and the usd exchange rate in nepal. the monthly time series data from january 2006 to june 2018 are used. time series properties of the data are diagnosed using the ng-perron unit root test and johansen's cointegration test. finally, the granger causality test based on the vector error correction model (vecm) is used to find the direction of causation, and to model the short and long-run relationships between the variables. the findings suggest that there exists a feedback relationship between the nepse index and the interest rate, and there exists a unidirectional causation from the gold price to both the exchange rate and the interest rate. there is also a unidirectional causation from the exchange rate to the nepse index during the sample period. these findings have implications for government agencies, investors, researchers, stakeholders, and others interested in the topic. k e y w o r d s: causality; cointegration; exchange rate; gold price; interest rate; nepse index j e l classification: c22; e00; e44 introduction the relationship between the stock exchange index and macroeconomic variables including the gold price has received considerable attention in the *correspondence to: mitra lal devkota, university of north georgia, department of management and marketing, mike cottrell college of business, 82 college circle, dahlonega, ga 30597, united states, e-mail: mldevkota@ung.edu; humnath panta, brenau university, college of business and communication, 500 washington street se, gainesville, ga 30501, united states, e-mail: hpanta@brenau.edu. mitra lal devkota, humnath panta dynamic econometric models 18 (2018) 49–65 50 literature (fama, 1981; gunes, 2007; pilinkus and boguslauskas, 2009). different researchers have used different set of variables in their studies. for example, smyth and nandha (2003) and nieh and lee (2001), among others, have studied the relationship between stock prices and exchange rates, whereas other researchers have used several macroeconomic variables in their study (tursoy, gunsel and rjoub, 2008). the selection of variables, however, depends on the nature and structure of the economy as well as the size and significance of the stock market in the economy. empirical studies reveal that with financial deregulation, the stock market of a country has become more sensitive to both domestic and external factors (mishra, das, and mishra, 2010). several studies have examined the impact of macroeconomic variables on stock prices for developed as well as for developing countries. alam and uddhin (2009) document that stock prices and interest rate are the crucial factors which determine the economic growth of a country. the impact of the interest rate on stock prices provides important implications for monetary policy, risk management practices, financial securities valuation and government policy towards financial markets. several researchers have investigated the relationship between gold prices and stock exchange indices. according to sujit and kumar (2011), gold provides high liquidity, and investment on gold can also be used as a hedge against inflation and currency depreciation. from an economic and financial point of view, movements in the price of gold are both interesting and important, and hence, it is necessary to validate the dynamic relationship of gold price with other variables under study periodically. likewise, a number of empirical studies have investigated the relationship between stock prices and exchange rates. likewise, a number of empirical studies have investigated the relationship between stock prices and exchange rates. according to ratanapakorn and sharma (2007), "for several reasons, foreign exchange rates should not be ignored when modelling stock prices. first, the money supply is used to stabilize foreign exchange rates. second, the exchange rate movements may reinforce the link from money to inflation. third, exchange rates may influence stock prices through interest rate effects. finally, foreign exchange rates are important for investors deciding whether they should invest in the foreign exchange market or in the stock market." ratanapakorn and sharma (2007) are correct that some countries target exchange rates with monetary policy. other countries target inflation rates with monetary policy, but, even so, monetary policy affects exchange rates, regardless of which target monetary authorities choose. most of the studies in this area focus either on large economy such as india (upadhyaya, nag and mixon jr, 2018) or developed economies such as the usa (ratanapakorn and sharma, 2007). there is a relative dearth of such an enquiry into the effect of the interest rate, gold price, and the exchange… dynamic econometric models 18 (2018) 49–65 51 studies in small countries like nepal. the purpose of this study is to fill this void. using nepalese data, our research work differs from the existing literature in several ways. first, we have used the fairly-recently developed unit root test that has better size and power properties than the widely used phillips perron (pp) test and the augmented dickey-fuller (adf) test. second, our research uses recent data that covers a longer period of time than the existing literature and includes the exchange rate, the additional variable excluded in most of the existing papers in the context of nepalese data. we have found that this variable has causal relationship with the gold price and the nepse index. finally, we examine both short and long-run causal relationships between the nepse index, the interest rate, gold price, and the usd exchange rate. the rest of the paper is organized as follows. a review of previous empirical studies is carried out in section 1. a detailed description of the data and the variables used in the study are presented in section 2. the econometric methodology used in the study and discussion of the empirical results are presented in section 3. the last section concludes the paper. 1. literature review the study of the causal relationship between stock prices and macroeconomic variables has received considerable attention in the literature. these studies have used different macroeconomic variables and data from both developed and developing countries. in this section, we review a selected number of research articles from a plethora of publications. alam and uddin (2009) examined the relationship between stock prices and interest rates for fifteen developed and developing countries: australia, bangladesh, canada, chile, colombia, germany, italy, jamaica, japan, malaysia, mexico, philippine, south africa, spain, and venezuela based on monthly data from january 1988 to march 2003. for all the countries in their sample, they found a significant negative relationship between interest rate and share price, and for six countries, they found a significant negative relationship between changes of interest rate and changes of share price. so, a considerable control in interest rate would be of a great benefit to these countries’ stock exchange through a demand-pull mechanism, by way of more investors in share market, and supply-push mechanism, by way of more investment by companies. ratanapakorn and sharma (2007) studied the long-term and short-term relationships among the us stock price index (s&p 500) and macroeconomic variables from the first quarter of 1975 to the fourth quarter of 1999. they document that the s&p 500 average and long-term interest rates are negatively correlated while the money supply, industrial production index, inflation rate, mitra lal devkota, humnath panta dynamic econometric models 18 (2018) 49–65 52 exchange rate, and short-term interest rates are positively correlated. their causality analysis revealed that every macroeconomic variable considered caused the stock price in the long-run but not in the short-run. graham (2001) examined the relationship between the price of gold and stock price for the us over the period from january, 1991 to october, 2001 using four gold prices and six stock price indices. his analysis revealed an evidence of a unidirectional causality from stock price to returns on the gold price set in the london morning fixing and the closing price. however, for the price set in the afternoon fixing, his analysis shows an evidence of feedback relationship between the gold price and the stock price. levin, montagnoli and wright, (2006) presented the short-run and long-run determinants of the price of gold using the theoretical framework of supply and demand. they revealed that total supply of gold is a function of the gold price. they also concluded that fluctuations in the gold price are caused by political stability, financial turmoil, changes in exchange rates and real interest rates. smith (2001) investigated the short-term and long-term relationships between the gold price and stock exchange price index using daily, weekly and monthly time series data from 1991 to 2001. four gold prices and six stock exchange indices were included in the study. he found no bilateral long-run relationship, or cointegration, between a gold price series and a stock market index. while there was some evidence of negative short-term granger causality running from us stock index returns to gold returns, the reverse was not the case. moore (1990) examined the link between anticipated inflation and gold returns, using a leading index of us inflation from 1970 to 1988 compiled by the colombia university business school. he found that gold price is negatively correlated with stock/bond markets. he further added that gold was an alternative investment tool for turkish investors. this result is consistent with the finding of buyuksalvarci (2010), who analyzed the effects of seven macroeconomic variables (the consumer price index, money market interest rate, gold price, industrial production index, oil price, foreign exchange rate, and money supply) on the turkish stock exchange market. tsoukalas (2003) studied the relationship between stock prices and the macroeconomic variables in cyprus. the results of the study found a strong relationship between stock prices and exchange rates. this is because the cypriot economy depends on services (import sector) such as tourism, off shore banking, etc. vygodina (2006) examined the relationship between exchange rates and stock prices nexus for large cap and small cap stocks for the period from 1987 to 2005 in the usa. he found that large cap stocks granger cause exchange rates. however, there was no causality for small cap stocks. stock prices and exchange rates were affected by the same macroeconomic variables and changes in federal monetary policy in the usa had a significant effect on an enquiry into the effect of the interest rate, gold price, and the exchange… dynamic econometric models 18 (2018) 49–65 53 the nature of these relationships. smyth and nandha (2003) examined the relationship between exchange rates and stock prices in bangladesh, india, pakistan and sri lanka using daily data over a six-year period from 1995 to 2001. they found that there is no long-run equilibrium relationship between the financial variables in any of the four countries. also, the empirical results revealed unidirectional causality running from exchange rates to stock prices for only india and sri lanka, but no evidence of any causality was found between exchange rates and stock prices in bangladesh and pakistan. nieh and lee (2001) studied the relationship between stock prices and exchange rates for g-7 countries taking the daily closing stock market indices and foreign exchange rates for the period from october 1, 1993 to february 15, 1996. they found that there is no long-run equilibrium relationship between stock prices and exchange rates for each of g-7 countries. they did find a significant relationship between stock prices and exchange rates for a single day in some g-7 countries but found no significant correlation in the united states. these results might be explained by the difference between the countries’ economic development stages, government policies, expectation patterns, etc. wongbampo and sharma (2002) employed a vecm model to investigate the relationship between stock prices and five macroeconomic variables such as gnp, inflation, money supply, interest rate, and exchange rate in five asian countries, namely, malaysia, indonesia, philippines, singapore and thailand. they used monthly data for the period from 1985 to 1996, and found that, there exists both a short-term and long-term relationship between the stock prices and the macroeconomic variables. they also found a feedback relationship between the stock prices and the macroeconomic variables in all the countries in their study. similarly, mukherjee and naka (1995) also employed a vecm model to examine the relationship between stock market returns and a set of six macroeconomic variables such as exchange rate, inflation, money supply, industrial production index, the long-term government bond rate, and call money rate in japan. their analysis suggests that there exists a long-run equilibrium relationship between the stock prices and the macroeconomic variables in japan. on the other hand, srinivasan (2014) used the autoregressive distributed lag (ardl) bounds testing approach and the granger causality test on monthly time series data from june 1990 to april 2014 to investigate the causal nexus between the gold price, stock price, and the exchange rate in india. the results revealed that the gold price and stock price tend to have a long-run relationship with the exchange rate in india. however, there was no evidence of a stable long-run or short-run causal relationship between the stock price and gold price in india. in the nepalese context, the study of gaire (2016) is the only one in the area of our interest. he examined cointegration and causality between nepse mitra lal devkota, humnath panta dynamic econometric models 18 (2018) 49–65 54 index with regard to short-term interest rates and gold prices in nepal. he analyzed monthly time series data from january 2006 to december 2016. he found that there is a long-run equilibrium relationship between the nepse index, short-term interest rate, and gold price in nepal. he further added that the short-term interest rate could be one of the predictors of stock prices in the secondary market of nepal. although a pioneering study in the nepalese context, his research suffers from some weaknesses in the adopted methodologies. for instance, he used the augmented dickey fuller (adf) test which is known to have low power and size properties. in addition, the adf test is known to have a severe size distortion (in the direction of over-rejecting the null when it is true) when the series has a large negative moving average root. moreover, our research uses more recent data that covers a longer period of time and includes the exchange rate. we have found that the exchange rate, the variable excluded in his paper, has a causal relationship with the gold price and the nepse index. 2. data this study is based on secondary data for the period between january, 2006 to august, 2018. these are obtained from various sources including nepal rastra bank (nrb), the central bank of nepal, nepalese stock exchange (nepse) limited, and nepal gold and silver dealers’ association (negosida). it consists of monthly time series data with the variables nepse index, interest rate, gold price, and exchange rate (usd exchange rate expressed as the amount of nepalese rupees per unit of usd). statistical software packages r and eviews are used for arranging the data and conducting econometric analysis. nepse index: the transaction index published at the end of the day by the nepal stock exchange. the nepse index data are collected from various reports of nepal stock exchange ltd and current macroeconomic and financial situation dataset from the nrb. the nepse index on the last day of the month is considered for the analysis. interest rate: the interest rate, also known as interbank rate, is the rate of interest for short-term lending/borrowing among commercial banks. the monthly average interest rate data are obtained from the quarterly economic bulletin of nrb. gold price: end of month’s gold prices per 10-gram data are obtained from the website of nepal gold and silver dealers’ association (negosida). exchange rate: the exchange rate is monthly average rate of exchange between us dollars (usd) and nepalese rupees (npr). the exchange rates an enquiry into the effect of the interest rate, gold price, and the exchange… dynamic econometric models 18 (2018) 49–65 55 data are obtained from the nrb website and are computed by taking the average of the buying rates and selling rates. these were the only variables with sufficient monthly data available to the authors for the time period under study. for example, even though we wanted to include variables such as industrial production index and consumer price index, the unavailability of the data on a monthly basis prevented us from including them into our analyses. 3. methodology and empirical results 3.1 test of correlation and multicollinearity the correlation coefficients between the variables are summarized in table 1. the figures indicate that the nepse index has a negative correlation with the interest rate, but positive correlation with the gold price and the exchange rate. we also observe that there is a very strong correlation between the exchange rate and the gold price. this suggests that multicollinearity might be an issue in the time series data. table 1. correlation matrix ni ir gp er ni 1 ir −0.15 1 gp 0.22 −0.36 1 er 0.57 −0.53 0.79 1 note: ni, gp, er, and ir stand for nepse index, gold price, exchange rate, and interest rate, respectively. wooldridge (2011) states that multicollinearity is likely to exist if the t-statistics corresponding to the parameter estimates of independent variables in an ordinary least square (ols) regression model are not statistically significant, whereas the overall f statistic is statistically significant.wooldridge (2011) further adds that multicollinearity is a serious problem if the vif is greater than 10. in this regard, an ols regression model is fitted using nepse index as the dependent variable and the interest rate, gold price, and exchange rate as independent variables and the results are reported in table 2. furthermore, the variance inflation factors (vif) for the coefficients of each variable are estimated and the results are reported in table 3. the ols results indicate that all the parameters as well as the overall f statistic of the regression model are statistically significant at 5% level of significance. mitra lal devkota, humnath panta dynamic econometric models 18 (2018) 49–65 56 table 2. results from ordinary least squares regression model estimate standard error 𝑡-ratio pr (> |𝑡|) intercept −0.675 0.400 − 1.687 0.094 ir 0.115 0.030 3.914 0.000* gp −0.789 0.116 −6.826 0.000* er 3.642 0.310 11.764 0.000* residual standard error: 0.1586 on 146 degrees of freedom multiple r-squared: 0.5153, adjusted r-squared: 0.5053 f-statistic: 51.74 on 3 and 146 df, p-value: <2.2e-16 note: * – statistical significance at 5% level of significance. in addition, the vif results in table 3 suggest that the vifs for all the variables are smaller than 10. these results suggest that multicollinearity is not an issue in the time series data. table 3. variance inflation factor interest rate gold price exchange rate 1.4133 2.6958 3.2663 for the econometric analyses of the time series data, the ng-perron test (ng and perron, 2001) is used to test the stationarity both at the levels and the first differences. then the johansen cointegration method is used to investigate the long-term relationship among the variables and to determine the number of cointegrating vectors. the granger causality test based on the vector error correction model (vecm) is used to find the direction of causation and to model the short and long-run relationships between the variables. 3.2 unit root test for testing stationarity (ng-perron test) according to ng-perron (2001), the widely used adf test suffers from low power, especially when the moving-average polynomial of the first differenced series has a large negative root. the adf test seems to over-reject the null hypothesis when it is true and fails to reject the null hypothesis when it is false. to overcome this issue, they proposed a new test known as the ngperron test. compared to the adf and pp unit root tests, this test possesses better power and size properties, so its results are more reliable when applied to small data sets (harris and sollis, 2003). the ng-perron test has the null hypothesis of non-stationarity of the time series. there are four test statistics, (mz0,mz2,msb,mpt) associated with this test. the first two test statistics (mz0,mz2) are efficient versions of the z0 and z2 test statistics, and are usually reported more often for interpretation of empirical results (gregorioui, kontonikas and montagnoli, 2006; cuestas and harrison, 2008; cuestas and staehr, 2013; raihan et al., 2017). these statistics are given by: an enquiry into the effect of the interest rate, gold price, and the exchange… dynamic econometric models 18 (2018) 49–65 57 mz0 = :;<(=>)?@ab cd (1) m𝑍f = mz0msb (2) where k = ∑ i=j;< : k c ,:2lc msb = i d ab k m cn , f0 is the spectral density at frequency zero, and y:is the generalized least squares (gls) de-trended value of the variable. these statistics are based on a specification for 𝑥f and a method for estimating f0. the test uses a gls detrended series to improve the power properties and uses modified lag selection criteria to address the size distortion. we use the ng-perron test to test the stationarity of the variables in logarithmic scales at levels, and their first differences. we considered intercept as well as intercept and trend while testing at levels and their first differences. this analysis is performed using eviews 10. as reported in table 4 and table 5, each series is non-stationary at levels, and then stationary at the first differences, suggesting that all the variables are individually integrated of order 1, that is 𝐼(1). after establishing the stationarity of the time series data, we proceed to conduct the test for cointegration. 3.3. the johansen test for cointegration to further investigate the long-term relationships, the johansen (1988, 1991, 1992) and johansen and juselius (1990) maximum likelihood cointegration technique is used. this technique also determines for the number of cointegrating vectors and is based on granger’s (1981) ecm representation. the multivariate cointegration test can be expressed as follows: y2 = kb + km∆y2@m + kc∆y2@c + ⋯+ ky@m∆y2@y + πy2@y + ϵ2 (3) where: y2 = (nepse index, interest rate, gold price, exchange rate)’ and are cointegrated of order one [i.e. , i(1)], k = a 4 × 4 matrix of coefficients, ∆=a difference operator, π = a 4 × 4 matrix of parameters, and ϵ2 = a vector of normally and identically distributed error terms. mitra lal devkota, humnath panta dynamic econometric models 18 (2018) 49–65 58 table 4. ng-perron test results-levels variable intercept intercept and trend mz0 mz: mz0 mz: ni −0.176 −0.127 −2.529 −1.124 gp −0.682 −0.763 −4.453 −1.414 ir −1.929 −0.307 −9.971 −2.131 er −0.329 −0.021 −9.146 −0.214 critical value −8.100 −1.980 −17.3 −2.910 note: critical values are for 5% level of significance; ni, gp, er, and ir stand for nepse index, gold price, exchange rate, and interest rate, respectively. table 5. ng-perron test results-first differences variable intercept intercept and trend mz0 mz: mz0 mz: ni −21.170* −3.163* −37.446* −4.321* gp −73.886* −6.078* −72.906* −6.036* ir −9.574* −2.16* −73.185* −6.048* er −65.761* −5.733* −67.054* −5.790* critical value −21.170* −3.163* −37.446* −4.321* note: * – statistical significance at 5% level of significance; ni, gp, er, and ir stand for nepse index, gold price, exchange rate, and interest rate, respectively. the presence of r cointegrating vectors between the elements of y implies that π is of rank r (0 < r < 4). to determine the number of cointegrating vectors, there are two likelihood ratio tests available. these are trace test (λ-trace) and maximum eigenvalue test (λ-max). we conducted the johansen cointegration test with all the variables in their logarithmic scales and used both the λ-trace and λ-max statistics options in eviews. for both the directions, 3 lags were used which is consistent with bhattacharjee, et al., (2014). the results for both the λ-trace and λ-max statistics are summarized in table 6. we see that the λ-trace statistic identified one cointegrating relationship among the nepse index and the three macroeconomic variables, while the λ-max statistic identified no cointegrating relationship among the variables at α = 0.05 level of significance. since the trace statistic takes into account all of the smallest eigenvalues, it possesses more power than the maximum eigenvalue statistic (kasa, 1992; serletis and king, 1997). furthermore, according to cheung and lai (1993), the λ-trace statistic is more robust than the λ-max statistic, and hence, we conclude that there is at least one cointegrating relationship between nepse index, the interest rate, gold price, and the usd exchange rate in nepal. in other words, there exists a long run equilibrium relationship between the variables. furthermore, bruesch-godfrey serial correlation lm test results in a chisquare test statistic of 3.741 with a p-value of 0.154. this suggests that the an enquiry into the effect of the interest rate, gold price, and the exchange… dynamic econometric models 18 (2018) 49–65 59 null hypothesis of no serial correlation is not rejected at 5% level of significance, and thus, the adequacy of the model is confirmed. table 6. johansen cointegration test results (trace and max. eigenvalue) null hypotheses λ2mnop stat 5% critical value p-value λqnr stat 5% critical value p-value r = 0 56.62 47.86 0.006* 27.37 27.58 0.0532 r ≤ 1 29.25 29.80 0.058 20.43 21.13 0.062 r ≤ 2 8.81 15.49 0.383 6.97 14.26 0.492 r ≤ 3 1.84 3.84 0.175 1.84 3.84 0.175 notes: *: statistically significant at 5% level of significance; r =hypothesized number of cointegrating equations; the cointegration model is based on the vector autoregression model (var) with 3 lags as determined by the likelihood ratio test; the critical values for trace and max-eigen statistics are calculated by eviews (10). 3.4 granger causality and vector error correction model (vecm) the results from the johansen test for cointegration indicate that causality exists between the cointegrated variables. the granger causality test (1987) is a statistical procedure used to determine if one time series is helpful in forecasting another. according to engle and granger (1987), if two variables x2 and y2 are cointegrated, there exists an error correction model given by ∆x2 = γm + θmect2@m + ∑ δm∆x2@z q zlm + ∑ τm∆y2@z | zlm + ϵm2 (4) ∆y2 = γc + θcect2@m + ∑ δc∆y2@z q zlm + ∑ τc∆x2@z | zlm + ϵc2 (5) where ∆ is the difference operator, 𝑚 and 𝑛 are the lag lengths of the variables, ect refers to the error correction term(s) derived from the long-run cointegration relationship via johansen maximum likelihood procedure, γ,θ,δ,τ are the parameters to be estimated, and ϵm2 and ϵc2 are the white opens up an additional channel for granger causality to emerge that is completely ignored by the standard granger and sims tests. the granger causality can be tested by examining the statistical significance of the lagged ects using a t-test or by a joined test applied to significance of the sum of the lags of each explanatory variable by an f-or wald χc test1. the johansen test of cointegration in section 3.3 shows that there is cointegration between the nepse index, the interest rate, gold price, and the usd 1 we tested the long-run causality through the statistical significance of each error correction term by an individual t-test, and the short-run granger causality through the joint significance of the lags of each explanatory variable by a wald 𝜒c test. a variable x is said to granger cause a variable y, if addition of lagged values of x in the regression model describing y can improve quality of the model and/or forecasts (see, syczewska, 2014; osinska, 2011). mitra lal devkota, humnath panta dynamic econometric models 18 (2018) 49–65 60 exchange rate. we now proceed to fit the vecm model to test the existence of short and long-run causal relationships. vecm includes lags of the dependent variables, in addition to its own lags (upadhyaya, nag and franklin jr, 2018). in addition to indicating the direction of causality amongst the variables, the vecm allows us to distinguish between short-run and long-run granger causality because it can capture both the short-run dynamics between time series and their long-run equilibrium relationship (mashi and mashi, 1996)2. table 7. the long-run and short-run granger causality null hypotheses ect (𝑡 -stat) χc-stat nature of causality direction of causality gp⇒ni −0.420 0.522 none none ni ⇒ gp 0.923 1.826 none gp⇒er 2.666** 3.704** short and long-run unidirectional er⇒gp −0.800 0.041 none gp⇒ir −2.844** 0.434 long-run unidirectional ir⇒gp 0.713 0.030 none er ⇒ir 0.076 0.408 none none ir⇒er −0.484 0.333 none ni ⇒ir 0.040 2.980** short-run feedback ir⇒ni −1.787* 0.128 long-run er ⇒ni 1.969* 3.420** short and long-run unidirectional ni⇒er −.219 0.360 none notes: 𝐻b: x⇒y represents the null hypothesis that x does not granger cause y; ** – statistically significant at 1% level of significance, * – statistically significant at 10% level of significance; ni, gp, er, and ir stand for nepse index, gold price, exchange rate, and interest rate, respectively. the short and long-run causality results from table (7) indicate that there are two short and long-run causal relationships between the nepse index, the interest rate, gold price, and the exchange rate. these causal relationships run from the gold price to the exchange rate, and from the exchange rate to the nepse index. in addition, there are two long-run causal relationships between the variables. these causalities run from the gold price to the interest rate, and from the interest rate to the nepse index. there is only one shortrun causal relationship. this runs from the nepse index to the interest rate. there is one feedback relationship between the nepse index and the interest rate. thus, we can conclude that, for our dataset, there is unidirectional causality running from the gold price to the exchange rate and the interest rate, 2 number of lags of the vecm estimation was selected using the likelihood ratio test. an enquiry into the effect of the interest rate, gold price, and the exchange… dynamic econometric models 18 (2018) 49–65 61 and from the exchange rate to the nepse index3. it means that the gold price granger causes both the exchange rate and the interest rate, and finally, both the interest rate and the exchange rate granger cause the nepse index. no causality exists between the rest of the pairs. summary, conclusions and discussion the present study investigated the causal relationships between the nepse index, the interest rate, gold price, and the usd exchange rate in nepal. the analysis used the monthly data for the period between january, 2006 to august, 2018 which are obtained from various sources including nepal rastra bank (nrb), the central bank of nepal, nepalese stock exchange limited, and nepal gold and silver dealers’ association (negosida). the nepse index is used to represent the nepalese stock market index. it is believed that, the selected variables, among others, represent the state of the economy of nepal. we used the ng-perron unit root test to check the stationarity of the variables. this test possesses better power and size properties, due to which the results are more reliable when applied to small data sets. our results indicate that each series is non-stationary at levels, and then stationary in the first differences. to further investigate the long-run relationship among the variables, we used the johansen cointegration test to determine the number of cointegrating vectors. we conducted this test with all the variables in their logarithmic scales and used both the λ-trace and λ-max statistics options. we find that there is only one cointegrating relationship between the variables. in other words, there exists a long run equilibrium relationship between the variables. we then employed granger causality test based on vecm framework to determine the existence of both short and long-run causal relationships between the variables. our results indicate that there are two short and long-run causal relationships which run from the gold price to the exchange rate, and from the exchange rate to the nepse index. in addition, there are two longrun causal relationships which run from the gold price to the interest rate, and from the interest rate to the nepse index. there is only one short-run causal relationship which runs from the nepse index to the interest rate. also, there is a feedback relationship between the nepse index and the interest rate. thus, for our dataset, there is a unidirectional causality running from the gold 3 if the null hypothesis that 𝑥 does not cause 𝑦 is rejected, but 𝑦 does not cause 𝑥 is not rejected, it is called a unidirectional causation. however, if both tests are rejected, then a feedback or a bidirectional relationship is established between 𝑥 and 𝑦. if both tests fail to reject the null hypothesis, then a contemporaneous relationship is established. mitra lal devkota, humnath panta dynamic econometric models 18 (2018) 49–65 62 price to the exchange rate and a unidirectional causality running from the gold price to the interest rate, and from the exchange rate to the nepse index. it means that the gold price granger causes both the exchange rate and the interest rate, and finally, both the interest rate and the exchange rate granger cause the nepse index. no causality exists between the rest of the pairs. our finding is in line with smyth and nandha (2003) who concluded that the exchange rate granger causes stock price in india and sri lanka, and with abdalla and murinde (1997) who found that the exchange rate granger causes stock market prices in pakistan. similarly, our finding of no causality (in either direction) between the gold price and the nepse index is consistent with the finding of mishra et al., (2010) and of gaire (2016). thus, while our finding is consistent with that of most previous studies, it is contrary to the result of gaire (2016) with respect to causality from the gold price to the interest rate, as he concluded that there is no causality from the gold price to the interest rate. in addition, he found a unidirectional causality from the interest rate to the nepse index, which is also contrary to our finding of a feedback relationship between the two variables. in addition to his research, we included an additional macroeconomic variable-exchange rate, and concluded that there exists a unidirectional causality from the gold price to the exchange rate, and from to the exchange rate to the nepse index. as the study pertains to nepal, where capital account transactions are not open, our results are more relevant in countries where capital account transactions are not open. in addition, inflation and economic growth, the other fundamental macroeconomic variables, are not directly included in the study, except as they interact with our chosen variables. for future studies, we suggest including these variables to understand their effect and direction of causality with the ones included in this paper. according to ratanapakorn and sharma (2007), one should have at least 30 years of data to use cointegration analysis. clearly, our data does not cover a long enough time period for the analysis of cointegration. similar to those authors, our objective is to investigate a shorter time period, and hence, the results should be viewed with caution. however, even considering these limitations, our results have important policy implications. the present study has concluded that the interest rate and the usd exchange rate could be the important determinants of the nepalese stock exchange index. as such, understanding the stock market reaction to these variables over time should provide insight to practitioners, researchers, government agencies, and others interested in the topic. an enquiry into the effect of the interest rate, gold price, and the exchange… dynamic econometric models 18 (2018) 49–65 63 references abdalla, i. s., murinde, v. (1997), exchange rate and stock price interactions in emerging financial markets: evidence on india, korea, pakistan and the philippines, applied financial economics, 7(1), 25–35, doi: http://dx.doi.org/10.1080/096031097333826. alam, m., uddin, g. (2009), relationship between interest rate and stock price: empirical evidence from developed and developing countries, doi: http://dx.doi.org/10.5539/ijbm.v4n3p43. bhattacharjee, k., bang, n. p., mamidanna, s. (2014), transmission of pricing information between level iii adrs and their underlying domestic stocks: empirical evidence from india. journal of multinational financial management, 24, 43–59, doi: http://dx.doi.org/10.1016/j.mulfin.2013.12.001. buyuksalvarci, a. (2010), the effects of macroeconomics variables on stock returns: evidence from turkey, european journal of social sciences, 14(3), 404–416. cheung, y. w., lai, k. s. (1993), finite-sample sizes of johansen’s likelihood ratio tests for cointegration, oxford bulletin of economics and statistics, 55(3), 313–328, doi: http://dx.doi.org/10.1111/j.1468-0084.1993.mp55003003.x. cuestas, j. c., harrison, b. (2008) testing for stationarity of inflation in central and eastern european countries (no. 2008/13). cuestas, j. c., staehr, k. (2013), fiscal shocks and budget balance persistence in the eu countries from central and eastern europe, applied economics, 45(22), 3211–3219, doi: http://dx.doi.org/10.1080/00036846.2012.703316. engle, r. f., granger, c. w. (1987), co-integration and error correction: representation, estimation, and testing, econometrica: journal of the econometric society, 251–276, doi: http://dx.doi.org/10.2307/1913236, https://www.jstor.org/stable/1913236. fama, e. f. (1981), stock returns, real activity, inflation and money, american economic review, 71, 545–565, doi: https://www.jstor.org/stable/1806180. gaire, h. (2016), stock index, interest rate and gold price of nepal: cointegration and causality analysis, retrieved from https://nrb.org.np/ecorev/articles/2.nepseindex20180206.pdf. granger, c. w. (1981), some properties of time series data and their use in econometric model specification. journal of econometrics, 16(1), 121–130, doi: http://dx.doi.org/10.1016/0304-4076(81)90079-8. gregoriou, a., kontonikas, a., montagnoli, a. (2006), euro area inflation differentials: unit roots, structural breaks and nonlinear adjustment, university of glasgow, department of economics, working paper, (2007_13). gunes, s. (2007), functional income distribution in turkey: a cointegration and vecm analysis, journal of economic and social research, 9(2), 23–36. harris, r., sollis, r. (2003), applied time series modelling and forecasting, wiley. johansen, s. (1988), statistical analysis of cointegration vectors, journal of economic dynamics and control, 12, 231–254, doi: http://dx.doi.org/10.1016/0165-1889(88)90041-3. johansen, s. (1991), estimation and hypothesis testing of cointegration vectors in gaussian vector autoregressive models, econometrica, 59, 1551–1580, doi: https://www.jstor.org/stable/2938278. johansen, s. (1992), determination of cointegration rank in the presence of a linear trend. oxford bulletin of economics and statistics, 54(3), 383–397, doi: http://dx.doi.org/10.1111/j.1468-0084.1992.tb00008.x. mitra lal devkota, humnath panta dynamic econometric models 18 (2018) 49–65 64 johansen, s., juselius, k. (1990), maximum likelihood estimation and inference on cointegration–with applications to the demand for money, oxford bulletin of economics and statistics, 52, 169–210, doi: http://dx.doi.org/10.1111/j.1468-0084.1990.mp52002003.x. kasa, k. (1992), common stochastic trends in international stock markets, journal of monetary economics, 29(1), 95–124, doi: http://dx.doi.org/10.1016/0304-3932(92)90025-w. levin, e. j., montagnoli, a., wright, r. e. (2006), short-run and long-run determinants of the price of gold. mishra, p. k., das, j. r., mishra, s. k. (2010), gold price volatility andstock market returns in india. american journal of scientific research, 9(9), 47–55. moore, g. h. (1990), analysis: gold prices and a leading index of inflation. challenge, 33(4), 52–56, doi: http://dx.doi.org/10.1080/05775132.1990.11471444. mukherjee, t. k., naka, a. (1995), dynamic relations between macroeconomic variables and the japanese stock market: an application of a vector error correction model, journal of financial research, 18, 223–237, doi: http://dx.doi.org/10.1111/j.1475-6803.1995.tb00563.x. ng, s., perron, p. (2001), lag length selection and the construction of unit root tests with good size and power. econometrica, 69(6), 1519–1554, doi: https://www.jstor.org/stable/2692266. nieh, c. c., lee, c. f. (2001), dynamic relationship between stock prices and exchange rates for g-7 countries, the quarterly review of economics and finance, 41(4), 477–490, doi: http://dx.doi.org/10.1016/s1062-9769(01)00085-0. osińska, m. (2011), on the interpretation of causality in granger sense, dynamic econometric models, 11, 129-140, doi: http://dx.doi.org/10.12775/dem.2011.009. pilinkus, d., boguslauskas, v. (2009), the short-run relationship between stock market prices and macroeconomic variables in lithuania: an application of the impulse response function. engineering economics, 65(5). raihan, s., abdullah, s. m., barkat, a., siddiqua, s. (2017), mean reversion of the real exchange rate and the validity of ppp hypothesis in the context of bangladesh: a holistic approach. ratanapakorn, o., sharma, s. c. (2007), dynamic analysis between the us stock returns and the macroeconomic variables, applied financial economics, 17(5), 369–377, doi: http://dx.doi.org/10.1080/09603100600638944. serletis, a., king, m. (1997), common stochastic trends and convergence of european union stock markets. the manchester school, 65(1), 44–57, doi: http://dx.doi.org/10.1111/1467-9957.00042. smith, g. (2001), the price of gold and stock price indices for the united states, the world gold council, 8(1), 1–16. smyth, r., nandha, m. (2003), bivariate causality between exchange rates and stock prices in south asia. applied economics letters, 10(11), 699–704, doi: http://dx.doi.org/10.1080/1350485032000133282. srinivasan, p. (2014), gold price, stock price and exchange rate nexus: the case of india. romanian economic journal, 17(52). sujit, k. s., kumar, b. r. (2011), study on dynamic relationship among gold price, oil price, exchange rate and stock market returns. international journal of applied business and economic research, 9(2), 145–165. syczewska, e. m. (2014), the eurpln, dax and wig20: the granger causality tests before and during the crisis, dynamic econometric models, 14, 93–104, doi: http://dx.doi.org/10.12775/dem.2014.005. an enquiry into the effect of the interest rate, gold price, and the exchange… dynamic econometric models 18 (2018) 49–65 65 tsoukalas, d. (2003), macroeconomic factors and stock prices in the emerging cypriot equity market, managerial finance, 29(4), 87–92, doi: http://dx.doi.org/10.1108/03074350310768300. tursoy, t., gunsel, n., rjoub, h. (2008). macroeconomic factors, the apt and the istanbul stock market. international research journal of finance and economics, 22, 49–57. upadhyaya, k. p., nag, r., mixon jr, f. g. (2018). stock market capitalization and the macroeconomics of transition economies: the case of india. dynamic econometric models, 18, 35–47, doi: http://dx.doi.org/10.12775/dem.2018.002. vygodina, a. v. (2006), effects of size and international exposure of the us firms on the relationship between stock prices and exchange rates. global finance journal, 17(2), 214–223, doi: http://dx.doi.org/10.1016/j.gfj.2006.05.001. wongbangpo, p., sharma, s. c. (2002), stock market and macroeconomic fundamental dynamic interaction: asean-5 countries, journal of asian economics, 13, 27–51, doi: http://dx.doi.org/10.1016/s1049-0078(01)00111-7. wooldridge, m. (2011). modern econometrics. mcgraw hill. microsoft word dem_2018_81to97.docx © 2018 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2018.005 vol. 18 (2018) 81−97 submitted november 25, 2018 issn (online) 2450-7067 accepted december 17, 2018 issn (print) 1234-3862 dominik krężołek* testing day of the week effect on precious metals market a b s t r a c t. market efficiency assumes that asset prices should be characterized by randomness and unpredictability, so that potential market participants are not able to generate aboveaverage profits. this means that there should be no seasonal phenomenon in time series, which clearly projects a certain pattern of behavior of financial assets. the paper is an attempt to verify some specific seasonal effect called “the day of the week” on the precious metals market using ar-garch(aparch) models. the selection of this area is not accidental. precious metals are an alternative to classic capital investments, especially in the case of financial and economic crises. in addition, the literature shows a gap in this area in terms of dynamics analysis on commodity markets, if compared to capital market. the results are not unambiguous and the effect of seasonality was observed for the volatility of gold returns in the entire period and in the period of upward trend (positive returns on fridays). the returns of palladium suffer from seasonality during entire period and in the period of upward trend (positive returns on fridays, negative on mondays, tuesdays and thursdays). moreover, it was observed that the araparch models is more appropriate when taking into account the heavy-tail distributions of residuals. k e y w o r d s: aparch model; day of the week; garch model; precious metals; time series. j e l classification: c22; g01; g14. introduction in the theory of economics and finance the term “market efficiency” assumes that at any time asset prices accurately reflect the full available * correspondence to: dominik krężołek, department of demography and economic statistics, university of economics in katowice, 1 maja 50 street, 40-287 katowice, poland, e-mail: dominik.krezolek@ue.katowice.pl. dominik krężołek dynamic econometric models 18 (2018) 81–97 82 information about them. the term "effective market" first appeared in bachelier's paper (bachelier, 1900), but it did not get much interest from the scientific community. in the second half of xx century, the american economist eugene fama discussed this problem and considered so-called efficientmarket hypothesis (fama, 1970) in the context of weak, semi-strong and strong efficiency. according to his perception, week market efficiency assumes that currently realized prices reflect the historical data and it is not possible to forecast future level of prices from historical data. the hypothesis of semi-strong efficiency proclaims that current asset prices reflect all public information available, including historical data, reports and any available economic forecasts the last one – strong market efficiency assumes the existence of full public and non-public information about assets considered, and this information is available for everyone at any given time. this hypothesis is the most restrictive and means that all kind of analyzes carried out on financial market, including technical and fundamental analysis, would be unnecessary, as it wouldn’t be possible to obtain above-average profits based on full available information. as one can see, the efficiency of the market doesn’t go hand in hand with the main goal of investment activity, which is multiplying assets. in this paper some selected calendar anomalies are considered. as “calendar anomalies” we define statistically significant differences in levels of prices/returns for a given financial assets depending on the adopted period. 1. calendar effects on financial market – literature review when introducing the problem of calendar anomalies, it is worth to define those which are most often considered in the analysis of financial market data. one can mention here, among others, such anomalies like effect of the month in the year, effect of the turn of the month, holidays effect, effect of the day of the week or even the effect of the hour on the day. each of these anomalies undermines the efficiency of the market and is the topic of analyzes in scientific communities around the world. the effect of the month in the year reflects significant differences in the average values of returns of financial assets in particular months of the year. the most popular anomalies in this group are the effect of january, may or september. in the first case, it was observed that average level of returns in january is significantly higher than these observed in other months. in turn, may and september effects are associated with semi-low level of returns. the turn of the month is related by the phenomenon when we can observe the average higher prices (returns) at the end of month if compared to their values in the first days of the following month. this situation may result from the allocation of funds obtained in the middle of the month in investment funds testing day of the week effect on precious metals market dynamic econometric models 18 (2018) 81–97 83 (fiszeder et al., 2013). in the case of the holidays effect, which s related to the effect of the month, increases of financial assets returns are observed before holidays, while declines in the period just after holidays. a similar relationship can be observed in the case of holiday periods, when no trading sessions are held. if the anomalies during the week are observed, it means so-called the day of the week effect. it can be described as significant differences of the level of returns on financial assets depending on the day of the week (most often fiveday week is considered, however, there are some assets listed in the seven-day trading system (e.g. the returns of electricity prices (ganczarek-gamrot, 2013)). the phenomenon of volatility in daily returns during a week is the problem analyzed not only by practitioners but also by scientists. k. french (1980), while analyzing the u.s. market, observed high, statistically significant volatility in stocks returns. in his research he showed that the returns realized on friday were significantly higher comparing to the other days, while the returns realized on monday – respectively lower, negative on average. other researchers also came to similar conclusions, including lakonishok and smidt (1988). jaffe and westerfield analyzed the capital markets in japan and australia and they observed average negative, statistically significantly different returns on tuesdays comparing with other days of the week (jaffe et al., 1985). on the polish capital market that kind of research was conducted, among others, by szyszka (1999). in his research based on data from the warsaw stock exchange, he observed positive returns on mondays and negative on tuesdays. similar results were obtained by landmesser (2006), witkowska, kompa (2007). additionally, they pointed out positive returns on fridays. a slightly different study was conducted by a fiszeder and kożuchowska (2013). they studied anomalies of indices wig and wig20. the obtained results indicating the turn of the month effect, no seasonal and mild disturbances during the week. when analyzing the precious metals market, there is few papers in which the issue of seasonality in the context of anomalies would be discussed. ma (1986) in his research showed positive returns on fridays and negative mondays for gold. more recent studies, taking into account the last 30 years, result in slightly different conclusions. aksoy (2013) observed negative returns form gold on mondays and fridays on istanbul gold exchange. arora et al. (2013) discovered positive, statistically significant returns on tuesdays and fridays. in research of kohli (2012) the positive returns on fridays and wednesdays for gold and positive returns on wednesdays for silver were detected. dominik krężołek dynamic econometric models 18 (2018) 81–97 84 the use of appropriate statistical tools allows to create models that adequately reflect such kind of seasonality. the results will allow to effectively forecast level of returns or prices in economic (financial) time series. 2. methodology and statistical tools based on the regressive approach, the model describing the seasonality of the returns, taking into account different days of the week, can be written in the following way: 𝑟" = ∑ 𝜔&𝑑&"(&)* + 𝜀" (1) where 𝑟" is the return on the asset at the moment 𝑡, while 𝑑&" is the binary variable representing the 𝑘 − 𝑡ℎ day of the week. to describe volatility on financial time series one of the most popular model is the one proposed by engle (1982) – the arch model extended by bollerslev (1986) into the garch model. one of the main assumptions of arch model is the time-independence of returns. the expected return and variance can be described by historical information and presented by formulas: 𝜇" = 𝐸(𝑟"|𝐼"6*) (2) 𝜎"9 = 𝜎9(𝑟"|𝐼"6*) (3) where 𝜇" and 𝜎"9 define conditional expected return and conditional variance in time 𝑡, whereas 𝐼"6* defines the information set available in time 𝑡 − 1. the garch(𝑚,𝑠) model of bollerslev can be described using formulas (tsay, 2005): 𝑟" − 𝜇" = 𝑎" = 𝜎"𝜀" (4) 𝜎"9 = 𝛼@ + ∑ 𝛼a𝑎"6a 9b a)* + ∑ 𝛽d𝜎"6d 9e d)* (5) where 𝛼@ > 0. 𝛼a ≥ 0 for 𝑖 > 0. 𝛽d ≥ 0 and ∑ (𝛼a + 𝛽a) bkl(b,e) a)* < 1. the family of garch models is comprehensively described in the literature where many interesting properties are highlighted, i.e. the ability of modelling heavy-tailed distribution. the main disadvantage of garch models is that do not describe asymmetry observed in the and neither both leverage and long-memory effects. to solve this problems ding et al. (1993) proposed new family of models describing these stylized facts observed in financial time series – the class of aparch (asymmetric power arch) models. mathematical formula takes a form (karanasos et al., 2006): 𝜎"n = 𝛼@ + ∑ 𝛼a(|𝑎"6a| − 𝛾a𝑎"6a)n p a)* + ∑ 𝛽d𝜎"6d nq d)* (6) testing day of the week effect on precious metals market dynamic econometric models 18 (2018) 81–97 85 where −1 < 𝛾a < 1 and 𝛿 > 0. the parameter 𝛾 describes the leverage effect. a positive value of 𝛾 means that past negative shocks have a deeper impact on current conditional volatility than past positive shocks. a negative value means the opposite (negative value means that positive information has stronger impact than the negative information on the price volatility). moreover, arch and garch models are special cases of aparch (arch(q) for 𝛿 = 2, 𝛾a = 0. 𝛽a = 0. garch (p, q) for 𝛿 = 2, 𝛾a = 0). the estimation of unknown parameters of aparch model is usually conducted using the mle. in this paper, to model volatility, we used the combination of two models: ar and garch (aparch). the first one represents conditional expected return (with a part related to the seasonality effect) whereas the second one is related to conditional variance. therefore, we can write the formulas for ar(m)-garch(p, q) and ar(m)-aparch(p, q) models as below: ar-garch: 𝑟" − 𝜇 − ∑ 𝜑a𝑟"6a b a)* − ∑ 𝜔&𝑑&" ( &)* = 𝑎" = 𝜎"𝜀" (7) 𝜎"9 = 𝛼@ + ∑ 𝛼a𝑎"6a 9p a)* + ∑ 𝛽d𝜎"6d 9q d)* (8) ar-aparch: 𝑟" − 𝜇 − ∑ 𝜑a𝑟"6a b a)* − ∑ 𝜔&𝑑&" ( &)* = 𝑎" = 𝜎"𝜀" (9) 𝜎"n = 𝛼@ + ∑ 𝛼a(|𝑎"6a| − 𝛾a𝑎"6a)n p a)* + ∑ 𝛽d𝜎"6d nq d)* (10) where 𝜑a, 𝑖 = 1,…,𝑚 stands form autoregressive model’s parameters and 𝑑&" is dummy variable representing the 𝑘 − 𝑡ℎ day of the week in the equation of conditional expected return. to avoid the effect of collinearity between intercept and dummy variables representing 𝑘 − 𝑡ℎ day of the week in equations (7) and (9) while estimating unknown parameters, one of the dummy variables has to be omitted and its missing value should be assessed using appropriates identities (fiszeder et al., 2013) from the literature we know that the correct financial time series analysis requires to examine simultaneously the conditional mean, conditional variance and standardised residuals described by the proper probability function. the most popular distributions describing residuals 𝜀" are conditional standard normal, t-student or generalized error distribution (ged) (piontek, 2002). this t-student and ged distributions have gained popularity because of some specific features as high level of kurtosis or heavy tails (similar to distributions based on extreme value theory). the conditional standard normal, t-student and ged distributions for residuals can be described in terms of probability distribution functions as follow: dominik krężołek dynamic econometric models 18 (2018) 81–97 86 𝑓x(𝜀",𝜎"9;𝜃) = * [\√9^ 𝑒𝑥𝑝b− c\ d 9[\ de (11) 𝑓f"gh(𝜀",𝜎"9;𝜃) = ijklm d n opij k d nq^(r69) j1 + c\ d (r69)[\ dn klm d (12) 𝑓stu(𝜀",𝜎"9;𝜃) = 2 6 klm k r [\v wxkymz wx{kymz 9y d ki(rym) 𝑒𝑥𝑝 ⎩ ⎪ ⎨ ⎪ ⎧ −* 9 � � c\ [\v wxkymz wx{kymz 9y d k �� r ⎭ ⎪ ⎬ ⎪ ⎫ (13) where {𝜀"} stands for the sequence of iid random variables, 𝜎"9 is the conditional variance of the process, 𝜃 is the vector of estimated parameters, 𝑣 is the number of degrees of freedom, and γ(𝑘) = ∫ 𝑥&6*𝑒6* 𝑑𝑥 �� @ is the gamma function with parameter 𝑘. if the model is fitted, the next step is to assess the goodness of this fitting. we can use the information criteria of: akaike (aic), schwarz (bic) and hannan-quinn (hqc). the values of these criteria are calculated using formulas as below: 𝐴𝐼𝐶 = −2𝑙𝑛[𝐿𝐿𝐹(𝜃�)] + 2𝑘 (14) 𝐵𝐼𝐶 = −2𝑙𝑛[𝐿𝐿𝐹(𝜃�)] + 𝑘𝑙𝑛(𝑛) (15) 𝐻𝑄𝐶 = −2𝑙𝑛[𝐿𝐿𝐹(𝜃�)] + 2𝑘𝑙𝑛[𝑙𝑛(𝑛)] (16) where 𝐿𝐿𝐹(𝜃�) is the log-likelihood function of the parameters vector 𝜃�, 𝑘 is the number of estimated parameters and 𝑛 is the number of observations. the final selection is based on the values of one the information criteria: the lower values, the better the model. 3. empirical analysis on precious metals market the precious metal market is part of the commodity market – an alternative area for classical financial investments. its popularity has increased due to the opportunities that offers to investors. the main reason for investing funds on the metals market is to hedge against financial crises. the empirical research shows a negative correlation of precious metals prices and stock exchange indices (draper et al., 2010). precious metals have a wide range of applications. they are used in many industries (e.g. automotive, space industry etc.), medicine, biology, jewelry, cosmetology and even in gastronomy. it is difficult to synthetically indicate the exact use of precious metals, but it is worth mentioning about their psychological aspects: precious metals are testing day of the week effect on precious metals market dynamic econometric models 18 (2018) 81–97 87 usually associated with wealth and economic stability, especially during crisis. therefore, the analysis of that kind of assets is justified. the hypothesis says that precious metals market is characterized by calendar anomalies (in this case: the day of the week effect). the daily log-returns for gold, silver, platinum and palladium from the london metal exchange are considered. the period of analysis is january 2000 – december 2016, finally divided into two sub-periods: a period of upward trend (january 2000 – august 2011) and a downward trend respectively (september 2011 – december 2016). a comparison of the volatility of gold and silver prices with the volatility of global stock indices seems worth commenting. for example, the s&p500 and nasdaq composite quotes together with gold and silver are presented in the same time period (fig. 1). figure 1. quotations of gold (top-left), silver (top-right), s&p500 (bottom-left) and nasdaq composite (bottom-right) – all period it is clearly visible that the upward trend after the crisis on the financial markets (2007–2009) coincides with a significant downward trend observed for gold/silver. in addition, when considering the period of the crisis, it is worth mentioning that then were recorded the highest gold prices comparing to the entire period. the driving force behind this is the withdrawal of funds from the capital market and allocation in an alternative way (e.g. on metals market) in order to hedge against price drops. the main goal of the analysis is the volatility of returns observed within precious metals. based on the data from the lme, daily log-returns were dominik krężołek dynamic econometric models 18 (2018) 81–97 88 estimated. the time series of the analyzed data during entire period are presented in figure 2. figure 2. log-returns of gold (top-left), silver (top-right), platinum (bottom-left) and palladium (bottom-right) – all period table 1. average levels of returns for days of the week all entire period monday tuesday wednesday thursday friday gold 0.00006 –0.00011 –0.00005 0.00018 0.00151 silver 0.00021 –0.00006 0.00026 –0.00049 0.00134 palladium –0.00071 0.00015 –0.00013 –0.00028 0.00147 platinum 0.00016 –0.00029 0.00009 0.00015 0.00074 period 1 period of upward trend monday tuesday wednesday thursday friday gold 0.00038 0.00016 0.00045 0.00021 0.00193 silver 0.00064 0.00014 0.00111 –0.00042 0.00202 palladium –0.00044 0.00023 0.00027 –0.00068 0.00155 platinum 0.00082 –0.00034 0.00049 0.00031 0.00118 period 2 period of downward trend monday tuesday wednesday thursday friday gold –0.00064 –0.00071 –0.00114 0.00012 0.00060 silver –0.00073 –0.00051 –0.00163 –0.00062 –0.00014 palladium –0.00130 –0.00004 –0.00101 0.00059 0.00129 platinum –0.00126 –0.00019 –0.00080 –0.00019 –0.00023 the highest volatility was observed for silver and palladium return rates. data clusters and grouping of variances are clearly visible. in addition, all figures show a significant increase in the level of volatility corresponding to the beginning of the upward trend on global economy. in table 1 the average testing day of the week effect on precious metals market dynamic econometric models 18 (2018) 81–97 89 levels of returns for every day of the week for entire period and all subperiods are shown. the results show that during entire period the average positive returns on fridays were observed, regardless of the type of metal. gold realized negative returns on tuesdays and wednesdays, silver on tuesdays and thursdays, palladium on mondays, wednesdays and thursdays, while platinum only on tuesdays. positive returns were observed for other days. during the upward trend, the average returns on fridays were much higher comparing to the other days of the week. all metals generated on average positive returns and the differences determined by the day of the week are clearly visible. similar conclusions can be drawn in the case of a downward trend, with the average negative returns observed on all days. nevertheless, we can still see the diversity due to the day of the week. in the case of gold and silver, the lowest returns were observed on wednesdays, while for platinum and palladium on mondays. summarizing this results we can suppose that the day of the week is observed on metals market. table 2. ar-garch and ar-aparch models – gold – entire period parameter ar(1)-garch(1,1)-n p-value ar(1)-garch(1,1)-s p-value ar(1)-garch(1,1)-ged p-value 𝜑@ 0.00158 0.000*** 0.00132 0.000*** 0.00124 0.001*** 𝜑* –0.00111 0.950 –0.02239 0.097* –0.02077 0.192 𝜔��� –0.00164 0.001*** –0.00109 0.005*** –0.00106 0.035** 𝜔�g� –0.00159 0.002*** –0.00129 0.002*** –0.00113 0.034** 𝜔��h –0.00169 0.001*** –0.00112 0.007*** –0.00100 0.042** 𝜔��g� –0.00145 0.007*** –0.00121 0.005*** –0.00118 0.017** 𝜔��a 0.00158 0.000*** 0.00132 0.000*** 0.00124 0.000*** 𝛼@ 0.00000 0.031** 0.00000 0.001*** 0.00000 0.003*** 𝛼* 0.03952 0.002*** 0.04504 0.000*** 0.04256 0.000*** 𝛽* 0.94563 0.000*** 0.94754 0.000*** 0.94636 0.000*** 𝜈 – – 4.71052 0.000*** 1.18248 0.000*** parameter ar(1)-aparch(1,1)-n p-value ar(1)-aparch(1,1)-s p-value ar(1)-aparch(1,1)-ged p-value 𝜑@ 0.00164 0.000*** 0.00138 0.000*** 0.00130 0.000*** 𝜑* –0.00220 0.357 –0.02688 0.057* –0.02415 0.000*** 𝜔��� –0.00164 0.001*** –0.00112 0.004*** –0.00111 0.001*** 𝜔�g� –0.00161 0.002*** –0.00131 0.001*** –0.00117 0.000*** 𝜔��h –0.00170 0.001*** –0.00118 0.003*** –0.00104 0.002*** 𝜔��g� –0.00151 0.005*** –0.00127 0.002*** –0.00125 0.000*** 𝜔��a 0.00164 0.000*** 0.00138 0.000*** 0.00130 0.000*** 𝛼@ 0.00000 0.030** 0.00000 0.001*** 0.00000 0.001*** 𝛼* 0.04547 0.000*** 0.05035 0.000*** 0.04965 0.000*** 𝛽* –0.07825 0.503 –0.28993 0.004*** –0.19508 0.090** 𝛾* 0.94625 0.000*** 0.95404 0.000*** 0.95079 0.000*** 𝛿 1.66345 0.000*** 1.24743 0.000*** 1.36498 0.000*** 𝜈 – – 4.74212 0.000*** 1.18170 0.000*** in the next stage of the analysis, the effect of the day of the week was verified using the models presented in theoretical part of this paper. the dominik krężołek dynamic econometric models 18 (2018) 81–97 90 seasonality effect was included in the conditional mean equation. at the beginning, the assumption of normality of returns was examined. the results showed that this hypothesis has to be rejected. in the next step, based on the appropriate statistical tests (the information criteria, jung-box autocorrelation test and the arch effect test), the following models were finally selected for given lags: ar(1)-garch(1,1) and ar(1)-aparch(1,1). it was also assumed that the rest of the model follow normal (n), t-student (s) and ged distributions. the stability of model parameters was assessed using the chow and nyblom tests. in tables 2–4, as example, the results obtained for testing the day of the week effect for gold (entire period), palladium (a period of upward trend) and platinum (a period of downward trend) are presented. table 3. ar-garch and ar-aparch models – palladium – period of upward trend parameter ar(1)-garch(1,1)-n p-value ar(1)-garch(1,1)-s p-value ar(1)-garch(1,1)-ged p-value 𝜑@ 0.00151 0.037** 0.00118 0.044** 0.04351 0.000*** 𝜑* 0.11368 0.000*** 0.08499 0.000*** 0.00041 0.000*** 𝜔��� –0.00228 0.028** –0.00173 0.038** –0.00053 0.000*** 𝜔�g� –0.00166 0.102 –0.00103 0.210 –0.00041 0.000*** 𝜔��h –0.00128 0.196 –0.00040 0.636 –0.00012 0.905 𝜔��g� –0.00157 0.139 –0.00105 0.223 –0.00041 0.000*** 𝜔��a 0.00151 0.037** 0.00118 0.044** 0.00041 0.551 𝛼@ 0.00002 0.006*** 0.00001 0.013** 0.00001 0.009*** 𝛼* 0.13384 0.000*** 0.17310 0.000*** 0.15462 0.000*** 𝛽* 0.84171 0.000*** 0.82803 0.000*** 0.83330 0.000*** 𝜈 – – 4.20127 0.000*** 1.08528 0.000*** parameter ar(1)-aparch(1,1)-n p-value ar(1)-aparch(1,1)-s p-value ar(1)-aparch(1,1)-ged p-value 𝜑@ 0.00156 0.056* 0.00122 0.162 0.00044 0.000*** 𝜑* 0.11159 0.000*** 0.08166 0.000*** 0.04126 0.000*** 𝜔��� –0.00222 0.048** –0.00173 0.112 –0.00055 0.000*** 𝜔�g� –0.00159 0.144 –0.00101 0.332 –0.00044 0.000*** 𝜔��h –0.00124 0.263 –0.00038 0.742 –0.00014 0.000*** 𝜔��g� –0.00154 0.183 –0.00100 0.406 –0.00044 0.000*** 𝜔��a 0.00156 0.045** 0.00122 0.042** 0.00044 0.000*** 𝛼@ 0.00002 0.008*** 0.00002 0.015** 0.00002 0.010** 𝛼* 0.14094 0.000*** 0.17941 0.000*** 0.16495 0.000*** 𝛽* –0.05296 0.409 –0.06803 0.224 –0.06944 0.150 𝛾* 0.84864 0.000*** 0.84015 0.000*** 0.84345 0.000*** 𝛿 1.67146 0.000*** 1.35168 0.000*** 1.38839 0.000*** 𝜈 – – 4.15511 0.000*** 1.08185 0.000*** discussing the results presented in the tables above, it was indicated that for the analyzed metals, the statistically significant effects of the week were mainly observed during the entire period or during the upward trend period. in the case of a downward trend, these results can’t be unambiguously confirmed. similar results were obtained for other metals in the examined subperiods. the values of 𝛾* for each model are positive which means that testing day of the week effect on precious metals market dynamic econometric models 18 (2018) 81–97 91 negative information has stronger impact than the positive information on the returns volatility of analyzed metals. the figures below present synthetic results obtained for all metals tested for every day of the week. the sign (+) indicates the positive return, while the sign (–) – negative respectively. shaded cells indicate statistically significant returns. table 4. ar-garch and ar-aparch models – platinum – period of downward trend parameter ar(1)-garch(1,1)-n p-value ar(1)-garch(1,1)-s p-value ar(1)-garch(1,1)-ged p-value 𝜑@ –0.00022 0.761 –0.00030 0.663 –0.00030 0.873 𝜑* 0.07461 0.021** 0.06262 0.038 0.06995 0.023** 𝜔��� –0.00110 0.282 –0.00078 0.431 –0.00074 0.670 𝜔�g� –0.00016 0.869 –0.00001 0.993 0.00008 0.969 𝜔��h –0.00024 0.809 –0.00028 0.774 –0.00024 0.935 𝜔��g� –0.00014 0.893 –0.00010 0.919 –0.00016 0.946 𝜔��a –0.00022 0.761 –0.00030 0.663 –0.00030 0.682 𝛼@ 0.00000 0.164 0.00000 0.142 0.00000 0.140 𝛼* 0.03465 0.003*** 0.03374 0.001*** 0.03441 0.001*** 𝛽* 0.95579 0.000*** 0.95879 0.000*** 0.95664 0.000*** 𝜈 – – 13.18740 0.001*** 1.67895 0.000*** parameter ar(1)-aparch(1,1)-n p-value ar(1)-aparch(1,1)-s p-value ar(1)-aparch(1,1)-ged p-value 𝜑@ –0.00044 0.489 –0.00042 0.491 –0.00045 0.276 𝜑* 0.06781 0.018** 0.05963 0.051* 0.06531 0.040** 𝜔��� –0.00095 0.340 –0.00069 0.420 –0.00065 0.434 𝜔�g� 0.00001 0.989 0.00007 0.941 0.00017 0.500 𝜔��h 0.00004 0.970 –0.00010 0.920 –0.00002 0.985 𝜔��g� 0.00002 0.988 0.00000 0.997 –0.00003 0.971 𝜔��a –0.00044 0.561 –0.00042 0.535 –0.00045 0.522 𝛼@ 0.00000 0.256 0.00000 0.275 0.00000 0.262 𝛼* 0.02989 0.018** 0.03046 0.003*** 0.03036 0.013** 𝛽* 0.41536 0.391 0.28854 0.636 0.37204 0.489 𝛾* 0.96999 0.000*** 0.96847 0.000*** 0.96935 0.000*** 𝛿 1.25298 0.166 1.50355 0.434 1.30223 0.280 𝜈 – – 14.20030 0.002*** 1.70171 0.000*** the results confirm the existence of day of the week effect, mainly in the case of gold and palladium in the whole period and in the period of upward trend. statistically significant positive returns were observed on fridays, while during the other days the returns were negative. different results were obtained for the period of downward trend. there were statistically significant negative returns on tuesdays and wednesdays. taking into account the models used, the statistical significance of the day of the week effect was primarily observed for models with the distribution of residuals described by the t-student or ged distribution. it may be the result of clustering in variance and existence of outliers. the selection of final model depends on the values of information criteria given by the formulas (14)–(16). the results are below. dominik krężołek dynamic econometric models 18 (2018) 81–97 92 table 5. day of the week effect – statistically significant results – entire period silver ar-garch-n ar-garch-s ar-garch-ged ar-aparch-n ar-aparch-s ar-aparch-ged monday – – – – – – tuesday – – – – – – wednesday – – – – – – thursday – – + – – + friday + + + + + + gold ar-garch-n ar-garch-s ar-garch-ged ar-aparch-n ar-aparch-s ar-aparch-ged monday – – – – – – tuesday – – – – – – wednesday – – – – – – thursday – – – – – – friday + + + + + + palladium ar-garch-n ar-garch-s ar-garch-ged ar-aparch-n ar-aparch-s ar-aparch-ged monday – – – – – – tuesday – – – – – – wednesday – – – – – – thursday – – – – – – friday + + + + + + platinum ar-garch-n ar-garch-s ar-garch-ged ar-aparch-n ar-aparch-s ar-aparch-ged monday – – – – – – tuesday – – – – – – wednesday – – + – + + thursday – – – – – – friday + + + + + + table 6. day of the week effect – statistically significant results – period of upward trend silver ar-garch-n ar-garch-s ar-garch-ged ar-aparch-n ar-aparch-s ar-aparch-ged monday – – – – – – tuesday – – – – – – wednesday – – – – – – thursday – – – – – – friday + + + + + + gold ar-garch-n ar-garch-s ar-garch-ged ar-aparch-n ar-aparch-s ar-aparch-ged monday – – – – – – tuesday – – – – – – wednesday – – – – – – thursday – – – – – – friday + + + + + + palladium ar-garch-n ar-garch-s ar-garch-ged ar-aparch-n ar-aparch-s ar-aparch-ged monday – – – – – – tuesday – – – – – – wednesday – – – – – – thursday – – – – – – friday + + + + + + platinum ar-garch-n ar-garch-s ar-garch-ged ar-aparch-n ar-aparch-s ar-aparch-ged monday – – + – + + tuesday – – – – – – wednesday – + + – + + thursday – – – – + + friday + + + + + + testing day of the week effect on precious metals market dynamic econometric models 18 (2018) 81–97 93 table 7. day of the week effect – statistically significant results – period of downward trend silver ar-garch-n ar-garch-s ar-garch-ged ar-aparch-n ar-aparch-s ar-aparch-ged monday – – + – – + tuesday – – – – – – wednesday – – – + – – thursday – + + – + + friday – – – + – – gold ar-garch-n ar-garch-s ar-garch-ged ar-aparch-n ar-aparch-s ar-aparch-ged monday – – – – – – tuesday – – – – – – wednesday – – – – – – thursday – – – – – – friday + + + + + + palladium ar-garch-n ar-garch-s ar-garch-ged ar-aparch-n ar-aparch-s ar-aparch-ged monday – – – – – – tuesday – – – – – – wednesday – – – – – – thursday – – – – + + friday + + + + + + platinum ar-garch-n ar-garch-s ar-garch-ged ar-aparch-n ar-aparch-s ar-aparch-ged monday – – – – – – tuesday – – + + + + wednesday – – – + – – thursday – – – + + – friday – – – – – – considering the entire period we noticed that for gold and silver, the best fitted model was the ar-aparch model with conditional t-student distribution of residuals. for palladium and platinum information criteria give different assessments, however, these are still models with residuals described by conditional t-student distribution. similar results were obtained for gold and silver during the upward trend. in the case of palladium, the ar-garch model with the ged distribution for residuals was proposed, while for platinum the ar-aparch model with t-student distribution. the last subperiod represents the downward trend. the results suggest different models: for silver, ar-garch model with ged distribution for residuals, ar-garch model for gold with residuals described by t-student distribution, araparch model for palladium with t-student distribution for residuals, and finally ar-garch model for platinum with t-student distribution for residuals as well. we didn’t observe any model with conditional normal distribution for residuals. in figure 3 the ar(1)-aparch(1, 1)-s model for gold and silver is presented (entire period). dominik krężołek dynamic econometric models 18 (2018) 81–97 94 table 8. information criteria for estimated models metal model entire period period of upward trend period of downward trend aic bic hqc aic bic hqc aic bic hqc silver ar-garch-n –23385 –23327 –23365 –16079 –16025 –16060 –7312 –7265 –7294 ar-garch-s –23843 –23779 –23820 –16320 –16260 –16299 –7535 –7483 –7515 ar-garch-ged –23829 –23765 –23807 –16301 –16240 –16279 –7542 –7490 –7522 ar-aparch-n –23400 –23330 –23375 –16119 –16053 –16095 –7310 –7253 –7289 ar-aparch-s –23873 –23797 –23846 –16352 –16280 –16326 –7531 –7468 –7508 ar-aparch-ged –23850 –23774 –23823 –16330 –16258 –16304 –7539 –7476 –7516 gold ar-garch-n –27574 –27517 –27554 –18875 –18821 –18856 –8694 –8647 –8677 ar-garch-s –27987 –27923 –27964 –19115 –19055 –19094 –8873 –8820 –8853 ar-garch-ged –27975 –27911 –27952 –19112 –19052 –19091 –8864 –8812 –8844 ar-aparch-n –27575 –27505 –27550 –18908 –18842 –18884 –8707 –8649 –8685 ar-aparch-s –28005 –27928 –27977 –19139 –19067 –19113 –8876 –8813 –8852 ar-aparch-ged –27983 –27907 –27956 –19134 –19062 –19108 –8866 –8803 –8842 palladium ar-garch-n –22449 –22391 –22429 –14987 –14933 –14968 –7468 –7421 –7450 ar-garch-s –22755 –22692 –22733 –15275 –15215 –15253 –7491 –7439 –7472 ar-garch-ged –22753 –22689 –22731 –15294 –15234 –15273 –7485 –7433 –7466 ar-aparch-n –22447 –22376 –22422 –14988 –14922 –14964 –7484 –7427 –7463 ar-aparch-s –22758 –22682 –22731 –15281 –15209 –15255 –7508 –7445 –7484 ar-aparch-ged –22754 –22677 –22727 –15299 –15226 –15273 –7492 –7439 –7476 platinum ar-garch-n –25876 –25819 –25856 –17649 –17595 –17629 –8246 –8199 –8228 ar-garch-s –26065 –26001 –26042 –17833 –17772 –17811 –8258 –8205 –8238 ar-garch-ged –26046 –25982 –26024 –17822 –17762 –17801 –8254 –8202 –8235 ar-aparch-n –25889 –25819 –25864 –17671 –17604 –17647 –8247 –8190 –8226 ar-aparch-s –26075 –25998 –26048 –17848 –17776 –17822 –8256 –8194 –8233 ar-aparch-ged –26056 –25979 –26029 –17838 –17766 –17812 –8254 –8191 –8231 figure 3. ar(1)-aparch(1,1)-s model for gold (left) and silver (right) – entire period in conclusion, it can be said that the models describing conditional mean and conditional variance should be described by heavy-tails distributions for residuals. it is also reasonable to use models that take into account the asymmetry observed in a data. testing day of the week effect on precious metals market dynamic econometric models 18 (2018) 81–97 95 conclusions the paper attempts to describe calendar anomalies in the case of precious metals market. that kind of anomalies deny the problem of market efficiency and their detection allows to obtain additional information about volatility of financial assets. the research area of precious metals is part of commodity market (metals market), an alternative to the capital one. diversification of the structure of the financial portfolio with components from different markets allows to protect against unpredictable events that may affect the broadly understood economic situation. the entire research period was divided into two sub-periods: period of upward trend and period of downward trend. an attempt was to verify whether general economic trends affect the occurrence of the day of the week effect. as a time series models the class of ar-aparch models with conditional residual distributions (normal, t-student and ged) were proposed. the seasonality effect was included in the conditional mean equation. the selection of models was made on the basis of appropriate diagnostic tests. the results show that in the entire period the returns on fridays (positive) were significantly different from these obtained for the other days of the week. similar results were observed during the period of upward trend. negative results for individual days of the week were observed during the period of downward trend. as we can see, the results are not unambiguous. in summary, the day of the week effect was observed for the volatility of gold returns in the entire period and in the period of upward trend (positive returns on fridays, negative for the other days). the day of the week effect was also observed for the volatility of the palladium returns in the entire period and in the period of upward trend (positive returns on fridays, negative on mondays, tuesdays and thursdays). moreover, it was pointed out that the ar-aparch models should be used when taking into account the heavy-tail distributions for describing model residuals (mainly the student's t-distribution). references aksoy, m. (2013), day of the week anomaly for istanbul gold exchange: gold and silver data, muhasebe finansman dergisi, ocak 2013, 149–164. arora, s., garg, n. (2013), day of the week effect on gold returns, doi: http://dx.doi.org/10.2139/ssrn.2229290. bachelier l. (1900), théorie de la spéculation, annales scientifiques de l’école normale supérieure, 3(17), 21–86. bollerslev, t. (1986), generalised autoregressive conditional heteroskedasticity, journal of econometrics, 31, 307–327, doi: https://doi.org/10.1016/0304-4076(86)90063-1. dominik krężołek dynamic econometric models 18 (2018) 81–97 96 ding, z., granger, c. w. j., engle, r. f. (1993), a long memory property of stock market returns and a new model, journal of empirical finance, 1, 83–106, doi: https://doi.org/10.1016/0927-5398(93)90006-d. draper, p., faff r.w., hillier, d. (2006), do precious metals shine? an investment perspective, financial analysts journal, 62(2), 98–106, doi: https://doi.org/10.2469/faj.v62.n2.4085. fama, e. (1970), efficient capital markets: a review of theory and empirical work, journal of finance, 25(2), 383–417, doi: https://doi.org/10.2307/2325486.jstor 2325486. fiszeder, p., kożuchowska, j. (2013), testowanie występowania wybranych anomalii kalendarzowych na gpw w warszawie, [in:] a. s. barczak, p. tworek, (ed.), „zastosowanie metod ilościowych w zarządzaniu ryzykiem w działalności inwestycyjnej, polskie towarzystwo ekonomiczne oddział katowice”, wydawnictwo uniwersytetu ekonomicznego w katowicach, katowice, 217–229. french, k. (1980), stock returns and weekend effect, journal of financial economics, 8, 55–69, doi: https://doi.org/10.1016/0304-405x(80)90021-5. ganczarek-gamrot, a. (2013), metody stochastyczne w badaniach porównawczych wybranych rynków energii elektrycznej, wydawnictwo uniwersytetu ekonomicznego w katowicach, katowice jaffe, j., westerfield, r. (1985), the week‐end effect in common stock returns: the international evidence, the journal of finance, 40(2), 433–454, doi: https://doi.org/10.1111/j.1540-6261.1985.tb04966.x. karanasos, m., kim, j. (2006), a re‑examination of the asymmetric power arch model, journal of empirical finance, 13, 113–128, doi: https://doi.org/10.1016/j.jempfin.2005.05.002. lakonishok, j., smidt, s., (1988), are seasonal anomalies real? a ninety-year perspective, review of financial studies,1(4), 403–425, doi: https://doi.org/10.1093/rfs/1.4.403. landmesser, j., (2006), efekt dnia tygodnia na giełdzie papierów wartościowych w warszawie, zeszyty naukowe ekonomika i organizacja gospodarki żywnościowej, 60, 187–196. ma, c. (1986), a further investigation of the day-of-the-week effect in the gold market, journal of futures markets, 6, 409–419, doi: https://doi.org/10.1002/fut.3990060306. piontek, k. (2002), pomiar ryzyka metodą var a modele ar‑garch ze składnikiem losowym o warunkowym rozkładzie z “grubymi ogonami”, materiały konferencyjne uniwersytetu szczecińskiego, część ii, 467–484. raj, k. kohli (2012), day-of-the-week effect and january effect examined in gold and silver metals insurance markets and companies, 3(2), 21–26. szyszka, a. (1999), efektywność rynku a anomalie w rozkładzie stóp zwrotu w czasie, nasz rynek kapitałowy, 108, 55–61. tsay, r., 2005, analysis of financial time series, chicago, wiley & sons. witkowska, d., kompa, k. (2007), analiza własności stóp zwrotu akcji wybranych spółek, zeszyty naukowe uniwersytetu szczecińskiego. finanse, rynki finansowe, ubezpieczenia, 6, 255–266. testowanie efektu dnia tygodnia na rynku metali szlachetnych z a r y s t r e ś c i. efektywność rynku zakłada, że ceny aktywów powinny cechować się losowością i nieprzewidywalnością tak, aby potencjalni uczestnicy rynku nie byli wstanie generować ponadprzeciętnych zysków. oznacza to, że w szeregach czasowych nie powinno występować zjawisko sezonowości, które jednoznacznie wyznacza pewien wzorzec zachowań testing day of the week effect on precious metals market dynamic econometric models 18 (2018) 81–97 97 aktywów finansowych. w referacie podjęto próbę weryfikacji efektu dnia tygodnia na rynku metali szlachetnych. wybór obszaru badawczego nie jest akcydentalny. metale szlachetne stanowią alternatywę dla klasycznych inwestycji kapitałowych, zwłaszcza w przypadku kryzysów finansowych i gospodarczych. ponadto literatura przedmiotu wykazuje lukę w obszarze analiz dynamiki na rynkach towarowych w porównaniu z aktywami rynku kapitałowego. wyniki nie są jednoznaczne. efekt dnia tygodnia zaobserwowano przede wszystkim dla stóp zwrotu złota i palladu (cały okres i okres wzrostu) oraz sporadycznie dla stóp zwrotu srebra (okres trendu spadkowego). badanie wykazało, że w kontekście kryteriów informacyjnych należy stosować modele ar-aparch z gruboogonowymi rozkładami prawdopodobieństwa dla reszt s ł o w a k l u c z o w e: efekt dnia tygodnia; metale szlachetne; model aparch; model garch; szeregi czasowe. microsoft word 00_tresc.docx © 2013 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2013.010 vol. 13 (2013) 175−194 submitted november 8, 2013 issn accepted december 30, 2013 1234-3862 andrzej geise, mariola piłatowska* synchronization of crude oil prices cycle and business cycle for the central eastern european economies∗∗ a b s t r a c t. the main purpose of the paper is to study the degree to which the brent crude oil price cycle is correlated and synchronized with business cycle in a set of chosen central eastern european (cee) economies. to indentify the oil price cycle and business cycles for chosen individual countries the markov-switching autoregressive model (ms-ar) is used. the identification of the smoothed probabilities of being in regime 1 and regime 2 enables the calculation of correlation coefficients between those probabilities and the concordance index to evaluate the synchronization of oil price cycle and business cycles for the cee economies. k e y w o r d s: markov switching model, crude oil prices, business cycle, price cycle. j e l classification: c22, c32, e23, q40. introduction the energy prices have the significant impact on the economic and civilization development. rising energy prices observed in recent years accelerated the realization of well-balanced energy consumption programs and the reduction of global energy intensity. energy prices can be affected not only by economic factors but to high extent also by unpredictable and non * correspondence to: andrzej geise, department of econometrics and statistics, 13a gagarina street, 87-100 toruń, poland, e-mail: a.geise@doktorant.umk.pl, mariola piłatowska, department of econometrics and statistics, 13a gagarina street, 87-100 toruń, poland, e-mail: mariola.pilatowska@umk.pl. ∗∗ this work was financed from grant no 1482-e. andrzej geise, mariola piłatowska dynamic econometric models vol.13 (2013) 175–194 176 economic events, such as wars, political situation in regions of fossil fuels mining or devastating weather. specifically, the ecology and the international environmental protection should be taken into account when having in mind the future development of fossil fuel market. special significance in the energy consumption structure has the crude oil which is treated as a major determinant of world economic activity. the economic and social significance of crude oil did not diminish along with the development of alternative energy source, like wind, water and sun energy. irrespective of the potentials of renewable energy the impact of crude oil on the economy is still strong. therefore, it is expected that the relation between economic growth and crude oil prices volatility exists. the main purpose of the paper is to study the degree to which the crude oil price cycle is correlated and synchronized with business cycle in czech republic, hungary, poland and slovenia. first, the markov-switching autoregressive model (ms-ar) is used to identify oil price cycle and the business cycle for chosen individual countries. next, the correlation coefficients between smoothed probabilities of a recession for business cycle in given countries and crude oil price cycle are calculated in order to evaluate the extent to which turning points in the two series occur near each other. finally, the concordance index is computed to reflect the degree to which two series are in the same state or in other words are synchronized. the concept of a research study is presented in scheme 1. scheme 1. the concept of studying the synchronization of oil price cycle and business cycle in the analysis the monthly data of brent crude oil prices from january 1995 to april 2013 (220 observations) and industry production index for a set of central eastern european (cee) economies from january 1995 to april analysis of oil price cycle analysis of business cycle for chosen cee countries analysis of probabilities of being in regime 1 application of markov-switching autoregressive models to identify the dating of cycles and calculate probabilities of recession application of correlation coefficient and concordance index to evaluate the degree of cycle synchronization synchronization of crude oil price cycle and business cycle... dynamic econometric models vol. 13 (2013) 175–194 177 2013 were used1. the data were taken from the u.s. energy information administration database (www.eia.gov) and the oecd database (www.stats.oecd.org). the analysis of relations between crude oil prices and economic activity is an important issue. the volatility of crude oil prices has become one of the most important indicator of economic growth due to the significant share of crude oil in the energy consumption structure. moreover, the government policy is driven by the crude oil market as the tax revenues (e.g. oil excise, oil vat, oil fee) from oil sector are significant. table 1 is as an example of the significance of oil sector to the polish budget. table 1. tax revenues from oil sector in poland in 2012 tax categories tax category revenue as percentage of total oil tax revenue (pln 52bn) oil tax as percentage of total tax category2 oil excise 51.93% 41.00% oil vat 40.38% 15.85% oil fee 7.69% 1.39% source: elaborated on the base of popihn, przemysł i handel naftowy – raport roczny 2012 (industry and oil trade − annual report 2012), warszawa, 2013. tax revenues from oil sector are an important part of state income in poland (in 2012 tax revenue reached ca. pln 52 billion. the structure of taxes coming from oil sector was dominated by an oil excise and oil vat which constituted respectively: 41% of total excise and 16% of total vat. this confirms that the oil sector is of importance to the budget of poland. consequently, the relation between crude oil prices volatility and business cycle is expected. therefore, the analysis directed to evaluate the degree to which the crude oil price cycle is correlated and synchronized with business cycle is needed. 1. the relationship between crude oil prices and economic activity − the literature review the economic theory suggests that assets prices should be determined by their expected, discounted cash flows. then, every factor influencing the discounted cash flows should have a significant impact on assets prices (fisher, 1930). expanding this rule to the crude oil prices we have that every 1 the brent crude oil constitutes the global benchmark of crude oil coming from north eastern europe and reflects changes in crude oil prices in the european region. 2 e.g. oil excise as percentage of total excise, oil vat as percentage of total vat, etc. andrzej geise, mariola piłatowska dynamic econometric models vol.13 (2013) 175–194 178 rise in crude oil prices entails the increase of production costs which causes the profits and capital value to drop. hence, every rise of oil prices brings on the decrease of stock prices. this effect should have occurred both for crude oil exporters and importers. however, many economists discuss whether the influence of oil price dynamics on the stock price dynamics results from an indirect effect which is escalated by the impact of macroeconomic indicators. according to bjornland (2008) and jimenez-rodriguez, sanchez (2005) the rise of oil prices has a positive effect on domestic income in oil exporting countries. also, the increase in expenditure and investment is expected which as a result will cause the increase of productivity and reduction of unemployment. by contrast, in oil importing countries every rise in oil prices will bring on the opposite effect, in the form of decrease in income (bacon, 2005; hooker, 2002). rising oil prices lead to an increase in production costs in oil-importing countries because the crude oil is the important input in the production process (arouri, nguyen, 2010; backus, crucini, 2000; kim, loungani, 1992). the increase in costs is transferred to consumers by increasing consumer prices, and this leads to the reduction of demand and consumer expenditure. the decrease in demand may result in an decrease in production, and hence in an increase in unemployment. consequently, the reaction of stock market will be negative. however, it should be remembered that the influence of oil shocks on stock market depends on the type of shock (demand or supply shock). in the case of demand shock the stock market is supposed to react positively, and in the case of supply shock − negatively (hamilton, 1988, 2006; abel, bernanke, 2001; brown, yucel, 1999, 2002). summing up, rising oil prices may lead to positive economic effects (like an increase in domestic income) in oil exporting countries, and negative ones (like a decrease in domestic income) in oil importing countries. studying relationship between the oil price volatility and economic activity many economists have suggested that the indirect transmission mechanism may have a fundamental impact in indentifying which oil price shocks are important from a macroeconomic point of view. for example, loungani (1986) and davis, haltiwanger (2001) investigated the relationship between oil prices and employment market; bernanke (1983) and dixit, pindyck (1994) analyzed connections of oil prices and uncertainty of investment; hamilton (1988) and lee, ni (2002) studied the ability of oil price volatility to smooth the consumption in durable goods sector; pierce, enzler (1974) and bruno, sachs (1982) analyzed the impact of oil price volatility on inflation. it should be emphasized that these studies did not decide clearly neither about the impact of crude oil price on economic growth, nor the impact of synchronization of crude oil price cycle and business cycle... dynamic econometric models vol. 13 (2013) 175–194 179 economic growth on oil price levels. therefore, it is important to investigate how the oil shocks are transmitted into economic activity. 2. univariate markov switching model in recent years many theoretical and empirical business cycle studies have analyzed the co-movements of macroeconomic time series and the regime-switching nature of macroeconomic activity. as the statistical measurement of macroeconomic fluctuations the markov-switching autoregressive time series model has become increasingly popular since hamilton's (1989) application of this technique to measure the us business cycle. in the markov-switching models the recessions and expansions are modeled as switching regimes of the stochastic process generating the growth rate of economic activity indicator, e.g. gdp (artis et al., 2004). the general form of switching model for growth rate of gross output tyδ can be written (krolzig, toro, 2005): ,)(...)( 111 tsqtqstst qttt yyy εμαμαμ +−δ++−δ=−δ −− −− (1) where ( )2,0~ σε nt is an independent and identically distributed series with zero mean and finite variance, { }mst ,...,1∈ − regime variable. the regimes are associated with different conditional distributions of the growth rate of real output, where the mean ts μ depends on the value of a discrete state variable .ts for instance, in two-regime model the mean may be negative in the first regime (recession), ,01 <μ and positive in the second regime (expansion), .02 >μ the variance of the disturbance term tε is assumed to be the same in both regimes (artis et al., 2004). the general idea behind the class of regime-switching models is that the parameters of equation (1) depend upon a stochastic, non-observable regime variable { }.,...,1 mst ∈ the stochastic process generating the unobservable regimes is an ergodic markov chain defined by the transition probabilities (artis et al., 2004; kośko, 2006; stawicki, 2004): ( ),|pr 1 isjsp ttij === + 1 1 =∑ = m j ijp { }.,...,1, mji ∈∀ (2) to highlight the stylized fact of growth output, i.e. greater volatility in recessions than expansions, the approach with time-dependence in the variances of the growth output depending on the regime can be used. andrzej geise, mariola piłatowska dynamic econometric models vol.13 (2013) 175–194 180 in this paper the markov two regime-switching autoregressive model with regime-dependent intercepts, variances and autoregressive parameters will be applied. 3. brent crude oil price cycle in 1995−2013 period the data correspond to monthly brent crude oil prices from january 1995 to april 2013 (220 observations in us dollars per barrel) and were drawn from the u.s. energy information administration database. based on these data the two-regime markov switching model was estimated where regime 1 stands for the drops in crude oil prices and regime 2 stands for the rises in crude oil prices. we consider a two-regime switching markov model where the logs of crude oil prices are modeled in first differences: ),,0(~|, 2 1 , ttt sttt q j jtsjst iidscc σξξγμ +δ+=δ ∑ = − (3) where tcδ stands for the log first differences of brent crude oil prices 3, 2 , ,, ttt ssjs σγμ − denote regime-dependent intercept, autoregressive parameters and variance of disturbance term tξ respectively. the unobserved variable 1=ts in the first regime (drop in oil prices) and 2=ts in the second regime (rise in oil prices). the markov-switching autoregressive models allowing the switches between regimes in different parameters according to a hidden markov chain were chosen based on the akaike information criterion (aic) and loglikelihood function (ll). we considered four types of markov switching models introducing switches in the intercept (msi), in intercept and variance (msih), in intercept and autoregressive parameters (msia) and in intercept, variance and autoregressive parameters (msiha). the value of aic and loglik for different ms-ar models4 are given in table 2. 3 to test the non-stationarity of the logs of brent crude oil prices the adf test was used. the adf statistic for log-levels amounts 0.85 (p=0.89) − without intercept, and −3.6 (p=0.03) − with intercept and trend, and for log-first differences: −6.76 (p=0.00) − without intercept, and −6.92 (p=0.00) − with intercept and trend. this indicates that the log-levels of crude oil prices are non-stationary and become stationary when they are first differenced. therefore the log-differences of crude oil prices are used in further analysis. 4 in this paper we focus on the maximum likelihood estimation of parameters. all the calculations in the paper were carried out in oxmetrics6. synchronization of crude oil price cycle and business cycle... dynamic econometric models vol. 13 (2013) 175–194 181 table 2. the akaike information criterion (aic) and log-likelihood function (ll) for the univariate markov switching model for the crude oil prices, january 1995 to april 2013 markov switching model aic ll msi(2)-ar(1) –674.27 343.13 msih(2)-ar(1) –672.48 343.24 msia(2)-ar(1) –687.23 350.61 msiha(2)-ar(1) –678.62 347.31 msi(2)-ar(2) –715.86 364.86 msih(2)-ar(2) –722.10 369.05 msia(2)-ar(2) –716.98 368.49 msiha(2)-ar(2) –719.35 368.57 table 3. univariate msih(2)-ar(2) model for brent crude oil prices parameters parameters estimates p-value regime-dependent intercepts μ1 –0.0063 (0.0149) 0.102 μ2 0.0156 (0.0069) 0.004 autoregressive parameters γ1 0.8678 (0.0629) 0.000 γ2 –0.4611 (0.0627) 0.000 regime-dependent variances σ21 0.0645 (0.0083) 0.000 σ22 0.0359 (0.0033) 0.000 ll 369.05 aic –722.1 lr test 15.63 0.0036 jarque-bera test 0.8013 0.6699 rcm 37.99 transition probabilities across regimes p1i p2i regime 1 0.9321 0.0292 regime 2 0.0679 0.9708 note: in parentheses the standard errors of estimates are given. using the aic and the ll function the markov switching model with regime-dependent intercept and variance (msih(2)-ar(2)) was found to be the best. the detailed estimation results for the period january 1995 to april 2013 are displayed in table 3. andrzej geise, mariola piłatowska dynamic econometric models vol.13 (2013) 175–194 182 the transition matrix allows us to observe the asymmetry of oil price cycle in terms of the duration of drops (regime 1) and rises (regime 2) in oil prices. drops have the duration of approximately 15 months and rises have an average duration of two times the regime 1 (34 months). both states are characterized by high probability of remaining in regime ( ,93.011 =p )97.022 =p − see table 3 and 4. in the regime 1 the oil prices tend to decrease )0( 1 <μ while in regime 2 they tend to increase );0( 2 >μ the volatility of oil prices is higher in the regime 1 (drops in oil prices) than in regime 2 (rises in oil prices) )( 22 2 1 σσ > − see table 3. the rcm (regime classification measure5) statistic indicates that the msih(2)-ar(2) model is able to confidently distinguish which regimes are occurring at each point in time (rcm < 50). in the studied period the first serious oil prices fall was observed in 1997−1999 period (table 4 and figure 1) which may be connected with financial (currency) crisis in south eastern asia and then in russia. the situation in russia has made the market react by falling oil prices because russia is the greatest crude oil producer (with average daily production about 10 millions of barrels; see international energy agency, 2013). the next two oil prices falls in 2001 and 2008−2009 period may be influenced by the economic crises in the u.s. (the third largest oil producing and the largest oil consuming country in the world). the decrease in oil prices in 2001 corresponds to the recession in the u.s. that occurred in march 2000 when the nasdaq crashed following the collapse of the dot-com bubble (figure 1). consequently, the decline of stock prices occurred in many countries (galbraith, hale, 2004). the second decline in oil prices, i.e. in 2008−2009 period, is connected with the subprime mortgage crisis that broke out in the latter half of 2007. this financial crisis spread to the oil market (and other commodity markets) which is driven by global changes in supply and demand along with a number of other geopolitical factors. 5 the rcm statistic was proposed by ang, bekaert (2002). for two regimes it takes the form: ),1( 1 400 1∑ = −= t t tt pp t rcm where tp is a smoothed probability of being in a certain regime at time t. the constant serves to normalize the statistic to be between 0 and 100. the rcm is a summary point statistic which describes the quality of regime classifications. an ideal model is that classifying regimes sharply and having smoothed probabilities which are either close to zero or one. if the rcm is close to zero, the regime classification is perfect whereas a value of 100 means that no information about regimes is revealed. the cut-off value of rcm statistic is 50 which is often used as a benchmark (chan, et al., 2011). synchronization of crude oil price cycle and business cycle... dynamic econometric models vol. 13 (2013) 175–194 183 table 4. dating of crude oil price cycle regime 1 regime 2 1995(04)–1997(10) 1997(11)–2001(11) 2001(12) –2006(07) 2006(08)–2006(12) 2007(01)–2008(06) 2008(07)–2009(02) 2009(03)–2013(04) 62 months (28.57%) out of 217 are in regime 1 average duration d(i) 14.73 months 155 months (71.43%) out of 217 are in regime 2 average duration d(i) 34.24 months figure 1. smoothed probabilities of being in regime 1 and regime 2 for the univariate markov switching model for the crude oil prices, january 1995 to april 2013 the oil price decline in 2006 is difficult to explain. in the second half of 2006 the oil price decline indeed occurred but it was rather small, however the ms-ar model captured it as a recession. hence, this behavior of oil prices may be rather combined with general economic situation of oil exporters and importers and uncertainty on financial markets (although there were some inflammatory events in 2006, like the attack on shell pipeline in nigeria, worries over iranian nuclear plans or israel's war against hezbollah). andrzej geise, mariola piłatowska dynamic econometric models vol.13 (2013) 175–194 184 in the next section the business cycle of chosen central eastern european economies (poland, czech republic, hungary, slovenia) is identified. 4. identification of business cycle in central eastern european (cee) economies in 1995−2013 period the data consists of seasonally adjusted6 monthly industry production index for a set of cee economies from january 1995 to april 2013. these monthly series were taken from the oecd database. our analysis is restricted to a subset of four cee countries: czech republic, hungary, poland, slovenia7. log-transformed data along with seasonally adjusted data are displayed in figure 2. it is seen that the economies experienced the trend break in 2008 referring to the raging financial crisis (subprime mortgage crisis). specifically, the serious decline in the industry production index is apparent in the case of czech republic, hungary and slovenia. in the case of poland the impact of financial crisis on the industry production index was not so evident. table 5. the adf test results for industry production index (ipi) in a set of cee countries time series lag adf test for ipi adf test for δipi without intercept with intercept and trend without intercept with intercept and trend czech republic 5 1.3983 (0.96) –2.7621 (0.2116) –4.4468 (0.0000) –4.7682 (0.0005) hungary 5 2.3805 (0.9962) –0.8783 (0.9569) –4.5967 (0.0000) –9.7907 (0.0000) poland 5 3.9040 (0.9999) –2.7320 (0.2235) –32319 (0.0012) –8.7368 (0.0000) slovenia 5 1.0068 (0.9178) –1.9891 (0.6067) –6.3661 (0.0000) –6.4719 (0.0000) note: in parentheses the p-values are given. to test the non-stationarity of series the adf test was applied. table 5 reports the results of the adf unit root test for the logs of industry production index (ipi) for levels and the first differences. for all ipi series in levels the null hypothesis of a unit root cannot be rejected ).05.0value( =>− αp the adf results for first differences, δipi, allow to reject the null hypothe 6 for seasonal adjustment the tramo/seats procedure was used. 7 the intention of a research was to include all cee economies. however, the final choice of countries was determined by the availability of data, i.e. sufficient length of time series. synchronization of crude oil price cycle and business cycle... dynamic econometric models vol. 13 (2013) 175–194 185 sis δipi~i(1) ).05.0value( =<− αp the results of the adf test indicate that all industry production indexes under consideration are non-stationary in their levels and become stationary when they are first differenced. hence, in further analysis the first differences of ipi for given cee economies were used. figure 2. industry production index in the cee countries to identify the business cycle of different cee countries we applied the markov switching model. the two-regime markov switching model for industry production index modeled in first differences takes the form: ),,0(~|, 1 , ttt sttt q j jtsjst iidsyy σεεφμ +δ+=δ ∑ = − (4) where tyδ stands for first differences of the natural log of industry production index for a given country (economy), 2,, ,t t ts j s sμ ϕ σ − denote regimedependent intercept, autoregressive parameters and variance of disturbance term tε respectively. the unobserved variable 1=ts in the first regime (recession) and 2=ts in the second regime (expansion). andrzej geise, mariola piłatowska dynamic econometric models vol.13 (2013) 175–194 186 like in the case of the ms-ar models for crude oil price, four type of markov switching models were considered, i.e. msi, msih, msia, msiha. the value of aic and ll for different ms-ar models are reported in table 6. table 6. the akaike information criterion (aic) and log-likelihood function (ll) for the univariate markov switching model for industry production index markov switching model czech republic poland hungary slovenia ll aic ll aic ll aic ll aic msi(2)–ar(1) 774.99 –1537.9 849.35 –1686.7 798.03 –1584.1 1055.4 –2098.9 msih(2)–ar(1) 778.55 –1543.1 850.91 –1687.8 801.15 –1588.2 1056.0 –2088.0 msia(2)–ar(1) 780.04 –1546.1 852.42 –1690.9 802.51 –1591.0 1055.6 –2097.2 msiha(2)– ar(1) 784.92 1553.85 852.53 –1689.1 808.31 –1602.6 1061.1 –2106.2 msi(2)–ar(2) 804.30 –1594.6 876.61 –1739.2 807.19 –1600.4 1078.2 –2142.3 msih(2)–ar(2) 811.06 –1606.1 879.28 –1742.6 822.95 –1629.9 1088.6 –2161.2 msia(2)–ar(2) 814.48 –1611.6 893.47 –1768.9 828.67 –1639.3 1086.0 –2153.9 msiha(2)– ar(2) 815.77 –1611.6 886.97 –1753.9 824.23 –1628.5 1097.5 –2174.9 the aic criterion and ll function found the msiha(2)-ar(2) model to be the best for czech republic and slovenia, and the msia(2)-ar(2) model − for poland and hungary (table 6). the detailed estimation results for these models for the period january 1995 to april 2013 are displayed in table 7. while the parameter estimation is successfully made by the markov switching models, it is worthwhile to make a more formal assessment of the performance of these models. therefore, the formal testing for nonlinearity in the context of markov switching models is carried out. the results of the likelihood ratio (lr) test indicate that the significant evidence of nonlinear behavior in series under consideration has been found (see table 7). therefore the markov switching model are considered to be attractive in addressing the nonlinear nature of studied series (especially their asymmetric behavior). the state of low economic activity (regime 1) is highly persistent in case of poland )93.0( 11 =p and relatively persistent in case of czech republic and hungary 73.0( 11 =p and 0.70 respectively) − see table 7, with exception of slovenia for which the probability of remaining in regime 1 is equal 5.011 =p what along with 49.012 =p may indicate a poor model specifica synchronization of crude oil price cycle and business cycle... dynamic econometric models vol. 13 (2013) 175–194 187 table 7. the univariate markov switching model for industry production index for chosen cee countries parameters czech rep. poland hungary slovenia type of markov switching model msiha(2)– ar(2) msia(2)–ar(2) msia(2)–ar(2) msiha(2)– ar(2) regime-dependent intercept μ1 –0.0036 (0.0012) –0.0014 (0.0005) –0.0011 (0.0011) –0.0017 (0.001) μ2 0.0037 (0.001) 0.0051 (0.0006) 0.0045 (0.0009) 0.0025 (0.0005) regime-dependent autoregressive parameters φ1–1 1.2943 (0.0827) 0.7356 (0.0974) 1.0516 (0.1199) 1.0107 (0.1772) φ1–2 0.8437 (0.0894) 0.9876 (0.0754) 0.8048 (0.0799) 1.3040 (0.0588) φ2–1 –0.2759 (0.0951) –0.4848 (0.0829) –0.0706 (0.0527) –0.8902 (0.1948) φ2–2 –0.6406 (0.0698) –0.6155 (0.0739) –0.6188 (0.0928) –0,4957 (0.0567) regime-dependent variances σ21 0.0056 (0.0005) 0.0034 (0.0002) 0.0049 (0.0003) 0.005 (0.0015) σ22 0.0042 (0.0005) – – 0.0014 (0.0001) ll 815.77 893.47 828.67 1097.5 aic –1611.6 –1768.9 –1639.3 –2174.9 lr test 22.95 (p=0.0008) 33.72 (p=0.000) 31.28 (p=0.000) 34.47 (p=0.000) jarque–bera test 5.549 (p=0.0624) 3.360 (p=0.186) 14.84 (p=0.0006) 3.516 (p=0.172) rcm 62.28 10.16 58.37 6.66 transition probabilities p11 0.7309 0.9205 0.7026 0.5005 p22 0.8094 0.9619 0.8648 0.9782 p12 0.2691 0.0795 0.2974 0.4995 p21 0.1906 0.0381 0.1352 0.0218 average duration of regime d(i)1 3.72 12.58 3.36 2.00 d(i)2 5.25 26.25 7.40 45.87 note: in parentheses the standard errors of parameter estimates are given. andrzej geise, mariola piłatowska dynamic econometric models vol.13 (2013) 175–194 188 tion8. what is interesting the rcm statistic (see table 7, rcm=6.66 is substantially below 50) seems to indicate something opposite, i.e. that msiha(2)-ar(2) model for slovenia performs well in distinguishing between two regimes. however, it should be noted that a low value of the rcm statistic does not necessarily imply that the switches are correctly predicted, but only that they are sharp, with a small frequency of periods of uncertainty on the nature of regimes (guidolin, 2011). the regime 2 (expansion) is highly persistent for all countries. in the recessionary periods the industry production index tends to decrease )0( 1 <μ whereas in the expansionary periods − tends to increase 2( 0)μ > for all considered countries. the volatility of industry production index is higher in the regime 1 (recession) than in regime 2 (expansion) for czech republic and slovenia ).( 22 2 1 σσ > the duration of recessions and expansions is asymmetric for all countries, e.g. the duration of expansions is about twice as long as the recession state (except slovenia) − see table 7. smoothed probabilities of being in a given regime are calculated (see figure 3) and every observation is assigned to a given regime according to the highest smoothed probability. for the case of two regimes, the rule reduces to assigning the observation to the first regime if 5.0)|1pr( >= tt ys and assigning it to the second regime if .5.0)|1pr( <= tt ys corresponding dating of business cycle in studied countries is given in table 8. according to the classical methodology of nber (national bureau of economic research) the u.s. economic recession occurred in 2001 (collapse of the dot-com bubble, march 2001 to december 2001) and in 2007−2009 period (subprime mortgage crisis, december 2007 to july 2009)9. the smoothed probabilities of being in a recession (figure 3) obtained with the markov switching models indicate that in case of hungary and poland the financial crisis affected these economies. the impact of the dot-com crisis has been demonstrated in the case of hungary and poland (see figure 3 and table 8). generally, dating of business cycle obtained with the ms-ar models is partly consistent with the dating of world economic recession ob 8 in order to find a better specification with more plausible values of transition probabilities we tried out different model specifications (see table 6). only for the msia(2)-ar(1) model the reasonable values of p11 and p12 were obtained, i.e. 0.74 and 0.26 respectively. however, simultaneously the rcm statistic took the higher value, rcm=36.06, and what it is more important the switches between regimes occurred very often what indicates the difficulties in distinguishing between two regimes. therefore the results for slovenia seem rather implausible and should be taken with caution. 9 see: http://www.nber.org/-cycles/cyclesmain.html. synchronization of crude oil price cycle and business cycle... dynamic econometric models vol. 13 (2013) 175–194 189 tained with the nber methodology, however there are other recessionary periods which are country specific, e.g. poland. figure 3. smoothed probabilities of being in a recession for industry production index in chosen cee countries whereas for poland the univariate ms-ar model seems to capture relatively well the different recessionary periods, in the case of czech republic and hungary the ms-ar model delivers the worst fit, with difficulties distinguishing clearly the recessionary periods (switches between regimes occur very often). besides, the rcm statistic (higher than 50, see table 7) in the case of czech republic and hungary suggests rather that the chosen markov-switching models are not able to distinguish between two regimes and as a result the existence of two regimes is doubtful (additionally, the model evaluation is weaken by the fact that the standarized residuals are not standard normally distributed − see table 7). only in the case of poland the markov-switching model can clearly differentiate one regime in each period considered (rcm statistic substantially below 50 and normal distribution of residuals − see table 7). andrzej geise, mariola piłatowska dynamic econometric models vol.13 (2013) 175–194 190 table 8. dating of business cycle (recessions) for considered central eastern european economies and crude oil price cycle (drops in oil prices), 1995−2013 oil price cycle (drops in oil prices) business cycle (regime 1 − recession) czech rep. poland hungary slovenia 1997(11)–2001(11) 2006(08)–2006(12) 2008(07)–2009(02) 1995(4)–1995(5) 1995(9)–1995(11) 1997(7)–1997(9) 1997(12)–1998(10) 1999(8)–1999(11) 2000(3)–2000(6) 2001(2)–2001(10) 2003(2)–2003(2) 2003(10)–2004(3) 2004(7)–2004(10) 2005(8)–2005(10) 2006(5)–2006(12) 2007(12)–2008(2) 2008(5)–2008(11) 2009(1)–2009(2) 2010(5)–2010(5) 2012(2)–2012(10) 1998(5)–1999(1) 2000(10)– 2002(4) 2004(5)–2005(2) 2008(2)–2009(1) 2012(1)–2013(4) 1995(9)–1995(10) 1997(8)–1997(10) 1999(7)–1999(9) 2000(4)–2000(8) 2001(5)–2001(6) 2001(9)–2001(11) 2008(3)–2008(12) 2009(2)–2009(4) 2011(3)–2011(6) 2012(3)–2012(7) 2012(9)–2012(10 1996(9)–1996(9) 1997(9)–1997(9) 2008(11)–2009(2) in order to evaluate the extent to which turning points in the two series occur near each other and to evaluate synchronization of cycles the correlation of business cycle in given countries and crude oil price cycle and concordance index respectively have been calculated10. for the comparison purposes these calculations are carried out for all chosen countries, although only in the case of poland they are sensible. the correlation coefficients between probabilities of being in regime 1 (recession and drops in oil prices for business cycle and crude oil price cycle respectively) are displayed in figure 4. generally, the correlations between probabilities of being in regime 1 (figure 4) are weak what means that turning points of business cycle in the recessionary periods does not occur close to turning points of crude oil price cycle. the highest correlation of business cycle and oil price cycle is observed in the case of poland (32%), and the lowest − in the case of slovenia (10%). 10 harding, pagan (2006) and konopczak (2009) indicate that the correlation coefficient and concordance index are the most frequently used measures of the synchronization of cycles identified by the markov-switching model. synchronization of crude oil price cycle and business cycle... dynamic econometric models vol. 13 (2013) 175–194 191 figure 4. correlation coefficients for probabilities of being in regime 1 for business cycle in given countries and crude oil price market in 1995–2013 period table 9. concordance index for oil price cycle and business cycle for given economies crude oil price cycle vs. business cycle czech republic poland hungary slovenia concordance index 0.7143 0.6866 0.7051 0.7235 to evaluate synchronization of business cycle for different economies and oil price cycle the concordance index was applied. this index (for two series ,tx ty and a sample size of t ) takes the form (harding, pagan, 2006): 1 1 1 (1 )(1 ) , t t xt yt xt yt t t i s s s s t = = ⎡ ⎤ = + − −⎢ ⎥ ⎣ ⎦ ∑ ∑ (5) where xts and yts denote binary variables that takes the value unity in case of recession regime and zero − in case of expansion regime at time .t values of concordance index measuring the synchronization of cycles (in terms of being in the same state) indicate a high degree of concordance of the business cycles and oil price cycle (table 9). however, having in mind earlier remarks on the quality of regime classification, the above conclusion refers only to the case of poland. this means that business cycle in poland in 1995−2013 period has the same state (recession or expansion) as the brent oil price cycle (drops or rises in oil prices). even in that case the caution   0,2928 0,3174 0,2784 0,1012 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 czech republic poland hungary slovenia andrzej geise, mariola piłatowska dynamic econometric models vol.13 (2013) 175–194 192 should be taking when interpreting the results because a low value of correlation coefficients corresponds to a high degree of concordance of business cycles and oil price cycle. this may suggest the overestimated degree of concordance. conclusions in this paper we used the approach innovated by hamilton in his analysis of the us business cycle to identify the brent crude oil price cycle and business cycles in a set of cee economies in 1995−2013 period. the obtained results support to a varying degree our modelling approach based upon markov-switching. the msih(2)-ar(2) model produced the sharpest classification of regimes (drops or rises in oil prices) was obtained in the case of crude oil price cycle. while the msia(2)-ar(2) model produced the sharpest classification of regimes (recession on expansion) for poland. in the case of czech republic and hungary the ms-ar models were not able to distinguish between two regimes (switches between regimes occurred very often) and as a result the existence of two regimes is doubtful. the ms-ar models obtained for slovenia indicated either a poor model specification or difficulties in distinguishing between two regimes. therefore the results for slovenia should be taken with caution. our results indicated that the ms-ar models captured the different regimes of brent crude oil price cycle and business cycle for the case of poland in a satisfactory way. the correlation of oil price cycle and business cycle in poland turned out to be rather weak, however the synchronization of cycles occurred (a high degree of concordance) what indicates that there are co-movements in crude oil cycle and business cycles. week correlation of oil price cycle and business cycle may come from the specificity of poland, i.e. the size of crude oil imported by these countries from the north sea region is small (in poland the import of crude oil from norway amounts to 3.5% of total crude oil import; popihn, 2013). poland but also the other cee countries are still dependent on the crude oil imported from russia which is cheaper but worse in quality (it contains a high amount of the impurity sulfur and has high density). references abel, b. a., bernanke, b. s. (2001), macroeconomics, addison wesley longman inc. ang, a., bekaert, g. (2002), regime switches in interest rates, journal of business & economic statistics, 20(2), 163–182. synchronization of crude oil price cycle and business cycle... dynamic econometric models vol. 13 (2013) 175–194 193 arouri, m., nguyen, d. (2010), oil prices, stock markets and portfolio investment: evidence from sector analysis in europe over the last decade, energy policy, 38(8), 4528–4539. artis, m., krolzig, h. m., toro, j. (2004), the european business cycle, oxford economic papers, 56, 1–44, doi: http://dx.doi.org/10.1093/oep/56.1.1. backus, d., crucini, m. (2000), oil prices and the terms of trade, journal of international economics, 50, 185–213, doi: http://dx.doi.org/10.1016/s0022-1996(98)00064-6. bjornland, h. (2008), oil price shocks and stock market booms in an oil exporting country, norges bank working paper, 16, 1–33. doi: http://dx.doi.org/10.1111/j.1467-9485.2009.00482.x. brown, s., yucel, m. (1999), oil prices and u.s. aggregate economic activity: a question of neutrality, economic and financial review, 2, 16–23. brown, s., yucel, m. (2002), energy prices and aggregate economic activity: an interpretative survey, quarterly review of economics and finance, 42, 193–208, doi: http://dx.doi.org/10.1016/s1062-9769(02)00138-2. bruno, m., sachs, j. (1982), input price shocks and the slowdown in economic growth: the case of u.k. manufacturing, the review of economic studies, 49(5), 679−705, doi: http://dx.doi.org/10.2307/2297185. chan, k. f., treepongkaruna, s., brooks, r., gray, s. (2011), asset market linkages: evidence from financial commodity and real estate assets, journal of banking and finance, 35(6), 1415–1426, doi: http://dx.doi.org/10.1016/j.econmod.2011.10.006. davis, j. s., haltiwanger, j. (2001), sectoral job creation and destruction responses to oil price changes, journal of monetary economics, 48, 465–512, doi: http://dx.doi.org/10.1016/s0304-3932(01)00086-1. dixit, a. k., pindyck, r. s. (1994), investment under uncertainty, princeton university press, new jersey. fisher, i. (1930), the theory of interest, macmillan, new york. galbraith, j. k., hale, t. (2004), income distribution and the information technology bubble. university of texas inequality project, http://utip.gov.utexas.edu/papers/utip_27.pdf, (16.08.2013). guidolin, m. (2011), markov switching model in empirical finance, innocenzo gasparini institute for economic research, bocconi university, working paper, no. 415, 1–60. hamilton, j. (1988), are the macroeconomic effects of oil-price changes symmetric? a comment, carnegie-rochester conference series on public policy, 28, 369–378, doi: http://dx.doi.org/10.1016/0167-2231(88)90031-0. hamilton, j. (1989), a new approach to the economic analysis of nonstationary time series and the business cycle, econometrica, 57(2), 357–384, doi: http://dx.doi.org/10.2307/1912559. hamilton, j. (1996), this is what happened to the oil price macroeconomy relationship, journal of monetary economics, 38, 215–220, doi: http://dx.doi.org/10.1016/s0304-3932(96)01282-2. harding, d., pagan, a. (2006), synchronization of cycles, journal of econometrics, 132(1), 59–79. hooker, m. (1999), are oil shock inflationary? asymetric and nonlinear specifications versus changes in regime, journal of money, credit and banking, 34(2), 540–561, doi: http://dx.doi.org/10.1353/mcb.2002.0041. international energy agency (iea), http://omrpublic.iea.org/currentissues/full.pdf, (16.08.2013). andrzej geise, mariola piłatowska dynamic econometric models vol.13 (2013) 175–194 194 jimenez-rodrigues, r., sanchez, m. (2005), oil price shocks and business cycles in major oecd economies, applied economics, 37(2), 201–228. kim, i. m., loungani, p. (1992), the role of energy in real business cycle models, journal of monetary economics, 29, 173–189, doi: http://dx.doi.org/10.1016/0304-3932(92)90011-p. konopczak, k. (2009), analiza zbieżności cykli koniunkturalnych gospodarki polskiej ze strefą euro na tle krajów europy środkowo-wschodniej oraz państw członkowskich strefy euro (business cycle synchronization in poland, the euro area and eastern european countries), [in:] nbp. raport na temat pełnego uczestnictwa rzeczypospolitej polskiej w trzecim etapie unii gospodarczej i walutowej: projekty badawcze część trzecia (nbp. report on the full participation of poland in the third stage of economic and monetary union), warszawa. kośko, m. (2006), application of markov-switching model to stock returns analysis, dynamic econometric models, 7, 259–268. krolzig, h-m., toro, j. (2005), classical and modern business cycle measurement: the european case, spanish economic review, 7, 1–21, doi: http://dx.doi.org/10.1007/s10108-004-0088-0. leblanck, m., chinn, m. (2004), do high oil prices presage inflation? the evidence from g–5 countries, sccie working paper, 4, 1–27, doi: http://dx.doi.org/10.2139/ssrn.509262. lee, k., ni, s. (2002), on the dynamic effects of oil price shock: a study using industry level data, journal of monetary economics, 49, 823–852, doi: http://dx.doi.org/10.1016/s0304-3932(02)00114-9. loungani, p. (1986), oil price shocks and dispersion hypothesis, review of economics and statistics, 58, 536−549, doi: http://dx.doi.org/10.2307/1926035. nber, http://www.nber.org/cycles/cyclesmain.html (16.08.2013 r.). pierce, j. l., enzler, j.j. (1974), the effects of external inflationary shocks, brooking paper on economic activity, 1, 13–61. popihn (2013), przemysł i handel naftowy. roczny raport 2012, warszawa. stawicki, j. (2004), wykorzystanie łancuchów markowa w analizie rynku kapitałowego (application of markov chains in stock market analysis), wydawnictwo uniwersytetu mikołaja kopernika, toruń. analiza zbieżności cykli cenowych rynku ropy naftowej z cyklami koniunkturalnymi gospodarek europy środkowo-wschodniej z a r y s t r e ś c i. głównym celem artykułu jest zbadanie, w jakim stopniu cenowe cykle ropy naftowej (brent) są skorelowane i zsynchronizowane z cyklem koniunkturalnym dla wybranych gospodarek państw europy środkowo-wschodniej (eśw). w celu identyfikacji cyklu cenowego ropy naftowej i cyklu koniunkturalnego został zastosowany przełącznikowy model markowa (ms-ar). określenie wygładzonych prawdopodobieństw w zależności od reżimu umożliwiło wyznaczenie korelacji tych prawdopodobieństw oraz obliczenie indeksu konkordancji dla oceny stopnia synchronizacji cyklu cenowego ropy naftowej i cyklu koniunkturalnego w badanych państwach eśw. s ł o w a k l u c z o w e: przełącznikowe modele markowa, ceny ropy naftowej, cykle koniunkturalne, cykle cenowe. microsoft word 00_tresc.docx dynamic econometric models vol. 10 – nicolaus copernicus university – toruń – 2010 jacek kwiatkowski nicholas copernicus university in toruń unobserved component model for forecasting polish inflation† a b s t r a c t. this paper aims to use the local level models with garch and sv errors to predict polish inflation. the series to be forecast, measured monthly, is consumer price index (cpi) in poland during 1992-2008. we selected three forecasting models i.e. ll-garch(1,1) with normal or student errors and ll-sv. a simple ar(2)-sv model is used as a benchmark to assess the accuracy of prediction. the presented results indicate, that there is no clear advantage of ll models in forecasting polish inflation over standard ar(2)-sv model, although all the models give satisfactory results. k e y w o r d s: local level model, inflation, conditional heteroscedasticity. 1. introduction econometric models, both of the structural and a-theoretical ones, are widely used to provide forecasts of inflation. in a recent study stock and watson (2007), found that a local level model with stochastic volatility gives the most accurate forecasts of quarterly inflation in the united states. in their paper they compare the accuracy of inflation forecasts of wide class of models including standard arima time series models, time-varying parameters models (tvp) and the phillips curve-based models. in this paper, we examine several types of inflation forecasts in poland, which are based on time-varying parameters model and subject them to tests for accuracy. the paper is organized as follows. section 2 presents the models which are used to forecast monthly inflation in poland: ll-sv and ll-garch with two † this work was financed from the polish science budget resources in the years 2009-2011 as the research project no. n n111 431737. i would like to thank grzegorz szafrański and one anonymous referee for helpful comments and suggestions; e-mail: jkwiat@umk.pl. jacek kwiatkowski 122 different distributions of the disturbances (normal, student's t) and standard ar(2)-sv model. in section 3, we compare forecast accuracy of the mentioned above models. we use two forecast accuracy measures, namely: the sign test and wilcoxon signed rank test (see diebold and mariano, 1995). the predictive distribution calculated for the future observables enable us to provide a detailed analysis of the inflation forecasts, therefore we also present the predictive medians and interquartile range. it is also well known, that polish monetary authorities conduct the policy under inflation targeting regime between 1.5% and 3.5% and the inflation prediction is one of the inputs to the monetary policy council’s decision-making process. then, it is worth to consider what the posterior probability for the hypothesis is that inflation will stay inside the targeting bound and how this posterior probability changes as the forecasts horizon grows. section 4 concludes the paper. 2. unobserved component model with time-varying conditional variance stock and watson (2007) used unobserved component model (ll-sv), which is very effective for forecasting inflation: ttty εδ += , ( )2,0~ tt n σε , tt ...,,2,1= (1) ttt ηδδ += −1 , ( )2,0~ tt n ωη , (2) where the irregular and level disturbances, tε and tη , respectively, are mutually independent, tδ denotes underlying stochastic level. consider now that tε and tη are stationary sv processes, where: tirregtirregirregtirreg hh ,1,, ζρ += − , (3) tleveltlevelleveltlevel, hh ,1, ζρ += − , (4) and ( )2,0~ tt n σε , ( )2,0~ tt n ωη , ( )tirregt h ,2 exp=σ , ( )tlevelt h ,2 exp=ω , ( )1,1)( −∈levelirregρ and ( )2 )()( ,0~ levelirreglevelirreg n γζ . it’s easy to see that the reduced form of unobserved component model (1)(2) is a local level model (ll) with restrictions in parameters: 22 σσ =t and 22 ωω =t . the local level model is a well-known model and it has a long tradition in economic time series. the literature that considers its properties is very extensive and previously interested many authors (see for example muth, 1960; harvey, 1989; west and harrison, 1989; durbin and koopman, 2001; koop, 2003). unobserved component model for forecasting polish inflation 123 consider the ll model when the disturbances follow normal garch or student-t garch. assuming that each noise is a conditionally normal process, we have: ( )2,0~ tt n σε and ( )2,0~ tt n ωη . (5) the equivalent student-t disturbances are denoted as: ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − irreg tirreg irreg t vv v t , 2 ,0~ 2σ ε and ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − level tlevel level t vv v t , 2 ,0~ 2ω η , (6) where ( )vpat ,, denotes student-t density with expectation a , precision p and v degrees of freedom. for garch(1,1) process, the variance of the observation equation (1) and state equation (2) varies over time according to (see bos, 2001): tirregt h , 2 =σ and tlevelt h , 2 =ω , (7) ( )21,1,01,,1, −− ++= tirregirregtirregirregtirreg eaahbh ε , (8) ( )21,1,01,,1, −− ++= tlevelleveltlevelleveltlevel eaahbh η , (9) with the parametric constraints that are sufficient for positivity and stationarity of the conditional variance: )(,1)(,1)(,0 1 levelirreglevelirreglevelirreg aba −−≡ , 0)(,1 ≥levelirregb , 0)(,1 ≥levelirrega , 1)(1)(,1 <+ levelirreglevelirreg ba . the properties of the ll model, when the disturbances follow garch(1,1) process, are presented in pellegrini, ruiz and espasa (2007, 2008). the last model is a simple ar(2)-sv model, which is used as a benchmark model. it has the following form: tttt yyy εδδδ +δ+δ+=δ −− 22110 , ( )2,0~ tt n σε . (10) 3. forecasting inflation in this section, we examined whether the proposed models successfully predict the value of the inflation rate. we considered 204=t monthly observations on the logarithm of cpi from january 1992 till december 2008. the data set employed in this study consists of logarithmic transformations of the original series cpi, computed as ( )tt cpiy ln100= . all data has been seasonally adjusted, using the moving average method implemented in eviews 6. jacek kwiatkowski 124 before starting the analysis, we briefly describe the processes that influence inflation in poland over the past twenty years. during the first years of 90’s the polish economy has been transformed to a market economy. due to marketization and stabilization program there were deep economic and social changes including the elimination of the state control of prices and liberalizing trade, investment and capital flow (fallenbuchl, 1994). in 1990-1992 the polish economy was in an early stage of transition and inflationary processes visibly accelerated reaching 685.8 % in 1990. therefore we begin our analysis from january 1992 to avoid the unusual effects of hyperinflation. in the years 1993-1997, monetary policy was focused on neutralizing the powerful inflationary forces discernible within the polish economy and then to achieve a further reduction in inflation1. the next years (1998–2004) increased poland's central bank independence and monetary policy was focused on maintaining price stability and preparation for integration with the eu. after 2004 the main goal of monetary policy was to achieve the maastricht price stability criterion in the coming future. the data previously discussed are presented in figure 1. the vertical line indicates the limit between sample and forecast-period. figure 1. the values of logarithms of cpi (seasonally adjusted data). the vertical line indicates the limit between sample and forecast-period using bayes' rule and monte carlo techniques, we calculated four competing bayesian models – ll-sv, ll-garch(1,1), ll-garch(1,1)student and ar(2)-sv – based on dataset ( )ty for 192...,,1=t . for each model and 12 months in 2008, as a result we obtained predictive distributions in the following form: ( )( )ith myyp ,|192+ , for 4...,,1=i , 12...,,1=h , (11) 1 nbp annual report, http://www.nbp.pl/ -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 1992 1994 1996 1998 2000 2002 2004 2006 2008 ln c p i unobserved component model for forecasting polish inflation 125 where ll-garch(1,1) – 1m , ll-garch(1,1)-student – 2m , ll-sv – 3m and finally ar(2)-sv – 4m . due to number of models and months, the entire procedure should be performed four times, giving a total of 48 predictive distributions. table 1. the quantiles of the predictive distribution date model ll-garch(1,1) ll-garch(1,1)-student ll-sv ar(2)-sv 2008m01 [-0,0584] 0.3429 (0.0367, 0.6564) 0.3535 (0.1118, 0.6075) 0.3214 (0.0894, 0.5398) 0.2825 (0.0437, 0.5265) 2008m02 [0,4363] 0.3429 (-0.0187, 0.7177) 0.3535 (0.0791, 0.6499) 0.3214 (0.0742, 0.5513) 0.4195 (0.1303, 0.7037) 2008m03 [0,4509] 0.3429 (-0.0662, 0.7587) 0.3535 (0.0482, 0.6693) 0.3214 (0.0520, 0.5732) 0.3364 (0.0242, 0.6593) 2008m04 [0,1780] 0.3429 (-0.0974, 0.8076) 0.3535 (0.0160, 0.7123) 0.3214 (0.0443, 0.5923) 0.3181 (-0.0413, 0.6756) 2008m05 [0,8002] 0.3429 (-0.1545, 0.8569) 0.3535 (-0.0098, 0.7290) 0.3214 (0.0329, 0.6017) 0.3396 (-0.0584, 0.7361) 2008m06 [0,4702] 0.3429 (-0.1728, 0.8888) 0.3535 (-0.0348, 0.7577) 0.3214 (0.0290, 0.6110) 0.3399 (-0.0947, 0.7633) 2008m07 [0,8979] 0.3429 (-0.2305, 0.9164) 0.3535 (-0.0471, 0.7721) 0.3214 (0.0128, 0.6237) 0.3223 (-0.1432, 0.8031) 2008m08 [0,1819] 0.3429 (-0.2603, 0.9458) 0.3535 (-0.0671, 0.7965) 0.3214 (-0.0159, 0.6441) 0.3221 (-0.1724, 0.8319) 2008m09 [-0,2893] 0.3429 (-0.2777, 0.9822) 0.3535 (-0.0910, 0.8156) 0.3214 (-0.0228, 0.6456) 0.3245 (-0.1968, 0.8558) 2008m10 [0,2222] 0.3429 (-0.3148, 1.0157) 0.3535 (-0.1182, 0.8377) 0.3214 (-0.0341, 0.6670) 0.3257 (-0.2442, 0.8905) 2008m11 [0,1811] 0.3429 (-0.3148, 1.0157) 0.3535 (-0.1182, 0.8377) 0.3214 (-0.0341, 0.6670) 0.3257 (-0.2442, 0.8905) 2008m12 [-0,1699] 0.3429 (-0.3944, 1.0672) 0.3535 (-0.1724, 0.8619) 0.3214 (-0.0571, 0.7093) 0.3056 (-0.3194, 0.9236) note: all data are expressed as percentage. table 1 contains a selection of the corresponding out-of-sample forecasting results for the selected models with constant and time-varying mean. the data in brackets represent the seasonally adjusted consumer price index. for four models and for each month of 2008, the medians and the first and third quartiles (on the second line) of the predictive distributions are reported. a forecast will be said to be accurate if true value falls within the interquartile range. we can observe that during 2008 year the ll-garch(1,1) model produces the most accurate predictive distribution of cpi. in this case the true values fall within the interquartile range in ten cases of twelve. for the other models i.e. the llgarch(1,1)-student, ll-sv and ar(2)-sv, true values lie between the first and third quartiles only in eight cases of twelve. for the ll-garch(1,1) model, when inflation forecast is an accurate, the true values of cpi lie mostly jacek kwiatkowski 126 between the median and third quartile. in other cases, predictive densities underestimate or overestimate the actual observation, both for the models with the constant and time-varying mean. it is also worthwhile to check formally the accuracy of point forecasts of the mentioned above models. we measure forecast accuracy using two exact finitesample test, namely: the sign test and wilcoxon signed rank test (see diebold and mariano, 1995). these tests allow us to analyze of statistical differences between predictions generated by competing specifications when only a small number of forecasts are available. we test the null hypothesis of that the forecasting performance of the two different models is equally well (poor). the results are summarized in the table 2, which includes p-values from the sign test (the first line) and wilcoxon signed rank test (in the second line). in our case we use quadratic loss function. the observed loss differentials are free of serial correlation. table 2. results from the sign test and wilcoxon signed rank test model model ll-garch(1,1) ll-garch(1,1)-student ll-sv ar(2)-sv llgarch(1,1) 0.7744 0.8501 1.0000 0.6772 1.0000 0.9697 llgarch(1,1)student 0.7744 0.7910 1.0000 0.9697 ll-sv 0.7744 0.1514 ar(2)-sv note: the first line denotes p-values from the sign test. the second line includes p-values from the wilcoxon signed rank test according to the results given in the table 2, we do not reject at conventional levels the hypothesis of equal expected quadratic loss. in other words there is no significant difference between the accuracy of point forecasts of the competing specifications. the standard ar(2)-sv model is not a significantly worse (better) predictor of the polish cpi than the ll model. it is well known, that polish monetary authorities conduct the policy under inflation targeting regime, with the medium and long term target for cpi index fixed at 2.5% and with one percentage point of accepted deviation. therefore it is interesting to consider what is the posterior probability of the hypothesis, that inflation will stay inside the targeting bound and how this posterior probability changes as the forecasts horizon grows. bayesian methodology provides a direct way to predict different scenarios of inflation. the predictive distribution depicts the probability of various outcomes for cpi inflation in the future and allows to assess uncertainty of monetary policy. unlike classical (sampletheory) approach we do not need carry out stochastic simulations since our unobserved component model for forecasting polish inflation 127 approach follows from the basic rules of probability. tables 3 and 4 present some characteristics of the probability distribution of the inflation path. table 3. posterior probability that inflation will remain within the targeting regime date probability of inflation ll-garch(1,1) ll-garch(1,1)-student ll-sv ar(2)-sv within (1.5%; 3.5%) within (1.5%; 3.5%) within (1.5%; 3.5%) within (1.5%; 3.5%) 2008m01 0.0052 0.0052 0.0104 0.0052 2008m02 0.0833 0.0365 0.0417 0.0208 2008m03 0.2083 0.0833 0.0938 0.1198 2008m04 0.2656 0.1719 0.1771 0.2292 2008m05 0.3333 0.2240 0.2760 0.3177 2008m06 0.3385 0.2552 0.3021 0.3438 2008m07 0.4010 0.3333 0.4479 0.4063 2008m08 0.3854 0.2813 0.3438 0.3802 2008m09 0.3906 0.2865 0.3542 0.3750 2008m10 0.4167 0.3021 0.3958 0.3958 2008m11 0.4635 0.3385 0.4635 0.4323 2008m12 0.4688 0.3281 0.4635 0.4271 table 4. posterior probability that inflation will be above the upper bound of targeting regime date probability of inflation ll-garch(1,1) ll-garch(1,1)-student ll-sv ar(2)-sv above 3.5% above 3.5% above 3.5% above 3.5% 2008m01 0.9948 0.9948 0.9896 0.9948 2008m02 0.9167 0.9635 0.9583 0.9792 2008m03 0.7917 0.9167 0.9063 0.8802 2008m04 0.7344 0.8281 0.8229 0.7708 2008m05 0.6667 0.7760 0.7240 0.6823 2008m06 0.6615 0.7448 0.6979 0.6563 2008m07 0.5990 0.6667 0.5521 0.5938 2008m08 0.6146 0.7188 0.6563 0.6198 2008m09 0.6094 0.7135 0.6458 0.6250 2008m10 0.5833 0.6979 0.6042 0.6042 2008m11 0.5365 0.6615 0.5365 0.5677 2008m12 0.5313 0.6719 0.5365 0.5729 tables 3 and 4 include assessment of the risks around central projections for prices of consumer goods and services. the predictive results indicate that inflation was more likely to be above target in 2008 than below target. the predictive probability for the hypothesis, that inflation will stay inside the targeting bound ranges from 0.0052 to 0.2656 in the period january–april, from 0.224 to 0.4479 in the period may–august and from 0.2865 to 0.4688 in the jacek kwiatkowski 128 period september–december, whereas the predictive probability of the hypothesis that inflation will be above the upper limit of the tolerance band (3.5%), for all months and models, ranges from 0.5313 to 0.9948. these forecasts are consistent with true values of annual cpi because in 2008, according the gus data2, inflation rose above the upper limit of the tolerance band. during first eight months of 2008, cpi inflation showed a rising tendency – from 4.0% in january to 4.8% in august. in the period september – december we observed decline in the annual growth from 4.3% to 3.3%. according to reports published by nbp, the main factor conducing to higher level of inflation was the prices of energy commodities in the world market3. thus, it seems that all models have ability to assess correctly the risk associated with cpi inflation. 4. conclusions in this paper the local level models are analyzed and compared from point of view of their ability to forecast monthly inflation in poland. the data concern the consumer price index and they range from january 1992 till december 2008. for each model and for each month of 2008 we constructed the predictive distributions. according to the sign test and wilcoxon signed rank test, there is no significant difference between the accuracy of point forecasts of the competing specifications. also all models show correctly a rising tendency of annual cpi inflation. analysis of forecast accuracy of competing specifications does not lead to decisive conclusion about superiority of any of the considered specifications. it seems that we can identify at least two reasons why unobserved component model could not satisfactorily predict the inflation. firstly, it is known that the first differences of local level model display the same correlation structure as the ima(1,1) model – that is, one in which the first order autocorrelation is negative for the first difference of series and all other autocorrelations are zero (see west and harrison, 1989). this is a very restrictive assumption which is in practice very difficult to obtain. from preliminary studies it was known that in our case not only first but also second order autocorrelation is negative and statistically significant, which may indicate a more complicated correlation structure of analyzed process. for this reason we consider a second order autoregressive process. secondly, the recent publication by grassi and proietti (2008) showed the strong evidence in favor of the local level model with heteroscedastic disturbances only in the core component of the u.s. inflation, whereas the transitory component was time invariant. the volatility of the disturbances driving only one i.e. core component may improve accuracy of forecasts. 2 central statistical office (gus), http://www.stat.gov.pl/gus/index_eng_html.htm 3 inflation report, http://www.nbp.pl/ unobserved component model for forecasting polish inflation 129 references bos, c. (2001), time varying parameter models for inflation and exchange rates, webdoc. diebold, f.x., mariano, r.s. (1995), comparing predictive accuracy, journal of business and economic statistics, 13, 253–63. durbin, j., koopman, s.j. (2001), time series analysis by state space methods, oxford university press, oxford. fallenbuchl, z.m. (1994), foreign trade in the process of transformation in poland, atlantic economic journal, 22, 2, 51–60. grassi, s., proietti, t. (2008), has the volatility of u.s. inflation changed and how?, mpra paper 11453. harvey, a.c. (1989), forecasting, structural time series models and the kalman filter, cambridge university press, cambridge. koop, g. (2003), bayesian econometrics, john wiley & sons. koop, g., potter, s. (2001), are apparent findings of nonlinearity due to structural instability in economic time series?, the econometrics journal, 4, 1, 37–55. muth, j.f. (1960), optimal properties of exponentially weighted forecasts, journal of the american statistical association, 55, 299–306. pellegrini, s., ruiz, e., espasa, a. (2007), the relationship between arima-garch and unobserved component models with garch disturbances, working paper, ws072706. pellegrini, s., ruiz, e., espasa, a. (2008), arima-garch and unobserved component models with garch disturbances: are their prediction intervals different? job market paper. stock, j.h., watson, m.w. (2007), why has u.s. inflation become harder to forecast? journal of money, credit, and banking, 39, 3–33. stock, j.h., watson, m.w. (2008), phillips curve inflation forecasts, working paper, 14322. west, m., harrison, j. (1989), bayesian forecasting and dynamic models, springer. prognozowanie inflacji w polsce przy użyciu modelu lokalnego poziomu z a r y s t r e ś c i. w artykule przeprowadzono badania dotyczące trafności prognoz otrzymanych za pomocą modelu lokalnego poziomu w wersji stocka i watsona (2008). rozważono różne postacie tego modelu i zbadano, które z nich dają możliwość uzyskania najtrafniejszej prognozy. badania empiryczne dotyczyły inflacji w polsce w latach 1992-2008. ostatni rok posłużył do oceny jakości prognoz. badania przeprowadzono na podstawie wskaźnika cen konsumenta cpi. uzyskane wyniki nie potwierdzają jednoznacznej przewagi modelu lokalnego poziomu, w prognozowaniu inflacji, nad standardowym modelem autoregresyjnym. wszystkie modele uzyskały zadowalającą dokładność prognozy. s ł o w a k l u c z o w e: model lokalnego poziomu, prognozowanie, inflacja. microsoft word 00_tresc.docx dynamic econometric models vol. 10 – nicolaus copernicus university – toruń – 2010 ryszard doman adam mickiewicz university in poznań modeling the dependence structure of the wig20 portfolio using a pair-copula construction† a b s t r a c t. elliptical distributions commonly applied to modeling the returns of stocks in highdimensional portfolio are not capable of adequate describing the dependence between the components when their statistical properties are very diverse. the mgarch and standard dynamic copula models are often of little usefulness in such cases. in this paper, we apply a methodology called the pair-copula decomposition to model the joint conditional distribution of the returns on stocks constituting the wig20 index, and show some advantage of this construction over the approach using the t student dcc model. k e y w o r d s: dependence, portfolio, copula, pair-copula construction. 1. introduction elliptical distributions commonly applied to modeling the returns of stocks in high-dimensional portfolio are not capable of adequate describing the dependence between the components of the return vector when their statistical properties are very diverse. except of a huge number of parameters necessary to estimate, this is why multidimensional garch models are often of little practical usefulness in such cases. also the standard dynamic copula models which are successfully applied in the bivariate case are not so useful for large dimension because of a small number of classical multivariate copulas that are flexible enough to fit the data. in this context, a new approach to modeling multivariate data which exhibit complex patterns, proposed recently by aas et al. (2009), has triggered off much interest. it was inspired by the work of joe (1996), bedford and cooke (2002), and kurowicka and cooke (2006). the main idea of this methodology is to model dependence by decomposing higher-dimensional copula densities into bivariate ones (pair-copula densities) arising from the condi † this work was financed from the polish science budget resources in the years 2007-2010 as the research project nn 111 1256 33. ryszard doman 32 tional and unconditional distribution functions of the modeled variables. a further simplification of the decomposition can be then obtained basing on conditional independence. in this paper, we apply the pair-copula decomposition approach to model the joint conditional distribution of the returns on stocks constituting the wig20 index. results of our investigation support a view that this construction not only is superior over the approach that uses multidimensional t copula model but also represents a promising technique of building flexible and accessible multivariate extensions of classical bivariate copulas which can be of great importance for optimal portfolio allocation and quantitative risk management. 2. dependence and copulas consider a multivariate return series 1, , ( , , )t t n tr r ′=r … decomposed as t t t= +r μ y , where 1( | )t t te −= ωμ r and 1t−ω is the set of information available up to time t . in standard multivariate garch models it is assumed that 1/2 t t t=y h ε , ~ ( , )t niidε 0 i , and thus th is the conditional covariance matrix of tr . a specific parameterization for the dynamics of th defines an element of the family of mgarch models (bauwens et al., 2006). one of the main difficulties when dealing with these models is a problem of dimensionality because the number of parameters to be estimated increases very fast with the number k. moreover, it is usually postulated that 1| ~ ( , )t t tn−ωy 0 h or, slightly generally, that the conditional distribution of ty is elliptical. the dynamic linear correlation which can be obtained from mgarch models still plays a central role in financial theory. one should realize, however, that this tool for measuring dependence is appropriate only in the case of elliptical distributions. an alternative concept that allows for modeling the dependence in general situation is copula. copulas were initially introduced by sklar (1959). formally, an n-dimensional copula is a distribution function c on n-cube n]1 ,0[ with standard uniform marginal distributions (nelsen, 2006). assume that x is an n-dimensional random vector with joint distribution f and univariate marginal distributions if . the importance of copulas in studying of multivariate distribution functions is summarized by sklar’s theorem which states that the f can be written as 1 1 1( , , ) ( ( ), , ( ))n n nf x x c f x f x=… … ’ (1) for some copula c. if the marginal distribution functions are continuous then c is unique, and is called the copula of f or x . conversely, if c is a copula and nff ,,1 … are univariate distribution functions, then the function f defined in (1) modeling the dependence structure of the wig20 portfolio using a pair-copula… 33 is a joint distribution function with margins nff ,,1 … . an explicit representation of c in terms of f and its margins is given by ))(,),((),,( 11 1 11 nnn ufuffuuc −−= …… ’ (2) where })( :inf{)(1 iiiiii uxfxuf ≥= − . since the marginals and the dependence structure in (1) can be separated, it makes sense to interpret the copula c as the dependence structure of the random vector x. the simplest copula is defined by ∏ = = n i in uuuc 1 1 ),,( … , (3) and it corresponds to independence of marginal distributions. the next important example is the comonotonicity copula, +c , which takes the form },,min{),,( 11 nn uuuuc …… = + . (4) it corresponds to perfect dependence between the components of a random vector 1( , , )nx x ′=x … in the sense that ix is the image of 1x under some strictly increasing transformation for ni ,,2 …= . in the empirical part of this paper we will use the student t copula. it is defined as follows: student 1 1 , 1 , 1( , , ) ( ( ), , ( ))p n p nc u u t t u t uη η η η − −=… … , (5) where tη is the distribution function of a standard student t distribution withη degrees of freedom, and ,ptη is the joint distribution function of a multivariate student t distribution with η degrees of freedom and the correlation matrix p. if a copula c is absolutely continuous, its density c is, as usual, given by n n n uu uuc uuc ∂∂ ∂ = … … … 1 1 1 ),,( ),,( . (6) for a copula c of absolutely continuous joint distribution function f with marginal distribution functions nff ,,1 … , joint density f, and marginal densities nff ,,1 … , the following representation holds )()())(,),((),,( 11111 nnnnn xfxfxfxfcxxf …… = , (7) ryszard doman 34 3. pair-copula decompositions the density of a vector 1( , , )nx x ′=x … can be factorized as 1 1 2 1 1 2 ( , , ) ( ) ( | ) ( | , ) ( | , ) n n n n n n n n n f x x f x f x x f x x x f x x x− − −= … … . (8) the idea of a cascade of bivariate copulas or a pair-copula decomposition (aas et al. 2009) comes from the fact that each conditional density in (8) can be further decomposed into a product of the appropriate bivariate copula (paircopula) density times a conditional marginal density. for example, 1 2 12 1 1 2 2 1 1( | ) ( ( ), ( )) ( )f x x c f x f x f x= , (9) 1 2 3 12|3 1 1 3 2 2 3 1 3( | , ) ( ( | ), ( | )) ( | )f x x x c f x x f x x f x x= . (10) more generally, it holds that |( | ) ( ( | ), ( | )) ( | )j jxv j j j jf x c f x f v f x− − − −= vv v v v , (11) where v is a d-dimensional vector and j−v denotes the vector obtained from v by excluding the j-th component. as concerns the marginal conditional distributions of the form ( | )f x v , it was shown by joe (1996) that, for every j, the following holds , | ( ( , ), ( , )) ( | ) ( | ) j jx v j j j j j c f x f v f x f v − − − − ∂ = ∂ v v v v v . (12) in particular, if x and v are observations of variables uniform on [0,1] then , ( , )( | ) x v c x v f x v v ∂ = ∂ . (13) it follows from (7) and (11) that by applying iteration one can express a multivariate density as a product of the form 1 1 1 , |1, , 1 1 1 1 1 1 1 ( , , ) ( ), ( ( | , , ), ( | , , )) = −− + − − + − = = =∏ ∏∏ … … i … … n n k k n jn j j i j j j j i j j i f x x f x c f x x x f x x x , (14) or 1 1 1 , | 1, , 1 1 1 1 1 1 1 ( , , ) ( ), ( ( | , , ), ( | , , )) . = −− + + + − + + − + + + − = = =∏ ∏∏ … … i … … n n k k n jn i i j i i j i i i j i j i i j j i f x x f x c f x x x f x x x (15) modeling the dependence structure of the wig20 portfolio using a pair-copula… 35 decompositions such as (14) and (15) are called pair-copula constructions. in fact, for high-dimensional distributions there are many possible pair-copula decompositions. some methods that help organize them are described by bedford and cooke (2001, 2002) in the language of the so-called regular vines. in what follows, we use some very basic notions concerning graphs, which can be found, for instance, in (kurowicka, cooke, 2006). we start with the definition of a regular vine on n variables. it is the structure composed of 1n − trees 1( , , )nt t… in which 1t is a tree with the set of nodes 1 {1, , }n n= … and the set of edges 1e , and for 2, , 1i n= −… , the it is a tree with the set of nodes 1i in e −= . moreover, it should hold that if some nodes 1 2{ , }a a a= , 1 2{ , }b b b= are connected by an edge then exactly one ia is equal to exactly one ib . in financial applications, the most important are two special cases of regular vines: canonical vines and the d-vines. a regular vine is called a canonical vine (or c-vine) if in each tree it ( 1)i n< − there exists exactly one node with degree n i− . the node in 1t that has maximal degree is called the root. a regular vine is called a d-vine if each node in 1t has a degree of at most 2. examples of cand d-vines are shown in figures 1 and 2. figure 1. a canonical vine on 5 variables 13 t1 t2 t3 t4 14 15 12 1 2 3 4 5 24|1 25|1 23|1 12 13 14 15 45|123 34|12 35|12 35|12 34|12 23|1 24|1 25|1 ryszard doman 36 figure 2. a d-vine on 5 variables it is not very hard to observe that formula (14) gives a pair-copula construction corresponding to a canonical vine, and formula (15) defines a pair-copula construction that can be described by a d-vine. it should be mention here that starting from n nodes one can construct !/ 2n different canonical vines and !/ 2n different d-vines (aas et al., 2009). when some components ix and jx of the vector x are conditionally independent given a subvector v of x then , | ( ( | ), ( | )) 1i j v i jc f x v f x v = , and thus the pair-copula decomposition in (14) or (15) simplifies. this property is of great importance from a practical point of view. it shows how a careful selection of variables and a proper choice of their ordering can affect the model complexity. the canonical vines and d-vines can be estimated by maximum likelihood method. if we assume that the data 1, ,( , , )t t n tx x x= … , 1, ,t t= … , are observations of variables that are independent over time then the log-likelihood for the canonical vine is given by 1 , |1, , 1 , 1, 1, 1 1 1 , 1, 1, , ( ; ) log[ ( ( | , , ), ( | , , ); )] , n jn t j j i j j t t j t j i t j i t t j t j i l x c f x x x f x x x −− + − − = = = + − θ = θ ∑∑∑ … … … (16) and for the d-vine it has the form , | 1, , 1 , 1, 1, 1 1 1 , 1, 1, , ( , ) log[ ( ( | , , ), ( | , ); ))] . n jn t i i j i i j i t i t i j t j i t i j t i t i j t i j l x c f x x x f x x x − + + + − + + − = = = + + + − θ = … θ ∑∑∑ … … (17) 12 34 23 45 13|2 34|5 24|3 1 3 2 4 5 t1 t2 t3 t4 15|234 14|23 25|34 modeling the dependence structure of the wig20 portfolio using a pair-copula… 37 the number of parameters depends on copula types used in the model specification. in the presence of temporal dependence, as is in the case of real data, usually some arma-garch models are fitted to the margins, and the estimation is performed for the standardized residuals. thus in fact, the estimation method is that of maximum pseudo-likelihood. the consistency and asymptotic normality of the estimators obtained in such a way is discussed by genest et al. (1995) and joe (1997). 4. the data and model specification the data we use in this paper consist of daily returns on the stocks of the companies that constituted the wig20 index of the warsaw stock exchange during the period from september 23, 2005 to may 29, 2009.the tickers of the securities under scrutiny are as follows: acp, bhw, bio, bre, bsk, bzw, cst, gtc, gtn, kgh, lts, orb, pbg, peo, pgn, pkn, pko, pxm, tps, tvn. the returns are calculated as 1100(ln ln )t t tr p p−= − , where tp is the closing quotation on day t. the return series showed some autocorrelation and in all cases conditional homoskedasticity was strongly rejected by the engle test. following a commonly accepted approach, we first estimated the arma-gjr-garch models for the marginal returns. in each case a standardized skewed student’s t distribution (lambert, laurent, 2001) was applied as the error distribution. next, the standardized residuals series were transformed into uniform on [0,1] by using the corresponding probability integral transforms. for the transformed data, we estimated a d-vine, described by the decomposition (15). prior to choosing an ordering of the univariate series we estimated a bivariate student’s t copula for each of the possible 190 pairs. next, we analyzed the pairs with respect the estimated number of degrees of freedom, which was assumed to be a risk factor. we decided to apply bivariate student’s t copulas for all pairs in estimated pair-copula decomposition. the final ordering of the marginal univariate series was chosen in such a way that the numbers of degrees of freedom of copulas connecting the consecutive series in tree 1 of the d-vine formed a non-decreasing sequence. 5. empirical results in applications of pair-copula constructions it is of great importance to carefully consider the selection of specific factorization, and the choice of bivariate copula types. thus in the first stage of our investigation we tried to fit bivariate copulas of diverse type to each of the possible pairs of the series from our dataset. finally we decided to use student’s t copulas, and focus on their numbers of degrees of freedom considered as risk factors. the obtained estimates of the numbers of degrees of freedom are presented in table 1. table 1. estimates for the number of degrees of freedom in bivariate student’s t copulas fitted to the pairs of the return series, period sept. 23, 2005 – may 29, 2009 acp bhw bio bre bsk bzw cst gtc gtn bhw 300.0 bio 36.7 40.9 bre 8.9 13.5 13.6 bsk 17.4 24.6 83.4 21.6 bzw 6.9 25.7 15.1 11.1 10.5 cst 21.5 300.0 286.7 272.1 49.3 14.7 gtc 10.3 24.1 33.6 19.4 43.4 16.1 23.2 gtn 13.7 19.4 19.0 21.4 12.5 10.9 153.6 14.1 kgh 8.0 11.4 22.2 11.2 19.3 13.2 44.3 25.3 17.6 lts 7.0 54.3 300.0 15.7 29.0 15.1 67.2 24.0 15.6 orb 7.9 300.0 233.1 300.0 300.0 17.2 32.1 18.8 25.7 pbg 13.0 300.0 19.6 11.5 28.6 13.4 17.2 10.2 10.6 peo 9.6 300.0 11.0 7.8 21.5 53.2 85.8 16.1 27.1 pgn 13.1 11.4 26.5 18.4 48.1 13.0 7.1 18.5 13.0 pkn 10.9 22.3 300.0 20.7 68.0 13.4 300.0 84.9 8.1 pko 12.5 13.5 36.7 12.0 13.8 30.2 63.8 194.6 12.2 pxm 9.4 11.8 12.3 9.2 21.3 8.4 13.5 7.5 11.4 tps 15.0 23.1 21.5 70.9 148.8 11.2 42.6 21.8 300.0 tvn 30.9 49.9 158.0 19.0 300.0 15.9 23.0 19.2 17.3 kgh lts orb pbg peo pgn pkn pko pxm tps lts 24.2 orb 20.5 10.1 pbg 8.8 16.4 29.5 peo 17.6 16.3 24.4 300.0 pgn 15.1 10.6 15.8 13.7 11.5 pkn 27.7 5.1 14.5 85.0 13.0 13.6 pko 14.6 10.0 11.8 22.6 7.3 6.2 22.7 pxm 8.7 33.7 8.8 58.7 10.2 18.7 28.2 14.2 tps 56.6 13.2 9.5 47.4 25.3 17.3 26.3 25.2 17.0 tvn 21.4 9.6 119.1 18.1 12.8 21.5 13.3 15.7 12.3 108.4 note: we estimated the number of degrees of freedom parameter subject to upper bound equal to 300. in practice, the value 300 means that the gaussian copula is the proper one. modeling the dependence structure of the wig20 portfolio using a pair-copula… 39 table 2. estimates for the number of degrees of freedom in the fitted d-vine 1 2 3 4 5 6 7 8 9 1 5.1 26.4 25.1 38.6 300.0 300.0 300.0 20.6 14.3 2 13.6 54.2 53.8 13.7 300.0 35.9 300.0 53.2 300.0 3 6.2 22.1 32.5 14.0 300.0 272.2 45.0 46.8 300.0 4 12.5 17.4 300.0 8.6 21.1 22.8 36.7 223.1 24.2 5 6.9 300.0 18.8 13.3 15.1 31.0 77.2 31.8 43.2 6 14.7 66.5 13.0 23.3 300.0 37.3 28.5 20.1 8.1 7 85.8 300.0 27.9 31.6 14.5 74.9 18.0 300.0 46.2 8 7.8 300.0 38.5 21.1 19.8 25.6 36.1 17.1 15.0 9 300.0 20.5 14.1 31.9 12.2 294.0 260.6 44.8 39.1 10 18.8 11.2 29.1 300.0 15.6 187.2 300.0 300.0 300.0 11 7.5 65.0 11.7 20.0 37.8 35.9 63.3 59.4 36.6 12 8.7 279.5 29.8 20.8 33.4 300.0 19.3 38.1 13 8.8 300.0 247.0 300.0 285.1 13.3 34.5 14 47.4 16.6 23.2 16.3 153.7 42.6 15 111.6 20.8 300.0 38.5 300.0 16 158.0 13.4 78.1 300.0 17 19.0 50.4 300.0 18 19.4 15.6 19 24.6 10 11 12 13 14 15 16 17 18 19 1 300.0 300.0 300.0 42.5 42.4 14.6 300.0 108.1 300.0 300.0 2 97.3 19.3 300.0 36.5 19.3 300.0 9.2 300.0 300.0 3 44.1 30.4 76.6 300.0 300.0 100.4 30.6 300.0 4 14.2 82.3 113.4 300.0 24.5 22.1 300.0 5 300.0 73.7 275.0 39.5 300.0 27.7 6 22.9 15.2 36.9 300.0 11.8 7 300.0 29.8 300.0 300.0 8 16.2 300.0 151.2 9 23.7 44.8 10 300.0 note: the numbers of degrees of freedom for the copulas appearing in trees 1-19 of the estimated d-vine are presented in columns. we estimated the number of degrees of freedom parameter subject to upper bound equal to 300. in practice, the value 300 means that the gaussian copula is the proper one. we used the obtained estimates of the number of degrees of freedom for the bivariate return series to chose an efficient ordering of the variables included in ryszard doman 40 the decomposition (15). our choice was the following: 1. lts, 2. pkn, 3. pgn, 4. pko, 5. acp, 6. bzw, 7. cst, 8. peo, 9. bre, 10. orb, 11. gtc, 12. pxm, 13. kgh, 14. pbg, 15. tps, 16. tvn, 17. bio, 18. gtn, 19. bhw, 20. bsk. the motivation was that in that case, for the 19 bivariate copulas fitted to the pairs of the returns in accordance to the formula of tree 1 of the estimated d-vine, the sequence of the corresponding numbers of degrees of freedom is non-decreasing. the d-vine estimation results are presented in table 2. for a comparison we fitted to the investigated vector return series a standard 20-dimensional student’s t copula. as an estimate for the number of degrees of freedom we obtained 34.4265. looking at the estimates for the bivariate copulas in tree 1 of the estimated d-vine, which vary from 5.1 to 300, we can state that the superiority of the pair-copula construction approach over the standard multidimensional copula approach is strongly supported. a significantly better fit of the d-vine model has been also indicated by the akaike information criterion. our next objective was to use the estimated d-vine to compute in-sample var estimates for long and short positions (giot, laurent, 2003) for the portfolio composed of the considered stocks, and compare them to the ones obtained by using engle’s (2002) dcc model with multivariate student’s t distribution. in the dcc model case we could use the well-known formulas for a portfolio var (see e.g. giot, laurent, 2003), having the estimates of the conditional covariances and the degree of freedom for the conditional student’s t distribution, which was estimated as 13.7881. for the approach using the pair-copula construction, we simulated for each day 1000 20-dimensional vectors from the fitted d-vine. then we transformed them into the one-dimensional standardized residuals, and, finally, into the daily returns of the portfolio components. after that we obtained the daily var estimates as the corresponding quantiles. the var calculation was performed for significance levels 0.01, 0.025, and 0.05. an algorithm for sampling from an n-dimensional d-vine proceeds as follows (see aas et al., 2009). start with sampling variates nuu ,,1 … independent uniform on [0, 1]. then set 11 uw = , )|( 12 1 2 wufw −= , ),|( 213 1 3 wwufw −= , ),,|( 11 1 − −= nnn wwufw … . the general formula for the functions ),,|( 11 −jj xxxf … involves (12) and (13), and it is given in the paper by aas et al. (2009), where one can also find explicit formulas for the inverse in the case in which all the bivariate copulas of a pair copula construction are student’s t copulas. mod our r assess the dence tes good resu tions. fo dcc mo the cover level are figure 1. figure 2. 5. conc in thi eling the index. w dent’s t c results of large. th superiorit copula co of the wi deling the depe results conce e quality of sts by chris ults for long or long posit del definitel rage test wa shown in fig in-sample v tolerance le in-sample v t conditional clusions is paper we a dependence we focused on copulas ente f our investi his fact supp ty over the c onstruction f ig20 index p endence structu erning in-sam the var est stoffersen (1 g trading pos tions, howev ly, especially as close to 1 gures 1 and 2 var calculated evel equal to 0 var calculated l distribution. applied the p structure of n the numbe ring the con igation show ports usefuln classical app for simulatio portfolio, an ure of the wig2 mple var cal timates we a 998). gener sitions, and ver, the pair y at toleranc 1. the result 2. d by means o 0.05 d by means o tolerance lev pair-copula c f the returns er of degrees nstruction, co w that the ra ness of the proaches. in n from the j d applied the 20 portfolio usi lculation are applied the c rally speakin rather poor r-copula mo e level 0.05 ts correspon of the fitted p f engle’s dc vel equal to 0. construction on stocks co s of freedom onsidering it ange of the e new method addition, we oint multidim e simulation ing a pair-copu e not unambi coverage and ng, we obta results for s odel outperfo where the p nding to this pair-copula co cc model with 05 methodolog onstituting th m of the biva t as a risk fa estimates ca dology and p e used the f mensional di output for c ula… 41 guous. to d indepenained very short posiormed the p-value of tolerance onstruction. h student’s gy to modhe wig20 ariate stuactor. the an be very proves its fitted pair istribution calculating ryszard doman 42 value-at-risk. our results show that the var estimates for long trading positions at tolerance level 0.05 obtained in this way definitely outperform the ones calculated by using a fitted dcc model with student’s t conditional distribution. references aas, k., czado, c., frigessi, a., bakken, h. (2009), pair-copula constructions of multiple dependence, insurance: mathematics and finance, 44, 182–198. bauwens, e. laurent, s. rombouts, j.v.k. (2006), multivariate garch models: a survey, journal of applied econometrics, 21, 79–109. bedford, t., cooke, r.m. (2001), probability density decomposition for conditionally dependent random variables modeled by vines, annals of mathematics and artificial intelligence, 32, 245–268. bedford, t., cooke, r.m. (2002), vines – a new graphical model for dependent random variables, annals of statistics, 30, 1031–1068. christoffersen, p. f. (1998), evaluating interval forecasts, international economic review, 39, 841–862. engle, r. f., (2002), dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models, journal of business & economic statistics, 20, 339–350. genest, c. ghoudi k. rivest, l.-p. (1995), a semiparametric estimation procedure of dependence parameters in multivariate families of distributions, biometrika, 82, 543–552. giot, p., laurent, s. (2003), value-at-risk for long and short trading positions, journal of applied econometrics, 18, 641–664. joe, h. (1996), families of m-variate distributions with given margins and m(m 1)/2 bivariate dependence parameters. in: rüschendorf, l., schweizer, b.,taylor, m.d. (eds.), distributions with fixed marginals and related topics, ims lecture notes monograph series 28, institute of mathematical statistics, hayward, ca, 120–141. joe, h. (1997), multivariate models and dependence concepts, chapman & hall, london. kurowicka, d., cooke, r. m., (2006), uncertainty analysis with high dimensional dependence modelling, wiley, new york. lambert, p., laurent, s. (2001), modelling financial time series using garch-type models with a skewed student distribution for the innovations, institut de statistique, université catholique de louvain, discussion paper 0125. nelsen, r. b. (2006) an introduction to copulas (2nd ed.), springer, new york sklar, a. (1959), fonctions de rérpartition à n dimensions et leurs marges, publications de l’institut statistique de l’université de paris, 8, 229–231. modelowanie struktury zależności portfela indeksu wig20 za pomocą kaskady kopuli dwuwymiarowych z a r y s t r e ś c i. w artykule przedstawiono wyniki zastosowania nowej metodologii modelowania zależności pomiędzy zwrotami składników portfela wysokowymiarowego. idea tego podejścia polega na dekompozycji gęstości rozkładu łącznego na iloczyn, w którym występują jedynie gęstości kopuli dwuwymiarowych pewnych rozkładów warunkowych wyznaczonych przez modelowane zmienne. badania dotyczą stóp zwrotu z akcji wchodzących w skład indeksu wig20 i potwierdzają pewną przewagę nowej metodologii nad podejściem, w którym stosowany jest model dcc engle’a z wielowymiarowym rozkładem t studenta. s ł o w a k l u c z o w e: zależność, portfel, kopula, kaskada kopuli dwuwymiarowych. microsoft word 00_tresc.docx © 2013 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2013.001 vol. 13 ( 2013) 5−31 submitted june 16, 2013 issn accepted december 5, 2013 1234-3862 małgorzata doman, ryszard doman* the dynamics and strength of linkages between the stock markets in the czech republic, hungary and poland after their eu accession∗∗ a b s t r a c t. we analyze the dynamics and strength of linkages between the czech, hungarian and polish stock markets after the eu accession of the corresponding countries. in addition, we examine linkages between each of the markets and developed markets (european and us). the analysis is based on the daily quotations of the main representative stock indices (px, bux, wig20, dax, s&p 500) and includes the period from may 5, 2004 to july 20, 2012. the dynamics of dependencies is modeled by means of markov-switching copula models, and the applied measures of the strength of the linkages are dynamic spearman’s rho and tail dependence coefficients. the results show that dependencies between the considered emerging markets are very sensitive on market situation, but the linkages of these markets with the developed ones are stable. k e y w o r d s: central european stock market, conditional dependence, markov-switching copula model, spearman’s rho, tail dependence, model confidence set, stock index. j e l classification: g15; g01; c32; c58. * correspondence to: małgorzata doman, poznań university of economics, department of applied mathematics, al. niepodległości 10, 61-875 poznań, poland, e-mail: malgorzata.doman@ue.poznan.pl; ryszard doman, adam mickiewicz university in poznań, faculty of mathematics and computer science, umultowska 87, 61-614 poznań, poland, e-mail: rydoman@amu.edu.pl. ∗∗ this work was financed from the polish science budget resources in the years 20102013 as the research project n n111 035139. małgorzata doman, ryszard doman dynamic econometric models 13 (2013) 5–31 6 introduction in the paper, we analyze the dynamics and strength of linkages between the czech (prague stock exchange, pse), hungarian (budapest stock exchange, bse) and polish (warsaw stock exchange, wse) stock markets after the eu accession of the corresponding countries. in addition, we examine the linkages between each of the markets and developed markets (european and us). the analysis is based on the daily quotations of the main representative stock indices (px, bux,wig20, dax, s&p 500). the dynamics of dependencies is modeled by means of markov-switching copula models and the applied measures of the strength of the linkages are dynamic spearman’s rho and tail dependence coefficients. we investigate the period from may 5, 2004 to july 20, 2012. it begins shortly after the three considered central european countries joined the european union and includes subperiods of two recent financial crises (the us subprime and european debt crises). the czech republic, hungary and poland are often perceived by investors as one block of very similar markets. in particular, bad news concerning one of these countries strongly impacts all the markets. the history of the three economies looks similar. in 1989 all they started the transformation process, in may 2004 joined the european union, and now all they are getting ready to adopt the euro. nevertheless, their specific paths of development differ from each other. the impact of the recent crisis on the economies, measured with the evolution of the gdp annual growth rate, was dissimilar to that exerted on the stock markets (see federation of european securities exchange, 2013; trading economics, 2013). the stock mar-kets themselves are unlike each other as well. the warsaw stock exchange plays the special role in the region as the largest stock exchange market (see federation of european securities exchange, 2013). a comparison of the quotations of the representing stock indices with the evolution of gdp annual growth rate in the considered countries (see federation of european securities exchange, 2013; trading economics, 2013) shows that during the subprime crisis of 2007–2009 all the markets experienced significant decline, and the same applies to their gdp growth rate, though the polish gdp growth rate remained positive. there exists a quite extensive literature on linkages between the three central european stock markets. many investigations were performed for the period before the recent crises (see for instance scheicher, 2001; gilmore et al., 2008; caporale, spagnolo, 2011). some newer papers on the subject, mostly concerning the impact of the recent crises, can also be found. for the dynamics and strength of linkages between the stock markets… dynamic econometric models 3 (2013) 5–31 7 instance, ülkü and demirci (2012) present a thorough analysis of the joint dynamics of foreign exchange and stock markets in emerging european countries for the period 2003–2010 and document the key role of global developed market returns in driving the stock market–exchange rate interaction in emerging economies. their study uses impulse response functions from a cointegrating structural vector autoregressive model as the research tool. another methodology, involving a dynamical conditional correlation model, has been applied by kiss (2011) who examines the transmission mechanism of interbank, stock and currency market crashes for the czech republic, hungary and poland as well as for the us and euro area, using daily data from january 2002 to october 2010. the main novelty in our analysis is a detailed description of the dynamics of two types of dependence between the considered markets: an average dependence measured by spear-man’s rho, and tail dependence estimated by means of upper and lower tail dependence coefficients. unlike pearson’s correlation, which is invariant only under increasing affine trans-formations of the margins of a bivariate random vector, the dependence measures applied by us are invariant with respect to strictly increasing transformations of the margins. hence, the measures are more suitable in detecting dependencies of non-linear type between analyzed stock returns. in fact, we use a copula regime-switching approach to model the conditional bivariate distributions of the returns. so, the aforementioned properties of the applied depend-ence measures can be drawn from the fact that the measures are copula-based. what is more, using copulas models allows to leave behind the class of elliptical distributions. this is because, in this approach, description of the behavior of the marginals is separated from model-ing their dependence structure. the advantages of our approach stem also from the fact that the tail dependence coefficients reflect the dependence in extreme values, which is very fruitful in describing the market linkages during crisis periods. additionally, a markov regime switching mechanism we use provides a possibility to model the temporal fluctuation of the strength and characteristics of the modeled dependencies. we also compare the estimates of the measures of the dependencies between the czech, polish and hungarian stock markets with the corresponding quantities obtained for the dependencies between these markets and western european and american stock markets represented by the stock indices dax and s&p 500. our results show that the dynamics of dependencies in the two cases are completely different, and that the dependencies between the emerging markets are more sensitive to new information flow. however, the linkages of these markets with the main world markets are rather stable. małgorzata doman, ryszard doman dynamic econometric models 13 (2013) 5–31 8 1. markov-switching copula models modeling the dynamics of dependencies between financial returns is not an easy task because of interaction between the volatility of the individual returns and the dependence structure. the interaction is especially apparent during the periods of financial crises and turbulence in financial markets (ang, bekaert, 2002; forbes, rigobon, 2002). in such circumstances, the investigation of the dynamics of the linkages becomes more difficult due to different types of asymmetries and structural breaks which are likely to arise (ang, bekaert, 2002; ang, chen, 2002; patton, 2004). from the point of view of the choice of a suitable multivariate statistical model for the conditional distribution of the returns, this means that elliptical distributions should hardly be assumed. as a consequence, multivariate garch models (see a survey by bauwens et al., 2006) can be an inappropriate tool for modeling the dynamics of linkages. in such a situation, a concept of copula can provide an alternative solution. a bivariate copula is a mapping :[0,1] [01] [0,1]c × → from the unit square into the unit interval, which is a distribution function with standard uniform marginal distributions. assume that ),( yx is a 2-dimensional random vector with joint distribution h and marginal distributions f and g. then, by a theorem by sklar (1959), h can be written as: )).(),((),( ygxfcyxh = (1) if f and g are continuous then the function c is uniquely given by the formula: )),(),((),( 11 vgufhvuc −−= (2) for ]1 ,0[, ∈vu , where 1 ( ) inf{ : ( ) }f u x f x u− = ≥ (nelsen, 2006). in this case, c is called the copula of h or of ( , )x y . since the marginal distributions in the decomposition (1) are separated, it makes sense to interpret c as the dependence structure of the vector ( , )x y . we refer to patton (2009) and references therein for an overview of applications of copulas in analysis of financial time series. the simplest copula is defined by uvvuc =π ),( and it corresponds to independence of marginal distributions. in the empirical part of this paper we will also use the gaussian, clayton, and joe-clayton copulas. they are defined as follows: the dynamics and strength of linkages between the stock markets… dynamic econometric models 3 (2013) 5–31 9 ( ), )(),(),( 11 vuvuc gauss −− φφφ= ρρ (3) ,)1(),( 1 γγγγ −−− −+= vuvuc clayton (4) ( ) ,)1])1(1[])1(1([11 ),( /1/1 , κγγκγκ γκ −−− − −−−+−−−−= = vu vuc claytonjoe (5) where ρφ denotes the distribution of a 2-dimensional standardized normal vector with the linear correlation coefficient ρ , φ stands for the standard normal distribution function, and 1≥κ , 0>γ . the clayton copula claytonc γ is a special case of the joe-clayton copula claytonjoec −γκ , for 1=κ . in the limit case 0=γ , the clayton copula approaches the independent copula πc (nelsen, 2006). the density associated to an absolutely continuous copula c is a function c defined by: . ),( ),( 2 vu vuc vuc ∂∂ ∂ = (6) for an absolutely continuous random vector, the copula density c is related to its joint density function h by the following canonical representation: ),()())(),((),( ygxfygxfcyxh = (7) where f and g are the marginal distributions, and f and g are the marginal density functions. in the case of nonelliptical distributions, the most well-known copulabased dependence measures, which are more appropriate than the linear correlation coefficient, are kendall’s tau and spearman’s rho (embrechts et al., 2002). since the dynamics of spearman’s rho is exploited in this paper, we recall a suitable definition. if ( , )x y is a random vector with marginal distribution functions f and g, then spearman’s rho for ( , )x y can be defined as: )),(),((),( ygxfyxs ρρ = (8) where ρ denotes the usual pearson correlation. for continuous marginal distributions, spearman’s rho, ( , )s x yρ , depends only on the copula c linking x and y, and, in particular, is given by the formula: 3),(12),( 2]1,0[ −== ∫∫ dudvvucyx cs ρρ (9) małgorzata doman, ryszard doman dynamic econometric models 13 (2013) 5–31 10 (nelsen, 2006). it follows from (9) that if a copula c is a mixture of copulas 1c and 2c : 21 )1( ccc αα −+= , 10 ≤≤α , then: .)1( 21 ccc ρααρρ −+= (10) for the gaussian copula, gausscρ , spearman’s rho equals ρπ 2 16 arcsin . for the clayton and joe-clayton copulas it can be computed numerically using (9). another concept of dependence, which depends only on the copula of a random vector ),( yx , is tail dependence. it measures the dependence between extreme values of the variables. if f and g are the cumulative distribution functions of x and y, then the coefficient of upper tail dependence is defined as follows: )),(|)((lim 11 1 qfxqgyp qu −− → >>= −λ (11) provided a limit ]1,0[∈uλ exists. analogously, the coefficient of lower tail dependence is defined as: )),(|)((lim 11 0 qfxqgyp ql −− → ≤≤= +λ (12) provided that a limit [0,1]lλ ∈ exists. if (0,1]uλ ∈ ( (0,1]lλ ∈ ), then x and y are said to exhibit upper (lower) tail dependence. upper (lower) tail dependence quantifies the likelihood to observe a large (low) value of y given a large (low) value of x. the coefficients of tail dependence can be expressed in terms of the copula c of x and y in the following way: , ),( lim 0 q qqc ql +→ =λ (13) , ),(ˆ lim 0 q qqc qu +→ =λ (14) where )1,1(1),(ˆ vucvuvuc −−+−+= . for the gaussian copula it holds 0== lu λλ (see embrechts et al., 2002), meaning asymptotic independence in the tails. the clayton copula claytoncγ has lower tail dependence: 1/2 .l γλ −= in the joe-clayton copula case, κλ /122 −=u and γλ /12−=l for 0γ > (patton, 2006). thus both upper and lower tail dependence can be nonzero, and, moreover, they can change independently of each other. in applications of copulas to financial time series the notion of conditional copula introduced by patton (2004) is usually employed. it allows to the dynamics and strength of linkages between the stock markets… dynamic econometric models 3 (2013) 5–31 11 model the joint conditional distribution 1| −ωttr , where ),( ,2 ,1 ttt rr=r is a bivariate vector of financial returns, and 1−ωt is the information set available up to time 1−t . in this paper we consider the following general conditional copula model: ),( ~| 1,1 ⋅ω − ttt fr ),( ~| 1,2 ⋅ω − ttt gr (15) ),|)(),((~| 11 −− ω⋅⋅ω tttttt gfcr (16) where tc is the conditional copula linking the marginal conditional distributions, and the information set 1−ωt remains the same for the copula and marginals. further, we assume that: ,ttt yμr += ),|( 1−ω= ttt e rμ (17) ,,,, tititiy εσ= ),|var( 1, 2 , −ω= ttiti rσ (18) ),,,1 ,0(_ iid~, iiti tskew ηξε (19) where ),,1 ,0(_ ηξtskew denotes the standardized skewed student t distribution with 2>η degrees of freedom, and skewness coefficient 0>ξ (lambert, laurent, 2001). moreover, the marginal return series tir , , 2 ,1=i , are modeled as arma-garch processes. when the conditional copula tc is allowed to fluctuate over time, some model for its evolution has to be specified. commonly, the functional form of the conditional copula is fixed but its parameters evolve through time (patton, 2004, 2006). in this paper, however, we apply an alternative approach (okimoto, 2008; garcia, tsafack, 2011) assuming the existence of regimes where a fixed copula prevails, which are switched according to some homogeneous markov chain. thus, in the applied markov-switching copula model (msc model) the joint conditional distribution has the following form: ),|)(),((~| 11 −− ω⋅⋅ω tttstt gfc tr (20) where ts is a homogeneous markov chain with state space }2 ,1{ . the parameters of the model are the parameters of the univariate models for the marginal distributions, the parameters of the copulas 1c and 2c , and the transition probabilities: małgorzata doman, ryszard doman dynamic econometric models 13 (2013) 5–31 12 11 1( 1| 1),t tp p s s −= = = 22 1( 2| 2)t tp p s s −= = = (21) of the markov chain. we estimate the msc models by the maximum likelihood method. the main by-product of the estimation are the probabilities ),|( 1−ω= tt jsp )|( tt jsp ω= , 2 ,1=j , which are calculated by means of hamilton’s filter (hamilton, 1994): ,)|()|( 2 1 111 ∑ = −−− ω==ω= i ttijtt isppjsp (22) , )|(),|( )|(),|( )|( 2 1 11 11 ∑ = −− −− ω=ω= ω=ω= =ω= i ttttti tttttj tt ispisc jspjsc jsp u u (23) where: 12 1 11( 2 | 1) 1t tp p s s p−= = = = − , 21 1 22( 1| 2) 1t tp p s s p−= = = = − , ),,( ,2,1 ttt uu=u )( ,1,1 ttt rfu = , )( ,2,2 ttt rgu = , (24) and ),|( 1−ω=⋅ ttj jsc is the density of the conditional copula coupling the conditional marginal distributions in regime j. by the arguments of hamilton (1994), the maximized log-likelihood function has the form: ( ) ( )∑∑ ∑ ∑ = − = − = − = − ω+ω+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ω=ω== t t ttt t t ttt t t tt j tttj rgrf jspjscl 1 21,2 1 11,1 1 1 2 1 1 );|(ln);|(ln );|();,|(ln θθ θθu , (25) where tf and tg are the density functions corresponding to tf and tg , fitted using arma-garch models. in the empirical results presented in this paper we apply the so-called smoothed probabilities which can be obtained from the probabilities (23) and (24) by using the backward recursion: , )|( )|( )|()|( 2 1 1 1∑ = + + ω= ω= ω==ω= i tt ttji tttt isp ispp jspjsp .1,,1…−=tt (26) the dynamics and strength of linkages between the stock markets… dynamic econometric models 3 (2013) 5–31 13 the series )|( tt jsp ω= indicates which regime prevails at each date, using all the information in the sample. 2. the model confidence set methodology the model confidence set (mcs) methodology, introduced by hansen et al. (2003), is usually applied to select in a set of forecasting models a subset of models which are better than the others in terms of forecasting ability, with a given level of confidence. more precisely, for some collection 0 of models for which the subset of superior models, * , is defined, the mcs procedure produces a subset ˆ of 0 that contains the set * with a given confidence level α−1 . the collection 0 can be any finite set of objects indexed by 01, ,i m= … and evaluated in terms of a loss function that assigns to the object i in period t the loss ,i tl , 1, ,t n= … . if the relative performance is defined as , , ,ij t i t j td l l≡ − for 0,i j ∈ and the mean ,( )ij ij te dμ ≡ is finite and does not depend on t, then the set of superior objects is defined by: { }* 0 0: 0 for all iji jμ≡ ∈ ≤ ∈ . the mcs procedure is based on an equivalence test δ that tests the hypothesis: 0, : 0ijh μ = for all ,i j ∈ , at levelα for any 0⊂ . when 0,h is rejected, an elimination rule e is applied to identify the object of that is to be removed from . the mcs algorithm proposed by hansen et al. (2003) is as follows: − step 0: set 0 . − step 1: test 0,h using δ at significance level α . − step 2: if 0,h is not rejected define, * 1 ˆ α− = , otherwise use e to eliminate an object from and repeat the procedure from step 1. it is proved in hansen et al. (2011) that under some standard requirements for the equivalence test and the elimination rule it holds that: * * 1 ˆlim inf ( ) 1 ,n p α α→∞ −⊂ ≥ − (27) małgorzata doman, ryszard doman dynamic econometric models 13 (2013) 5–31 14 * 1 ˆlim ( ) 0n p i α→∞ −∈ = for all *,i ∉ (28) * * 1 ˆlim ( ) 1n p α→∞ −= = if * is a singleton. (29) if 0 , ih p denotes the p-value associated with the null hypothesis 0, i h (with convention that 0 0 , 1mhp = ) , then the mcs p-value for model 0 j e ∈m is defined as 0 ,ˆ max . j i e i j hp p≤≡ it is shown in hansen et al. (2011) that 0,h can be tested using traditional quadratic-form statistics or multiple t-statistics. in this paper we apply the t-statistic ,rt defined as: , , max | |,r ij i j t t ∈ = (30) where: ( )ˆvar ij ij ij d t d = for ,i j ∈ , (31) and 1 ,1 n ij ij tt d n d− = = ∑ . the asymptotic distribution of the statistic ,rt is nonstandard. the bootstrap implementation of the mcs procedure involving this statistic is described in appendix b in the paper by hansen et al. (2003). the estimated bootstrap distribution of ,rt , under the null hypothesis, is given by the empirical distribution of: ( ) * ,* , , | | max ˆvar b ij ij b r i j ij d d t d∈ − = , (32) where * 1, ,1 bt n b ij ijn t d d τ== ∑ is the bootstrap resample average, and ( ) ( ) 2 *1 ,1 ˆvar b ij b ij ijb b d d d = = −∑ . for generating b resamples, the block bootstrap is used. the p-value of 0,h is given by: 0 , * , , 1 1 1{ } b h b r r b p t t b = = >∑ . (33) even though the mcs methodology was originally designed in hansen et al. (2003) to select the best volatility forecasting models, its applications are not limited to comparisons of models. it can be used as well to compare the dynamics and strength of linkages between the stock markets… dynamic econometric models 3 (2013) 5–31 15 the means of two or more populations. in this paper, we apply the mcs procedure to establish rankings of strength of conditional dependencies between the investigated stock markets, calculated by means of the discussed copulabased measures of dependence. 3. the data the dataset used in the analysis includes closing quotations of the following stock indices: the hungarian bux, the czech px, the polish wig20, the dax, and the s&p 500, from the period may 5, 2004 to july 20, 2012. the period under scrutiny begins after the czech republic, hungary and poland joining the european union. this allows us to avoid a possible structural break in the data caused by that event. the quotations of the px index were obtained from the website of the prague stock exchange, and those of the other indices come from the website stooq.pl. figure 1. time plots of daily quotations of bux/10, px, wig20, dax/5 and s&p500 from may 5, 2004 to july 20, 2012. scaling in the case of bux and dax is performed for easier comparison since the patterns of non-trading days in the national stock markets differ, for the purposes of modeling the dependencies, the dates of observations for all the indices were checked and observations not corresponding to ones in the other index quotation series were removed. the plots of the adjusted series of quotations are presented in figure 1. the modeled time series are percentage logarithmic daily returns calculated by the formula: 0 500 1000 1500 2000 2500 3000 3500 4000 4500 05/04/2004 02/24/2005 12/27/2005 10/19/2006 08/23/2007 06/23/2008 04/20/2009 02/16/2010 12/13/2010 10/07/2011 bux/10 px wig20 dax/5 s&p500 małgorzata doman, ryszard doman dynamic econometric models 13 (2013) 5–31 16 1100 (ln ln )t t tr p p−= − , (34) where tp denotes the closing index value on day t. figure 2. evolution of gdp annual growth rate in the czech republic, hungry, poland, the ue, and the usa, 2004–2012. data source: trading economics (2013) figure 3. market capitalization of the stock exchanges wse, bse, and pse in millions euro, 2004–2012 data source: federation of european securities exchanges (2013). ‐10 ‐8 ‐6 ‐4 ‐2 0 2 4 6 8 10 czech republic hungary poland  eu usa 0 20 000 40 000 60 000 80 000 100 000 120 000 140 000 160 000 2004 2005 2006 2007 2008 2009 2010 2011 wse bse pse the dynamics and strength of linkages between the stock markets… dynamic econometric models 3 (2013) 5–31 17 in figure 2, we show time plots displaying the evolution of gdp annual growth rate in the czech republic, hungry, poland, the ue, and the usa during the period under scrutiny. the market capitalization of the analyzed stock exchanges is presented in figure 3 and shows the leading position of the warsaw stock exchange in the region. from the plots in figures 1–3, it is evident that during the subprime crisis of 2007–2009 all the markets experienced significant decline, and that the same applies to their gdp growth rate, though the polish gdp growth rate remained positive. as in many papers dealing with dynamic copula models, we separate the estimation of models for marginal distributions from the estimation of the model for the dependence structure. the resulting multi-stage maximum likelihood estimation method, known as „inference functions for margins” (joe, xu 1996; joe, 1997), is not fully efficient but there are simulation results that motivate this approach (patton, 2009). to describe marginal distributions, we use arma-garch models. the estimation of the parameters of the marginal distributions (arma-garch) was performed using g@rch6.2 package (laurent, 2009; doornik, 2006). figure 4. volatility estimates for the indices dax, s&p 500, bux, wig20, and px for the period may 5, 2004 – july 20, 2012 in table 1, the parameter estimates of best fitted models are presented. in the case of the wig20, the best fitted model is garch(3,1) with the innovation term following a standardized student t distribution. the result indicates lack of asymmetry in volatility dynamics. for all the other indices, the chosen volatility models are arma-gjr-garch models specified as: 0 10 20 30 40 50 60 70 80 05/05/2004 12/15/2004 07/29/2005 03/08/2006 10/20/2006 06/11/2007 01/25/2008 09/04/2008 04/21/2009 12/03/2009 07/19/2010 02/25/2011 10/10/2011 05/25/2012 dax s&p500 bux wig20 px małgorzata doman, ryszard doman dynamic econometric models 13 (2013) 5–31 18 ,)( 11 t q i iti p i itit yybrar ++−=− ∑∑ = − = − μμ (35) ,ttty εσ= (36) ( )( )2 2 2 2 1 1 0 , q p t i t i i t i t i j t j i j y i y yσ ω α γ β σ− − − − = = = + + < +∑ ∑ (37) where i is the indicator function. table 1. parameter estimates for fitted gjr-garch models parameters bux px wig20 dax s&p500 μ 0.0542 (0.0294) 0.0586 (0.0239) 0.0563 (0.0278) 0.0313 (0.157) 1a 0 0.0257 (0.0249) 0.6577 (0.1084) 2a –0.0550 (0.0226) –0.005 (0.0236) 1b –0.7240 (0.0987) ω 0.0743 (0.0221) 0.0622 (0.0166) 0.0309 (0.0099) 0.0146 (0.0053) 1α 0.0581 (0.0206) 0.0919 (0.0193) 0 –0.0214 (0.0095) –0.0860 (0.0097) 2α 0 0 3α 0.1040 (0.0202) 0.0828 (0.0153) 1γ 0.0859 (0.0255) 0.0898 (0.0332) 0.1784 (0.0376) 0.0952 (0.0317) 2γ 0.1876 (0.0568) 3γ –0.1644 (0.0283) 1β 0.8741 (0.0255) 0.8351 (0.0227) 0.8833 (0.0212) 0.9177 (0.0182) 0.9106 (0.0204) ξln –0.0842 (0.0289) –0.1475 (0.0263) –0.1644 (0.0283) df 10.2987 (2.0213) 6.9791 (1.0306) 8,8613 (1.5857) 8.7099 (1.7851) 7.0120 (1.1411) figure 4 presents the plots of volatility (conditional variance) estimates from fitted models for the period from may 5, 2004 to july 20, 2012. the plots allow to easy identify periods of turmoil in financial markets. in the considered period, usually the px index exhibits the highest volatility level. an explosion of volatility visible in figure 4 is the result of a panic after the dynamics and strength of linkages between the stock markets… dynamic econometric models 3 (2013) 5–31 19 bankruptcy of lehman brothers. the peaks in the volatility of the bux and px indices appearing just after october 22, 2008 are likely to be a reaction to hungary’s request for financial support directed to the international monetary fund. 4. the analysis of dependencies modeling the dependencies between the european and the us market one meets the problem of non-synchronic observations due to different trading hours. there is no a really good solution to this difficulty. the most frequently applied approaches include using weekly data (thus losing much of the available information), interpolating the quotations beyond the trading hours to get simultaneous observations (thus introducing additional noise), or analyzing the observations that are coincident but locally come from different phases of a trading day (thus approving of noise caused by microstructure effects). in our research, we decided to use the close to close returns, but in the cases involving the s&p 500 index we investigate two types of dependencies. in the first case the analysis is performed for the returns corresponding to the same date, and in the second – for the returns of the s&p 500 one-day lagged with respect to the returns on the european indices, denoted as s&p 500(–1). a part of the results concerning the wig20 index was first published in polish (doman, doman, 2013). we quote them here for the reader’s convenience. tables 2–4 show parameter estimates of the msc models fitted to the returns of each of the considered pairs of the indices. a large set of copula specifications was considered during the fitting procedure. taking into account the results by doman and doman (2012), we considered msc models with 3 and 2 regimes, and static copula models. the best models are selected on the basis of information criteria and the results of likelihood ratio tests. apart from the pair (wig20, s&p 500), in all cases we apply 2-regime msc models with the first regime described by the gaussian copula and the second characterized by the clayton or the joe-clayton copula. so, in the first regime there is no dependence in tails, and in the second at least dependence in lower tail is observed. the plots of estimated dynamic dependence measures are presented in figures 5–8. the first impression from the pictures is that the dependencies within the group of the central european markets are stronger and exhibit stronger dynamics, in comparison with their linkages with the considered developed markets. it means that the linkages within the group of the central european markets are more sensitive on new information. małgorzata doman, ryszard doman dynamic econometric models 13 (2013) 5–31 20 table 2. linkages of the bux: parameter estimates for the msc models. standard errors in parentheses bux and px wig20 dax s&p 500 s&p 500(–1) 11p 0.9960 0.9895 0.9994 0.9982 0.9991 22p 0.8957 0.9702 0.9987 0.9964 0.9991 copula in regime 1 gauss ρc gauss ρc gauss ρc gauss ρc gauss ρc ρ 0.6658 (0.0183) 0.6799 (0.0216) 0.5937 (0.0157) 0.4260 (0.0275) 0.3089 (0.0343) spearman’s rho 0.6482 0.6624 0.5756 0.4100 0.2962 copula in regime 2 clayton γc claytonjoe, − γκc clayton γc clayton γc clayton γc κ 1.3925 (0.1281) γ 0.4769 (0.1179) 0.4144 (0.1872) 0.3533 (0.0811) 0.1680 (0.0854) 0.1382 (0.0401) lλ 0.2338 0.1878 0.1406 0.0162 0.0066 uλ 0.3549 spearman’s rho 0.2843 0.4265 0.2230 0.1159 0.0968 figure 5. dependencies between the bux and the other indices. estimates of spearman’s rho 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 05/05/2004 12/15/2004 07/29/2005 03/08/2006 10/20/2006 06/11/2007 01/25/2008 09/04/2008 04/21/2009 12/03/2009 07/19/2010 02/25/2011 10/10/2011 05/25/2012 px wig20 dax s&p500 s&p lag the dynamics and strength of linkages between the stock markets… dynamic econometric models 3 (2013) 5–31 21 figure 6. dependencies between the bux and the other indices. estimates of lower tail dependence coefficient the first observation concerning models describing the linkages of the bux with the other indices (table 2) is that in each case the value of spearman’s rho is higher in the regime with the gaussian copula. figure 5 indicates an increase in the strength of connection between the bux and dax, which can be interpreted as an effect of the eu joining. during the period of the subprime crisis we observe a drop in the strength of dependencies between the returns on the bux and lagged returns on the s&p 500. at the same time, however, there is an increase in the strength of dependencies between the returns with the same date. taking into account different impact of the hungarian and the us market on the global economy, we can formulate a conjecture that during the crisis period the dependencies between the markets were affected by a common factor driving the prices’ dynamics. in figure 6, the plots of the conditional lower tail dependence coefficients are presented. this type of dependence is observed for the pairs (bux, px) and (bux,wig20). it does not occur when we analyze the linkages for the pair (bux, s&p 500(–1)). some very weak dependence is visible for the pairs (bux, dax) and (bux, s&p 500), but it disappears at the beginning of 2006. in the case of the px index, the results are similar to those for the bux in the part dealing with the emerging markets, but different when considering the dax and s&p 500 (table 3). first observation is that the dependence between the px and dax, measured by means of spearman’s rho (figure 7), is very strong during almost all the period under scrutiny (from 0 0,05 0,1 0,15 0,2 0,25 05/05/2004 02/25/2005 12/28/2005 10/20/2006 08/24/2007 06/24/2008 04/21/2009 02/17/2010 12/14/2010 10/10/2011 px wig20 dax s&p500 s&p lag małgorzata doman, ryszard doman dynamic econometric models 13 (2013) 5–31 22 the end of 2005). some decrease in the strength of dependencies is visible from july 2009 to february 2010. the dependence between the returns on the px and s&p 500 is quite significant for both the returns from the same day and in the case of lagged returns on the s&p 500. the pair (px, s&p 500) is the only one for which the value of spearman’s rho in the regime with tail dependence is higher than in that with the gaussian copula. lower tail dependence coefficient for (px, s&p 500) is higher than for (px, dax). the periods of stronger dependencies in lower tail for the pair (px, dax) occur in the beginning of the sample period and in late 2009. table 3. linkages of the px: parameter estimates for the msc models. standard errors in parentheses px and wig20 dax s&p500 s&p 500(–1) 11p 0.9589 0.9960 0.9903 0.9932 22p 0.9702 0.9849 0.9909 0.9916 copula in regime 1 gauss ρc gauss ρc gauss ρc gauss ρc ρ 0.6796 (0.0216) 0.6028 (0.0219) 0.2305 (0.0504) 0.3209 (0.0471) spearman’s rho 0.6621 0.5848 0.2206 0.3078 copula in regime 2 clayton γc claytonγc claytonjoe, −γκc claytonγc κ 1.2935 (0.0733) γ 0.6991 (0.1286) 0.3991 (0.1227) 0.4484 (0.0866) 0.2832 (0.0637) lλ 0.3710 0.1761 0.2132 0.0865 uλ 0.2911 spearman’s rho 0.3779 0.2465 0.4013 0.1848 table 4 and figures 9–10 present the estimation results for dependencies with the wig20. the estimates for (wig20, dax) are very similar to those for (wig20, bux) and (wig20, px). the linkages between the wig20 and dax, however, seem to be stronger than for (px, dax) or (bux, dax). the results concerning dependencies between the returns on the wig20 and s&p 500 (table 4) could seem surprising, but are in agreement with practitioners’ beliefs. the best models describing the dependencies are in this case the static gaussian copula for the returns from the same day, and the static joe-clayton copula in the case of lagged s&p 500 returns. this supports the established opinion about the stability of the us market impact on the warsaw stock exchange. the plots in figure 9 show that the estimates of dyna the dynamics and strength of linkages between the stock markets… dynamic econometric models 3 (2013) 5–31 23 figure 7. dependencies between the px and the other indices. estimates of spearman’s rho figure 8. dependencies between the px and the other indices. estimates of lower tail dependence coefficient mic spearman’s rho for the pairs (wig20, bux) and (wig20, px) are changing between 0.4 and 0.7. in the case of linkages with the dax, in principle only values 0.34 (until july 2006) and 0.63 (from december 2006) are observed. tail dependence for the wig20 and dax has disappeared since december 2006, and tail dependence between the returns on the wig20 and lagged s&p 500 is present during all the sample period, but is very weak. 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 05/05/2004 02/25/2005 12/28/2005 10/20/2006 08/24/2007 06/24/2008 04/21/2009 02/17/2010 12/14/2010 10/10/2011 bux wig20 dax s&p500 s&p500 lag 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 05/05/2004 02/25/2005 12/28/2005 10/20/2006 08/24/2007 06/24/2008 04/21/2009 02/17/2010 12/14/2010 10/10/2011 bux wig20 dax s&p500 s&p500 lag małgorzata doman, ryszard doman dynamic econometric models 13 (2013) 5–31 24 table 4. linkages of the wig20: parameter estimates for the msc models. standard errors in parentheses wig20 and dax s&p 500 s&p 500(–1) 11p 0.9995 22p 0.9962 copula in regime 1 gausscρ gausscρ ρ 0.6529 (0.0127) 0.3936 (0.0177) spearman’s rho 0.6531 0.3783 copula in regime 2 clayton γc claytonjoe, −γκc κ 1.0406 (0.0205) γ 0.6132 (0.0737) 0.1895 (0.0291) lλ 0.3229 0.0258 uλ 0.0534 spearman’s rho 0.3440 0.1570 as it was mentioned earlier, we used the model confidence set methodology, described in section 3, to compare the average strength of dependencies for the considered pairs of indices. table 5 presents the comparison results for the strength of dependencies in the form of a ranking, where a higher score means stronger average dependence. the mcs procedure is applied sequentially – after each run, the series constituting the mcs obtain the subsequent score and are excluded. in the case where the mcs consists of at least two elements, the arithmetic mean of the corresponding scores is assigned to each of them. the procedure is applied to the series of the smoothed values of spearman’s rho and tail dependence coefficients. all the mcss have been estimated (with the significance level 0.1α = ) using the package mulcom of hansen and lunde (2010). the ranking results for spearman’s rho differ from those concerning the lower tail dependence coefficient. generally, however, we can say that in average the linkages between the bux, px and wig20 are stronger than the dependencies between each of them and the dax or s&p 500. the only exception is for the px index where the mean levels of the lower tail dependence coefficient for (px, wig20) and (px, s&p 500) are statistically indistinguishable. the results indicate the importance of the wig20 as far as we measure average dependence using spearman’s rho. the pattern for dethe dynamics and strength of linkages between the stock markets… dynamic econometric models 3 (2013) 5–31 25 pendencies in lower tail is more complicated. in the case of the bux, the highest score is assigned to the pair (bux, px). accordingly, for the px this is so for the pairs (px, wig20) and (px, s&p 500), and for the wig20 – for the pair (wig20, bux). figure 9. dependencies between the wig20 and the other indices. estimates of spearman’s rho figure 10. dependencies between the wig20 and the other indices. estimates of lower tail dependence coefficient 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 05/05/2004 02/25/2005 12/28/2005 10/20/2006 08/24/2007 06/24/2008 04/21/2009 02/17/2010 12/14/2010 10/10/2011 bux px dax s&p500 s&p500 (‐1) 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 05/05/2004 12/15/2004 07/29/2005 03/08/2006 10/20/2006 06/11/2007 01/25/2008 09/04/2008 04/21/2009 12/03/2009 07/19/2010 02/25/2011 10/10/2011 05/25/2012 bux px dax s&p500(‐1) małgorzata doman, ryszard doman dynamic econometric models 13 (2013) 5–31 26 table 5. rankings of average strength of dependencies bux & spearman’s rho lambda_l px & spearman’s rho lambda_l wig20 & spearman’s rho lambda_l px 4 5 bux 4 3 bux 5 3 wig20 5 4 wig20 5 4.5 px 4 5 dax 3 3 dax 3 1.5 dax 3 4 s&p 500 2 2 s&p 500 2 4.5 s&p 500 2 1 s&p 500 (-1) 1 1 s&p 500 (-1) 1 1.5 s&p 500 (-1) 1 2 figure 11. smoothed probabilities of the regime with tail dependence for pairs (bux, px), (bux,wig20) and (px,wig20) usually in literature, the presence of tail dependence is linked with the turmoil periods in markets. the plots presented in figures 11–13 allow to determine the moments of switching on the regime with tail dependence. first observation is that the changes of regimes are very frequent as far as we consider the three pairs of the analyzed emerging markets (figure 9). if tail dependence is present for the pair (bux, wig20) it is usually present for the two other pairs. however, for the pairs (bux, px) and (px, wig20), there occur some additional periods with tail dependence. tail dependence between the considered emerging markets and the german market disappeared in 2006. in the case of the pair (px, dax), it appeared again during a short period from july 2009 to february 2010. for the pair (bux, s&p 500(–1)), tail dependence has occurred since september 2008 (the crisis effect), and for (wig20, s&p500(–1)), it is present during all the sample period. in the 0 0,2 0,4 0,6 0,8 1 1,2 05/05/2004 12/15/2004 07/29/2005 03/08/2006 10/20/2006 06/11/2007 01/25/2008 09/04/2008 04/21/2009 12/03/2009 07/19/2010 02/25/2011 10/10/2011 05/25/2012 bux‐px bux‐wig20 px‐wig20 the dynamics and strength of linkages between the stock markets… dynamic econometric models 3 (2013) 5–31 27 case of (px, s&p 500(–1)), the pattern is more complicated because periods with tail dependence appear and disappear cyclically. tail dependence in the case of (wig20, s&p 500) is absent. for the pair (bux, s&p 500) it occurs in pre-crisis period, and for (px, s&p 500) it is present during most of the time in the recent crisis years. figure 12. smoothed probabilities of the regime with tail dependence for pairs (bux, dax), (px, dax) and (wig20, dax) our findings can be sum up as follows. dependencies between the three considered emerging markets are stronger than the one of each of them with the german or the us markets, regardless of the applied dependence measure. in almost all cases the dynamics of the dependencies is rich, indicating sensitivity on information process. the periods with tails dependence occur very often and they are not necessary connected with crisis periods, though this type of dependence is present during these periods too. the linkages between the considered emerging markets and the developed ones are very stable and mostly do not change during the crises. the extreme example involves the wig20 index for which the dependencies with contemporary and lagged s&p 500 returns are described by means of static copulas. as regards the dynamics of dependencies between the czech, hungarian and polish markets, the crucial relation is that between the bux and wig20. the importance of the influence of major european markets and the us market on the analyzed emerging markets can be perceived as comparable if one takes into account that due to the no-synchronization effect the impact of the us market is divided between two days. 0 0,2 0,4 0,6 0,8 1 1,2 05/05/2004 12/15/2004 07/29/2005 03/08/2006 10/20/2006 06/11/2007 01/25/2008 09/04/2008 04/21/2009 12/03/2009 07/19/2010 02/25/2011 10/10/2011 05/25/2012 bux px wig20 małgorzata doman, ryszard doman dynamic econometric models 13 (2013) 5–31 28 figure 13. smoothed probabilities of the regime with tail dependence for pairs (bux, s&p 500), (px, s&p 500) and (wig20, s&p 500) figure 14. smoothed probabilities of the regime with tail dependence for pairs (bux, s&p 500(–1)), (px, s&p500(–1)) and (wig20, s&p500(–1)) conclusions in the paper we analyzed the dynamics and strength of linkages between the czech, hungarian and polish stock markets. in addition, we examined linkages between each of these markets and developed markets (european and us). the analysis was based on the daily quotations of the main representative stock indices (px, bux, wig20, dax, s&p 500). the period 0 0,2 0,4 0,6 0,8 1 1,2 05/05/2004 12/15/2004 07/29/2005 03/08/2006 10/20/2006 06/11/2007 01/25/2008 09/04/2008 04/21/2009 12/03/2009 07/19/2010 02/25/2011 10/10/2011 05/25/2012 bux px wig20 0 0,2 0,4 0,6 0,8 1 1,2 05/05/2004 12/15/2004 07/29/2005 03/08/2006 10/20/2006 06/11/2007 01/25/2008 09/04/2008 04/21/2009 12/03/2009 07/19/2010 02/25/2011 10/10/2011 05/25/2012 bux px wig20 the dynamics and strength of linkages between the stock markets… dynamic econometric models 3 (2013) 5–31 29 under scrutiny was from may 5, 2004 to july 20, 2012. so, it starts after the eu accession of the corresponding countries and includes recent financial crises. our tool to model the dynamics of dependencies were markovswitching copula models. due to this choice, we were able to use two kinds of measure of the strength of the linkages: dynamic spearman’s rho and tail dependence coefficients. our results show that the dependencies between the three considered emerging markets are stronger than those between each of them and the german or the us markets, regardless of the applied measure. the dynamics of dependence in almost all cases is rich, clearly indicating sensitivity on information process. the tail dependence appears and disappears cyclically and is not necessary connected with the crisis periods, though this type of dependence is present during these periods too. the linkages of the examined emerging markets with the considered developed markets are very stable, not sensitive on present information, and mostly do not change during the crises. the crucial relation for the considered region is the dependence between the hungarian and polish markets. the importance of the impact of the developed european market and the us market on the analyzed emerging markets can be viewed as comparable. references ang, a., bekaert, g. (2002), international asset allocation with regime shifts, review of financial studies, 15, 1137–1187. ang, a., chen, j. (2002), asymmetric correlations of equity portfolios, journal of financial economics 63, 443–494, doi: http://dx.doi.org/10.1016/s0304-405x(02)00068-5. bauwens, e., laurent s., rombouts, j.v.k. (2006), multivariate garch models: a survey, journal of applied econometrics, 21, 79–109, doi: http://dx.doi.org/10.1002/jae.842. caporale, g.m., spagnolo, n. (2011), stock market integration between three ceecs, russia, and the uk, review of international economics, 19, 158–169, doi: http://dx.doi.org/10.1111/j.1467-9396.2010.00938.x. doman, m., doman, r. (2012), dependencies between stock markets during the period including the late-2000s financial crisis, procedia economics and finance, 1, 108–117, doi: http://dx.doi.org/10.1016/s2212-5671(12)00014-7. doman, m., doman, r. (2013), dynamika powiązań polskiego rynku kapitałowego z rynkami czech i węgier oraz głównymi rynkami światowymi (the dynamics of linkages between the polish stock market and the markets of czech republic, hungary and major world markets), studia oeconomica posnaniensia, 9. doornik, j.a. (2006), an object-oriented matrix programming language – ox™, timberlake consultants ltd, london. embrechts, p., mcneil, a., straumann, d. (2002), correlation and dependence in risk management: properties and pitfalls, in dempster m. (ed.) risk management: value at risk and beyond, 176–223, cambridge, cambridge university press. małgorzata doman, ryszard doman dynamic econometric models 13 (2013) 5–31 30 federation of european securities exchanges (2013), statistics & market research, brussels (available at http://fese.eu, 16.06.2013). forbes, k., rigobon, r. (2002), no contagion, only interdependence: measuring stock market comovements, journal of finance, 57, 2223–2261, doi: http://dx.doi.org/10.1111/0022-1082.00494. garcia r., tsafack g. (2011), dependence structure and extreme comovements in international equity and bond markets, journal of banking and finance, 35, 1954–1970, doi: http://dx.doi.org/10.1016/j.jbankfin.2011.01.003. gilmore, c.g., lucey, b. m., mcmanus, g.m. (2008), the dynamics of central european equity market comovements, the quarterly review of economics and finance, 48, 605–622, doi: http://dx.doi.org/10.1016/j.qref.2006.06.005. hamilton, j.d. (1994), time series analysis, princeton university press, princeton. hansen, p.r., lunde, a., nason, j.m. (2003), choosing the best volatility models: the model confidence set approach, oxford bulletin of economics and statistics, 65, 839–861. hansen, p.r., lunde, a. (2010), mulcom 2.00, an ox™ software package for multiple comparisons, http://mit.econ.au.dk/vip_htm/alunde/mulcom/mulcom.htm. hansen, p.r., lunde, a., nason, j.m. (2011), the model confidence set, econometrica, 79, 453–497. joe, h., xu, j.j. (1996), the estimation method of inference functions for margins for multivariate models, technical report no. 166, department of statistics, university of british columbia. joe, h. (1997), multivariate models and dependence concepts, chapman and hall: london. kiss, g.d. (2011), the impact of financial interdependence on the czech, hungarian and polish interbank, stock and currency market, international journal of management cases 13, 555–565. lambert, p., laurent, s. (2001), modelling financial time series using garch-type models with a skewed student distribution for the innovations, discussion paper 0125, institut de statistique, université catholique de louvain. laurent, s. (2009), estimating and forecasting arch models using g@rch™ 6, timberlake consultants ltd: london. nelsen, r.b. (2006), an introduction to copulas, new york, springer. okimoto, t. (2008), new evidence of asymmetric dependence structures in international equity markets, journal of financial and quantitative analysis, 43, 787–815, doi: http://dx.doi.org/10.1017/s0022109000004294. patton, a.j. (2004), on the out-of-sample importance of skewness and asymmetric dependence for asset allocation, journal of financial econometrics, 2, 130–168, doi: http://dx.doi.org/10.1093/jjfinec/nbh006. patton, a.j. (2006), modelling asymmetric exchange rate dependence, international economic review 47, 527–556, doi: http://dx.doi.org/10.1111/j.1468-2354.2006.00387.x. patton, a.j. (2009), copula-based models for financial time series, in t. g. andersen, r. a. davies, j.-p. kreiss, and t. mikosch (ed.) handbook of financial time series, 767–785, berlin: springer. scheicher, m. (2001), the comovements of stock markets in hungary, poland and the czech republic, international journal of finance & economics, 6, 27–39, doi: http://dx.doi.org/10.1002/ijfe.141. sklar, a. (1959), fonctions de répartition à n dimensions et leurs marges, publications de l’institut statistique de l’université de paris, 8, 229–231. the dynamics and strength of linkages between the stock markets… dynamic econometric models 3 (2013) 5–31 31 trading economic (2013), indicators, new york. (available at http://tradingeconomics.com, 16.06.2013). ülkü, n., demirci, e. (2012), joint dynamics of foreign exchange and stock markets in emerging europe, journal of international financial markets, institutions & money, 22, 55–86. dynamika i siła zależności pomiędzy rynkami giełdowymi czech, węgier i polski po ich wejściu do unii europejskiej z a r y s t r e ś c i. w pracy analizowana jest dynamika i siła powiązań pomiędzy rynkami giełdowymi czech, węgier i polski. badanie obejmuje okres po wejściu wymienionych krajów do unii europejskiej. ponadto oceniamy powiązania pomiędzy każdym z wymienionych rynków a wybranymi rynkami rozwiniętymi (europejskim i amerykańskim). analiza oparta jest na dziennych notowaniach głównych indeksów giełdowych (px, bux, wig20, dax, s&p 500) i obejmuje okres od 5 maja 2004 r. do 20 lipca 2012 r. dynamika zależności jest opisywana za pomocą modeli kopuli z przełączaniem typu markowa, a stosowanymi miarami siły powiązań są dynamiczne współczynniki rho spearmana i dynamiczne współczynniki zależności w ogonach rozkładów. uzyskane wyniki pokazują, że zależności pomiędzy rozważanymi rynkami wschodzącymi są bardzo wrażliwe na sytuację rynkową, natomiast powiązania z rynkami rozwiniętymi są stabilne. s ł o w a k l u c z o w e: środkowoeuropejski rynek giełdowy, zależność warunkowa, modele kopuli z przełączaniem typu markowa, współczynnik rho spearmana, zależność w ogonie rozkładu, zbiór ufności modeli, indeks giełdowy. microsoft word 00_tresc.docx dynamic econometric models vol. 10 – nicolaus copernicus university – toruń – 2010 mariola piłatowska nicolaus copernicus university in toruń choosing a model and strategy of model selection by accumulated prediction error a b s t r a c t. the purpose of the paper is to present and apply the accumulative one-step-ahead prediction error (ape) not only as a method (strategy) of model selection, but also as a tool of model selection strategy (meta-selection). the ape method is compared with the information approach to model selection (aic and bic information criteria), supported by empirical examples. obtained results indicated that the ape method may be of considerable practical importance. k e y w o r d s: model selection, meta-selection, information criteria, accumulative prediction error. 1. introduction in the literature different methods (strategies) of model selection are available, among others: strategies based on sequences of tests (forward/backward selection), strategies related to information criteria of akaike type, strategies based on predictive criteria (out-of-sample validation), which can be treated as mainstream directions in model selection. for the reason that the true generating model is unknown in practice, the focus in model selection strategies is being moved from the issue of selection the only one, true model to the issue of selection the best model among the set of candidate models fitted to the data or selection of several plausible models, where the best model may have relatively weak support against others models (burnham, anderson, 2002). selection of the best model or multi-model inference assumes that the set of models has been well founded, because even the relatively best model in a set might be poor in an absolute sense. associated with each strategy is an algorithm to be specified which within given data enables to choose the best (in some sense) model among the candidate models (generally they may be nested or non-nested models, different models based on different scientific theories or modeling assumptions). howevmariola piłatowska 108 er, the problem of model selection implies not only the choice of model in the framework of a given strategy but also the choice of model selection strategy. the focus in the literature is mainly on the choice of model or the comparison of different model selection strategies with regard to the choice of the best model, without touching the issue of model selection strategy. the choice of model selection strategy and its suitability and properties may depend on the goals of an analysis (estimation, prediction), sample size (some strategies perform in different way in small and large samples), characteristics of the data generating model (dgm)1. in practice, there is a need to propose a data-driven framework which allows to help choosing a model selection strategy without making any reference to the actual dgm. this identification is called the meta-selection of a model (de luna, skouras, 2003). the metaselection framework obeys the ‘prequential’ principle (dawid, 1984)2 which abandons the goal of selecting the true model in favor of seeking as small a predictive error as possible by comparing obtained predictions from each strategy and the actual values observed for the data independent on which model was used to forecast (clarke, 2001). the essential point for this approach is that the adequacy of a model must be reflected in accurate prediction regardless of the goals of an analysis, i.e. if the goal of analysis is model estimation (model identification or hypothesis testing), then the best model should give the best predictions. the purpose of the paper is to present and apply the accumulative one-stepahead prediction error (ape) not only as a method (strategy) of model selection, but also as a tool of model selection strategy (meta-selection). 2. accumulative one-step-ahead prediction error the choice of model according to the accumulative prediction error (ape) consists in evaluating how well the models in the set are able to predict the next unseen data point 1+nx . in other words, according to the ape method the most useful model is the model with the smallest out-of-sample one-step-ahead prediction error. the prediction error cannot be calculated because 1+nx has not been observed. what can be calculated, however, are the prediction error for 1+ix based on the previous ix )0( ni << by the sum of the previous one-stepahead prediction errors for data that are available. let us consider a time series of n observations, ),...,,( 21 n n xxxx = . 1 some strategies are optimal depending on whether the data generating model is one of the candidate models or not (shao, 1997). 2 ‘prequential’ is from predictive sequential (dawid, 1984). choosing a model and model selection strategy by… 109 the ape method proceeds by calculating sequential one-step-ahead forecasts based on a gradually increasing part of the data. for model jm the ape is calculated as follows (wagenmaker, grunwald, steyvers, 2006): 1. determine the smallest number s of observations that makes the model identifiable. set 1+= si , so that si =−1 . 2. based on the first 1−i observations, calculate a prediction ip̂ for the next observation i . 3. calculate the prediction error for observation i , e.g. squared difference between the predicted value ip̂ and the observed value ix . 4. increase i by 1 and repeat steps 2 and 3 until ni = . 5. sum all of the one-step-ahead prediction errors as calculated in step 3. the result is the ape. for model jm the accumulative prediction terror is given by: 1 1 ˆape( ) [ , ( )], n i j i ii s m d x p x − = + = ∑ where d indicates the specific loss function that quantifies the discrepancy between observed and predicted values. applying the ape method the form of prediction should be considered: whether to predict using a single value (skouras, dawid, 1998) or a probability distribution (aitchison, dunsmore, 1975). in the first case, the predictions ip̂ are predictions for the mean value of ith outcome ix . in the latter case, ip̂ is a distribution on the set of possible outcomes ix . the choice of the loss function should be considered in order to quantify the discrepancy between predicted values and observed values. this can be measured in a variety of different ways. for a single-value predictions, one typically uses the squared error 2)ˆ( ii px − . another choice would be to compute the absolute value loss ii px ˆ− or more generally an α-loss function α ii px ˆ− , where ]2,1[∈α (rissanen, 2003). for probabilistic predictions, one typically uses the logarithmic loss function )(ˆln ii xp− , thus the loss depends on the probability mass or density that ip̂ assigns to the actually observed outcome .ix the larger the probability, the smaller the loss 3. 3 taking the logarithmic loss function makes the ape method compatible with maximum likelihood, bayesian inference and minimum description length (mdl) (amongst others: wagenmaker, grünwald, steyvers, 2006). mariola piłatowska 110 the ape method can be also applied to select the model selection strategy (de luna, skouras, 2003). let qsss ,...,, 21 qk ,...,2,1= be q potential model selection strategies applicable to a given set of model )( ppp θ , mp ,...,2,1= which approximate the data generating model. the parameters pθ assigned to each model have to be estimated. if each strategy leads to an identical choice of model p , there is no real reason for selecting a given strategy. in the case of disagreement, however, a strategy ks , qk ,...,2,1= , is selected for which the accumulated prediction error 1ˆape( ) ( , ( ), n i k i ki m s l x x s− = = ∑ reaches the minimum, where )(ˆ 1 k i sx − is the prediction )(ˆ 1 pxi− resulting from the choice of model p made by the ks strategy based on the sub-sample 121 ,...,, −ixxx . hence the ape( )ks measures the predictive performance when strategy ks was used to form predictions sequentially, by updating not only the estimated parameters in each step but the choice of model as well (the meta-selection method computes ape for model selection method instead of models). the metaselection should not just focus on the minimization of ape( )ks , but also on its evolution for increasing sample sizes. 3. empirical example to present the predictive performance of accumulated one-step-ahead prediction error (ape) in model selection and the choice of model selection strategy the data from maddison base4 have been taken. it includes annual time series of gdp for 36 countries. in the study, as an example, the gdp for france (1947-2003) and poland (1952-2003) have been used. data are expressed in millions of us dollars in constant prices from 1990 having taken into account purchasing power parity. the essential point in model selection is the identification of initial set of candidate models. in this study the set of models consist of two models: arima(1,1,0) and linear trend with autoregression of second order (t+ar(2)). this choice of models is justified by the traditional approach to the analysis of gdp fluctuations. during last thirty years this analysis focused on either the verification of unit root hypothesis (what means that gdp is nonstationary in variance or has stochastic trend and the arima model is more appropriate) or testing hypothesis of stationary deviations around deterministic trend (what 4 in the paper the updated maddison base is used which is available on website www.ggdc.net, see also: maddison (2001). choosing a model and model selection strategy by… 111 means that gdp is nonstationary in mean and model with deterministic trend is more appropriate). in spite of huge literature devoted to the distinguishing of these alternative hypothesizes, this dispute has not be settled upon yet5. model arima(1,1,0) was selected from different specification of arima(p, d, q) model, for p, q = 0, 1, 2, d = 0, 1, by the means of aic differences6, i.e. minaic aici iδ = − , where aici denotes the aic value for i-th model, minaic – aic value for the best model. models were estimated on the same sample length, i.e. 1947-2000 (gdp in france) and 1952-200 (gdp in poland). the larger iδ is, the less plausible the fitted model is the good model in the k-l information sense7, given the data. in practice, the models with 4<δ i are accepted (burnham, anderson, 2002). having iδ the akaike weights (evidence ratios) can be obtained which are useful in calculating the relative evidence for the best model (with the biggest weight) versus the rest of r-models in the set. the akaike weights are given by (burnham, anderson, 2002; piłatowska, 2009, 2010): ,)5,0exp(/)5,0exp( 1∑ = δ−δ−= r r rii w ∑ = = r i i w 1 1 . for arima(1,1,0) the difference iδ was equal to zero, i.e. this model was the best, and for the rest of models 3<δ i , so, they were plausible in the k-l information sense. however, the support for the arima(1,1,0) was substantial (i.e. it had the dominating weight equal to 0.55). 5 to papers concerning the choice of stochastic trend (nonstationarity in variance) versus deterministic trend (nonstationarity in mean) for gdp series belong among others: nelson, plosser, 1982; stock, watson, 1986; quah, 1987; perron, phillips,1987; christiano, eichenbaum,1990; rudebusch,1993; diebold, senhadji, 1996; murray, nelson, 1998. it is pointed out ((haubrich, lo, 2001) that the reason of no settlement in this dispute is the false assumption that one of the above hypothesizes is true. as a result, only the possibility of persistent fluctuations (shocks to gdp are persistent and there is no trend reversion at all) or transitory fluctuations (shocks are transitory and trend reversion occurs) is taken into account, but the indirect fluctuations, i.e. long memory dependence, are omitted, and the latter can be described by different model than previously, i.e. arfima model. 6 in the paper the modified aic (second-order variant of aic) was applied, i.e. 2 ( 1) aic aic , 1c k k n k + = + − − where aic 2 ln 2 ,l k= − + k denotes the number of estimated parameters, n – sample size. standard aic may perform poorly (may indicate not parsimonious model), if there are too many parameter in relation of the size of the sample. the use of aicc is advocated when the ratio kn / is small, say < 40, (sugiura, 1978). for the purposes of presentation further only ‘aic’ notation is used. 7 the kullback-leibler (k-l) distance or information is the measure of discrepancy between true (but unknown) model and fitted model. akaike (1973) showed that the choice of model with minimum relative expected information loss (i.e. model with minimum k-l information) is asymptotically equivalent to the choice of model with minimum aic. mariola piłatowska 112 in similar way the specification of an alternative model to arima was chosen, i.e. model of linear trend with autoregression of second order t+ar(2), where maximum lag length was equal to 3. to make a choice between arima(1,1,0) model and t+ar(2) model three model selection strategies were used: information criteria: aic and bic, and also accumulated one-step-ahead prediction error (ape). in the latter case the squared error (ape_se) and absolute error (ape_ae) were taken as a loss function8. the estimation9 of models has been starting with minimum sample size equal to 11 observations, then the sample size has been increased by one until n (until the year 2000) and the estimation was repeated. at each stage criteria: aic and bic, the forecasts from both types of models and accumulated one-step-ahead prediction error (ape_se and ape_ae) were calculated. results in form of differences among aic, bic and ape for both types of models depending on sample size are presented in figures 1 (gdp in france) and 2 (gdp in poland). figure 1 (panel a and b) shows that as the sample size increases the criteria aic and bic give a general support for the t+ar(2) model, because the difference of criteria: aic(arima)-aic(t+ar(2)) and bic(arima)bic(t+ar(2)) is positive (what denotes smaller value of aic and bic for model t+ar(2)); only for a few periods: 18th (a year 1975), 28th and 29th (a year 1985 and 1986) the difference of criteria is negative, what gives a preference for the arima(1,1,0) model in these periods. however, observing the evolution of difference in ape (ape_se and ape_ae) for both types of models no support for the t+ar(2) model as in the case of aic and bic is obtained – see panel c and d. almost in the whole forecast period the difference in ape_se for both models10 is negative what leads to a general preference for the arima(1,1,0) model when the gdp for france is to be forecast – see panel c (with exception of first 3 observations referring to 1958-1960 period, 12th and 13th observations referring to 1969-1970). different performance shows the difference in ape_ae for both models (figure 1, panel d), i.e. it favors the arima(1,1,0) model from 4th observation up until the data set has increased to n = 35 (what refers to 1961-1993 period), after which it starts to prefer the t+ar(2) model11. this means that the 8 accumulated prediction error (ape) was calculated using gretl script written by author for that purpose. 9 model arima(1,1,0) has been estimated by maximum likelihood method, and model t+ar(2) – least squares method. 10 the notion ape_se(arima(1,1,0)-ape_se(t+ar(2)) stands for the difference in ape_se calculated for both models – see figure 1. 11 the negative difference in ape_ae denotes better predictive performance (smaller onestep-ahead prediction errors) of the arima(1,1,0) model than the t+ar(2) model, and the positive difference in ape_ae – on the contrary. choosing a model and model selection strategy by… 113 choice of model will depend on the loss function taken to calculate the accumulated prediction error. figure 1. difference between choice criteria for the arima(1,1,0) model and the t+ar(2) model using to obtain forecasts of gdp in france. panel a – aic, panel b – bic, panel c – ape_se, panel d – ape_ae figure 2. difference between choice criteria for the arima(1,1,0) model and the t+ar(2) model using to obtain forecasts of gdp in poland. panel a –aic, panel b – bic, panel c – ape_se, panel d – ape_ae ‐10 0 10 1 11 21 31 41 a ic (a r im a (1 ,1 ,0 )) ‐ a ic (t +a r (2 )) n a)                         aic ‐10 0 10 1 11 21 31 41 b ic (a r im a (1 ,1 ,0 )) ‐ b ic (t +a r (2 )) n b)                         bic ‐1,e+09 ‐5,e+08 0,e+00 5,e+08 1 11 21 31 41 a p e_ se (a r im a (1 ,1 ,0 )) ‐ a p e_ se (t +a r (2 )) n c)                         ape_se ‐20000 0 20000 1 11 21 31 41 a p e_ a e( a r im a (1 ,1 ,0 )) ‐ a p e_ a e( t+ a r (2 )) n d)                       ape_ae 0 5 10 15 1 11 21 31 41a ic (a r im a (1 ,1 ,0 )) ‐ a ic (t +a r (2 )) n a)                         aic ‐5 0 5 10 15 20 1 11 21 31 41bi c (a r im a (1 ,1 ,0 )) ‐ b ic (t +a r (2 )) n b)                             bic ‐1,e+09 ‐5,e+08 0,e+00 5,e+08 1 11 21 31 41 a p e_ se (a r im a (1 ,1 ,0 )) ‐a p e_ se (t +a r (2 )) n c)                         ape_se ‐80000 ‐60000 ‐40000 ‐20000 0 20000 1 11 21 31 41 a p e_ a e( a r im a (1 ,1 ,0 )) ‐ a p e_ a e( t+ a r (2 )) n d)                       ape_ae mariola piłatowska 114 when forecasting the gdp in poland – see figure 2, panel a and b – the positive difference of aic and bic criteria for alternative models indicates that the t+ar(2) model is to be preferred over the arima(1,1,0) model. however, in the case of bic the support for the t+ar(2) model decreases as the sample size increases what is seen in decreasing difference in bic for both models. the opposite pattern shows the difference in ape for both models (ape_se, ape_ae – see panel c and d), i.e. it indicates the substantial preference for the arima(1,1,0) model (negative difference in ape_se and also ape_ae for both models) and better predictive performance (smaller one-step-ahead prediction errors) almost in entire data set except the 2nd and 9th observations (1960 and 1968 periods). an alternative method in assessing the performance for model selection methods is to quantify their predictive performance through a model metaselection procedure. the aim of this procedure is to evaluate predictive value not of the models (e.g. arima, arma), but the model selection methods (aic, bic, ape). just as in the calculation of ape earlier, the meta-selection procedure requires to fit the arima(1,1,0) and t+ar(2) models (in above case) for each of an increasing (by one) number of observations. the predictive value of, say aic, is then quantified by the accumulative prediction error for the models chosen by aic. for instance, suppose that for a particular time series, aic prefers the arima model up until the data set has increased to n = 20, after which aic starts to prefer the t+ar(q) model. then the accumulative prediction error for the aic model selection procedure is a sum of the prediction errors made by the arima and t+ar(q) models (for the first and second half of the time series respectively). having calculated the difference in ape for different model selection procedures (strategies), the relative value of model selection tools as e.g. aic is obtained. figure 3 depicts the differences in accumulated prediction errors (ape) for various model selection procedures, i.e. aic, bic, ape_se, ape_ae. for particular time series (gdp in france) panel a in figure 3 demonstrates that the use of aic for model selection results in smaller one-step-ahead prediction error than the use of bic (because the difference in ape_se for aic and bic model selection methods (ape_se(aic)-ape_se(bic)) is negative)12. note that horizontal stretches in figure 3 indicate that the difference in accumulated prediction errors between two model selection strategies does not change (e.g. aic and bic, panel a). this occurs when two model selection strategies prefer the same model. the results are about the same when the absolute error (ae) was used as a loss function (panel d). 12 the abbreviation, e.g. ape_se(aic) stands for the accumulated prediction error (with squared error, se, as a loss function) calculated when the aic procedure was used to select a model from two ones: arima or t+ar(2) in the example at hand. choosing a model and model selection strategy by… 115 figure 3. model meta-selection as a function of the number of observations. each panels shows the difference in ape for pairs of various model selection methods: aic, bic, ape_se and ape_ae for gdp in france comparing the performance for pairs of model selection strategies, i.e. aic and ape_se, bic and ape_se (panel b and c, figure 3) evidently smaller prediction error are obtained when the ape_se strategy was used to select a model than aic and bic strategies13. similar results are observed when the absolute error was taken as a loss function (panel e and f) except first ten periods when the difference in ape_ae is constant what denotes that both strategies (aic vs. ape_ae and bic vs. ape_ae) perform about the same. generally, the use of ape_se (or ape_ae) strategy leads to smaller accumulated prediction error than aic or bic strategy. 13 the differences ape_se(aic)-ape_se(ape_se) are positive in entire data set what leads to a preference of ape_se strategy. ‐2,0e+08 ‐1,0e+08 0,0e+00 1,0e+08 1 11 21 31 41 a p e_ se (a ic )‐ a p e_ se (b ic ) n a)                aic vs. bic 0,0e+00 5,0e+08 1,0e+09 1,5e+09 1 11 21 31 41 a p e_ se (a ic )‐ a p e_ se (a p e_ se n b)             aic vs. ape_se 0,0e+00 5,0e+08 1,0e+09 1,5e+09 1 11 21 31 41 a p e_ se (b ic )‐ a p e_ se (a p e_ se ) n c)                bic vs. ape_se ‐15000 ‐10000 ‐5000 0 5000 1 11 21 31 41 a p e_ a e( a ic )‐ a p e_ a e( b ic n d)                 aic vs. bic 0 20000 40000 60000 1 11 21 31 41 a p e_ a e( a ic )‐ a p e_ a e( a p e_ a e) n e)               aic vs. ape_ae 0 20000 40000 60000 1 11 21 31 41 a p e_ a e( b ic )‐ a p e_ a e( a p e_ a e) n f)              bic vs. ape_ae mariola piłatowska 116 figure 4. model meta-selection for various model selection methods: aic, bic, ape_se and ape_ae for gdp in poland for another series, gdp in poland, the performance of aic and bic strategies is the same for the first 30 periods of data set (referring to 1960-1990 period) because the difference ape_se(aic)-ape_se(bic) is equal to zero (figure 4) – but for the rest of data set the use of bic strategy for model selection results in relatively smaller one-step-ahead prediction errors than in the case of aic strategy (panel a, figure 4), because the difference in ape_se for aic and bic strategies is positive. however, when the absolute error (ae) is used as a loss function, the results are opposite, i.e. the strategy aic is to be preferred (the difference in ape_ae for aic and bic strategies is negative, see panel d). this confirms earlier conclusion that the choice of model as well the choice of model selection strategy depends on the form of loss function. comparing the performance for pairs of model selection strategies, i.e. aic and ape_se, bic and ape_se (panel b and c, figure 4) results that the ape_se strategy performs better in model selection (i.e. gives smaller accumulated prediction errors) than aic or bic strategy almost in the entire data set 0,0e+00 2,0e+07 4,0e+07 6,0e+07 8,0e+07 1 11 21 31 41 a p e_ se (a ic )‐ a p e_ se (b ic n a)                aic vs. bic ‐1,0e+08 0,0e+00 1,0e+08 2,0e+08 3,0e+08 1 11 21 31 41 a p e_ se (a ic )‐ a p e_ se (a p e_ se n b)             aic vs. ape_se ‐1,0e+08 0,0e+00 1,0e+08 2,0e+08 1 11 21 31 41a p e_ se (b ic )‐ a p e_ se (a p e_ se ) n c)                bic vs. ape_se ‐4000 ‐2000 0 2000 1 11 21 31 41 a p e_ a e( a ic )‐ a p e_ a e( b ic ) n d)               aic vs. bic ‐10000 ‐5000 0 5000 10000 15000 1 11 21 31 41a p e_ a e( a ic )‐ a p e_ a e( a p e_ a e) n e)              aic vs. ape_ae ‐10000 ‐5000 0 5000 10000 15000 1 11 21 31 41a p e_ a e( b ic )‐ a p e_ a e( a p e_ a e) n f)              bic vs. ape_ae choosing a model and model selection strategy by… 117 except first 15 periods (1959-1965 period) when the difference in ape_se for various pairs of strategies (aic vs. ape_se and bic vs. ape_se) are negative, and then the aic and bic strategies respectively are preferred. about the same results are obtained when the performance of aic vs. ape_ae and bic vs. ape_ae is compared (panel e and f, figure 4) except the end of data set when the relative decrease for support of ape_ae strategy is noticed (the difference in ape_ae for various pairs of strategies is positive, but decreasing). table 1. one-step-ahead forecasts of gdp in france made by arima(1,1,0) model and t+ar(2) model with prediction errors forecast period realization model: arima(1,1,0) model: t+ar(2) forecast δt δ*t forecast δt δ*t 2001 1289387 1297071 -7684.3 -0.60% 1292864 -3477.0 -0.27% 2002 1305136 1312186 -7050.8 -0.54% 1309083 -3947.3 -0.30% 2003 1315601 1323622 -8021.1 -0.61% 1321281 -5680.3 -0.43% table 2. one-step-ahead forecasts of gdp in poland made by arima(1,1,0) model and t+ar(2) model with prediction errors forecast period realization model: arima(1,1,0) model: t+ar(2) forecast δt δ*t forecast δt δ*t 2001 281508 286913 -5406 -1.92% 286307 -4798.8 -1.70% 2002 285365 284901 464 0.16% 283789 1575.8 0.55% 2003 296237 289382 6856 2.31% 288394 7843.2 2.65% to check the choice of model (arima or t+ar(2)) made by the accumulated prediction error (ape_se and ape_ae) one-step-ahead forecasts of gdp in france and poland were calculated in out-of-sample (i.e. 2001-2003 period). these forecasts with prediction errors (absolute δt and relative δ*t) are showed in table 1 and 2. it is seen in table 1 that one-step-ahead prediction errors are smaller when forecasts of gdp in france are made from t+ar(2) model what confirms the choice of model by the ape_se method (see figure 1, panel d). however, the prediction errors from the arima(1,1,0) model are only slightly higher what would suggest the predictive value also for that model. this means that although the t+ar(2) model is preferred, the arima model may be also useful in forecasting. forecasting the gdp in poland the smaller one-step-ahead prediction errors are obtained when forecasts are made from arima(1,1,0) model which was indicated by the ape method (see figure 2, panel c and d). mariola piłatowska 118 4. conclusions the presented empirical example indicates the usefulness of one-step-ahead accumulated prediction error (ape) as a method of model selection. the ape method is conceptually straightforward, as it accumulates ‘honest’ one-stepahead prediction errors, i.e. its predictions always concern unseen data. additionally, observing the evolution of ape as the number of observations is increased, suggests that the choice of best model should be referred to the number of observations, that is, the best model in given sample size may be replace with another model which has better prediction value. the ape method can be applied to nested and non-nested models alike and it is sensitive to the functional form of the model parameters (myung, pitt, 1997), and not just to their number as in aic and bic method. also, the ape is a data-driven method that does not rely on the accuracy of asymptotic approximations. in particular, the use of ape does not require to include the true (data generating process) model into the set of candidate models. unquestionable advantage of ape is that can be used not only for the selection of models, but also for the selection of model selection methods, and thus, the comparison of various model selection methods can be carried out. hence the ape method enhances the issue of model selection and therefore may be of considerable practical importance. references aitchison, j., dunsmore, i. r. (1975), statistical prediction analysis, cambridge university press, cambridge. akaike, h. (1973), information theory and an extension of the maximum likelihood principle, [in:] petrov b. n., csaki f., second international symposium on information theory, kiado academy, budapest. burnham, k. p., anderson, d. r. (2002), model selection and multimodel inference, springer, christiano, l. j., eichenbaum, m. (1990), unit roots in real gnp: do we know and do we care?, carnegie-rochester conference series on public policy, no. 32, 7–61. clarke, b. (2001), combining model selection procedures for online prediction, sahkhya: the indian journal of statistics, 63, series a, 229–249. dawid, a. p. (1984), statistical theory: the prequential approach, journal of royal statistical society series b, 147, 278–292. de luna, x., skouras, k. (2003), choosing a model selection strategy, scandinavian journal of statistics, 30, 113–128. diebold, f. x., senhadji, a. (1996), deterministic vs. stochastic trend in u.s. gnp. yet again, nber working papers, nr 5481. haubrich, j. g., lo, a. w. (2001), the source and nature of long-term memory in aggregate output, federal reserve bank of cleveland „economic review”, qii, 15–30. maddison, a. (2001), the world economy – a millennial perspective, oecd development centre, paris. murray, c., nelson, c. (1998), the uncertain trend in u.s. gnp, discussion papers in economics at the university of washington, nr 0074. myung, i. j., pitt, m. a. (1997), applying occam’s razor in modeling cognition: a bayesian approach, psychonomic bulletin and review, 4, 79–95. choosing a model and model selection strategy by… 119 nelson, c. r. , plosser, c. i. (1982), trends and random walks in macroeconomic time series: some evidence and implications, journal of monetary economics, 10(2), 139–162. perron, p., phillips, p. c. b. (1987), does gnp have a unit root?, economics letters, 23, 129– 145. piłatowska, m. (2009), prognozy kombinowane z wykorzystaniem wag akaike’a (combined forecasts using akaike weights), acta universitatis nicolai copernici, ekonomia, xxxix, 51–62. piłatowska, m. (2010), kryteria informacyjne w wyborze modelu ekonometrycznego (information criteria in model selection), studia i prace uniwersytetu ekonomicznego w krakowie, 25–37. quah, d. (1987), what do we learn from unit roots in macroeconomic series?, nber working papers, nr 2450. rissanen, j. (2003), complexity of simple nonlogarithmic loss function, ieee transactions on information theory, 49, 476–484. rudebusch, g. d. (1993), the uncertain unit root in real gnp, american economic review, 83(1), 264–272. shao, j. (1997), an asymptotic theory for linear model selection, statistica sinica, 7, 221–264. skouras, k., dawid, a. p. (1998), on efficient point prediction systems, journal of royal statistical society b, 60, 765–780. sugiura, n. (1978), further analysis of the data by akaike’s information criterion and the finite corrections, communications in statistics, theory and methods, a7, 13–26. stock, j., watson, m. (1986), does gnp have a unit root?, economics letters, 22(2/3), 147–151. wagenmaker, e-j., grünwald, p., steyvers, m. (2006), accumulative prediction error and the selection of time series models, journal of mathematical psychology, 50, 149–166. wybór modelu i strategii selekcji modelu za pomocą skumulowanego błędu predykcji z a r y s t r e ś c i. celem artykułu jest prezentacja i wykorzystanie skumulowanego błędu prognoz na jeden okres naprzód (ape) nie tylko jako metody (strategii) wyboru modelu, ale również jako narzędzie do wyboru samej strategii (meta-wybór). na przykładach empirycznych metoda ape jest porównywana z metodami wykorzystującymi kryteria informacyjne (aic i bic). otrzymane wyniki wskazują na dużą praktyczną przydatność metody ape. s ł o w a k l u c z o w e: wybór modelu, meta-wybór, kryteria informacyjne, skumulowany błąd prognoz microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 mariola piłatowska nicolaus copernicus university in toruń combined forecasts using the akaike weights a b s t r a c t. the focus in the paper is on the information criteria approach and especially the akaike information criterion which is used to obtain the akaike weights. this approach enables to receive not one best model, but several plausible models for which the ranking can be built using the akaike weights. this set of candidate models is the basis of calculating individual forecasts, and then for combining forecasts using the akaike weights. the procedure of obtaining the combined forecasts using the aic weights is proposed. the performance of combining forecasts with the aic weights and equal weights with regard to individual forecasts obtained from models selected by the aic criterion and the a posteriori selection method is compared in simulation experiment. the conditions when the akaike weights are worth to use in combining forecasts were indicated. the use of the information criteria approach to obtain combined forecasts as an alternative to formal hypothesis testing was recommended. k e y w o r d s: combining forecasts, weighting schemes, information criteria. 1. introduction the development of time series analysis and computing power of computers made that many different forecasts can be obtained when forecasting the same economic variable with different methods. many selection criteria based on the performance of ex post forecasts are used to choose the best forecast (armstrong, 2001). combining forecasts can be treated as an alternative approach to the selection of the best individual forecast. since the seminal paper of bates and granger (1969) has been known that combining forecasts can produce a forecast superior to any element in the combined set1. hence, instead of seeking the best forecasting model the combined forecasts based on competing models are received. moreover, the reason for combining forecasts (or model averaging) is that the data generating model (true model) is unknown. therefore each model 1 the paper of bates and granger (1969) caused the development of research on combining forecasts (for overview see timmermann, 2006). mariola piłatowska 6 should be treated as an approximation of unknown data generating model. these models may be incomplete (or incorrectly specified) in different ways; forecast based on them might be biased. even if forecasts are unbiased, there will be covariances between forecasts which should be taken into account. then, combining forecasts produced by misspecified models may improve the forecast in comparison to any individual forecast obtained from the underlying models. as a consequence, the problem of selecting the individual forecasts over the set of available forecasts and the weighting schemes is occurred. especially, the selection of weighting scheme is important. most frequently the following weighting schemes can be distinguished: equal weights (stock, watson, 2004, 2006; marcellino, 2004), akaike weights (atkinson, 1980; swanson, zeng, 2001; kapetanios et al., 2008), optimized and constrained weights (jagammathan, ma, 2003), bayesian weights (min, zellner, 1993; diebold, pauly, 1980; wright, 2003). in the paper the focus is on the information criteria approach, especially the akaike information criterion which is used to produce the akaike weights. this approach enables to obtain not only one, but several plausible models for which the ranking can be built using the akaike weights. the individual forecasts, calculated from the considered models, are aggregated with the akaike weights. the paper propagates the application of the akaike weights, previously unknown in polish literature, and evidence ratios in selecting a model over the underlying set of models and in producing the combined forecasts. the purpose of the paper is to propose the procedure of combining forecasts using the akaike weights, and also to compare the combined forecasts (with the akaike weights and equal weights) with individual forecasts obtained from the best model selected according to: (1) the akaike information criterion and (2) traditional hypothesis testing. the analysis will be conducted in the simulation study in which autoregressive models and causal models are taken as approximating models provided that the data generating model is unknown. the structure of the paper is as follows. in section 2 the construction of the akaike weights will be presented. in section 3 the procedure of combining forecasts using the akaike weights will be depicted. next, the results of simulation experiment will be showed and at the very end – some conclusions. 2. the akaike weights the akaike information criterion is applied to select the best model from among the candidate models considered. the akaike’s (1973) seminal paper proposed the use of the kullback-leibler information or distance as a fundamental basis for model selection. the kullback-leibler (k-l) information between models f (true model or probability distribution) and g (approximating model in terms of a probability distribution) is defined for continuous functions as the integral: combined forecasts using the akaike weights 7 ∫= ,)|( )( ln)(),( dx xg xf xfgfi θ (1) where ,0),( ≥gfi 0),( =gfi only if ,gf = ).,(),( fgigfi ≠ ),( gfi denotes the information lost when the model g is used to approximate the model f. the purpose is to seek an approximating model that loses as little information as possible. this is equivalent to minimizing ),( gfi over g. akaike (1973) found a rigorous way to estimate k-l information based on the empirical log-likelihood function at its maximum point. this result took the form of an information criterion: ,2)ˆ(ln2aic kl +−= θ (2) where )ˆ(θl is the maximum likelihood for the candidate model ,i which is corrected by k the number of estimated parameters. akaike has showed that choosing the model with the lowest expected information loss (i.e. the model which minimizes the expected kullback-leibler discrepancy) is asymptotically equivalent to choosing the model im ),...,2,1( ri = that has the lowest aic value. to obtain the akaike weights a simple transformation of the raw aic values should be performed. for each model the difference in aic with respect to the aic of the best candidate model is computed: .aicaic min−=δ ii (3) these iδ are easy to interpret and allow a quick comparison and ranking of candidate models. the best model over the candidate models has 0min =δ≡δ i . the larger iδ is, the less plausible is that the fitted model im is the k-l best model, given the data. for nested model some rough rules of thumb are available in selecting the model (burnham, anderson, 2002), i.e. models with 2<δ i have substantial support, models with 74 <δ< i – considerable less support. models with 10>δ i have either essentially no support and might be omitted from further consideration, because they fail to explain some substantial explainable variation in the data. in empirical data analysis the models with 4<δ i are accepted. from the differences iδ we can obtain the relative plausibility of model im over the set of candidate models by estimating the relative likelihood )|( xml i of model im given the data x (akaike, 1983): ),5.0exp()|( ii xml δ−∝ (4) where ∝ stands for „is proportional to”. mariola piłatowska 8 finally, the relative model likelihoods are normalized (divided by the sum of the likelihoods of all models) and the akaike weights iw are obtained: , )5.0exp( )5.0exp( 1 ∑ = δ− δ− = r r r i iw .1 1 =∑ = r i iw (5) weight iw can be interpreted as the probability that im is the best model (in the aic sense, i.e. the model minimizing the k-l information) given the data and the set of candidate models. additionally weights iw can be useful in evaluating the relative strength of evidence for the best model (with biggest weight) over the other in the set of r models. thus, the evidence ratios or the ratio of akaike weights ji ww / (in particular the ratio ,/1 jww where 1w is the weight for the best model, and jw – weights for models in the set) are calculated. it is worth pointing out that this approach does not assume that any of the candidate models is necessarily true, but rather the ranking of models in the sense of k-l information is considered2. the aic weights (5) can be generalized into the form (burhnham, anderson, 2002, 2004): , )5.0exp( )5.0exp( 1 r r r r ii i q q w ∑ = δ− δ− = (6) where iq is a prior probability of our prior information (or lack thereof) about which of the r models is the k-l best model for the data. by the aic weights we mean the expression (6) with the equal prior probabilities, i.e. ./1 rqi = the inclusion of prior probabilities in (6) makes that the aic weights can be treated as an approximation of the bayesian posterior model probability (burnham, anderson, 2002, 2004). however, it is not a true bayesian approach. the full bayesian approach to model selection requires both the prior iq on the model and a prior probability distribution on the parameter θ in model im for each model. then the derivation of posterior results requires integration (usually achievable only by markov chain monte carlo methods). in that context the aic weight seem to be useful because they are much easier to compute and additionally the researcher is not required to determine prior densities for the parameters. 2 it is the main difference in the comparison with the bayesian model averaging which assumes that the true generation model is in the set of candidate models and measures the degree of belief that a certain model is the true data-generating model. combined forecasts using the akaike weights 9 3. the procedure of obtaining the combined forecasts using the akaike weights when calculating combined forecasts using the akaike weights some conditions should be satisfied. namely, all models in the set of candidate models should be fitted to exactly the same set of data because the inference based on information criteria is conditional on the data in hand. moreover, all models in the set should represent the same response variable. a common type of mistake is to compare models of ty with models of transformed variable, e.g. tyln or .tyδ the steps in the procedure of obtaining forecasts aggregated with the akaike weights are following. step 1. establishing the initial set of r models describing a given variable and their specification. the guidelines on specifying causal models should be derived from an economic theory explaining the phenomenon in interest. in the case of large number of variables it is not recommended to run all possible regressions because the set of candidate models should be plausible with respect to the economic theory, and not be automatically selected. the true generation model does not have to be included in the set of models. step 2. fitting the models )...,,2,1( ri = to the data, calculating the aic values and differences .iδ models should satisfy statistical and economic requirements. step 3. creating the reduced set of models )...,,2,1( *ri = for which ,4<δ i i.e. models plausible in the sense of k-l information. step 4. calculating the akaike weights (eq. (4)) and combined forecasts according to formula: ∑ = ++ = * 1 ,,, ,ˆ r i htihihtt fwy ,1 * 1 , =∑ = r i hiw (7) where htty +,ˆ – combined forecast, hiw , – the weight assigned to the forecast httif +,, based on the ith individual model. when combining forecasts the problem is to estimate the weights ,, hiw so as to minimize a penalty function depending on the forecast errors. very often, the penalty function is simply the mean square forecast error (msfe). 4. simulation experiment results the purpose of simulation experiment is to compare combining forecasts (using the akaike weights and equal weights) with individual forecasts obtained from the best model selected according to: (1) the akaike information criterion mariola piłatowska 10 and (2) traditional hypothesis testing. in experiment the autoregressive models and causal models are taken as approximating models provided that the data generating model is unknown. simulation scenario is following. the data-generating model of ty has the form: ,,33,22,110 ttttt xxxy εββββ ++++= ),,0(~ εσε nt ,3,2,1=εσ with parameters: 2,2.1,5.1,10 3210 ==== ββββ , for samples: 50,100=n (number of replications 1000=m ). processes tt xx ,2,1 , and tx ,3 have following structure: ,6.08.012 11,1,1 −− +++= tttt xx ζζ ),1,0(~ ntζ ,8.07.014 11,2,2 −− +++= tttt xx ηη ),1,0(~ ntη ,4.02.18 2,31,3,3 tttt xxx ξ+−+= −− ).1,0(~ ntξ as approximating models are taken: the autoregressive models: ,...110 qtqtt yyy −− +++= γγγ ,4,3,2,1=q and causal models3: tttttttt vxxyyyyy +++++++= −−−−− 1,12,11443322110 γγααααα (m1) ttttttt vxxyyyy ++++++= −−−− 1,12,113322110 γγαααα (m2) tttttt vxxyyy +++++= −−− 1,12,1122110 γγααα (m3) .1,12,11110 ttttt vxxyy ++++= −− γγαα (m4) ,,1122110 ttttt vxyyy ++++= −− γααα (m5) .,11110 tttt vxyy +++= − γαα (m6) it is assumed that the true generation model is unknown, therefore specifying the causal models the variables tx ,2 and tx ,3 were omitted. the analysis was carried out separately for autoregressive models and causal models. in each replication only the models with 4<δi were taken. for those models the akaike weights and equal weight )/1( *r were received, and having calculated the individual4 forecasts, the h-period ahead combined forecasts were obtained. to compare forecasts the mean square forecast error (msfe) was calculated for combined forecasts and individual forecasts obtained from the 3 having conducted initial simulation, these models are accepted as plausible. 4 individual forecasts were dynamic, and as values of explanatory variables x1t in forecast period the generated values are taken. combined forecasts using the akaike weights 11 best model (selected by: the aic and a posteriori selection method applied to the causal model5 m1). the results present table 1 (for )100=n and table 2 (for ).50=n results presented in table 1 and 2 show that the differences between msfe obtained for combined forecasts (using aic weights and equal weights) and individual forecasts (from the best model selected by the minimum of aic and by a posteriori selection method) are small. however, certain regularities indicating the usefulness of combined forecasts can be observed. in the case of small size of disturbance (σε = 1) the combined forecasts (with the akaike weights, waic) obtained from causal models give smaller mean square forecast error (msfe) than forecasts combined with equal weights (weq.) at the whole forecast horizon (table 1). this slight dominance of combined forecast with the aic weights is hold for σε = 2, 3 at longer horizons (h > 5), and for shorter horizons – the forecasts combined with equal weights (weq.) have lower msfe. in general the combined forecasts (with aic weights and equal weights) outperform the individual forecasts obtained from the best model (selected by a posteriori method, msel.), because the msfe for combined forecasts are visibly lower than the msfe for individual forecasts; this occurs for all sizes of disturbance σε (table 1). such performance indicates the dominance of combined forecasts. however this dominance is not complete, because the lowest msfe are obtained for individual forecasts calculated from the best model selected by the minimum of aic (for σε =1). for bigger size of disturbance, i.e. σε =2 and 3, the lower msfes at the horizon h ≤ 5 give the forecasts combined using equal weights, and at the horizon h ≥ 6 – forecasts combined using the aic weights. these lower msfes for forecasts from models selected by the minimum of aic (for σε =1) refer to the cases when the set of candidate models is small (in the considered experiment it were models m3 and m4), and additionally one model in the set has the dominating aic weight (waic > 0.7). for the bigger size of disturbance, i.e. σε = 2 and 3, the set of competing models, being used in combining forecasts, consisted most frequently of models m3, m4, m5, m6, and none had the dominating aic weight. then, the combined forecasts (using the aic weights or equal weights) outperformed the individual forecasts from model selected by the minimum of aic, i.e. they gave the lower msfes. 5 variables elimination in a posteriori selection method was realized at the 5% significance level. table 1. mean square forecast errors (msfe) for sample n = 100 h causal models autoregressive models waic weq. minaic msel. waic weq. minaic σε = 1 1 2.950 2.973 2.935 3.050 3.102 3.117 3.096 2 2.290 2.301 2.287 2.357 2.530 2.547 2.523 3 2.471 2.500 2.456 2.604 2.568 2.595 2.548 4 3.280 3.320 3.254 3.436 2.569 2.602 2.537 5 4.064 4.097 4.043 4.192 3.706 3.748 3.653 6 3.949 3.985 3.926 4.092 3.413 3.452 3.367 7 3.807 3.842 3.785 3.944 3.192 3.225 3.156 8 3.589 3.622 3.568 3.717 3.038 3.073 3.001 9 3.503 3.528 3.488 3.607 2.931 2.960 2.906 10 3.903 3.915 3.900 3.961 3.266 3.280 3.265 σε = 2 1 1.958 1.944 2.008 2.035 2.220 2.217 2.229 2 2.200 2.175 2.280 2.207 3.111 3.105 3.121 3 3.236 3.221 3.284 3.266 2.890 2.885 2.899 4 4.795 4.794 4.823 4.901 3.200 3.198 3.204 5 6.334 6.342 6.350 6.497 4.696 4.698 4.697 6 6.333 6.341 6.347 6.484 4.836 4.838 4.835 7 5.971 5.978 5.984 6.107 4.596 4.598 4.595 8 5.675 5.681 5.689 5.792 4.359 4.361 4.358 9 5.397 5.403 5.410 5.508 4.172 4.173 4.171 10 5.224 5.229 5.237 5.324 4.087 4.088 4.086 σε = 3 1 3.172 3.177 3.168 3.222 4.430 4.415 4.459 2 3.456 3.455 3.459 3.477 5.061 5.048 5.087 3 5.283 5.251 5.321 5.211 6.934 6.921 6.961 4 7.320 7.320 7.331 7.361 9.907 9.897 9.933 5 8.008 8.014 8.010 8.066 11.199 11.189 11.225 6 7.621 7.624 7.625 7.670 10.966 10.957 10.992 7 7.171 7.172 7.178 7.215 10.380 10.371 10.405 8 6.794 6.796 6.802 6.836 9.846 9.837 9.870 9 6.514 6.518 6.520 6.563 9.339 9.331 9.362 10 6.260 6.262 6.267 6.305 8.993 8.985 9.015 note: in columns waic, weq. are the msfes for forecasts combined using the akaike weights and equal weights, and in columns minaic., msel – the msfes for individual forecasts obtained from model selected by the minimum of aic and model received after applying a posteriori selection method to model m1 at the 5% significance level. combined forecasts using the akaike weights 13 table 2. mean square forecast errors (msfe) for sample n = 50 h causal models autoregressive models waic weq. minaic msel. waic weq. minaic σε = 1 1 1.019 1.018 1.018 1.023 1.069 1.069 1.076 2 1.502 1.587 1.409 1.888 2.712 2.705 2.721 3 1.564 1.576 1.570 2.002 2.804 2.796 2.815 4 2.440 2.393 2.493 2.494 2.489 2.482 2.498 5 3.599 3.486 3.722 3.251 2.288 2.283 2.296 6 3.685 3.609 3.761 3.274 2.136 2.132 2.144 7 3.455 3.391 3.518 3.144 2.219 2.212 2.229 8 3.279 3.212 3.352 3.093 2.443 2.435 2.456 9 3.164 3.110 3.224 2.988 2.342 2.334 2.355 10 3.669 3.642 3.699 3.412 2.569 2.565 2.576 σε = 2 1 3.522 3.508 3.539 4.230 5.419 5.399 5.414 2 3.854 3.803 3.918 4.644 5.739 5.711 5.744 3 4.199 4.100 4.318 4.900 5.669 5.641 5.677 4 4.832 4.671 5.018 5.261 5.483 5.458 5.489 5 5.379 5.222 5.555 5.668 5.679 5.660 5.685 6 5.493 5.324 5.683 5.616 5.356 5.340 5.362 7 6.133 5.955 6.329 5.949 5.273 5.262 5.280 8 6.171 6.016 6.346 5.890 5.139 5.130 5.146 9 6.311 6.195 6.456 6.048 5.362 5.357 5.369 10 6.286 6.189 6.411 6.034 5.380 5.376 5.387 σε = 3 1 5.088 5.096 5.085 5.335 7.026 6.997 7.064 2 6.378 6.371 6.389 6.498 8.233 8.203 8.276 3 6.968 6.952 6.987 6.993 8.696 8.683 8.720 4 6.578 6.562 6.597 6.599 8.389 8.386 8.399 5 6.193 6.175 6.213 6.205 7.944 7.945 7.948 6 5.803 5.788 5.823 5.821 7.391 7.393 7.396 7 5.599 5.579 5.627 5.593 6.938 6.939 6.942 8 5.504 5.484 5.532 5.492 6.593 6.594 6.597 9 5.325 5.308 5.351 5.334 6.456 6.458 6.459 10 5.181 5.166 5.206 5.228 6.601 6.603 6.603 note: see table 1. in the case of autoregressive models the forecasts from models selected by the aic gave the lowest msfe for small size of disturbance σε =1 for the same reasons as in the case of causal models, i.e. the set of candidate models contained the small number of models (here ar(2), ar(3) and ar(4)) and one mariola piłatowska 14 model had a dominating aic weight (0.5 < waic < 0.6). hence, in the sense of the msfe, the individual forecasts outperformed the combined forecasts. for the bigger size of disturbance, i.e. σε = 2 (and bigger uncertainty) and at the shorter horizon (h ≤ 5) the dominance of forecasts combined using equal weights is observed, and at the longer horizons (h ≥ 6) the dominance of forecasts combined using the aic weights occurs. for the disturbance σε = 3 the forecasts combined using equal weights slightly outperform the forecast combined using the aic weights and individual forecast (minaic) – see table 1. generally, the combined forecasts gave the lower msfe than the individual forecasts. this refers to the cases when the set of models consisted of many autoregressive models of different order and none of them had the dominating aic weight. then, the lower msfes are received for combined forecasts. the results tabulated in table 2, for sample n = 50 indicate that for the causal models the msfes are lower for forecasts combined using equal weights than those using the aic weights for all size of disturbance (except σε = 1 and h = 1, 2). the dominance of combined forecasts (waic, weq.) or individual forecasts (minaic, msel.) depends on the forecast horizon and size of disturbance σε . for horizon h ≤ 4 and disturbance σε = 1 (also σε = 2 and h ≤ 7; σε = 3) the msfes for combined forecasts are lower than for forecasts from the best model selected by a posteriori method (msel.), but for longer horizon h ≥ 5 (for σε = 1 ) and h ≥ 7 (for σε = 2) the msfes are lower for forecasts from msel.. forecasts from models selected by the minimum of aic have the higher msfes than combined forecasts and in general also higher than forecasts from msel.. in the case of autoregressive models the msfes for combined forecasts are always lower than for individual forecasts (minaic) – see table 2. simultaneously the combined forecasts using equal weights (weq.) outperform those using the aic weights (waic). 5. summary from comparison of forecasts combined using the aic weights and equal weights results that in the case when the set of candidate models contains the model with dominating aic weight (waic > 0.7) the combination of forecasts using the aic weights is not effective. then the msfes are higher than those for forecasts combined using equal weights. however, in such case the aic weights can be useful in building ranking of models, and additionally in calculating the evidence ratios informing about the relative strength of evidence for the best model (with biggest weight) in the sense of aic over the other models in the set of candidate models. the benefits from applying the aic weights occur when the number of candidate models in the set is big and none has the dominating weight waic. the results of experiment indicate that the combined forecasts outperform the individual forecasts in the case of autoregressive models. for causal models combined forecasts using the akaike weights 15 this dominance of combined forecasts is hold at the shorter horizon for disturbance σε = 1, 2 and at the whole horizon for bigger size of disturbance σε = 3. summing up, the information criteria approach, particularly the use of aic weights to build the ranking of models and to calculate the combined forecasts, can be treated as alternative to the traditional hypothesis testing approach directed to select the best model and calculate individual forecasts. references akaike, h. (1973), information theory as an extension of the maximum likelihood principle, [in:] petrov, b. n., csaki, f., second international symposium on information theory, akademia kiado, budapest. akaike, h. (1978), on the likelihood of a time series model, the statistician, 27, 217–235. armstrong, j. s. (2001), principles of forecasting, springer. atkinson, a. c. (1980), a note on the generalized information criteria for choice of a model, biometrika, 67 (2), 413–418. bates, j. m., granger, c. w. j. (1969), the combinations of forecasts, operations research quarterly, 20, 415–468. burnham, k. p., anderson, d. r. (2002), model selection and multimodel inference, springer. burnham, k. p., anderson, d. r. (2004), multimodel inference. understanding aic and bic in model selection, sociological methods and research, vol. 33 (2), 261–304. jagannathan, r. ma, t. (2003), risk reduction in large portfolios: why imposing the wrong constraints helps, the journal of finance, 58 (4), 1651–1684. kapetanios, g., labhard, v., price, s. (2008), forecasting using bayesian and informationtheoretic model averaging: an application to u.k. inflation, journal of business and economics statistics, 26 (1), 33–41. kitchen, j., monaco, r. (2003), real-time forecasting in practice, business economics, 38 (4), 10–19. marcellino, m. (2004), forecast pooling for short time series of macroeconomic variables, oxford bulletin of economic and statistics, 66, 91–112. min, c. k., zellner, a. (1993), bayesian and non-bayesian methods for combining models and forecasts with applications to forecasting international growth rates, journal of econometrics, 53 (1–2), 89–118. stock, j. h., watson, m. (2004), combination forecasts of output growth in a seven-country data set, journal of forecasting, 8, 230–251. stock, j. h., watson, m. (2006), forecasting with many predictors, [in:] elliott, g., granger, c. w. j., timmermann, a. (ed.), handbook of economic forecasting, elsevier. swanson, n. r., zeng, t. (2001), choosing among competing econometric forecasts: regression-based forecast combination using model selection, journal of forecasting, 20, 425– 440. timmermann, a. (2006), forecast combinations, [in:] elliott g., granger c. w. j., timmermann a. (ed.), handbook of economic forecasting, ch. 4, elsevier. prognozy kombinowane z wykorzystaniem wag akaike’a z a r y s t r e ś c i. w artykule uwaga jest skupiona na podejściu wykorzystującym kryteria informacyjne, a w szczególności kryterium akaike’a, które jest wykorzystywane do wyznaczenia wag akaike’a. podejście to umożliwia otrzymanie nie jednego, a kilku wiarygodnych modeli, dla których można stworzyć ranking stosując wagi akaike’a. modele te stanowią podstawę obliczemariola piłatowska 16 nia prognoz indywidualnych, a te z kolei służą do wyznaczenia ostatecznej prognozy kombinowanej, przy formułowaniu której wykorzystuje się wagi akaike’a. s ł o w a k l u c z o w e: prognozy kombinowane, systemy wag, kryteria informacyjne. microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 elżbieta szulc nicolaus copernicus university in toruń modeling of dynamic spatial processes a b s t r a c t. the paper is concerned with econometric modeling of the dynamic spatial processes on the example of the gdp per capita in selected european countries. the considerations of the paper are focused on investigations of the structure of components of the spatiotemporal process. as a result of the analysis some specifications of the dynamic spatial models have been obtained. next the issues of the estimation and verification of the models are presented. the main conclusion from the analysis is that the econometric models of the spatio-temporal processes ought to be of the dynamic character, e.g. considering the spatial and spatio-temporal trends and spatial, temporal and spatio-temporal autodependence as well. k e y w o r d s: spatio-temporal trend, autocorrelation, spatial lag model, dynamic spatial model. 1. introduction the paper presents the methodology of econometric modeling of the internal structure of dynamic spatial processes. the considerations recapitulate the results of the previous analyses (see: szulc, 2008, 2009a, 2009b). an empirical illustration of the considerations is the spatio-temporal distribution of the gdp per capita in selected european countries. they are: austria, germany, the czech republic, slovakia and hungary. the data relate to the established regions according to the european classification system nuts and they are taken from the data released by eurostat. in szulc (2008) the gdp per capita across the separated area in 2004 was analysed. the componential structure of the single “pure” spatial process ( )iz s , observed on the plane at the spatial locations [ ]iii yx ,=s , where i = 1, 2, ..., 84, was investigated. in szulc (2009a) some time aspect was enclosed into the previous analysis, i.e. the changes of the gdp per capita across the separated area in the period: 2000–2006 were considered. the componential structure of the spatial process, in successive years, i.e. ( ) [ ]iiiit yxz ,, =ss , i =1, 2, ..., 84, t =1, 2, ..., 7, was investigated. thus the conditional, in relation to time, approach to elżbieta szulc 18 the analysis of the spatio-temporal process was undertaken. then the conclusions concerning the total spatio-temporal structure, leading to the appropriate empirical model, were formulated only in reference to the so-called spatiotemporal trend. moreover some probable specification of the dynamic spatial model was proposed. in szulc (2009b) the approaches, mentioned above, were connected with one another by presenting more extended models which described the componential structure of the spatio-temporal process ( )tz i ,s , were [ ]iii yx ,=s , i = 1, 2, ..., 84, t = 1, 2, ..., 7. in the investigations the following assumptions were received: 1. the economic spatial processes demonstrate spatial and/or spatio-temporal trends, which are identified as the mean value of the process, changing in space and/or in time. 2. they usually demonstrate autodependence too, which in the structure of the process creates the autoregressive component. 3. the autoregressive component creates the homogeneous/stationary spatial or/and spatio-temporal process. it means, that for the spatial process ( )iz s there is assumed the basic structure of components, which symbolically may be written down in the following general form: ( ) ( ) ( ) ( ) ( )iiii zapz sswss ε++= . (1) in turn, in the case of the spatio-temporal process ( )tz i ,s , the basic structure of components may be symbolically presented in the form as follows: ( ) ( ) ( ) ( ) ( )ttzuatptz iiii ,,,,, sswss ε++= . (2) the symbols in the formulas (1)–(2) signify: ( )ip s , ( )tp i ,s – respectively, spatial and spatio-temporal trend which is usually expressed in the form of the two-dimensional (three-dimensional) polynomial function of the co-ordinates of the location on plane (and of the time variable); ( )wa , ( )ua ,w – summable spatial and spatio-temporal shift operators, defined in such a way, that w (the matrix of spatial connections) causes the variable to be shifted in space, whereas u (the backwards shift operator) causes the lag of it in time; ( )isε , ( )ti ,sε – spatial and spatio-temporal white-noise processes. 2. investigating the trend structure in the investigations of spatial trends the hypothesis of two-dimensional polynomial trend was used. the expression of the form: modeling of dynamic spatial processes 19 ( ) ∑∑ = = = p k p m m i k imki yxp 0 0 ,θs , (3) where: [ ]iii yx ,=s – the co-ordinates of the location on the plane, i = 1, 2, ..., n – indexes of the investigated spatial units, k + m ≤ p, presents the spatial trend of degree p. the models with the trend of the 1st , 2nd and 3rd degree were estimated and verified successively. in all cases the models of the 1st degree appeared the best. the results of the estimation and verification of the models are presented in table 1. finally it was confirmed that the spatial trends occurred in all years of the investigated period. table 1. estimates of the parameters of the spatial trends of the 1st degree for the gdp per capita in the period: 2000–2006 parameters years 2000 20001 2002 2003 2004 2005 2006 00θ̂ 15510.4 15943.0 16455.2 17148.2 17925.6 18752.7 19795.4 ( )00θ̂s 561.122 586.790 597.896 615.903 632.881 680.731 702.129 00t 27.6418 27.1700 27.5218 27.8424 28.3238 27.5479 28.1934 10θ̂ -0.0170 -0.0169 -0.0168 -0.0177 -0.0177 -0.0182 -0.0192 ( )10θ̂s 0.0017 0.0018 0.0018 0.0019 0.0019 0.0021 0.0021 10t -10.000 -9.3890 -9.3333 -9.3158 -9.3158 -8.6667 -9.1429 01θ̂ -0.0080 -0.0084 -0.0089 -0.0095 -0.0100 -0.0103 -0.0101 ( )01θ̂s 0.0024 0.0025 0.0026 0.0027 0.0027 0.0029 0.0030 01t -3.3333 -3.3600 -3.4231 -3.5185 -3.7037 -3.5517 -3.3667 r2 0.5676 0.5450 0.5360 0.5470 0.5355 0.5144 0.5216 figure 1 presents the theoretical surfaces of the trend. almost parallel location of the surfaces show that the spatial trends of the gdp per capita across the investigated area in substance do not change with regard to the forms in the successive years. the surfaces referring to the consecutive periods are located higher and higher in relation to the axis of the gdp per capita values, which means, that the mean value of the gdp per capita in the regions grows in time. the results of investigating the spatial trends include the information on the spatio-temporal trend of the gdp. usually such a trend may be described with the three-dimensional polynomial function of the following general form: ( ) ∑∑∑ = = = = p k p m p l lm i k ilmki tyxtp 0 0 0 ,,, θs , (4) where: t – time variable, k + m + l ≤ p, other significations – like in (3). 20 in particu (p the mode successiv estimates (19800 145ˆ =θ conclusio the gdp formed ac the e presented dg r2 = 0.55 figure 1. t s 3. inves in the residuals pressed b some, no observed ular, the spati )ti 000, θ +=s els of the 1st ve years may of the con ) ( 412.44616.8 3.7122.512 + on 1 p per capita ccording to t empirical mo d by the equa ( pd ti 88.509 , 14367ˆ = 04. the surfaces o spatial trend m stigating th e investigati from the pre by the formul t all too high (see, table 2 io-temporal t xi 010100 θθ ++ t degree tren be treated as stants in the ) t 2 14 . other p a across the the spatio-tem odel of the s ation (6), i.e.: ) ( ) x 0007.02 0177.02.7 − of the theoret models in 200 e autoregr ons of the s eviously fitte la (7), was u h but statisti 2). elżbieta szulc trend of the tyi 0010 θ+ . nd, obtained s the conditio e models sh parameters al separated a mporal trend patio-tempor : ( ) yx ii 0010.0 0093.0− tical values of 0–2006 ressive str spatial autoc ed models o used. in all th ically signifi c 1st degree tak for the gdp onal trends i how the line lmost do not area in the i d of the 1st de ral trend of ( ) t 242.113 582.748+ f the gdp per ructure correlation of of the trend t he years of t icant, positiv kes the form p spatial proc n relation to ear trend of t change. investigated egree. the gdp pe , r capita accor f the 1st ord the test mora the investigat ve autocorrel : (5) cess in the time. the the form: period is r capita is (6) rding to the der for the an’s i, exted period lation was modeling of dynamic spatial processes 21 ( )[ ] ( )[ ] ( )[ ]∑ ∑∑ ∑∑ = = = = = − −− = n i i ji n i n j ij n i n j ij zz zzzzw w n i 1 2 1 1 1 1 s ss , (7) where: ( )iz s , ( )jz s – values of the process of interest at locations i and j, z − the mean value of the process, wij – the spatial weight of the link between i and j. table 2. testing of spatial autocorrelation year i e(i) var(i) 2000 0.234031 -0.012048 0.004699 standardized statistic i 3.6204 p-value = 0.00015 2001 0.195583 -0.012048 0.004610 standardized statistic i 3.0579 p-value = 0.001114 2002 0.173191 -0.012048 0.004624 standardized statistic i 2.724 p-value = 0.003225 2003 0.171139 -0.012048 0.004638 standardized statistic i 2.6899 p-value = 0.003573 2004 0.164096 -0.012048 0.004647 standardized statistic i 2.584 p-value = 0.004884 2005 0.138682 -0.012048 0.004637 standardized statistic i 2.2135 p-value = 0.01343 2006 0.130453 -0.012048 0.004659 standardized statistic i 2.0878 p-value = 0.01841 conclusion 2 the values of the gdp per capita in the neighbouring regions are similar to one another. in the successive years the values of the moran’s statistic were decreasing. conclusion 3 the resemblance among the values of the gdp per capita in the neighbouring regions decreases in time. for investigating the spatial range of the autocorrelation two methods were used. the first one consisted in calculating and verifying significance of the appropriate moran’s statistics, assuming the neighbourhood of different orders, while the second one consisted in using the classic correlation coefficient, calculated for each of the established spatial shift. the significance of the coeffielżbieta szulc 22 cients of the 1st and 5th order or of the 1st , 3rd , 4th and even 5th order was confirmed (according to the used method)1. conclusion 4 the spatial autocorrelation of the gdp per capita across the investigated area may relate not only to the so-called nearest neighbours. with regard to the diversity of meaning of the results concerning the spatial autocorrelation of the higher orders which were obtained with the help of different methods, the autocorrelation of the 1st order was admitted as the most possible. 4. modeling of the trend-autoregressive structure of the spatial process the analysis of the trend and autoregressive structure of the gdp per capita across the separated area in the successive years led to the following conclusion: conclusion 5 the following form of the spatial econometric model of the gdp per capita should be proposed: ( ) iiiii gdpyxgdp ερθθθ ++++= w011000 , (8) the same one for each year of the investigated period. the models of the form (8) are named spatial lag models with regard to the presence of the spatial shifted dependent variable w(gdpi). the variable measures the levels of the investigated phenomenon (of the dependent variable) in the neighbouring regions. the results of the estimation and verification of the models with the form (8) for the successive years of the period: 2000–2006 are presented in table 3. the obtained empirical models are characterized by significant parameters. the residuals of the models do not show any autocorrelation. thus, it should be admitted, that the dependence of the 1st order is sufficient to be taken into account in the autoregressive structure. 5. modeling of the trend-autoregressive structure of the spatiotemporal process the investigations allow to specify the model referring to the total spatiotemporal structure of the analyzed process. the successive versions of the spatio-temporal models of the gdp process are presented below. the model of the 1 with regard to the limited volume of the paper (caused by the editorial requirements) the results of estimations of the appropriate coefficients and of verification of their significance are not placed here (for details, see: szulc, 2009). modeling of dynamic spatial processes 23 form (9) is a direct result of the previous settlements, while the next models came into existence by respecification of this model. table 3. the results of the model (8) estimation and verification for the successive years of the period: 2000–2006 years parameters estimates of parameters standard errors statistics z pr ( > | z | ) 2000 θ00 7749 1971.2 3.9316 0.000084 θ10 -0.008797 0.002455 -3.5832 0.000339 θ01 -0.003669 0.002272 -1.6150 0.106319 ρ = 0.51452, test lr = 12.155, p-value = 0.00049 autocorrelation of residuals: test lm = 0.48187, p-value = 0.48758 2001 θ00 8931 2149.3 4.1552 0.00003 θ10 -0.009725 0.002611 -3.7250 0.000195 θ01 -0.004394 0.002462 -1.7843 0.074368 ρ = 0.45339, test lr = 8.8238, p-value = 0.0029732 autocorrelation of residuals: test lm = 0.049286, p-value = 0.82431 2002 θ00 9734.9 2275.5 4.2781 0.000019 θ10 -0.010189 0.002680 -3.8015 0.000144 θ01 -0.004963 0.002565 -1.9348 0.0530169 ρ = 0.42117, test lr = 7.2391, p-value = 0.007133 autocorrelation of residuals: test lm = 0.039654, p-value = 0.84216 2003 θ00 10148 2368.9 4.2840 0.000018 θ10 -0.010728 0.002798 -3.8349 0.000126 θ01 -0.005286 0.002655 -1.9907 0.046519 ρ = 0.42057, test lr = 7.1947, p-value = 0.007312 autocorrelation of residuals: test lm = 0.021646, p-value = 0.88303 2004 θ00 10898 2505.1 4.3505 0.000014 θ10 -0.010998 0.002855 -3.8526 0.000117 θ01 -0.005787 0.002763 -2.0943 0.036232 ρ = 0.40379, test lr = 6.5691, p-value = 0.01037 autocorrelation of residuals: test lm = 0.0052268, p-value = 0.94237 2005 θ00 12269 2716.7 4.5162 0.000006 θ10 -0.01215 0.003076 -3.9493 0.000078 θ01 -0.006447 0.003011 -2.1412 0.032260 ρ = 0.35696, test lr = 4.9313, p-value = 0.026374 autocorrelation of residuals: test lm = 0.15626, p-value = 0.69262 2006 θ00 13268 2891 4.5894 0.000004 θ10 -0.013137 0.003248 -4.0448 0.000052 θ01 0.006423 0.003096 -2.0746 0.038030 ρ = 0.33907, test lr = 4.4004, p-value = 0.035932 autocorrelation of residuals: lm = 0.12389, p-value = 0.72486 5.1. model with spatio-temporal trend and spatial autocorrelation the separated spatial analyses for each point in time and the comparison of the obtained results induced to formulate the general conclusion relating to the elżbieta szulc 24 total spatio-temporal structure of the investigated process in the form of the theoretical model as follows: ( ) titiiiti gdptyxgdp ,,001010100000, ερθθθθ +++++= w . (9) the results of the estimation and verification of the model (9) are presented in table 4. table 4. the results of the model (9) estimation and verification parameters estimates of parameters standard errors statistics z pr (>|z|) θ000 θ100 θ010 θ001 8584.5 -0.010805 -0.005248 447.98 875.05 0.001068 0.001021 112.72 9.8103 -10.1139 -5.1403 3.9742 0.000000 0.000000 0.000000 0.000071 ρ = 0.41449 test lr: 49.408, p-value: 0.000000 wald statistic: 70.115, p-value: 0.000000 aic: 11754 (aic for lm: 11801) autocorrelation of residuals test lm: 0.003232, p-value: 0.95467 the model with the spatio-temporal trend and spatial shifts is characterized by significant parameters; the residuals do not show any autocorrelation and it is better than the model which takes into consideration only the trend. 5.2. model with spatio-temporal trend and with spatial and also with time autoregression the existence of the very strong time autocorrelation of the gdp per capita (the coefficient of time autocorrelation of the 1st order for the residuals of the model with the spatio-temporal trend of the 1st degree equals 0.9951) justifies including the component gdpi,t-1 into the model which describes the structure of the gdp process. thus, the next specification of the model is following: ( ) tititiiiti gdpgdptyxgdp ,,1,001010100000, εραθθθθ ++++++= − w .(10) the results of the estimation and verification of the model (10) are presented in table 5. apart from the improvement in the general degree of the model fitting, it cannot be treated as the final one because the autocorrelation appeared in the residuals. 5.3. model with the spatio-temporal trend and with spatial, time and spatio-temporal autoregression just as the coefficients of the spatial and time autocorrelation, the coefficient of the spatio-temporal autocorrelation of the 1st order appeared significant. its value amounted to 0.1636. therefore the next model of the gdp spatiomodeling of dynamic spatial processes 25 temporal structure additionally takes into consideration the component w(gdpi,t-1). it has the following form: ( ) ( ) .,1,, 1,001010100000, tititi tiiiti gdpgdp gdptyxgdp εγρ αθθθθ +++ ++++= − − ww (11) table 5. the results of the model (10) estimation and verification parameters estimates of parameters standard errors statistics z pr (>|z|) θ000 θ100 θ010 θ001 α 161.68 -0.000219 -0.000305 125.35 1.0420 121.88 0.000145 0.000121 13.627 0.004225 1.3265 -1.5097 -2.5191 9.1985 246.6556 0.18467 0.13111 0.01176 0.00000 0.00000 ρ = -0.034921 test lr: 17.139, p-value: 0.000000 wald statistic: 17.519, p-value: 0.000000 aic: 7643.4 autocorrelation of residuals test lm: 49.851, p-value: 0.000000 the results of the estimation and verification of the model (11) are presented in table 6. table 6. the results of the model (11) estimation and verification parameters estimates of parameters standard errors statistics z pr (>|z|) θ000 θ100 θ010 θ001 α γ 252.99 -0.000290 -0.000275 78.610 1.0458 -0.42254 115.64 0.000137 0.00014 13.9317 0.004019 0.057249 2.1879 -2.1088 -2.4146 5.6485 260.1819 -7.3808 0.02868 0.03496 0.01575 0.00000 0.00000 0.00000 ρ = 0.37102 test lr: 37.425, p-value: 0.000000 wald statistic: 44.667, p-value: 0.000000 aic: 7600.3 autocorrelation of residuals test lm: 0.12572, p-value: 0.72291 the considered model is characterized by significant parameters. the residuals of the model do not show any autocorrelation. according to its general fitting to the data it is the best among all the models proposed in this paper. elżbieta szulc 26 5. final remarks the considerations of the paper confirm that investigating the properties and structures of spatial and spatio-temporal economic processes is important for modeling of them. the econometric models of the spatio-temporal processes should have the dynamic character. it is expressed in the appropriate specification of the trendautoregressive structure, characterizing temporal, spatial and spatio-temporal tendencies and the lags and spatial or/and spatio-temporal shifts of the observed dependence. the gdp per capita across the separated area in the investigated period realizes the spatio-temporal process, which shows the spatio-temporal trend and the spatial and spatio-temporal autodependence. the specification of the dynamic spatial model for the gdp per capita caused that a “good” empirical model was obtained. references kopczewska, k. (2006), ekonometria i statystyka przestrzenna z wykorzystaniem programu r cran (spatial econometrics and statistics with r cran), cedewu sp. z o. o. schabenberger, o., gotway, a. c. (2005), statistical methods for spatial data analysis, champion & hall/crc, new york. szulc, e. (2008), analiza struktury ekonomicznych procesów przestrzennych na przykładzie pkb w wybranych krajach europejskich (analysis of the structure of economic spatial processes on the example of gdp in chosen european countries), acta universitatis nicolai copernici, ekonomia (economy) xxxviii, no. 388, 7–20. szulc, e. (2009a), analiza zmian w czasie struktury ekonomicznych procesów przestrzennych na przykładzie pkb w wybranych krajach europejskich (analysis of the changes in time of the structure of economic spatial processes on the example of gdp in chosen european countries), a report presented during the 3rd professor aleksander zeliaś scientific conference on modelling and forecasting of socio-economic phenomena, zakopane, may 5th – 8th 2009, in print in: j. pociecha (ed.), współczesne problemy modelowania i prognozowania zjawisk społeczno-gospodarczych (modern problems of modelling and forecasting of socio-economic phenomena), studia i prace uniwersytetu ekonomicznego w krakowie (studies and works of economic university in cracow). szulc, e. (2009b), modelowanie dynamicznych procesów przestrzennych (modeling of dynamic spatial processes), acta universitatis nicolai copernici, ekonomia (economy) xxxix, no. 389, 63–70. modelowanie dynamicznych procesów przestrzennych z a r y s t r e ś c i. artykuł przedstawia ekonometryczną analizę procesu przestrzennoczasowego na przykładzie pkb w wybranych krajach europejskich. przedmiotem rozważań są przestrzenne i przestrzenno-czasowe trendy oraz autozależności charakteryzujące składnikową strukturę badanego procesu. składniki te są podstawą do specyfikacji dynamicznych modeli przestrzennych. zaproponowane w artykule specyfikacje dynamicznych modeli przestrzennych poddaje się empirycznej weryfikacji. s ł o w a k l u c z o w e: trend przestrzenno-czasowy, autokorelacja, model przesunięć przestrzennych, dynamiczny model przestrzenny. microsoft word 00_tresc.docx dynamic econometric models vol. 10 – nicolaus copernicus university – toruń – 2010 anna michałek nicolaus copernicus university in toruń the importance of calculating the potential gross domestic product in the context of the taylor rule a b s t r a c t. taylor stated humorously that his rule was so easy that it could be written down on the back of a business card. the reality shows that the practical use of this type of rule implies accepting many assumptions about its final shape. the article mentions only the matter of influence of calculating the potential gdp and output gap on the empirical relevance of the taylor rule. two ways of calculating potential gdp were presented, i.e. the hp filter and linear trend of the current and the real gdp both seasonally adjusted (an additive model with seasonal dummies; tramo/seats procedure). k e y w o r d s: taylor rule, output gap. 1. introduction while the opinion that the central bank's actions can influence both inflation and the economy's real sphere is common, the economic reality shows, however, that precise specification of their effects is difficult for decision-makers. there is not one specific answer to the question how to run the monetary policy. the decision-makers should take into account the fact that good monetary policy must be more transparent, coherent and understandable for participants of the economic life. that is the reason why a possibility of basing the monetary policy on a certain rule has been recently often considered. it does not mean, however, that calculation of the interest rate should be carried out in the mechanical way. it is rather generally about outlining certain frames which could acquaint economic subjects with the essence of the interest rate policy. enhancing the monetary policy lies in taking into account the complementary elements of so called taylor's triangle, consisting of the direct inflation targeting strategy, the floating rate of exchange and the rule of the monetary policy understood as some plan of action (wojtyna, 2004). anna michałek 132 the taylor rule, which is most popular among researchers, belongs to the category of instrument rules1, which means that it determines the interest rate considering only current values of variables: the output gap and the difference between inflation and the inflationary target (żyżyński, 2006). the rule of the monetary policy considered in this paper is a classic taylor rule represented by the formula (taylor, 1993)2: ,5,0 ,21 ∗∗ −= +++= πα εβπβα r yi tttt (1) where: i – interest rate, π the rate of inflation, ∗π – the predetermined rate of inflation (inflation target), ∗r – the real interest rate responding to the level of full employment, y – the output gap as the percentage deviation of real gdp (y) from potential gdp (y*), hence: .100⋅⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = ∗ ∗ y yy y (2) parameters β1 and β2 should be positive, what more the parameter β1 should be higher than one (β1>1) in order to treat the taylor rule as a nominal anchor for inflation and expectations. in other words, the nominal interest rate should rise more than one-for-one with an increase in inflation above inflation target. only in that case, real interest rate is positive, when inflation is above the target. 2. seeking the optimal taylor rule generally, it is assumed that a good rule should characterize itself by its simplicity, be commonly comprehensible, durable and valid. it is also important to announce a rule in advance, in order to inform about it the highest possible number of economic subjects. according to j. b. taylor (2000) a monetary policy rule is just “some emergency plan defining in the most coherent way the circumstances in which the central bank should make changes in the monetary policy instrument”. the rule suggested by taylor has a specific form which has been modified for years. at present, there is no agreement among researchers on applied measures of inflation and output gap, types of data, degree of rule complexity or even weights attributed to inflation and the gap. thereby, the differences occurring in empirical studies make impossible to compare the results. the types of 1 basically, the rules of the monetary policy are divided into instrument rules and targeting rules which characterize decision makers’ aim in more general categories. 2 taylor has suggested a function of reaction which describes well the fed monetary policy in 1987-92, in which federal funds’ rate is lifted or lowered according to inflation’s deflection from the aim of inflation and the size of the output gap. the β1 and β2 parameters amounted to 1.5 and 0.5, respectively. the influence of calculating the potential gross domestic product… 133 problems with empirical verification of the taylor rule and some of its modifications are shown in figure 1. figure 1. types of the taylor rule’s modification it is important to mention that not all of the variants of the monetary policy rules were shown here. the chart was limited only to the most often estimated types of the taylor rule. there are researches, who consider different interest rate measures. economists use both the official interest rates of central banks and short-term market rates3. many different combinations regarding the performance of monetary policy rule exist. this paper is restricted only to the selected measure of output gap, taking the consumer price index, cpi, as a measure of inflation, and the central bank reference rate – as a measure of interest rate. in practice, the measuring of the level and growth rate of potential gdp can be inaccurate. there is no one recommendation how to estimate the potential gdp because of its unobservability. taking into account the unobservability of the potential gdp, there is no fixed and recommended measuring method. the evidence of that may be the numerous works on calculating the output gap for poland, in which the results vary greatly (compare: gradzewicz, kolasa, 2004; petru, mrowiec, 2005; białkowski, rosiak-lada, zwiernik, żochowski, 2007). the hp filter beside the trend function is one of the most used empirical techniques by researches who deal with the calculation of potential gdp in the context of the taylor rule. the hodrick-prescott filter and the deterministic trend are easier in comparison with structural methods, but they do not take into ac 3 while the rate of open market’s short-term operations influences mostly market interest rates short term maturity. long-term instruments are shaped under the market participants’ expectations. taylor rule’s modification measures inflation cpi core inflation inflation expectations output gap structural approach dynamic production function permanent profit analytical approach determini-stic trend hp filter types of data real-time data ex-post data additional variables interest rate lags currency exchange rate other features value of parameters reacting of the interest rate anna michałek 134 count structural changes occurring in the economy. however, the use of the cobb-douglas dynamic function requires availability of quarterly data on the real level of fixed assets which are not published by central statistical office (cso) and accepting many assumptions referring to data calculation, that may deform the final result (see: gradzewicz, kolasa, 2004). the aim of this paper is to evaluate the influence of different methods of calculating the potential gdp on estimation of the parameters of taylor rule. methods of estimating the potential gdp resting on the smoothing of the real gdp using hodrick-prescott filter and the linear trend are presented. author tests the hypothesis that poland’s monetary policy council sets the interest rate according to the taylor rule. 3. empirical analysis for poland in the study the following data were used: quarterly gdp data, cpi – the indicator of goods’ prices and consumer services (corresponding period of previous year=100) published by cso and the level of the nbp reference rate (from the end of a quarter). two not seasonally adjusted gdp series were based: current gdp (current prices in billions pln) and real gdp – nominal gdp corrected by the change in prices, expressed in annual average fixed prices from the previous year. the gdp deflator is utilized as a measure of the proportional change in prices of all goods and services and is published annually by cso. using this procedure presented in the paper cso information on quarterly gross domestic product estimate4 enabled to calculate the price deflators for the period i quarter of 1998 – iv quarter 2007, where: gdp nominal dynamics ith quarter 1998 (current prices) gdp delator ith quarter 1998 = gdp real dynamics ith quarter 1998 (annual average prices from the previous year) quarterly data consist of 40 observations. the sample covers the period from i quarter of 1998 to iv quarter 2007. data are presented in figure 2. 4 http://www.stat.gov.pl/gus/45_1437_plk_html.htm (18. 01. 09). it is important to notice that cso publishes the annual gdp deflator. the way of calculating the quarterly gdp price indicators presented in the work is correct because the geometric average from the quarterly gdp price indicators for following years is equal to the annual gdp price indicators published by cso. the influence of calculating the potential gross domestic product… 135 figure 2. reference rate srt and cpit, current gdpt, real gdpt in poland over the period 1998–2007 behavior of current and real gdp in figure 2 indicates the occurrence of seasonal fluctuations. there is no one recommended method of eliminating seasonal fluctuations. for seasonal adjustment of time series, the cso uses the tramo/seats method. in this paper the procedure recommended by cso as well as the method including seasonal dummies (the additive model with seasonal dummies, where qit*= qit – qmt) are used. only for the real gdp process slightly difference between the methods of seasonal adjustment in iv quarter of 1998 is observed (see figure 3). it is assumed that this difference will influence substantially the gdp gap series obtained at a later stage. seasonally adjusted real gdp time series are presented in figure 3. sr t 0 5 10 15 20 25 30 98 99 ooo1 o2 o3 o4 o5 o6 o7 % cpi t 0 5 10 15 98 99 oo o1 o2 o3 o4 o5 o6 o7 % current gdp t 100 150 200 250 300 350 98 99 ooo1 o2 o3 o4 o5 o6 o7 bi llio ns p ln real gdp t 100 150 200 250 300 350 98 99 ooo1 o2 o3 o4 o5 o6 o7 bi llio ns p ln anna michałek 136 figure 3. real gdpt seasonally adjusted in poland over the period 1998–2007 in order to apply the taylor rule, the assumptions concerning the real interest rate r*, the inflationary target and the method of estimating the output gap should be taken. the real interest rate is treated as fixed, i.e. during the period 1998–2007 it took the value from the interval (3%–6.6%) (brzoza-brzezina, 2003)5. the inflationary target in 1998 was equal to 9.5%. since 1999 an obligatory long-term target at 4 percent level was applied which was supposed to be reached by the end of 2003. since 2004 the inflationary target was equal to 2.5%. for calculating potential gdp the hodrick-prescott filter was used as well as the linear trend model for current and real gdp (seasonally adjusted using tramo/seats procedure and seasonal dummies). in the case of the hp filter the standard value of parameter λ was taken at the basic level for quarterly data, i.e. (hodrick, prescott, 1980). after obtaining the potential gdp the output gap (gdp gap) can be calculated. it is the rate of deviation of real gdp from potential gdp, showed in a formula (2), i.e.: gdphpt gdphptgdp gapgdp − −−= or , gdptrend gdptrendgdp gapgdp − −−= where: gdphpt _ – potential gdp estimated using the hodrick-prescott filter; gdptrend _ – potential gdp estimated using the linear trend model. the output gap informs about the inequality existing in the real economy and is treated as a factor influencing the inflationary processes. inflation usually decreases when real gdp is below potential gdp (negative output gap) and increases when real gdp is above potential gdp – positive output gap (solow, 5 according to the author there exists a way of determining the real rate which does not require using intricate econometric techniques when time series of a moderately long period of stable inflation is observed. it is assumed that the beginning of the stabilization period in poland have begun in 2002. real gdpt seasonally adjusted 130 180 230 280 330 98 99 oo o1 o2 o3 o4 o5 o6 o7 bi llio ns p ln real gdp (0-1) real gdp (tram o/seats) the influence of calculating the potential gross domestic product… 137 taylor, 2002)6. behavior of the output gap is presented in figure 4 and 5. the following notations are used: gap_n (method of seasonal adjustment; method of calculating potential gdp; the initial time series). figure 4. the gdp gap in poland over the period 1998–2007 (current gdp) figure 5. the gdp gap in poland over the period 1998–2007 (real gdp) 6 taylor suggests that this relation is rather short-term, nonetheless, he stresses that there is a necessity of choice between the size of inflation’s fluctuations and the size of deviation of real gdp from potential gdp. output gap current gdp -10 -5 0 5 10 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 % gap1 (0-1; hp; c) gap2 (0-1; t; c) gap3 (t/s; hp; c) gap4 (t/s; t; c) output gap real gdp -10 -5 0 5 10 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 % gap5 (0-1; hp; r) gap6 (0-1; t; r) gap7 (t/s; hp; r) gap8 (t/s; t; r) anna michałek 138 gap 1 the gdp gap as a percent deviation of nominal gdp seasonally adjusted (the model with dummy variables (0-1)) from potential gdp calculated by the hp filter (hp), used for series gdp in current prices (c); (0-1; hp; c) the gdp gap which is based on the gdp in current prices series takes the same value as the gap based on the gdp in constant prices series (average 1998=100); gap 2 the gdp gap as a percent deviation of nominal gdp seasonally adjusted (the model with dummy variables (0-1)) from potential gdp calculated as a trend of gdp (t) in current prices (c) seasonally adjusted (the model with dummy variables (0-1)); (0-1; t; c) gap 3 the gdp gap as a percent deviation of nominal gdp seasonally adjusted (t/s analysis) from potential gdp calculated by the hp filter (hp), used for the gdp series in current prices (c); (t/s;hp; c) gap 4 the gdp gap as a percent deviation of nominal gdp seasonally adjusted (t/s analysis) from potential gdp calculated as a gdp trend (t) in current prices (c) seasonally adjusted (t/s analysis); (t/s; t; c) gap 5 the gdp gap as a percent deviation of real gdp seasonally adjusted (the model with dummy variables (0-1)) from potential gdp calculated by the hp filter (hp) used for the real gdp series (r); (0-1; hp; r) gap 6 the gdp gap as a percent deviation of real gdp seasonally adjusted (the model with dummy variables (0-1)) from potential gdp calculated as a trend (t) of real gdp (r) seasonally adjusted (the model with dummy variables); (0-1; t; r) gap 7 the gdp gap as a percent deviation of real gdp seasonally adjusted (t/s analysis) from potential gdp calculated by the hp filter (hp) used for the real gdp series (r); (t/s; hp; r) gap 8 the gdp gap as a percent deviation of gdp seasonally adjusted (t/s analysis) from potential gdp calculated as the trend (t) of real gdp (r) seasonally adjusted (t/s analysis). (t/s; t; r) figure 4 and 5 shows that in 2002 the gdp fell below the potential level. when real gdp is placed below potential gdp the inflationary pressure does not exist. negative output gap was observed in 2003, 2004 and 2005, respectively. the situation where gdp was above the potential gdp occurred at the beginning of 1998 year and is maintained more or less until the beginning of 2002 with a certain exception at the turn of 1998 and 1999. the differences between the gdp gap based on the current gdp (gap 1, 2, 3 and 4) and the gap based on the real gdp are observed. the greatest distance between gap 7 (t/s;hp;r), gap 8 (t/s,t;r), which take negative values is noticed in 1998. the output gap based on the real gdp was closed in 2006 (gap 5, 6, 7 and 8). in the case of gdp gap based on the current (gap 1, 2, 3 and 4) the closing output gap was observed year later, in the second half of 2006 (compare figure 4 and 5)7. alternative methods of estimating the potential product lead to various calculations of the gdp gap. a much bigger fluctuations are observed, when the gap is calculating using the linear trend than the hp filter. it is especially relevant to observations of the end of year 2007. also the choice of methods elimi 7 the closure of the output gap means not only balance between real and potential gdp, but also the situation where real gdp grows faster than potential one. the influence of calculating the potential gross domestic product… 139 nating the seasonal fluctuations from the original gdp series impacts on the final results. in the case of output gaps calculated for series seasonally adjusted using dummies variables 0-1 (gaps 1, 2, 5, 6) the outlier values were observed, which are not likely from the point of view of economic reality (iv quarter of 1998). using the obtained series of the gdp gap, an estimation (ols) of the original taylor rule was carried out. its results were presented in table 1. table 1. the results of the original taylor rule’s estimation – types of output gap va ria bl e gap1t (0-1;hp;c) gap2t (0-1;t;c) gap3t (t/s;hp;c) gap4t (t/s;t;c) gap5t (0-1;hp;r) gap6t (0-1;t;r) gap7t (t/s;hp;r) gap8t (t/s;t;r) coeff. coeff. coeff. coeff. coeff. coeff. coeff. coeff. const 3.27*** 2.90*** 3.38*** 2.90*** 3.37*** 3.27*** 3.42*** 3.28*** cpi t 1.44*** 1.52*** 1.42*** 1.52*** 1.41*** 1.44*** 1.41*** 1.44*** gap t 0.08 -0.12 0.16 -0.13 0.37*** 0.19* 0.64*** 0.27* summary r2 89.42 89.71 89.56 89.67 91.23 90.13 92.53 90.41 s(u) 2.01 1.98 2.00 1.99 1.83 1.94 1.69 1.92 dw 0.27 0.3 0.27 0.29 0.45 0.28 0.29 0.25 11ρ 0.89 0.86 0.90 0.87 0.83 0.92 0.91 0.93 note: ***, **, * denotes significance at 1%, 5%, 10%. r2 is the determination coefficient, s(u) is standard error of residuals, dw is the durbin-watson statistic and 11ρ the partial autocorrelation coefficient of first order. the coefficients of inflation impact are significant in all variants, positive and higher than 1 ( )11 >β and what is more, close to the value pointed by taylor – 1.5. the choice of type of series like real gdp is crucial for estimation results. the impact of the gap turned out irrelevant in the models where the initial series for estimating the gap was gdp in current prices. in the models where the gap was based on the real gdp series significant parameters were received, however, for gap 6 and 8 (potential gdp calculated as real gdp’s trend) significance was at the 10% level. the coefficients of gap 5, 6, 7 and 8 have positive sign, as expected. however, these models do not meet the statistic requirements. low values of the dw statistics for all models show first order autocorrelation for residual process. coefficients of the first order autocorrelation turned out to be significant ( 11ˆ 0.35ρ > autocorrelation coefficient is distinctly higher that the corresponding critical value of the quenouille test ( n/2 ). the high, positive autocorrelation of a residual process may give evidence about the omission of important variables or the elements of internal structure for given process. additionally, the relation between r2 and dw indicates a spurious relationship. the correct statistical inference requires taking into account the information about the internal structure of the processes being modeled (trend, autoregression). anna michałek 140 4. detecting the internal structure of processes assuming the nonstationarity in mean, the level of a trend, occurrence of seasonality and order of autoregression were detected. the results of detecting the internal structure of the analyzed processes are presented in table 2. table 2. the results of the study of the internal structure of particular processes structure srt cpit gap1t (0-1;hp;cb) gap2t (0-1;t;cb) gap3t (t/s;hp;cb) gap4t (t/s;t;cb) gap5t (0-1;hp;r) gap6t (0-1;t;r) gap7t (t/s;hp;r) gap8t (t/s;t;r) r 1 1 s no no no no no no no no no no ar(q) 4 5 5 5 4 4 5 5 1 2 note: r – degree of polynomial trend, s – occurrence of seasonality, ar(q) – order of autoregression for given processes. the choice of trend degree was made using the f-test. this test is most often used when comparing linear trend models and testing significant differences in the variances. in all estimated trend and seasonality models the seasonal coefficients turned out to be insignificant, hence the lack of seasonality in the analyzed time series was found. the third stage of studying the internal structure of processes is the identification of lag length of particular processes; therefore the akaike information criterion (aic) was used. because of the small sample size, the corrected aic was used, in the form: 2 ( 1) aic aic , 1c k k n k + = + − − where k stands for the number of estimated parameters, including the constant and variance (burnham, anderson, 2004)8. the models with the lowest value of aicc were chosen. accepting the assumption of nonstationarity in mean, linear models of reference rate were built, in which the specification of an equation describing the monetary policy rule was broadened by information of the internal structure of processes. the inclusion of internal structure of given processes is the basis of congruent modelling, formulated by professor zieliński. the concept of congruent modelling is understood as the congruence of the harmonic structure of an endogenous process with the joint harmonic structure of explanatory processes and residual process which is independent from explanatory processes (talaga, zieliński, 1986). hence, the estimated equations have the form: 8 the authors point that there is too much usage of aic in research, while the requirement that n/k>40 instead of aicc. thus, aicc should be used regardless of the sample size because with the growth observations’ number, aicc converges to aic.  the influence of calculating the potential gross domestic product… 141 ∑ ∑∑ = = −− = − +++++= 5 0 0 4 1 10 , k q k tktykktk k ktikt gapcpisrtsr εβββαα π where q denotes the lag length depending on the method of estimating the output gap. table 3. the reduced dynamic congruent models for the reference rate depending on different gdp gap’s variants va ria bl e gap1t (0-1;hp;c) gap2t (0-1;t;c) gap3t (t/s;hp;c) gap4t (t/s;t;c) gap5t (0-1;hp;r) gap6t (0-1;t;r) gap7t (t/s;hp;r) gap8t (t/s;t;r) coeff. coeff. coeff. coeff. coeff. coeff. coeff. coeff. const 0.15 4.09** 0.15 5.59*** 11.61`*** 7.70*** 9.45*** 7.71*** t -0.08** -0.12** -0.20*** -0.16*** -0.17*** -0.16*** cpi t 0.59*** 0.42*** 0.59*** 0.38*** 0.35*** 0.35*** cpi t-1 0.73*** 0.60*** cpi t-2 -0.41*** -0.48*** -0.41*** -0.50*** cpi t-3 0.19* 0.30** 0.19* 0.32*** 0.37*** 0.40*** 0.49*** 0.44*** cpi t-4 cpi t-5 -0.15** -0.22*** -0.17** gap t 0.27*** 0.12** gap t-1 0.24*** 0.79*** 0.32*** gapt-2 0.12** 0.18*** 0.34*** 0.21*** sr t-1 0.80*** 0.83*** 0.80*** 0.81*** 0.47*** 0.34* 0.65*** sr t-2 -0.36** -0.37*** sr t-3 -0.19* sr t-4 -0.18** -0.22*** -0.23*** sumary r2 99.38 99.54 99.38 99.56 99.65 99.57 99.61 99.60 s(u) 0.43 0.40 0.43 0.38 0.35 0.39 0.36 0.37 dw 1.55 1.96 1.55 2.08 1.85 1.97 2.28 2.31 11ρ 0.21 -0.01 0.21 -0.08 0.06 -0.04 -0.14 -0.23 note: ***, **, * denotes significance at 1%, 5%, 10% level. r2 is the determination coefficient, s(u) is standard error of residuals, dw is the durbin-watson statistic and 11ρ the partial autocorrelation coefficient of first order. after estimation of the starting version of congruent models the insignificant variables were eliminated using a posteriori selection method. the estimation results are presented in table 3. the reduced congruent models contain significant parameters at 10% significance level (except the constant in models 1 and 3) and have the desired properties of residual process. in all estimated equations the inflation’s influence is significant. in models 1 and 3 (gap1 (0-1;hp;c), gap3 (t/s;hp;c)) the impact of gap was not significant, and what is more, the reduced model has the same form in both cases. the impact of current inflation and output gap is observed only in model 6 (gap6 (0-1;t;r)), but the coefficient of the impact of inflation on the reference rate is smaller than one. taking into consideration the lagged influence of inflation (its accumulated impact is equal to 0.6 percentage point) the model does not satisfy the stability condition called the anna michałek 142 taylor principle. in model 5 (gap5 (0-1;hp;r)), the accumulated impact of inflation and output gap amounted to 1.1 and 0.85 percentage point, respectively. moreover, in each variant of model the lagged reference rate is significant, what means that changes of reference rate are autoregressive distributed (the interest rates’ smoothing effect). it shows the partial reaction of monetary policy council on changes in the economy, which may result from the unobservability of output gap and hence problems with estimating it precisely. the estimated reaction’s functions of reference rate to changes in the output gap and inflation level, differ from the original taylor rule because they include the internal structure of processes (compare table 1 and table 3). parameter estimates of lagged output gap (gap 2 and 4) are statistically significant with expected positive signs. in the case of the remaining models the influence of estimating the potential gdp on the relevance of the taylor rule is not so clear. it can be noticed that regardless of the way of seasonal adjustment of real gdp the accumulated impact of inflation and the output gap is similar in models where potential gdp was estimated using linear trend. 4. conclusions based on the obtained results one cannot tell categorically which of the analyzed methods of estimating the potential gdp gives better results in the contest of estimation of the taylor rule. the inclusion of the internal structure of processes eliminates the autocorrelations of residual process, however it changes the coefficient estimates in comparison with the original taylor rule. in that case, the taylor-type rules were considered. the taylor condition (the impact of inflation on the reference rate is bigger than one) was satisfied only in model 5 (gap5 (0-1;hp;r)). however it should be noticed, that the recommended original time series for estimating the output gap is real gdp. in the presented analysis, the assumption of nonstationarity in mean was taken into consideration, however processes can also be nonstationary in variance. in such a case, the recommended method of estimating the output gap is the hp filter. references białkowski, p., rosiak-lada, k., zwiernik, p., żochowski, d. (2007), zintegrowane modelowanie pkb, stopy bezrobocia i inflacji w oparciu o wielokomponentowe wskaźniki koniunktury (integrated modelling gdp, unemployment and inflation based on market situation indicators), ogólnopolska konferencja naukowa koniunktura gospodarcza – 20 lat doświadczeń irg sgh, warszawa. brzoza-brzezina, m. (2003), rola naturalnej stopy w polskiej polityce pieniężnej (role of the natural rate of interest in the polish monetary policy), ekonomista (the economist), vol. 4, 593–612. burnham, k., anderson, d. (2004), multimodel inference: understanding aic and bic in model selection, amsterdam workshop on model selection, 2–56. the influence of calculating the potential gross domestic product… 143 gradzewicz, m., kolasa, m. (2004), szacowanie luki popytowej dla gospodarki polskiej przy wykorzystaniu metody vecm (estimating the output gap in the polish economy: the vecm approach), bank i kredyt (the bank and credit), february, 14–30. hodrick, r. j., prescott, e. (1980), post-war u.s. business cycles: an empirical investigation, discussion papers 451, northwestern university, 1–27. petru, r., mrowiec, m. (2005), jakie stopy za dwa lata? (what interest rates will be in two years?), nawigator (navigator), bank bph, 6–10. solow, r., taylor, j. b. (2002), inflacja, bezrobocie, a polityka monetarna (inflation, unemployment and monetary policy), cedewu, warszawa. talaga, l., zieliński, z. (1986), analiza spektralna w modelowaniu ekonometrycznym (spectral analysis in econometric modelling), pwn, warszawa. taylor, j. b. (1993), discretion versus policy rules in practice, carnegie-rochester series on public policy, vol. 39, 195–214. taylor, j. b. (2000), alternative views of the monetary transmission mechanism:what difference do they make for monetary policy, oxford review of economic policy, vol 4, 54–72. żyżyński, j. (2006), modele decyzyjne a mechanizm podejmowania decyzji w polityce pieniężnej (decision-making models and mechanism of the decision making in monetary policy), ekonomista (the economist), vol. 2, 169–182. wojtyna, a. (2004), szkice o polityce pieniężnej (sketches about monetarny policy), pwe, warszawa. znaczenie szacowania potencjalnego pkb w kontekście reguły taylora z a r y s t r e ś c i. taylor wypowiedział się żartobliwie o swojej regule, że jest tak prosta, iż można ją zapisać na odwrocie wizytówki. rzeczywistość pokazuje, że praktyczne wykorzystanie tego typu reguły implikuje przyjęcie wielu założeń, co do ostatecznego jej kształtu. artykuł porusza jedynie kwestię wpływu zastosowanych metod szacowania potencjalnego pkb, luki pkb na estymację parametrów reguły taylora. przedstawiono dwa sposoby szacowania pkb potencjalnego: filtr hp oraz trend liniowy, przy czym bazowano na wyrównanych sezonowo (model addytywny ze zm. 0-1; procedura tramo/seats) szeregach: pkb bieżący oraz pkb realny. s ł o w a k l u c z o w e: reguła taylora, luka popytowa. microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 monika kośko the university of computer science and economics in olsztyn markov switching models with application to contagion effect analysis in the capital markets a b s t r a c t. this article presents the analysis of the contagion effect in the capital markets on the basis of the markov switching models ms. the research is based on the return of the indexes. there is a distinction of two regimes with different volatility levels, the calm period and the crisis period. then the analysis of the period’s occurrence was conducted, in reference to global financial crisis. periods with a similar level of volatility occurrence in the same time. this analysis evidences the shocks transmission between financial markets, what confirms an occurrence of the contagion effect. k e y w o r d s: markov switching model, contagion effect. 1. introduction the aim of the article is an application of the markov switching model ms to contagion effect analysis in the capital markets. there are different definitions of contagion effect. the most popular definition affirms that shocks transmissions are caused by the herd behavior of investors and this is the most often assumed in the empirical research. there can be found three approaches in an application of the ms models to contagion effect analysis, such as: − univariate models with the switch in variance msh (moore, wang, 2007); − multivariate models with the switch in variance msh-var or both in the variance and mean msmh-var (linne, 2001; mandilaras, bird, 2005); − the garch models with the markov switching ms-garch (edwards, susmel, 2001). 2. a contagion effect definition a contagion effect definition the most often concerns the financial markets, but the transmission processes envelope an economic connections too. the monika kośko 74 word bank assumes three versions of the contagion effect definition1: broad, narrow and very narrow definition. according to broad definition the contagion effect is an international shocks transmission or wide-spread spillover effect. the transmission can refer to both good and bad periods and it’s not always identified as crisis. however in the crisis it can be more noticeable. in the narrow definition there is an assumption that contagion is a shocks transmission to other countries or the relations between economies except the fundamental connections and common shocks. this definition the most often is reduced to very similar changes in the financial markets that are usually explained by the herd behavior. the very narrow definition assumes that a contagion effect occurs when in the crisis period the correlation between economies is stronger than in the calm period. according to fiszeder (2009) the narrow definition of contagion effect is the shocks transmission between countries that cannot be explained fundamentally. these transmissions are real financial, economic and political connections. the most often cause of the contagion in narrow sense is the herd investors behavior. there can be noticed some specific group behavior of investors, what is more distinct in the crisis periods and causes crossing shocks over the financial markets. the understanding these behaviors could help to explain the transmission of the shocks. in the analysis of the contagion effect a transmission channels have the essential meaning. there can be found a three basic transmission channels: − real channel (international trade); − financial channel (global diversification of the investment portfolio); − herd behavior (a copy strategy in the investment); − international policy. 3. the markov switching model the markov switching model ar(p)-msmh(r) for stochastic process ty is given by: ( ) ( )( ) ( )( ) ( )( ) ,sy...sysysy tptptptttttt εμαμαμαμ +−+−+−+= −−−−−− 222111 (1) ( )( ),s,iid~ tt 20 εσε [ ],s,y|yey ttttt 1−−=ε where ( )tsμ is conditional expected value of ty [ ]( )ttt syy ,| 1−ε=μ and ( )ts2εσ is the variance of the disturbance term. 1 contagion of financial crises, world bank, http://www.worldbank.org. the markov switching model with application to contagion effect … 75 in the markov switching models the parameters ( ) ( ) jtt ,js,js πσμ ε == 2 2 are the unobserved variable ts realizations, that has a markov property 3. the conditional probabilities ( )tpij create the transition probabilities matrix p with the rr × dimension that is given by: , )t(p)t(p)t(p )t(p)t(p)t(p )t(p)t(p)t(p rrrrrr r r × ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = l momm l l 21 22221 11211 p (2) where r is the states (regimes) number of the ts variable process. the p matrix is the stochastic matrix, because its elements satisfy the following conditions: 0≥)t(pij , 1=∑ j ij )t(p . the homogeneous markov chain probabilities ( )tpij describing the one step change between states are constant and time independent. the markov switching models ms can generate the skew distribution (when the third central moment significantly differs from zero) and the leptokurtic distribution. in example, the model that can generate the skewness and the leptokurtosis of the distribution is the ar(0)-msm(2). the ar(0)-msh(2) model is given by: ( ) ( )( ),sisi,nid~,y ttttyt 210 2221 =+==− εε σσεεμ (3) where 22 2 1 εε σσ , the variance of the disturbance term, in the following first and second state ( ) ( )21 == tt si,si are dummies variables. the excess coefficient of the ty process for the markov switching ar(0)msh(2) model can be written as: ( )[ ] ( )[ ] ( ) ( ) , y y yt yt 22 22 2 11 22 2 2 121 22 4 3 3 εε εε σπσπ σσππ με με + − =− − − (4) 2 unconditional probabilities of the markov chain (ergodic) for two states are received from equations: 2211 22 1 2 1 pp p −− − =π , 2211 11 2 2 1 pp p −− − =π . 3 the finite homogeneous markov chain with the state space { }r,...,,21 is the stochastic process where for all { }rji ,...,2,1, ∈ the ( ) ==== − tpisjs ijtt )|pr( 1 ( )tp)is|jspr( ijtt ==== −1 equality is fulfilled. monika kośko 76 where ( )[ ]2yty με − is the second central moment and 21 ππ , are the ergodic probabilities in the following first and second states. the (4) coefficient significantly differs from zero when 22 2 1 εε σσ ≠ and when 10 1 << π . therefore the leptokurtosis is confirmed by the markov structure which has heteroscedastic disturbance term and different from zero the excess coefficient of the distribution. 4. the financial time series results in the empirical analysis the weekly return rates of the main stock exchange indexes were used, such as: rts (russian), sax (slovakia), his (china), px50 (czech republic), bux (hungary), cac40 (france), dax (germany), ftse100 (england). the analyzed series come from the period from september the 1th, 1995 to august 21th, 2009. the price series transformation into return series was achieved by calculating the week dynamics. the time series of the returns were multiplied by 100. then the adf test for unit root was applied and its results show that for all time series this test rejects the unit root hypothesis. in the next part of the empirical analysis the msh models with the variance switch and 1, 2 or 3 states were estimated. the appropriate order of autocorrelation and autoregression in these models were determined by the means of the durbin and watson test. for two states models one of the states is interpreted as low volatility periods and the second state as high volatility periods. for the msh(3) models with three states the additional state is characterized as periods with the moderated volatility. the switching models were checking for the presence of the arch effect (ljung and box test for the squares of return rates) and for the normality of distribution (jarque and bery test). the tests results are presented in the table 1. the distributions of the all residuals series are normal. in most models the arch effect doesn’t occur. the log-likelihood ratio analysis indicates that the ratios are higher for all models with three states msh(3) than ratios of the models with two states msh(2). moreover the log-likelihood test for the number of states4 was applied. the results of this test indicate the choice of the msh(3) models. the estimation results of the msh(2) and the msh(3) models are shown in the table 1 and in the table 2 appropriately. models with the highest values of probabilities are presented in the table 3. in msh(2) models the variance of high volatility state is about two times higher than the variance of the low volatility state. 4 the log-likelihood test is constructed on the basis of the ( ) ( )( )ahlhllr −−= 02 statistic that has a chi-square distribution ( )k2χ and k is a number of additional parameters of the alternative hypothesis model. the markov switching model with application to contagion effect … 77 table 1. estimation results of the msh(2) models series 11p 22p 1σ 2σ μ lr jarque and bery test (residual) wig 0.9695 0.9801 0.0289 0.0626 0.0123 [0.113] 1162.4 3.76 [0.152] rts 0.9871 0.9866 0.1149 0.0655 0.0337 [0.006] 795.1 3.18 [0.204] sax 0.9435 0.9608 0.0567 0.0327 0.0016 [0.797] 1242.8 6.30 [0.043] hsi 0.9949 0.9952 0.0290 0.0950 0.0143 [0.032] 1137.9 2.42 [0.298] px50 0.9846 0.9965 0.0775 0.0350 0.0111 [0.007] 1249.3 0.57 [0.751] bux 0.9557 0.9789 0.0835 0.0387 0.0215 [0.006] 1060.5 4.52 [0.104]] cac40 0.9953 0.9931 0.0488 0.0239 0.0134 [0.007] 1291.1 1.49 [0.474] dax 0.9778 0.9865 0.0653 0.0306 0.0146 [0.022] 1215.0 1.98 [0.371] ftse 0.9262 0.9864 0.0578 0.0249 0.0049 [0.225] 1473.4 1.68 [0.43] nikkei 0.9376 0.9961 0.0847 0.0370 -0.0006 [0.915] 1253.9 1.07 [0.586] sp500 0.9870 0.9789 0.0232 0.0462 0.0091 [0.029] 1430.2 1.49 [0.474] note: p-values have been presented in brackets. table 2. estimation results of the msh(3) models series 11p 22p 33p 1σ 2σ 3σ lr jarque and bery test (residual) wig 0.6032 0.9685 0.9721 0.0113 0.0628 0.0288 1164.7 6.50 [0.039] hsi 0.9949 0.9812 0.9836 0.0287 0.0561 0.0834 1146.0 5.29 [0.071] bux 0.9406 0.9827 0.9674 0.1034 0.0354 0.0566 1070.5 10.6 [0.005] dax 0.9281 0.9940 0.9835 0.0962 0.0286 0.0462 1230.4 5.95 [0.051] ftse 0.9913 0.9937 0.9081 0.0337 0.0177 0.1013 1513.3 7.89 [0.019] sp500 0.9930 0.9026 0.9642 0.0206 0.0600 0.0321 1441.9 5.24 [0.073] note: p-values have been presented in brackets. the high volatility periods that were pointed out on the basis of msh models are presented in the table 3. these periods represent the high variance regime. the common markets (indexes) periods are following (marked in the table 3): − 09.1998–12.2000 (the consequences of the russian crisis); − 02.2001–12.2002 (the beginning of this crisis is seen in the dax and sp500 indexes, then in the wig, rts and ftse indexes); − 01.2003–12.2004 (the beginning of this crisis is seen in the sp500 and ftse indexes, then in the wig, dax and rts indexes); − 07.2008–08.2009 (the beginning of this crisis is seen in the sp500 index, then in the rts, ftse, wig, px500, dax, sax and bux indexes). monika kośko 78 table 3. the high volatility periods pointed out on the basis of msh models wig rts sax hsi px50 bux cac40 dax ftse nikkei sp500 11.9512.00 10.9504.01 01.9602.96 04.9705.02 06.9805.99 01.9604.96 01.9705.03 07.9712.97 09.9711.97 03.02 09.9806.00 09.0102.02 11.0101.02 11.9602.97 07.0708.09 09.0808.09 12.9603.97 07.0708.09 07.9803.00 10.9802.99 10.0805.09 02.0112.01 07.0301.04 10.0312.03 12.9701.99 10.9701.98 02.0105.01 08.0111.01 05.0204.03 04.0609.06 04.9901.00 05.9801.00 08.0105.03 06.0209.00 01.0808.09 03.0703.08 03.0012.00 09.0001.01 09.0311.03 01.0305.03 09.0808.09 07.0808.09 03.0107.01 11.0512-05 10.0808.09 09.0805.09 05.02-07.02 05.0609.06 11.02-05.03 10.0808.09 01.04-03.04 09.04-04.05 10.08-08.09 the first period, shaded in the table 3, corresponds with the russian crisis and its consequences can be notice in the 1998 year. this crisis has sunk into a memory of worldwide economic recession. its beginnings were noticeable firstly in czech republic in march 1997 in the form of the banking crisis, and in markets of south-east asia. the analysis of these periods shows that the russian crisis periods coincide with the high volatility periods for the most index series. the next distinguished common markets periods of high volatility might be qualified as a derivative of economic crisis begun in the half of 2008 year in usa. these periods weren’t identified only in the case of the his and cac40 indexes. the contagion effect on financial markets in the crisis periods is seen more clearly. the high volatility periods occurrence usually at the beginning of the financial crisis. then the shocks are transmitted between markets. the analysis evidences these transmissions between financial markets, what confirms an occurrence of the contagion effect. the markov switching model with application to contagion effect … 79 5. summary the main aim of this article was an application of the markov switching models to identification and analysis the contagion effect in the capital markets. the research allowed to distinguish two regimes with different volatility levels, the calm period (low volatility) and the crisis period (high volatility). basing on this classification the identification of some interdependence pattern between markets was created. in the empirical part of the article the narrow contagion effect definition was assumed. according to this definition the cause of the shocks transmission are the herd behavior of investors mainly and the fundamental economic factors are not taken under attention. in these researches there were distinguished the high volatility periods (on the indicated capital markets) and the occurrence of these periods was analyzed. the conclusion is that in the case of the russian crisis and the crisis which begun in the 2008 year, the patterns of the high volatility periods were very similar. the capital markets are connected with themselves what is noticed clearly in investors behavior during the beginning of the crisis period. the high volatility periods usually occur with the price decrease and they coincide or come across on themselves. the more detailed analysis using the switching structure could be carried out on the basis of the multivariate markov switching models for two series for example, what would allow appointing the periods of common volatility. references dempster, a. p., laird, n. m., rubin, d. b. (1977), maximum likelihood from incomplete data via the em algorithm, journal of the royal statistical society, 39, 1–38. doman, m., doman, r. (2004), ekonometryczne modelowanie dynamiki polskiego rynku finansowego (an econometric modeling of polish financial market dynamic), wydawnictwo akademii ekonomicznej, poznań. hamilton, j. d. (1989), a new approach to the economic analysis of nonstationary time series and the business cycle, econometrics , 57, 357–384. edwards, s., susmel r. (2001), volatility dependence and contagion in emerging equity markets, journal of development economics, 66, 505–532. krolzig, h.-m. (1997), markov-switching vector autoregression. modeling statistical inference and application to business cycle analysis, springer verlag edition. stawicki, j. (2004), wykorzystanie łańcuchów markowa w analizie rynków kapitałowych (the markov chains in capital markets analysis), wydawnictwo umk, toruń. fiszeder, p. (2009), modele klasy garch w empirycznych badaniach finansowych (the garch models in the empirical financial research), wydawnictwo naukowe umk, toruń. moore, t., wang, p. (2007), volatility in stock returns for new eu member states: markov regime switching model, international review of financial analysis, 1, 282–292. przełącznikowy model typu markowa w badaniu efektu zarażania na rynkach kapitałowych z a r y s t r e ś c i. artykuł stanowi próbę analizy efektu zarażania na rynkach kapitałowych z wykorzystaniem przełącznikowego modelu typu markowa ms. badanie przeprowadzono monika kośko 80 w oparciu o indeksy giełdowe wybranych krajów. wyznaczono stany o niskiej i wysokiej zmienności dla poszczególnych szeregów oraz przeprowadzono analizę ich występowania w odniesieniu do globalnych kryzysów finansowych. stwierdzono występowanie wspólnych okresów zmienności dla badanych indeksy. okresy te wskazują na przenoszenie szoków pomiędzy rynkami, potwierdzając występowanie efektu zarażania na tych rynkach. s ł o w a k l u c z o w e: przełącznikowy model typu markowa, efekt zarażania. microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 dorota górecka, dominik śliwicki nicolaus copernicus university in toruń application of panel data models to exchange rates’ modeling for scandinavian and central and eastern european countries a b s t r a c t. in the paper the purchasing power parity (ppp) theory for 6 states belonging to oecd, namely denmark, norway, sweden, poland, czech republic and hungary, was examined. in order to do that the ips panel unit root test was employed. after establishing that the exchange rates permanently deviate from the long-term equilibrium rate and the ppp theory is at variance with the data, two panel models were estimated to identify factors that influence exchange rates of scandinavian and cefta countries. k e y w o r d s: purchasing power parity, long-term equilibrium exchange rate, panel models with fixed individual effects. 1. introduction literature referring to the exchange rates and calculation of their real equilibrium levels is very rich. a methodology related to this problems is depicted in the work of hinkel and montiel (1999) while a review of the results of the empirical investigations can be found in the article of edwards and savastano (1999). at least three concepts have been used so far in the analyses to determine the equilibrium exchange rate, namely: purchasing power parity theory (johansen, juselius, 1992; macdonald, nagayasu, 1998), fundamental theory (williamson, 1983, 1994) and behavioral theory (clark, macdonald, 1998, 2004). development of the econometric estimation methods for the nonstationary panel data has caused that cointegration models for this kind of data are used in many works (habermeier, mesquita, 1999; macdonald, ricci, 2001). into this stream an empirical investigation described in the article may be included. the investigation constitutes an attempt to determine dependencies between exchange rate and macroeconomic factors for three scandinavian (denmark, norway and sweden) and three cefta countries (czech republic, hungary and poland) by means of panel data models. such set of countries was dorota górecka, dominik śliwicki 52 selected in order to check if some differences concerning the exchange rate modeling between the developed and the developing european countries that have not adopted a common currency will occur. 2. purchasing power parity theory and the foreign exchange rate of oecd countries investigation concerns the inverse real exchange rate of the euro in relation to the currencies of three scandinavian countries (danish krone, norwegian krone and swedish krone) and in relation to the currencies of three cefta countries (czech krone, polish zloty, hungarian forint). the analysis is based on the data spanning the period from the first quarter of 1999 to the fourth quarter of 2008 (40 observations)1. the real exchange rate rerdc is calculated on the basis of formula: , d fdc dc p pe rer = (1) where: dce represents a simple nominal exchange rate, dp denotes a domestic price level, fp denotes a foreign price level. as price deflators the producer price indices (ppi) have been used. verification of the non-stationarity of the foreign exchange rates has been performed on the panel data2 (the first panel comprised scandinavian countries and the second one – central and eastern european countries) with the aid of ips unit root test (im, pesaran, shin, 1997, 2003) that has a relatively high power and satisfactory properties for short time series and small number of 1 quarterly data are used because the data concerning gdp (used in further analysis) are not available in monthly frequency. 2 until the moment of proposing the non-stationarity examination techniques for panels the analyses of bilateral exchange rates have provided very little evidence on the ppp theory. for instance the literature review made by edwards and savastano (1999) shows that in the case of developing countries the hypothesis that the real exchange rate series contain (at least) one unit root could not be rejected in 40 out of 54 individual country tests of rer stationarity. in turn, in the case of the empirical study of real effective exchange rates for the 51 largest economies in the world for the 1971-1997 period the relative version of the ppp theory was confirmed only for 14 countries at 10% significance level and merely for 2 at 1% significance level. research carried out for the panel data with the help of ips test have shaken the previous conclusions in favor of the ppp theory – at 5% significance level the null hypothesis of non-stationarity was rejected for all 51 countries (habermeier, mesquita, 1999). utilization of the panel non-stationarity examination techniques enables data range extension by adding the observations of the variables from other states and decreases ipso facto the risk of structural changes occurring. application of panel data models to exchange rates’ modeling … 53 cross-section data as the results of separate analysis of the currency exchange rate for each of six investigated oecd countries have not provided much evidence on the ppp theory – at 10% significance level there was no basis for rejection the null hypothesis of non-stationarity of the examined foreign exchange rates3. table 1. the results of the panel stationarity test (1999 1st quarter – 2008 4th quarter) countries deterministic component lm statistic p-value scandinavian constant 0.104 0.504 constant + trend 0.604 0.727 central and eastern european constant -1.358 0.087 constant + trend -0.405 0.343 on the basis of values of statistics presented in table 1 one may say that at 5% significance level there is no basis for rejection the null hypothesis of nonstationarity of the investigated foreign exchange rates. it may be treated as an evidence of the fact that purchasing power parity theory is false in the cases of scandinavian and cefta countries4. in order to verify if the global financial crisis has contributed to the failure of the ppp hypothesis the analysis has been carried out for datasets containing 32 and 36 observations. table 2. the results of the panel stationarity test (1999 1st quarter – 2006 4th quarter) countries deterministic component lm statistic p-value scandinavian constant -0.985 0.162 constant + trend 0.337 0.633 central and eastern european constant -1.253 0.105 constant + trend 0.238 0.594 table 3. the results of the panel stationarity test (1999 1st quarter – 2007 4th quarter) countries deterministic component lm statistic p-value scandinavian constant -0.623 0.267 constant + trend -0.459 0.323 central and eastern european constant -1.459 0.072 constant + trend -0.010 0.496 3 analysis has been also performed for monthly data and it has not provided much support for the ppp theory – the null hypothesis of non-stationarity was rejected only in the case of hungarian forint exchange rate (at 1% significance level). in the case of the monthly and quarterly data concerning the inverse real exchange rate of the u.s. dollar in relation to the currencies of six investigated oecd countries the null hypothesis was not rejected (at 10% significance level) even once. 4 empirical research with the help of the ips test has been also carried out for monthly data concerning the inverse real exchange rate of the euro and for monthly and quarterly data concerning the inverse real exchange rate of the u.s. dollar in relation to the currencies of six examined oecd countries. the ppp theory has not received much support from these studies as at 5% significance level the null hypothesis of non-stationarity was rejected only once (in the case of monthly exchange rates of central and eastern european countries against the euro). dorota górecka, dominik śliwicki 54 on the basis of values of the statistics presented in tables 2 and 3 one may say that the structural change that occurred at the end of the time series had no influence on the results of the research at 5% significance level there is no basis for rejection the null hypothesis of non-stationarity of the investigated foreign exchange rates. 3. reasons of the exchange rate volatility in connection with the conclusion presented in the previous part of this article the question of the reasons of the exchange rate deviations from the ppp should be raised. one of the most known conceptions explaining behavior of the real exchange rates in the long-run is so-called balassa-samuelson effect (balassa, 1964). the essence of the b-s effect is that the increase of the productivity in the tradable goods sector causes inflation in the non-tradable goods sector and raises ipso facto overall price index, which – in turn – leads to an appreciation of the real exchange rate. the b-s hypothesis concerns the catching-up economies, including transition countries entering the european union. the phenomenon occurring has been confirmed in many researches (rogoff, 1996), in which the significant positive influence of the economic growth on the real exchange rate has been demonstrated. another potential exchange rate determinant is government expenditure. it moves the internal demand towards the non-tradable items causing the increase of their prices and the real exchange rate appreciation (habermeier, mesquita, 1999). a different factor influencing the real exchange rate is terms of trade defined as a ratio between export and import prices. the rise of this index value (caused either by the increase of the export prices or by the decrease of the import prices) means a decline of the domestic products competitiveness and leads to a depreciation of the exchange rate (baffes, elbadawi, o’connell, 1997; habermeier, mesquita, 1999). the real exchange rate changes may be also explained by the behavior of the real interest rate. relatively higher national rate of interest, boosting the foreign currency supply, contributes to the drop of the national currency rate, that is its appreciation (brook, hagreaves, 2001; chortareas, driver, 2001). application of panel data models to exchange rates’ modeling … 55 4. panel data models for the real exchange rates of the oecd countries below two panel data models will be presented5. the first of them concerns scandinavian countries (denmark, norway and sweden) and the second one – central and eastern european countries (czech republic, hungary and poland). in both cases an explained variable is the real exchange rate of the examined countries against the euro (the nominal rate deflated by the producer price index – ppi). among the explanatory variables are: relative economic growth6 (representing the b-s effect), trade balance in relation to gdp (evidencing the competitiveness of a given economy) and relative real interest rate. it is expected that in accordance with the above-mentioned mechanisms the signs of the parameter estimates for all three explanatory variables will be negative. table 4. the results of the model estimation for the group of scandinavian countries variable parameter estimate student’s t-statistic p-value r2 constant 8.801 68.79 0.0000*** 80.12% interest rate 0.128 4.058 0.0000*** trade balance/gdp -6.682 -5.021 0.0000*** the results of estimation of the panel data model with fixed individual effects for developed scandinavian countries point to the existence of the significant relationship between real exchange rate and both the trade balance and the relative real interest rate. in the case of the relative real interest rate the sign of the parameter estimate is at variance with the predictions, which can be explained among other things by the influence of the world interest rates or different kinds of expectations. on the other hand, in accordance with the predictions the impact of the relative economic growth on the exchange rate in the case of scandinavian countries turned out to be statistically insignificant. in the case of the panel data model with fixed individual effects for developing countries belonging to cefta a direction of influence of the explanatory variables is in accordance with the earlier formulated expectations: the increase of the relative gdp as well as the trade surplus contribute to the decrease of the exchange rate, that is its appreciation. moreover, the influence of the relative real interest rate on the exchange rate is also negative, albeit statistically insignificant, which leads to the conclusion that in the case of central and eastern european countries behavior of the interest rates affects the exchange rate much more weakly than behavior of other macroeconomic factors. 5 panel data models, contrary to the time series models, allow one to investigate general relationships for fixed groups of chosen countries. 6 the term ‘relative’ means in this case comparison with the euro area. figure 1. fitted and actual values of the exchange rate of the danish krone (1999 1st quarter – 2008 4th quarter) figure 2. fitted and actual values of the exchange rate of the swedish krone (i quarter 1999 1st quarter – 2008 4th quarter) 6,9 7 7,1 7,2 7,3 7,4 7,5 7,6 7,7 7,8 7,9 8 i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 fitted actual 7 7,5 8 8,5 9 9,5 10 i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 fitted actual figure 3. fitted and actual values of the exchange rate of the norwegian krone (1999 1st quarter – 2008 4th quarter) table 5. the results of the model estimation for the group of central and eastern european countries variable parameter estimate student’s t-statistic p-value r2 constant 12.615 71.86 0.0000*** 99.07% relative economic growth -0.754 -3.607 0.0005*** trade balance/gdp -24.800 -5.507 0.0000*** figure 4. fitted and actual values of the exchange rate of the polish zloty (1999 1st quarter – 2008 4th quarter) 0 2 4 6 8 10 12 i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 fitted actual 0 1 2 3 4 5 6 7 i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 fitted actual dorota górecka, dominik śliwicki 58 figure 5. fitted and actual values of the exchange rate of the czech krone (1999 1st quarter – 2008 4th quarter) figure 6. fitted and actual values of the exchange rate of 100 hungarian forints (1999 1st quarter – 2008 4th quarter) summary the main conclusion drawn from the analysis carried out in this paper is rejection of the hypothesis that the exchange rates of the examined scandina0 5 10 15 20 25 30 35 40 i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 fitted actual 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv i ii ii i iv 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 fitted actual application of panel data models to exchange rates’ modeling … 59 vian and cefta countries in the years 1999-2008 were shaping according to the purchasing power parity theory. the results obtained indicate that the real exchange rate of six oecd countries is determined by such economic factors as economic growth, trade balance and interest rates. further research concerning exchange rates may, first of all, focus on widening the cross-section of the panels with other countries, for instance bulgaria, romania, croatia and ukraine in the case of central and eastern european countries. another possibility is taking into account in the models additional variables such as net foreign assets, real wages, private consumption, government spending, budget deficit, public debt or foreign direct investments (bęzabojanowska, macdonald, 2009). finally, these models may be employed in forecasting. references baffes, j., elbadawi, i.a., o’connell, s.a. (1997), single-equation estimation of the equilibrium real exchange rate, wb working paper 08/20/97, world bank, washington. balassa, b. (1964), the purchasing power parity doctrine: a reappraisal, journal of political economy, 72, 584–596. bęza-bojanowska, j., macdonald, r. (2009), the behavioural zloty/euro equilibrium exchange rate, cesifo working paper series no. 2568, university of glasgow, glasgow. brook, a., hagreaves, d. (2001), ppp-based analysis of new zealand’s equilibrium exchange rate, discussion paper series dp2001/01, reserve bank of new zealand, wellington. cassel, g. (1918), abnormal deviations in international exchanges, economic journal, blackwell publishers, oxford, 413–415. chortareas, g.e., driver, r.l. (2001), ppp and the real exchange rate – real interest rate differential puzzle revisited: evidence from non-stationary panel data, bank of england, london. clark, p.b., macdonald, r. (1998), exchange rates and economic fundamentals: a methodological comparison of beers and feers, imf working paper 98/67, international monetary fund, washington. clark, p.b., macdonald, r. (2004), filtering the beer: a permanent and transitory decomposition, global finance journal, 15, 29–56. edwards, s., savastano, m.a. (1999), exchange rates in emerging economies: what do we know? what do we need to know?, nber working paper no. 7228, national bureau of economic research, cambridge. habermeier, k., mesquita, m. (1999), long-run exchange rate dynamics: a panel data study, imf working paper wp/99/50, international monetary fund, washington. hinkle, l., montiel, p. (1999), exchange rates misalignment. concepts and measures for developing countries, oxford university press, new york. im, k.s., pesaran, m.h., shin, y. (1997), testing for unit roots in heterogeneous panels, discussion paper, university of cambridge, cambridge. im, k.s., pesaran, m.h., shin, y. (2003), testing for unit roots in heterogeneous panels, journal of econometrics, 115, 53–74. johansen, s., juselius, k. (1992), testing structural hypothesis in a multivariate cointegration analysis of the ppp and the uip for uk, journal of econometrics, 53, 211–244. macdonald, r., nagayasu, j. (1998), on the japanese yen-u.s. dollar exchange rate: a structural economic model based on real interest differentials, journal of the japanese and international economies, 12, 75–102. dorota górecka, dominik śliwicki 60 macdonald, r., ricci, l. (2001), ppp and balassa-samuelson effect: the role of the distribution sector, imf working paper wp/01/38, international monetary fund, washington. rogoff, k. (1996), the purchasing power parity puzzle, journal of economic literature, 34, 647–668. williamson, j. (1983), the exchange rate system, institute for international economics, washington. williamson, j. (1994), estimates of feers, [in:] williamson j. (ed.), estimating equilibrium exchange rates, institute for international economics, washington. zastosowanie modeli panelowych do modelowania kursów walutowych dla krajów skandynawskich i europy środkowo-wschodniej z a r y s t r e ś c i. w artykule podjęto próbę empirycznej weryfikacji teorii parytetu siły nabywczej w odniesieniu do sześciu krajów członkowskich oecd: danii, norwegii i szwecji oraz czech, polski i węgier. w związku z tym, że uzyskane wyniki nie potwierdziły prawdziwości weryfikowanej teorii, celem pracy stało się zidentyfikowanie czynników wpływających na poziom kursów walutowych państw skandynawskich oraz państw europy środkowo-wschodniej w latach 1999-2008 oraz wskazanie różnic między nimi. s ł o w a k l u c z o w e: teoria parytetu siły nabywczej (purchasing power parity, ppp), długookresowy kurs równowagi, modele panelowe z ustalonymi efektami indywidualnymi. microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 marek szajt technical university of częstochowa estimation of disproportions in patent activity of oecd countries using spatio-temporal methods a b s t r a c t. the article contains a presentation of possibility of using panel-based sample and modelling based on this sample as methods of determining indicators of patent activity. the research was conducted with the help of data from european countries. results in association with used methodology, which takes into account modern approach to stationary and cointegration for panel-based samples, indicate the usefulness of applied methods. k e y w o r d s: patent activity, panel model, decomposition of intercept. 1. introduction within the area of innovation, which enjoys an increasing interest of the economists, there are many ways of measurement. in the macro-economics conception – due to the requirements concerning the length of time series – space-time sample or panel sample are used frequently. their advantages include, apart from the opportunity to conduct research itself, the possibility of obtaining results which are comparable for various objects, which are received on the basis of decomposition of a random term or intercept. these study are directly connected with the patents based on the inventions understood as “original conception of technical innovation, which contains theoretical possibility of action” (budnikowski, 1995). the patent activity is one of the most accessible measures of innovation activity due to the possibility of obtaining fairly comparable data, which is a result of the legal framework behind the acceptance and granting patents. the available information comes mainly from the world intellectual property organization (wipo) and the european patent office (epo). putting aside the character of explanatory variables, the equation used to describe the patent activity with the use of panel data enables obtaining, as a result of decomposition, specific indices of patent activity. the differences between these values have a direct influence on the theomarek szajt 92 retical values of the dependent variable (depending on the model – additive or multiplicative) diversifying its value for various objects with the same basic assumptions. 2. assumptions in the present research the following assumptions were accepted: − the measure of patent activity is the number of patent applications submitted with the epo per one thousand professionally active persons, − the determinants of patent activity are gross outlays for the research and development activities as well as the researchers working within the research and development area, − the measurement (test) is of time cross-sectional character, and the data concern the periods from 1995–2005 and the european countries belonging to the oecd (together with latvia and estonia); on the one hand the use of longer sequences is impossible – lack of data, on the other hand there is a threat of disruption of the present relations by the introduced system changes, particularly in the central and eastern europe area, − the possibility of interpolation is accepted in the case of occurring incidental lack of data or reproducibility of collected results less frequent than annual. depending on the form of studied process, the segment method or a fitted trend function which has possibly most simplified analytical form (it concerns mainly a degree of a polynomial) are used, − source data coming from the analyses of the eurostat, oecd, wipo and national statistical offices is not directly corrected in the cases of suspected errors or inaccuracies. 3. introductory calculations at the initial stage the space-time sequences which were supposed to form the basis of the model construction, were taken into consideration. time series of 11 annual observations, despite they are short, seem to be sufficient to observe non-stationarity. what is more important, we want to treat the conclusions based on final calculations as independent of time factor. in this situation nonstationarity of these series should be researched, assuming that integration order is not higher than 2 in the case of annual data (gruszczyński, podgórska, 2004). in order to realize it, the procedures contained in the eviews package were used. these procedures enable a relatively fast evaluation of possible lack of stationarity or the evaluation of the integration order. the tables below present the results of a few unit root tests, which indicate the existence of unit root. estimation of disproportions in patent activity of oecd countries … 93 table 1. the results of unit root tests for levels (h0: δ = 0) variable estimator: method: newey-west andrews statistic p-value statistic p-value pet levin, lin & chu t* -7.1708 0.0000 -7.0058 0.0000 im, pesaran and shin wstatistic -1.9999 0.0228 -1.9999 0.0228 adf fisher χ2 69.8771 0.0331 69.8771 0.0331 pp fisher χ2 118.7770 0.0000 83.3070 0.0022 gerd levin, lin & chu t* -1.1929 0.1165 -8.4510 0.0000 im, pesaran and shin wstatistic 0.8544 0.8036 -5.4559 0.0000 adf fisher χ2 51.6001 0.4110 123.3190 0.0000 pp fisher χ2 54.5887 0.3044 121.3650 0.0000 rech levin, lin & chu t* -3.0271 0.0012 -9.5927 0.0000 im, pesaran and shin wstatistic 1.6752 0.9531 -5.5232 0.0000 adf fisher χ2 39.2213 0.8641 120.3260 0.0000 pp fisher χ2 42.0529 0.7804 117.9310 0.0000 note: * assumes common unit root process. the probabilities for fisher test are computed using an asymptotic chisquare distribution. all other tests assume asymptotic normal distribution. table 2. the unit root tests results for first difference (h0: δ = 0) variable estimator: method: newey-west andrews statistic p-value statistic p-value pet levin, lin & chu t* -10.2833 0.0000 -10.4350 0.0000 im, pesaran and shin wstatistic -6.2851 0.0000 -6.2851 0.0000 adf fisher χ2 133.2590 0.0000 133.2590 0.0000 pp fisher χ2 150.2810 0.0000 142.5140 0.0000 gerd levin, lin & chu t* -9.7824 0.0000 -8.4510 0.0000 im, pesaran and shin wstatistic -5.4559 0.0000 -5.4559 0.0000 adf fisher χ2 123.3190 0.0000 123.3190 0.0000 pp fisher χ2 144.5690 0.0000 121.3650 0.0000 rech levin, lin & chu t* -11.0622 0.0000 -9.5927 0.0000 im, pesaran and shin wstatistic -5.5232 0.0000 -5.5232 0.0000 adf fisher χ2 120.3260 0.0000 120.3260 0.0000 pp fisher χ2 131.2820 0.0000 117.9310 0.0000 note: * assumes common unit root process. regarding the endogenous variable (pet), all the tests results indicate stationarity. the remaining variables are characterized by different results, particularly the ones obtained with the use of newey-west estimator. the andrews estimator, produce more stable bandwidth estimates than the newey-west procedure (indicating the stationarity in this situation), which could be expected taking into account the pp test and the previous research conducted by yinmarek szajt 94 wong cheung and kon s. lai (1997). therefore, taking into account possible existence of unit roots, we can assume that our variables are integrated on order 1, ~ i (1) (what is suggested by the consistent results of all tests). hence assuming the integration order is common for all the variables, we try to test the existence of cointegration in the assumed system, i.e. equation with pet as dependent variable and gerd and rech as independent variables. the estimation with the use of eviews programme gives the possibility of obtaining (in the case of using summarised results) the evaluation of statistics for seven tests. however, the use of these tests is difficult, as they can give (and such is our case) different results. it is connected with the size of applied panel. pedroni (2004), who researched situations of this kind with the use of monte carlo simulation, indicated that the use of panel test-v and group test-rho gives bad results even in the case when the length of time series in the panel is smaller than 20 observations. in such cases, the group test – adf and panel test – adf are more appropriate. the test results are tabulated in table 3. table 3. the results of cointegration test for pedroni residuals in the model of pet on gerd and rech variables alternative hypothesis: common ar coefficients (within-dimension) model type no deterministic trend no deterministic intercept or trend test type statistic p-value statistic p-value panel v-statistic -0.8536 0.8033 0.9798 0.1636 panel rho-statistic 2.3876 0.9915 0.3700 0.6443 panel pp-statistic -0.0985 0.4608 -1.7103 0.0436 panel adf-statistic -0.4230 0.3361 -1.8798 0.0301 alternative hypothesis: individual ar coefficients (between-dimension) test type statistic p-value statistic p-value group rho-statistic 3.0116 0.9987 1.9235 0.9728 group pp-statistic -5.0759 0.0000 -3.5982 0.0002 group adf-statistic -2.5234 0.0058 -3.4899 0.0002 only the results based on “group” tests(recognized as being more powerful than “panel” tests when conducting a research on smaller samples (cf. pedroni, 1995)) – indicate the existence of cointegration. excluding the existence of intercept, tests based on v and rho do not reject the h0 of the lack of cointegration what is undesired from the point of view of this research. however, taking into account the remarks of pedroni, we find the results of adf tests as the more appropriate ones, which reject the h0. however, taking into account the remarks of pedroni, we find the results of adf test which indicate the rejection of h0 as more appropriate. hence, the existence of cointegrating vector can be stated. estimation of disproportions in patent activity of oecd countries … 95 it is worth emphasizing that the pp test also gives expected result. it should be remembered that the “group” tests, in contrast to the “panel” ones, assume that the autoregression coefficients do not have to be homogenous for all objects (hsu-ling and others, 2008). therefore, assuming the low power of the group test-rho, the results of remaining group tests indicating the existence of cointegration are accepted the achieved results do not offer the possibility of making an unambiguous decision by the researcher. on the one hand we can assume that the cointegration vector exists, if we exclude the intercept in our model. however, it should be remembered that this model, due to the panel construction, will have the decomposed intercept. this intercept, depending on the significance of its particular parts will be “complete” intercept (consisted of so many parts as many countries contains the model) or will be equivalent of a few dummies variables included in the model. on the other hand, having in mind the fact of pet stationarity, the recognition of pet variable as integrated of first order seems to be misused. in connection with the indicated doubts concerning the existence of cointegrating relations, the error correction model was proposed which in such cases is one of the most popular tools (strzała, 2005). ids additional advantage is taking into account both shortand long-term relationships. as a result, the interpretation of the decomposed intercepts (fundamental in this research) is more precise. the possible dynamic dependencies are more visible in the estimates of structural parameters (for independent variables). the following form of model was proposed: ,log log)log log)(log1('log 2 112 1111 itit itit ititiit recha gerdprecha gerdppataapata ξβ βδ δρ ++ δ+− −−+=δ − −− (1) where: a’i denotes the intercept decomposed into i = 25 objects – countries, patait – number of patents application, submitted by the residents of a given country i per number of professionally active persons in the period t, gerdpit – gross expenditures on research and development (r+d) activities per the r+d staff working on full-time basis in the country i in the period t, rechait – persons employed as researchers on full time basis in comparison to the number of professionally active persons in the country i in the period t, marek szajt 96 4. results in accordance with the accepted assumptions the estimated model (with the full decomposition of intercept) has showed dependence for logarithms of variables (szajt, 2006). due to the differences in the directions of dependencies between the particular variables for different countries, 16 countries were qualified to the final test. the gretl programme was used to estimate the model. at the beginning a test for variability of intercept was used. the test statistics f(15, 140) = 3.814 with the value p = 1.15252e-005 confirms the validity of estimation of panel model with fixed effects. during the estimation process the insignificant variable gerdt-1 was removed from the model. the final results are presented in the table 4. table 4. the values of structural parameter assessments in power model variable parameter parameter estimate t statistics p-value peta i,t-1 α1 0.5683 -6.9836 0.0000 recha i,t-1 δ2 0.7809 1.9086 0.0584 δgerdpi,t β1 1.0802 3.5093 0.0006 δrecha i,t β2 0.7617 2.6235 0.0097 bet αbe 0.3060 -2.9727 0.0035 czt αcz 0.1137 -5.1812 0.0000 dkt αdk 0.2999 -2.9377 0.0039 det αde 0.4010 -2.5089 0.0133 eet αee 0.0767 -5.0538 0.0000 iet αie 0.2281 -3.8027 0.0002 frt αfr 0.3007 -3.0255 0.0030 lvt αlv 0.0709 -5.4952 0.0000 hut αhu 0.1259 -5.1246 0.0000 nlt αnl 0.3877 -2.8611 0.0049 att αat 0.3329 -3.0377 0.0028 plt αpl 0.0597 -5.3160 0.0000 ptt αpt 0.0853 -5.2087 0.0000 fit αfi 0.2972 -2.4965 0.0137 set αse 0.3160 -2.6370 0.0093 not αno 0.1891 -3.3407 0.0011 it should be noted that all the estimates concerning the decomposed intercept are highly statistically significant, which is to a large degree the objective of this estimation. what is even more important, in connection with the form of function, the intercept has multiplicative character. estimation of disproportions in patent activity of oecd countries … 97 figure 1. the values of assessments of decomposed intercept (for simple countries) presented on the figure 1 constant level 0.1873 reflects the estimated in a test common intercept (an equivalent of the average level of patent activity) for the whole group. in comparison to it, such countries as germany or the netherlands turned out to be absolute leaders, whereas poland, latvia and estonia were outsiders. the consequents of such conclusions are important. in practice, with equal factors determining the patent activity, difference of final reaction – in the long -run will be close to differences presented on figure 1 therefore, a very high or low patent activity of particular countries can be found. as it is seen, this simple method (using panel sample construction) enables obtaining very valuable, comparable indicators. it is also important that, their estimates usually are not strongly sensitive on the changes of main determinants of studied process. in extreme cases, together with characteristic values (for chosen countries) we can obtain “typical” values represented by a common (for all countries) intercept. references budnikowski, f. (1995), ekonomia. innowacje ekonomiczne w gospodarce narodowej (economy. economic innovation in national economy), owpr, rzeszów. gruszczyński, m., podgórska, m. (2004), ekonometria (econometrics), oficyna wydawnicza sgh, warszawa. hsu-ling, ch., yahn-shir, ch., chi-wei, s., ya-wen, ch. (2008), the relationship between stock price and eps: evidence based on taiwan panel data, economics bulletin, vol. 3, no. 30. pedroni, p. (2004), panel cointegration: asymptotic and finite sample properties of pooled time series tests with an application to the ppp hypothesis, econometric theory, 20, 597–625. 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 po la nd la tv ia es to ni a po rtu ga l cz ec h re p. hu ng ar y no rw ay ire la nd fi nl an d de nm ar k fr an ce be lg iu m sw ed en au st ria ne th er la nd s g er m an y marek szajt 98 pedroni, p. (1995), panel cointegration, asymptotic and finite sample properties of pooled time series tests, with an application to the ppp hypothesis, indiana university, working paper in economics, no. 95–031. strzała, k. (2005), korelacja inwestycji i oszczędności w krajach unii europejskiej weryfikacja empiryczna z zastosowaniem podejścia panelowego (the correlation of investment and savings in countries of european union – the empirical verification with use of panel approach), prace i materiały wydziału zarządzania ug – „ekonometryczne modelowanie i prognozowanie wzrostu gospodarczego”, nr 1, gdańsk, 141–157. szajt, m. (2006), modeling of state innovativeness based on space-time models, in: dynamic econometric models, vol. 7, uniwersytet mikołaja kopernika w toruniu, toruń, 231–238. yin-wong, ch., kon, s. l. (1997), bandwidth selection, prewhitening, and the power of the phillips-perron test, econometric theory, 13, cambridge university press, 679–691. szacowanie dysproporcji w aktywności patentowej państw oecd z wykorzystaniem metod przestrzenno-czasowych z a r y s t r e ś c i. w artykule przedstawiono możliwość zastosowania próby panelowej i modelowania w oparciu o nią jako metody wyznaczenia wskaźników aktywności patentowej. badanie przeprowadzono z wykorzystaniem danych dla państw europejskich. otrzymane wyniki w zestawieniu z zastosowaną metodologią uwzględniającą nowoczesne podejście do badania stacjonarności i kointegracji dla prób panelowych, wskazują na użyteczność stosowanych metod. s ł o w a k l u c z o w e: aktywność patentowa, model panelowy, dekompozycja wyrazu wolnego. microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 joanna górka nicolaus copernicus university in toruń application of the family of sign rca models for obtaining the selected risk measures† a b s t r a c t. accurate modelling of risk is very important in finance. there are many alternative risk measures, however none of them is dominating. this paper proposes to use the family of sign rca models to obtain the value-at-risk (var) and expected shortfall (es) measures. for models from the family of sign rca models and ar-garch model the one-step forecasts of var were calculated based on rolling estimates from the given model using different window sizes. to obtain the var and es measures the filtered historical simulation was used in new version proposed by christoffersen. the results were verified using backtesting and the loss function. k e y w o r d s: family of sign rca models, risk measures, value at risk, expected shortfall. 1. introduction random coefficient autoregressive models (rca) are the straightforward generalization of the constant coefficient autoregressive models. a full description of this class of models including their properties, estimation methods and some application was originally presented by nicholls and quinn (1982). in later years, these models have been not so popular like garch models (bollerslev, 1986; engle, 1982) in general. garch models are easy to understand and estimate and they could describe a non-linear dynamics of financial time series. however, in the last decade one can see that rca models gained more interest again. as a result some of rca model were produced. the aim of this paper is to apply the family of sign rca models to obtain the selected risk measures for daily and weekly data. risk measures through different calculation method are obtained. † this work was financed from the polish science budget resources in the years 2008-2010 as the research project n n111 434034. joanna górka 40 2. the family of sign rca models in the table 1 equations of individual models from the family of sign rca models and their names were presented. table 1. the family of sign rca models (without conditions) model model equations equation rca(1) ( ) tttt yy εδφ ++= −1 i sign rca(1) ( ) ttttt ysy εδφ +φ++= −− 11 ii rca(1)-ma(1) ( ) 11 −− +++= ttttt yy θεεδφ iii sign rca(1)-ma(1) ( ) 111 −−− ++φ++= tttttt ysy θεεδφ iv rca(1)-garch(1,1) ( ) tttt yy εδφ ++= −1 , ttt zh=ε 11 2 110 −− ++= ttt hh βεαα v sign rca(1)-garch(1,1) ( ) ttttt ysy εδφ +φ++= −− 11 , ttt zh=ε 11 2 110 −− ++= ttt hh βεαα vi note: ts – sign function is described by equation (3); φ , θ , φ , iα , 1β – model parameters. to ensure the existence of the i-vi models the following assumption must be satisfied: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 2 2 0 0 , 0 0 ~ ε δ σ σ ε δ iid t t , (1) 122 <+ δσφ . (2) the sign function, described by following formula ⎪ ⎩ ⎪ ⎨ ⎧ <− = > = 0dla1 0dla0 0dla1 t t t t y y y s , (3) has the interpretation: if φ>+ tδφ , the negative value of φ means that the negative (positive) observation values at time 1−t correspond to a decrease (increase) of observation values at time t . in the case of stock returns it would suggest (for returns) that after a decrease of stock returns the higher decrease of stock returns occurs than expected, and in the case of the increase of stock returns the lower increase in stock returns occurs than expected. condition (2) is necessary and sufficient for the second-order stationarity of process described by equation i, however conditions (1)-(2) ensure the strict application of the family of sign rca models … 41 stationarity of this process. if conditions (1)-(2) are satisfied, then processes described by equations ii-iv are stationary in mean. theoretical properties of processes described by equations i-vi, satisfying conditions (1)-(2) can be found in several articles (appadoo, thavaneswaran, singh, 2006; aue, 2004; górka, 2008; thavaneswaran, appadoo, bector, 2006; thavaneswaran, appadoo, 2006). residuals from the rca model can be described by the garch model (thavaneswaran, peiris, appadoo, 2008; thavaneswaran, appadoo, ghahramani, 2009). then, the rca(1)-garch(p,q) model described by equation v, where ( )2,0~ zt nz σ , 00 >α , 0≥iα and 0≥jβ , is obtained. when the sign function is added to the rca-garch model, then the process described by equation vi is obtained (thavaneswaran, appadoo, ghahramani, 2009). the conditions ensuring the positive value of conditional variance of this process are following: ( )2,0~ zt nz σ , 00 >α , 0≥iα , 0≥jβ , 0α≤φ . 3. the selected risk measures in this paper two tools for measuring market risk were used, i. e. value-atrisk (var) and expected shortfall (es). value-at-risk is the maximum loss over a target horizon such that there is a low, prespecified probability that the actual loss will be larger. expected shortfall is a coherent alternative to valueat-risk (acerbi, tasche, 2002). it is the expected loss conditional on exceeding var. one-step-ahead conditional forecasts of value-at-risk are calculated in two ways. firstly, value-at-risk is calculated by formula: ( ) ,111 ασμα zvar tttt l t +++ += where tt |1+μ , tt |1+σ are one-step-ahead conditional forecasts of mean and volatility respectively. secondly, the formula proposed by christoffersen (2009) is used: ( ) ,111 t u tttt l t qvar ασμα +++ += where tuqα denotes the percentile of the set of standardization historical shocks tu . expected shortfall for the one-step forecast can be calculated as: − average of values exceeding var (yy) (yamai, yoshiba, 2002), − weighted average of values exceeding var (dowd) (dowd, 2002), joanna górka 42 − filtered historical simulation (fhs), in the version proposed by christoffersen (2009), i. e. ( ) ( ) ,11 1 1 1 11 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ <⋅= + + = ++ ∑ tt l t t t t tttt var uu t es σ α α σα where ( )∗1 denotes the indicator function returning a unit if the argument is true, and zero otherwise; α is tolerance level. to check the accuracy of risk measures the backtesting of var and es using the traditional var tests and the loss function was conducted. the traditional var tests used to compare results are following: − proportion of failures test1 – lrpof, − independence test – lrind, − time between failures test – lrtbf, the loss functions used to compare results are following: − regulatory loss function – rl − firm’s loss function – fl − loss function with the expected loss proposed by angelidis, degiannakis (2006). it can be described as msemaelfes += where ∑ = = n t tfn mae 1 |1 1 , ∑ = = n t tfn mse 1 |2 1 while: ⎪⎩ ⎪ ⎨ ⎧ −≤+ −> = ++ + + , ,0 ,1,1 ,1 1|1 trttrt trt t varresr varr f ( )⎪⎩ ⎪ ⎨ ⎧ −≤+ −> = ++ + + . ,0 ,1 2 ,1 ,1 1|2 trttrt trt t varresr varr f 4. empirical results the data used in the empirical application are eight stock exchange indexes and thirty three share prices of the polish firms’ from the warsaw stock exchange. it gives forty one time series. the data were obtained from bossa.pl for the period from november 30, 1998 to november 4, 2008, what gives 2490 daily percentage log returns and 493 weekly percentage log returns. the calculations were carried out in the gauss and microsoft excel. 1 other name of this test is the kupiec test. application of the family of sign rca models … 43 firstly, for each returns series the descriptive statistics and some tests were calculated. all series have positive kurtosis (leptokurtic). some of returns series are autocorrelated. secondly, parameters of six models from the sign rca family were estimated using maximum likelihood (mle). the number of models from the family sign rca models with statistically significant parameters for the total sample is presented in the table 2. table 2. the number of models with significant parameters from the family of sign rca models for percentage log returns (2490 daily data and 493 weekly data) model α = 5% α = 10% daily data weekly data daily data weekly data ar(1) 26 9 27 10 rca(1) 24 5 24 10 sign rca(1) 1 2 3 2 rca(1)-ma(1) 15 21 19 22 sign rca(1)-ma(1) 5 3 7 4 rca(1)-garch(1,1) 25 2 25 2 sign rca(1)-garch(1,1) 1 1 it is seen that the models like ar(1), rca(1), rca(1)-ma(1) and rca(1)garch(1,1) were found in about 50 percentage of cases. for smaller samples, similar results are obtained (see the table 3). models with sign function occur very seldom in empirical time series (see also the table 3) both at the 5% and 10% significance level, and also for different size of sample and different level of data aggregation (both daily and weekly data). table 3. the number of models with significant parameters from the family of sign rca models for percentage log returns (1500 daily data and 300 weekly data) model α = 5% α = 10% daily data weekly data daily data weekly data ar(1) 23 12 25 13 rca(1) 22 10 24 11 sign rca(1) 2 1 3 3 rca(1)-ma(1) 20 24 22 25 sign rca(1)-ma(1) 2 7 3 10 rca(1)-garch(1,1) 18 20 sign rca(1)-garch(1,1) 1 2 1 for example, the rca models for the selected indexes are presented in the table 4. joanna górka 44 on the basis of models from the family of sign rca models and argarch model fitted to the different window size, i.e, n =250, 500 and 1500 the one-step ahead forecasts of var and es were made forecasting one-stepahead from the end of window till the next 500 observations which were hold out. it should be pointed out that the last observation of each sample (n =250, 500 and 1500) is placed at the same point at time, hence these samples can be treated as overlapping. each one-step ahead forecast was generated from estimates of the given model2 using a sequence of rolling windows (with window size of 250, 500 and 1000 observations) which were moved 500 times by one observation on time axis. table 4. the rca model for the selected indexes wig-budow wig-spozyw daily data weekly data daily data weekly data φ 0.131 0.133 0.119 0.186 ( )φs 0.024 0.053 0.026 0.059 2 εσ 1.799 12.441 1.300 6.835 2 δσ 0.199 0.135 0.291 0.357 ln l -4480.33 -1348.43 -4146.82 -1244.88 q(3) 4.810 5.784 19.371* 2.728 arch(3) 73.649* 11.911* 90.960* 24.364* aic 8966.66 2702.86 8299.64 2495.76 bic 8984.12 2715.46 8317.10 2508.36 note: q(3) – the value of the ljung-box q-statistic up to 3 lags, arch(3) – the value of the engle arch test statistics up to 3 lags, aic – akaike information criterion, bic – bayesian information criterion. the backtesting of forecasts of var and es measures were carried out (the example results were presented in table 5–7). the empirically determined probability for the proportion of failures test for different window size is presented in table 5. almost all forecasts of var are underestimated. only for rca-ma with fhs method the var forecasts are overestimated. it is seen that as the window size decreases the scale of underestimation decreases, too. the results of traditional var tests and loss function for the var forecasts were carried out from rolling estimation of models using window size of 250 observations for one of selected index (see table 6). 2 forecasts were carried out form models with statistically significant parameters obtained in the basic sample. application of the family of sign rca models … 45 table 5. the empirically determined probability for the proportion of failures test for different window size (wig-spozyw) model empirically determined probability n = 1500 n = 500 n = 250 ar-garch 6.8% 6.8% 6.6% ar-garch (fhs) 10.8% 10.8% 8.8% rca 11,0% 9,0% 6.6% rca-garch 6.8% 6.6% 6.6% rca-garch (fhs) 10.8% 10,0% 8.8% rca (fhs) 13.4% 10.6% 7.4% rca-ma 11.8% 8.2% 6.4% rca-ma (fhs) 3.8% 3.6% 2.2% sign rca 11.4% 9.2% 6.8% sign rca-garch 6.8% 6.8% 7.2% sign rca-garch (fhs) 10.8% 10,0% 8.4% sign rca (fhs) 13.8% 10.6% 7.4% sign rca-ma 12,0% 8.6% 6,0% sign rca-ma (fhs) 9.8% 6.8% 4.8% note: n denotes the window size. table 6. results of traditional var tests for forecasts carried out from rolling estimation of models using window size of 250 observations for the wig-spozyw index model lrpof lrind lrtbf rl fl ar-garch 2.459 0.897 36.501 171.509 1386.669 ar-garch (fhs) 12.518*** 1.291 64.588** 205.050 1259.617 rca 2.459 0.897 49.856** 205.547 1281.037 rca-garch 2.459 0.897 31.090 190.525 1411.081 rca-garch (fhs) 12.518*** 1.291 57.746* 219.107 1289.683 rca (fhs) 5.317** 0.256 58.441** 215.699 1267.312 rca-ma 1.903 4.389** 43.849* 191.254 1287.360 rca-ma (fhs) 10.347*** 0.496 25.287*** 102.243 1927.830 sign rca 3.081 1.065 51.368*** 207.307 1279.988 sign rca-garch 4.511** 0.171 39.764 200.195 1389.469 sign rca-garch (fhs) 10.194*** 0.101 53.739 224.806 1289.986 sign rca (fhs) 5.317** 0.256 58.441 217.562 1272.351 sign rca-ma 0.992 3.840* 37.125 193.384 1289.115 sign rca-ma (fhs) 0.043 2.426 27.264 157.853 1412.718 sym. hist 9.1102*** 0.134 56.766 235.584 1232.198 note: *, **, *** indicate rejection of h0 at the 10% ,5% and 1% significant level, respectively, lrpof – the values of the proportion of failures test statistics, lrind – the values of the independence test statistics, lrtbf – the values of the time between failures test statistics, rl – regulatory loss function, fl – firm’s loss function. joanna górka 46 for ar-garch, rca-garch and sign rca-ma (with fhs method) models the null hypothesis for the traditional var tests is not rejected, what means that the proportion of failures is equal to the given tolerance level 5%, failures are serially independent and the time between failures is independent (see the table 6). slightly worse results are obtained for models: rca (with and without fhs method), rca-ma, sign rca and sign rca-ma. in that case the regulatory loss function is the smallest for the sign rcama (with fhs method) model. hence this model is preferred by the regulatory institution. however, the difference between this value of the regulatory loss function and their values calculated from other models are small, so these models are comparable. the firm’s loss function takes the smallest value for argarch (with fhs method), so this model is preferred from the firm point of view, but other values of the firm’s loss function are not much higher than for the best model (ar-garch, with fhs method). table 7. values of loss function with the expected loss (lfes) for different window sizes for the wig-spozyw index model lfes n = 1500 n = 500 n = 250 ar-garch (dowd) 0,238 0,238 0,252 ar-garch (fhs) 0,451 0,451 0,385 ar-garch (yy) 0,235 0,235 0,266 rca garch (dowd) 0,243 0,233 0,296 rca garch (fhs) 0,439 0,403 0,391 rca garch (yy) 0,241 0,248 0,317 rca (dowd) 0,415 0,363 0,313 rca (fhs) 0,567 0,440 0,389 rca (yy) 0,408 0,381 0,332 rca-ma (dowd) 0,401 0,353 0,301 rca-ma (fhs) 0,283 0,232 0,231 rca-ma (yy) 0,399 0,371 0,319 sign rca garch (dowd) 0,242 0,263 0,295 sign rca garch (fhs) 0,451 0,421 0,428 sign rca garch (yy) 0,240 0,277 0,316 sign rca (dowd) 0,422 0,368 0,311 sign rca (fhs) 0,580 0,445 0,385 sign rca (yy) 0,418 0,387 0,331 sign rca-ma (dowd) 0,411 0,370 0,306 sign rca-ma (fhs) 0,661 0,486 0,383 sign rca-ma (yy) 0,411 0,393 0,322 note: n denotes the window size. application of the family of sign rca models … 47 next, the values of loss function with the expected loss (lfes) for different window size of rolling estimation for example for one of indexes presented in table 7. for the window size of 500 and 250 observations the smallest value of the loss function with the expected loss is obtained for the rca-ma model (with fhs method), but for the window size of 1500 observations the best result is obtained from ar-garch model (with fhs method). it is seen that for the models without garch residuals the value of the lfes decreases as the window size decreases. models with garch errors prefer bigger windows. the comparison of results of the lfes for models with and without the sign function shows that introducing the sign function into the model causes the increase of the loss function with the expected loss. 5. summary in this paper, the family of sign rca models to obtain the selected risk measures was presented. empirical results showed that: − models with sign function occur seldom. it means that percentage log returns do not represent the asymmetric reaction to good or bad news coming from the market. − accuracy of the var measures for models with sign function and without garch errors depends on the size of window and almost all of them are underestimated (except the rca-ma model with fhs method). − accuracy of the var forecasts for models with garch errors (without using fhs method) does not depend on the size of window and all of them are underestimated. − filtered historical simulation (fhs) (christoffersen version) is sensitive to the size of window, i.e. for smaller window the empirically determined probability is closer to the nominal significance level for all models from the family of sign rca models. − using the sign rca-ma model with fhs method the empirical and nominal significance level are almost the same. − for rca-ma models with fhs method the forecasts of var are overestimated at the 5% significance level. − the smallest values of the regulatory loss function were obtained for the var forecasts from rca-ma model (fhs method). − the smallest values of the firm’s loss function were obtained for the var forecasts from the sign rca, rca, ar-garch models (all models with fhs method). − loss function with expected loss (lfes) takes the smallest values for ar-garch models (dowd and yy method) for each window size and joanna górka 48 from the rca-ma models (fhs method) for windows of 500 and 250 observations. − filtered historical simulation (christoffersen version) generates bigger value of lfes than other analyzed method (except the rca-ma model with fhs method). to sum up, some models from the family of sign rca models can generate useful results of the var and es measures only in some cases. references acerbi, c., tasche, d. (2002), expected shortfall: a natural coherent alternative to value at risk, economic notes, 31, 379–388. angelidis, t., degiannakis, s. (2006), backtesting var models: an expected shortfall approach, working paper series, http://econpapers.repec.org/paper/crtwpaper/0701.htm (2.09.2009). appadoo, s., thavaneswaran, a., singh j. (2006), rca models with correlated errors applied mathematics letters, 19, 824–829. aue, a. (2004), strong approximation for rca(1) time series with applications, statistics & probability letters, 68, 369–382. bollerslev, t. (1986), generalized autoregressive conditional heteroscedasticity, journal of econometrics, 31, 307–327. christoffersen, p. f. (2009), value-at-risk models, in andersen, t. g., davis, r. a., kreiss, j.-p., mikosch, t. (ed.), handbook of financial time series, springer verlag. dowd, k. (2002), measuring market risk, john wiley & sons, ltd. engle, r. f. (1982), autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation, econometrica, 50, 987–1006. górka, j. (2008), description the kurtosis of distributions by selected models with sing function, dynamic econometric models, vol.8, toruń nicholls, d., quinn, b. (1982), random coefficient autoregressive models: an introduction, springer, new york. thavaneswaran, a., appadoo, s. (2006), properties of a new family of volatility sing models, computers and mathematics with applications, 52, 809–818. thavaneswaran, a., appadoo, s., bector, c. (2006), recent developments in volatility modeling and application, journal of applied mathematics and decision sciences, 1–23. thavaneswaran, a., peiris, s., appadoo, s. (2008), random coefficient volatility models, statistics & probability letters, 78, 582–593. thavaneswaran, s., appadoo, s., i ghahramani, m. (2009), rca models with garch innovations, applied mathematics letters, 22, 110–114. yamai, y., yoshiba, t. (2002), comparative analyses of expected shortfall and value-at-risk: their estimation error, decomposition and optimization, monetary and economic studies, 20(1), 87–121. zastosowanie modeli klasy sign rca do wyznaczenia wybranych miar zagrożenia z a r y s t r e ś c i. w finansach bardzo ważne jest aby dokładnie ocenić ryzyko. istnieje wiele metod szacowania ryzyka jednak żadna z istniejących już metod nie jest najlepsza. w niniejszym artykule, do wyznaczenie takich miar ryzyka jak value at risk (var) i expected shortfall (es) zastosowano modele klasy sign rca obliczone zostały jednookresowe prognozy var oraz es application of the family of sign rca models … 49 dla ostatnich 500 obserwacji z wykorzystaniem modeli oszacowanych w oknach na podstawie prób wielkości 250, 500 i 1500 obserwacji. otrzymane wyniki zweryfikowano wykorzystując testowanie wsteczne oraz funkcje strat. s ł o w a k l u c z o w e: modele klasy sign rca models, miary ryzyka, value at risk, expected shortfall. microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 tomasz chruściński nicolaus copernicus university in toruń the study of interdependence between capital and currency markets using multivariate garch models a b s t r a c t. in the article an attempt was made to investigate the interaction among the various stock exchanges as well as various exchange rates and then to determine the direction of information flow between capital and currency markets. tools used in this study are multivariate garch models. presented results developed an earlier study of world stock exchange classification. these stock exchanges will be further analysed according to their interaction. k e y w o r d s: multivariate garch model, independence analysis, stock exchange, exchange rate. 1. introduction the research on the interaction of capital and currency markets in the world has been going on for many years, but this topic has not lost its popularity. especially in the last year, and when as a result of the rapid withdrawal of investors from the stock exchange, the inflow of speculative capital, and the bankruptcies of numerous well-known companies, discussions about interpenetrating influences of stock exchanges they have on one another were very hot. at the same time, there was considerable interest attracted into fluctuations in the currency market. the aim of this paper is to examine the relationship between stock exchanges of securities of eligible countries and exchange rates associated with them. the article presents a continuation of earlier studies by the author on the classification of stock exchanges in the world, which now will be analyzed in terms of the direction of information flow between the markets. multivariate garch models are applied to determine the influence on volatility. tomasz chruściński 112 2. methodology with a map-sharing of stock exchanges, resulting from the author’s previous studies1, an attempt was made to determine the direction in which information flows between the dissociated groups, and thus to examine what their spillover effects are. the study used a multivariate model mgarch, based on daily rates of return with the major indices and exchange rates. since it would be difficult to examine changes in all indices, representatives of groups have been chosen. therefore, for use of the specification of the model’s equations, logarithms adopted rate of return (1) of the following indices were accepted: nasdaq, dow jones, s & p500, dax, cac40, ftse250, sse comp., wig20, and the exchange rate representing the countries from which the selected indices. these are: eur/usd, eur/gbp, eur/pln, usd/gbp, usd/cny, cny/gbp, cny/pln, cny/usd, gbp/pln. the starting point for analysis are the concepts of conditional expected values, conditional variance-covariance matrix and conditional distribution of standardized residuals of the model, presented in the equation: ( ) .lnln100 1 tttttttt zhppr +=+=−= − μεμ (1) tp , 1−tp value of the index (exchange rate) at time t and t-1, tμ conditional expected value of rate of return rt at time t ( [ ]1| −ε= ttt irμ ), tε residual of the model, th conditional variance of returns at time t ( [ ]1|var −= ttt irh ), tz independent of the residuals of the model with zero mean and unit variance, it-1 information available at time t. more methodological information about the modelling of stochastic processes can be found in the literature (osińska, 2006; fiszeder, 2009). the paper focuses on key issues of mgarch structure model. although it has been over 20 years since engle and bollerslev (1986) introduced one-equation arch and garch models, their attitude to discrete-time variance of rates of return, is still valid and being developed. a natural extension of garch models for the analysis of financial markets was introduced by bollerslev (1988) multivariate garch model (mgarch). in the empirical part of this paper this model was used to describe the mutual influence of stock markets and exchange rates in the world. the general form of a multivariate model, which is equivalent to oneequation model garch (1,1) defined in the work of bollerslev, engle, wooldridge (1988) called vech-garch, is given by equation: ( ) ( ) ( ) ( ),11 1tτ hbawη −−− +⋅+= vechvechvechvech ttt εε (2) 1 chruściński (2008). the study of interdependence between capital and currency markets ... 113 where ( )⋅vech is an operator of symmetric vector. several conditions have to be met for an estimation of the model (2) to be properly conducted. a positive definiteness and stationary of matrix ht for each time t shall be provided, what requires the positive definiteness of matrix a and b and is associated with introducing very complicated nonlinear constraints. the primary consequence of the full form of vech equation is the need to estimate a large number of parameters, which even in the two-dimensional model amounts to 21. these problems resulted in little use the model has found in practice. models with diagonal matrices a and b, that reduced the number of estimated parameters and eliminated the transmittance variance effect by conditioning elements of matrix ht from its past values and error products at time t ( tjti ,, εε ), turned out to be a solution. the general form of the diagonal model vech (dvech) is: ( ) ,11 tt hbawh oo ++= −− ttt εε (3) where: ( )( )aa diagivech= , ( )( )bb diagivech= , intersection yx o is the hadamard product, and ( )⋅ivech is an inverse operator to ( )⋅vech . after reducing matrices a and b to the diagonal form, we obtain the final form of the model dvech, which is an extension of garch model (1.1) (see yang, 2001). a special variant of the vech model is the bekk model, which easily solves the problem of non-positive covariance matrix. however, since it is difficult to obtain stationarity of matrix ht as well as a small number of estimated parameters (see piontek, 2006), similarly applicable are diagonal matrices a and b, to obtain a model dbekk (2) .11 bhbaawwh 1t ttt t −−− ++= tt εε (4) diagonal forms can be used to estimate each equation of models (3) and (4) respectively, allowing to avoid numerous optimization problems of maximum likelihood method (ml) applied to several equations simultaneously. in the empirical example maximisation of the maximum likelihood function defined by the formula (5) was adopted: [ ].ln 2 1 1 '∑ = −+−= t t ttllf εε 1 tt hh (5) 3. empirical results the daily rate of return on the main stock indices of the selected countries and the related exchange rates were used for the analysis. time series were adjusted for comparability by removing rates of return for periods in which even only one variable was unknown. finally, 1865 observations for the period from tomasz chruściński 114 january 2001 to april 2009 were obtained. for such prepared data diagonal vech models, diagonal bekk models, and a constant conditional correlation model ccc have been built. the comparison was conducted to assess which of them matches reality the best. after estimating several dozen models, a decision to use the conditional student’s t-distribution was made. this has given better results of the parameters’ estimation than a normal distribution. table 1. comparison of estimation results for given models, according to the aic criterion model aic value optimal model dvech ccc dbekk rcny/gbp=f(rcac40, rdax); rdax=f(rs&p500, rcac40); rs&p500=f(rdjia) -19.6769 -19.5990 -19.6505 dvech rcny/pln=f(rdjia, rcac40); rcac40=f(rs&p500, rdax); rdax=f(rnasdaq) -20.1400 -20.0380 -20.1282 dvech rcny/eur=f(rdjia, rwig20); rcac40=f(rs&p500, rdax); rwig20=f(rftse250, rdax) -18.8546 -18.7995 -18.8423 dvech rgbp/pln=f(rs&p500, rdax); rdax=f(rftse250, rcac40); rwig20=f(rftse250, rdax) -18.5694 -18.5593 -18.5418 dvech reur/pln=f(rs&p500, rcac40); rs&p500=f(rnasdaq, rdjia); rcac40=f(rs&p500, rdax) -20.0290 -20.0298 -19.9953 ccc rwig20=f(rnasdaq, rdax); rdax=f(rs&p500, rcac40); r s&p500=f(rdjia) -18.1299 -18.1305 -18.1000 ccc rwig20=f(rs&p500, rdax); rdax=f(rs&p500, rcac40); rs&p500=f(rdjia) -18.1394 -18.1414 -18.1102 ccc rwig20=f(rdjia, rdax); rdax=f(rs&p500, rcac40); rdjia=f(rs&p500) -18.1705 -18.1764 -18.1469 ccc rwig20=f(rnasdaq, rftse250); rftse250=f(rs&p500, rcac40); rcac40=f(rs&p500, rdax) -19.0869 -19.0830 -19.0638 dvech rwig20=f(rs&p500, rftse250); rftse250=f(rs&p500, rcac40); rs&p500=f(rdjia) -18.8132 -18.8329 -18.7793 ccc rwig20=f(rdjia, rftse250); rftse250=f(rs&p500, rcac40); rdjia=f(rs&p500) -18.8622 -18.8839 -18.8324 ccc rsse=f(rftse250); rftse250=f(rs&p500); r s&p500=f(rnasdaq, rdjia) -18.5000 -18.5103 -18.4458 ccc rsse=f( rftse250); rftse250=f(rs&p500, rcac40); rwig20=f(rs&p500, rdax) -17.7720 -17.7780 -17.7281 ccc the purpose of this research was to determine the direction of the impact of a stock exchange representing one group on a representative of another stock exchange, the mutual influence of exchange rates on one another and, consequently, to examine the relationship between stock market and currency market. having available rates of return of 8 indices and 9 exchange rates, hundreds of multi-dimensional combinations have been analysed. because the publication has limited size only examples of the results of estimated models are presented. the value of aic has been chosen as the selection criterion. the results of the comparisons are shown in table 1. the study of interdependence between capital and currency markets ... 115 it is clearly visible while analysing the results of the estimation presented in table 1 that the best models to depict the interaction between stock indices and exchange rates are the diagonal vech model and the constant conditional correlation model ccc. the same structure of the optimal models occurs for all of the interdependence test results. an exemplary multivariate empirical dvech model is shown below: reur/pln(t) = -0.000332 – 0.042826·rs&p500(t-1) + 0.024953·rcac40(t-1) + εt rs&p500(t) = 0.000576 + 0.016579·rnasdaq(t-1) – 0.093922·rdjia(t-1) + εt rcac40(t) = 0.0007 + 0.420494·rs&p500(t-1) – 0.20429·rdax(t-1) + εt figure 1. the reciprocal influence of stock exchanges, represented by the major indexes in the period from january 2001 to april 2009 the more interesting part of the calculations’ results is an economic interpretation of the constructed multivariate models. basing on correctly evaluated exogenous variables of individual equations the direction of the information flow from the day before between the individual stock exchanges is clearly visible. an illustration of the mutual influence of stock exchanges used in the study is shown in figure 1, while figure 2 represents the results of studies using similar variables from the period from january 2001 to september 2008. a comparison of the two mentioned figures provides us with a clear picture of changes in a correlation between stock exchanges over a 7-month period of the economic crisis. tomasz chruściński 116 figure 2. the reciprocal influence of stock exchanges, represented by the major indexes in the period from january 2001 to september 2008 comparing the results of the analysis shown in figures 1 and 2, there is a noticeable change in the direction of information flow between the stock exchanges studied, which occurred during the period from october 2008 to april 2009. there is no longer a direct connection between djia and nasdaq indices on the u.s. market, however the london ftse250 index gained a more direct influence on german dax index on the european market. both stock exchanges (british and german) have had an impact on polish wig20 index. french stock floor which had previously provided the polish market with information contained in the s&p500 no longer fulfils this purpose and currently "yesterday's" changes of the american s&p500 directly affect the wig20. the warsaw stock exchange has lost its position, previously shared with the london stock exchange, and does not shape exchange rates in shanghai anymore. figure 3. mutual effect of exchange rates associated with the countries where the analysed stock exchanges come from the study of interdependence between capital and currency markets ... 117 the knowledge of dependencies inside the markets is desirable to examine the mutual influence of stock markets and exchange rates. in this case, similarly as for stock exchanges connections, dependencies of rates of return on exchange rates associated with the countries where the investigated stock exchange come from were determined. these dependencies for individual exchange rates are presented in figure 3. the analysis allowed to determine the direction of the influence the selected stock exchanges and exchange rates have on one another. the results of these interactions are shown in figure 4. figure 4. mutual effect of stock indexes and exchange rates associated with the countries where the analysed stock exchanges come from 4. summary the study confirmed the intuitive assumption about the interdependence of stock markets and exchange rates. in majority of cases, exchange rates depend on the stock exchanges indices changes, but not vice versa, what derives directly from investors’ tendency to generate profit. in the situation where stock floor is dominated by supply and shares prices of companies are dropping, players withdraw their capital, exacerbating the already declining trend, and invest in a foreign currency. therefore, the usual changes of stock indices and exchange rates are negatively correlated with one another. the s&p500 and the cac40 indices have the biggest influence on exchange rates in a selected group of stock exchanges. their combined impact shapes, inter alia, eur/pln exchange rate. the cny/pln exchange rate depends in turn on the djia and the cac40 index. this is due to the fact that the u.s. economy cooperates closely with the chinese economy and in consequence has a strong influence on the yuan. from the perspective of polish stock market it is interesting that the wig20 index has an impact tomasz chruściński 118 on the cny/usd exchange rate. this dependence may result from a shared impact of the non-distant, considering time, warsaw stock exchange and the london stock exchange on indices of the shanghai stock exchange. in such a case the connection with sse exchange is very normal. in this study, only eur/gbp exchange rate turned out to be independent of any stock exchange. on the other hand, its value depends on usd/gbp exchange rate. as far as the interdependence of capital markets and exchange rates is concerned, the analytical results presented in the paper are fragmentary. models computation is a complex and time-consuming process, therefore models comparisons and analysis presented in this paper have an illustrative value only and provide an example of mgarch models usage. references bollerslev, t., engle, r., wooldridge, j. (1988), a capital asset pricing model with timevarying covariance, journal of political economy, university of chicago press, 96(1). bollerslev, t. (1986), generalized autoregressive conditional heteroskedasticity, journal of econometrics, 31, 307–327. chruściński, t. (2008), analiza wielowymiarowa giełd papierów wartościowych na świecie (multivariate analysis of stock exchanges in the world), wiadomości statystyczne (statistical news), gus, warszawa, no. 9, 50–61. chruściński, t. (2009), wykorzystanie wielowymiarowych modeli klasy garch do badania wzajemnego wpływu rynków finansowych na świecie (using multivariate garch models to examine the mutual influence of financial markets in the world), equilibrium, wydawnictwo naukowe umk, toruń, 1(2), 61–68. engle, r. (1982), autoregressive conditional heteroskedasticity with estimates of the variance of uk inflation, econometrica, 50, 987–1008. fiszeder, p., (2009), modele klasy garch w empirycznych badaniach finansowych (garch models in empirical studies of financial), wydawnictwo umk, toruń. osińska, m. (2006), ekonometria finansowa (financial econometrics), pwe, warszawa. piontek, k. (2006), wyzwania praktyczne w modelowaniu wielowymiarowych procesów garch (the practical challenges in the modeling of multivariate garch processes), taksonomia (taxonomy), akademia ekonomiczna, wrocław, no. 13. yang, w. (2001), m-garch hedge ratios and hedging effectiveness in australian futures markets, edith cowan university. badanie współzależności rynków kapitałowych i walutowych z zastosowaniem modeli mgarch z a r y s t r e ś c i. w artykule podjęto próbę zbadania wzajemnych relacji pomiędzy wybranymi indeksami giełdowymi, relacji pomiędzy kursami walut, a następnie określono kierunek przepływu informacji pomiędzy rynkami kapitałowym i walutowym. do oszacowania współzależności instrumentów inwestycyjnych posłużył wielowymiarowy model klasy garch. praca jest kontynuacją poprzednich badań autora związanych z taksonometryczną klasyfikacją giełd na świecie. s ł o w a k l u c z o w e: wielowymiarowy model garch, analiza współzależności, giełdy papierów wartościowych, waluty. microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 sylwester bejger nicholas copernicus university in toruń econometric tools for detection of collusion equilibrium in the industry a b s t r a c t. the article presents the notion of detection of overt or tacit collusion equilibrium in the context of choice of the appropriate econometric method, which is determined by the amount of information that the observer possesses. there has been shown one of the collusion markers coherent with an equilibrium of the proper model of strategic interaction – the presence of structural disturbances in the price process variance for phases of collusion and competition. the markov switching model with switching of variance regimes has been proposed as a proper theoretical method detecting that type of changes without prior knowledge of switching moments. in order to verify the effectiveness of the method it has been applied to a series of lysine market prices throughout and after termination of its manufacturers’ collusion. k e y w o r d s: explicit and tacit collusion, collusive equilibrium, cartel detection, lysine, price variance, markov switching model 1. introduction the equilibrium of players’ collusion in the industry may occur as a consequence of players’ strategic interaction of the overt or tacit collusion nature. although these two types of interaction are described by means of different models of game theory with various informative assumptions, their equilibriums are characterized by similar consequences for the industry, i.e.: a) the occurrence players’ market power leading to the loss of social wealth (above-normal pcm), b) the limitation of competition and the impediment of the industry development . the article presents one of collusion markers resulting from the theoretical model of tacit collusion, which is price rigidity in the collusion phase, and proposes the application of the markov switching model of ms-ar-garch type in order to detect structural changes in market price variance, and thereby to sylwester bejger 28 verify the presence of the aforementioned marker. the preliminary verification of usefulness of the proposed econometric method constitutes a research problem. for this purpose, in the empirical section there has been applied the proposed model for a series of market prices of cartel players – lysine manufacturers. 2. collusion and its quantitative detection collusion constitutes a serious problem in the market economy. if we take into account only overt collusion (mainly in the area of english-speaking reporting) between 1990 – 2005, 283 cartels were confirmed (so called hard core cartels) with national and global scope of operation, however, it has been estimated that as little as 30% of overt collusion are detected and punished. there are no data concerning tacit collusion. the above mentioned 283 cartels influenced sales of 2.1 trillion usd, caused unjustifiable price overcharge of the value of 500 bn usd and were punished with pecuniary penalty of the nominal value amounting to 25.4 bn usd (connor, helmers, 2006). taking into account the fact how common and harmful such collusion is, it seems natural that it should be quickly detected. unfortunately, although theoretical models of overt or tacit collusion are described very well as research hypotheses concerning players’ behavior, their empirical verification presents great difficulties. it happens mainly due to the fact that the players participating in collusion have an advantageous position over the observer in the form of private information. moreover, the resources of public statistics are frequently very humble on the disaggregation level of the industry or individual players. thus, it is not difficult to ascertain that econometric methods, that are, on the one hand, economical as far as the use of statistical data is concerned, and, on the other hand, coherent with the model hypothesis, are particularly valuable. currently known econometric methods of detection can be divided into direct and indirect methods: a) direct the assessment of strategy profile in equilibrium that is consistent with the assumed collusion model, verification of the hypothesis on conformity with theoretical equilibrium, b) indirect measurement and/or identification of market power or detection of so called collusion markers (non-competitive behaviors), i.e, certain, characteristic for collusion, disorders concerning: − the relation between players’ prices and market demand changes, − price and market share stability, − the relation between players’ prices, − investment in production capacity. econometric tools for detection of collusion equilibrium in the industry 29 when it comes to statistical data, group a) methods are very demanding, their applications are very rare and they are possible only in specific circumstances1. group b) methods are much more common. the following basic methods are enumerated in the order of intensity of the use of statistical data: − the study of structural changes in price volatility , − non-parametric method based on revealed preferences, − the osborne – pitchik test, − the study of asymmetry of price reactions, − hall’s method, − the assessment of residual demand elasticity, − the panzar – rosse method, − cpm method. 3. price disorders characteristic for collusion one of the most promising collusion markers are markers that are based on the analysis of changes in price processes and/or market shares. in accordance with known tacit collusion models: 1. the player’s (players’) price and supply are negatively correlated, the price is ahead of the demand cycle, the stochastic process of market price undergoes changes of the regime type (green, porter, 1984; rotemberg, saloner, 1986; haltiwanger, harrington, 1991), 2. the price process variance is on average lower for collusion phases and may undergo changes of the regime type (athey, bagwell, sanchirico, 2004; connor, 2004; abrantes-metz, froeb, geweke, taylor, 2006; bolotova, connor, miller, 2008), the application marker 2 is particularly promising in practice. it is justified by the fact that the requirements for this marker concerning data are very little (market price is sufficient) and it has clear theoretical justification. lower price volatility in the collusion phase results directly from the equilibrium properties of sppe type (symmetric perfect public equilibrium) of the price super-game with standard assumption concerning sufficiently high discount rate. for the strategy profile in the equilibrium of this game, players: − achieve higher payments than in competitive equilibrium (cartel payments), − in the collusion phase, players’ prices are insensitive to costs shocks (players avoid price changes, even at the cost of effectiveness, not to cause switching to penalty phase). 1 for example, see slade (1992). sylwester bejger 30 it should be noted that the necessity of using a number of observations including both competition and collusion phases and high product homogeneity of the analyzed trade is one of the marker’s drawbacks. 4. econometric method verifying marker’s presence the works conducted so far that are connected with collusion detection on the basis of the detection of structural changes in variance included the application of the descriptive statistics methods for the comparison of variance level in collusion and competition phases (abrantes-metz, froeb, geweke, taylor, 2006) and the application of arch / garch specification for the market price process together with additional 0-1 variable describing collusion and competition phases (bolotova, connor, miller, 2008, further bcm ). this paper suggests using the markov switching model of ms(m)(ar(p))garch(p,q) type as an econometric method verifying the marker for the variance and/or the average (constant) of the price process 2. the application of this model has the following advantages: − it is theoretically coherent with the structure of equilibrium strategy of the super-game model, − it allows to model structural changes of process variance directly, without the use of additional artificial variables; such modeling is not possible in, e.g. arch / garch specification, − it is coherent with informative asymmetry between cartel members and the observer. ms(ar)garch specification does not require observation (knowledge) of the state variable, thus it may be used for actual detection of variance regimes and objective determination of switching moments, that is the detection of collusion and competition phases. the form of the general ms(m)(ar(p))garch(p,q) model is a development version of a wellknown ms model (hamilton, 1989; hamilton, susmel, 1994; krolzig, 1998; stawicki, 2004; davidson, 2004). the application of the model with regimes in variance refers mainly to high frequency data, such as currency rates, rates of return from financial instruments, prices of electric energy3. the general form of switching model which has been applied can be written down as4: 2 according to krolzig this type of model could be described as msi(m)h-ar(q) with garch(p,q) component, krolzig (1998). 3 for example, refer to fong (1998), włodarczyk, zawada (2005, 2007), kośko, pietrzak (2007). 4 there are various forms of notation, this one is from davidson (2004). econometric tools for detection of collusion equilibrium in the industry 31 ∑ = − ++= p m tmtmsst uyy tt 1 0 ,φα (1) where: ),1,0.(..~ oraz 2/1 diieehu tttt = , 1 2 0 ∑ ∞ = −+= m mtmsst uh tt ββ (2) the conditional variance equation (2) uses arch(∞) specification which also includes models of garch (p,q) class. in model (1),(2) each parameter may be potentially a random variable switched between the values from a finite set of values depending on the actual state of ts where mst ...,,1= . variable ts is assumed to be the exogenous, homogeneous markov process with fixed transition probabilities { }ijp where: )pr( 1 isjsp ttij === − . the probability that the observed process ty is in the state j in the period t is presented by means of the filtering (updating) equation: , )|pr(),|( )|pr(),|( )|pr( 11 1 11 −= − −− ω=ω= ω=ω= =ω= ∑ tt m i ttt ttttt tt isisyf jsjsyf js (3) where tω signifies the entire information available at moment t, and: ),|pr()|pr( 1111 −−=− ω==ω= ∑ tt m i ijtt ispjs (4) where transition probability ijp constitutes m(m-1) parameters to be estimated. the form of conditional density function of observed variable: ),,|(. 1−ω= tt jsf requires making an assumption concerning the type of distribution. estimation of model parameters may be obtained through maximum likelihood method. for this purpose a likelihood function is used: ),|pr(),|(prlog 11 1 1 −= − = ω=ω== ∑∑ tt m i ttt t t jsjsyfl (5) the maximization of function (5) is conducted by means of a very well known method, expectation maximization (em) algorithm (krolzig, 1998, p. 8). sylwester bejger 32 5. empirical verification the collusion of lysine producers5 was proved in 1996. the test includes monthly average lysine prices on the usa market in the period between 01/90 – 06/966. within this period, on the basis of collected evidence (connor, 2001) the following phases may be distinguished (table 1). table 1. the statistics of lysine price (prices per pound) phase months number average standard deviation coefficient of variation 1. competition (01.90 – 07.92) 31 102.90 16.22 15.8% 2. collusion (08.92 – 03.93) 8 90.13 9.83 10.9% 3. competition (04.93 – 07.93) 4 70.50 7.72 11% 4. collusion (08.93 – 06.95) 23 110.30 8.55 7.8% 5. competition (07.95 – 06.96) 12 102.50 9.51 9.3% the purpose of the empirical research is to check to what degree the model with the proposed specification may be used to detect changes of the regime type in the variance of the process generating data, and thereby detect collusion phases. such verification is possible due to the knowledge of the collusion history, provided that this case has been correctly determined (i.e., no significant evidence was omitted during the trial). initially, there was verified a hypothesis on variances equality for two comparable phases. table 2 summarizes this step. table 2. the value of statistics for the variance equality test in phase 1 and 4 bartlett 7.6064 (0.005) brown-forsythe 3.9206 (0.053) test f 3.2106 (0.003) note: p-values given in brackets. on the basis of the test it can be ascertained that variances in both phases are significantly different. next, the properties of the examined series were checked in terms of the distribution characteristics and autocorrelation, stationarity and homoscedasticity of residuals. the results of this part of research are included in table 3. 5 lysine is an basic amino acid required as a feed component in hog, poultry and fish production. 6 the prices are from connor, (2000), appendix a, table a2. econometric tools for detection of collusion equilibrium in the industry 33 table 3. characteristics of series skewness -0.765 jarque-bera normality test 7.634 (0.021) adf test -3.627* (0.007) kurtosis 3.081 ljung – box test for levels – q(5) 145.260 (0.000) kpss test 0.160** lm test for heteroscedasticity of residuals 9.685 (0.002) note: pvalues given in brackets, * value of t statistics (critical values for 1%, 5%, 10% sig. levels – (-3.519), (-2.900), (-2.587), ** value of lm statistics (asympt. critical values for 1%, 5%, 10% sig. levels – 0.739, 0.463, 0.347). the series is skewed, the hypothesis of normal distribution was rejected and by means of test with different configuration of hypotheses the series stationarity was confirmed. after removing autocorrelation, there is explicit heteroscedasticity of a random component, which indicates relations in the variance which were not included in the model. in the next stage of the research a number of models of ms(k)(ar(p))garch(p,q) type was constructed and estimated with the use of maximum likelihood method. the best results, when it comes to the model properties, were achieved for ms(2)(ar(2))garch(1;0) specification in the form of: ∑ = − ++= 2 1 0 , m tmtmst uyy t φα (6) where: ),1,0.(..~2/1 diieorazehu tttt = ,2 110 −+= tst uh t ββ (7) .2;1=ts this specification assumes controlling the observable price process through non-observable stochastic process of state variable st, which is assumed to be a homogeneous markov chain of two states and proper matrix of transition probabilities between states. a constant and unconditional error variance are parameter which depends on the regime. moreover, regardless of the regime, the average of a series is described by ar(2) process, whereas, garch(1;0) component is present in the variance. table 4 shows estimation results. sylwester bejger 34 table 4. estimation results ms(2)(ar(2))garch(1;0) parameter estimation p-value tests p11 0.945 ---l.og likelihood -204.670 p22 0.861 ---r2 0.961 φ1 1.290 0 skewness (residuals) -0.021 φ2 -0.485 0 kurtosis (residuals) 2.680 β1 0.405 0.094 jarque-bera test (residuals) 0.326 (0.849)* α01 111.216 0 ljung-box test (residuals, q(5)) 3.206 (0.668) α02 83.479 0 ljung-box (residuals2,q(5)) 4.699 (0.454) β01 2.104 (4.430)** ---lm autocorrelation test 0.042 (0.837) β02 3.497 (12.230)** ---lm neglected arch test 4.014 (0.547) note: * p-values in brackets, ** variance the most essential question is whether the proposed model may serve the assumed objective, i.e. collusion detection. it can be estimated on the basis of precision of regime detection. figure 1 shows the values of the observed variable and smoothed probabilities for regime 1 (i.e. conditional probabilities of the process is in state s1, while taking into account information from the entire sample) respectively, together with marked collusion phases. figure 1. lysine prices and smoothed probabilities econometric tools for detection of collusion equilibrium in the industry 35 one can notice significant conformity of changes detected by the model in the variance regime and the average with the observed collusion phases (particularly in the case of phase 2 which is detected almost ideally). the values of the average variance of an error term in regime 1 is more than 2.5 times bigger than in regime 2 which is consistent with the assumed theoretical hypothesis. probabilities pii of maintaining in the “competition” and “collusion” states are high, which well replicates the structure of profile of the equilibrium supergame strategies. unfortunately, this model does not detect a regime change for other periods, which is connected with the fact that both the constant and variance undergo changes. although the estimation of the value of the constant for the low variance regime (collusion phase) is significantly higher than for the competition regime, which may be confirmed by traditional understanding of price collusion, however, on the basis of regimes, generally speaking, the average price level, one cannot unambiguously draw conclusions concerning the type of equilibrium, without additional statistical information e.g. on demand level. figure 2 presents the comparison of price levels and smoothed probabilities for the model which is not so well fitted (with switching exclusively in variance) but which detects particular phases in more unambiguous manner7. figure 2. lysine prices and smoothed probabilities 7 autoregressive component in switching model could bias signal of filtered probabilities if corrects other then gaussian disturbances of error term. see lahiri, whang, (1994). sylwester bejger 36 specification for this model is ms(2)-ar(1), table 5 presents the estimation of parameters. table 5. estimation results ms(2)-(ar(2)) parameter estimation p-value tests p11 0.683 ---l.og likelihood -236.450 p22 0.967 ---r2 0.851 φ1 0.946 0 skewness (residuals) -0.115 α0 104.641 0 kurtosis (residuals) 2.452 β01 0.433 (0.187)** jarque-bera test (residuals) 1.134 (0.567) β02 5.911 (34.939)** ljung-box test (residuals, q(5)) 50.078 ( 0.000) ljung-box (residuals2,q(5)) 16.650 (0.005) lm autocorrelation test 11.832 (0.001) lm neglected arch test 12.179 (0.032) note: * p-values in brackets, **variance this model regarding the detection of equilibrium types on the basis of variance is closer to the actually observed cartel history. first of all, the average length of staying in each regime 1)1( −−= iist pd is consistent with the history as far as proportion is concerned, the probability of switching from the competition phase to the collusion phase is higher and collusion stability is lower. if we compare both models in term of value iip it turns out that model 2 is more consistent with the collusion history (the average collusion phase is 3.1 month long, whereas in the case of model 1 it is 16 months long) 6. summary the markov switching model with the switching component in variance and/or garch process parameters has unquestionable theoretical advantages when it comes collusion detection on the basis of variance changes in the market price. on the basis of the empirical research, the correctness of the detection may be found at least in respect to variance regimes. the model in specification that is more advantageous in terms of process replication in better adjusted to data than models used in bcm work (taking into account value of log likelihood). it must be remembered, however, that empirical verification was based on one, although unique, but relatively short series of data. at the next stage of research, the accepted method should be tested on other empirical series (which is difficult due to difficulties in obtaining data) or on the series generated for various collusion strategy profiles. econometric tools for detection of collusion equilibrium in the industry 37 references abrantes-metz, r., froeb, l., geweke, j., taylor, c. (2006), a variance screen for collusion, international journal of industrial organization 24, 467–486. athey, s., bagwell, k., sanchirico, c. (2004), collusion and price rigidity, review of economic studies 71, 317–349. bejger, s. (2004), identyfikacja, pomiar i ocena siły rynkowej podmiotów gospodarczych oraz stopnia konkurencyjności branż z wykorzystaniem metodologii teorii gier, (identification, measurement and estimation of market power of the firms and degree of competitiveness of the industries with application of game theory methodology), phd thesis. bolotova, y., connor, j.m., miller, d.j. (2008), the impact of collusion on price behavior: empirical results from two recent cases, international journal of industrial organization 26, 1290–1307. connor, j. (2000), archer daniels midland: price-fixer to the world, staff paper no. 00-11, department of agricultural economics, purdue university, west lafayette, in. connor, j. (2001), our customers are our enemies: the lysine cartel of 1992–1995, review of industrial organization 18, 5–21. davidson, j. (2004), forecasting markov-switching dynamic, conditionally heteroscedastic processes, statistic and probability letters, 68(2), 137–147. fong, w.m. (1998), the dynamics of dm/pound exchange rate volatility: a swarch analysis, international journal of finance and economics 3, 59 –71. fransens, p., h., van dijk, d. (2000), nonlinear time series models in empirical finance, cambridge university press. haltiwanger, j., harrington, j.e. (1991), the impact of cyclical demand movements on collusive behavior, rand journal of economics, 22 (1991), 89–106. hamilton, j. d. (1989), a new approach to the economic analysis of nonstationary time series and the business cycle, econometrica 57, 357–384. hamilton, j. d., r. susmel (1994), autoregressive conditional heteroscedasticity and changes in regime, journal of econometrics 64, 307–333 kośko, m., pietrzak, m. (2007), wykorzystanie przełącznikowych modeli typu markowa w modelowaniu zmienności finansowych szeregów czasowych, (an application of markovswitching model for modelling of variability of financial time series) (in) dynamiczne modele ekonometryczne, z. zieliński (ed.), wydawnictwo umk, toruń. krolzig, h. m. (1998), econometric modelling of markov-switching vector autoregressions using msvar for ox, working paper. lahiri, k., whang j. g. (1994), predicting cyclical turning points with leading index in the markov switching model, journal of forecasting, vol. 13, pp. 245–263. rotemberg, j., saloner, g. (1986), a supergame theoretic model of business cycles and price wars during booms, american economic review 76, 390–407 slade, m., e. (1992), vancouver's gasoline-price wars: an empirical exercise in uncovering supergame strategies, review of economic studies 59, 257–276. stawicki, j. (2004), wykorzystanie łańcuchów markowa w analizie rynków kapitałowych (the markov chains in capital markets analysis), wydawnictwo umk, toruń. włodarczyk, a., zawada, m. (2005), przełącznikowy model markowa jako przykład niestacjonarnego modelu kursu walutowego (markov switching model as an example of nonstationarity exchange rate model), [in:] dynamiczne modele ekonometryczne, z. zieliński (ed.) wydawnictwo umk, toruń, sylwester bejger 38 ekonometryczne narzędzia detekcji równowagi zmowy w branży z a r y s t r e ś c i. w artykule przedstawiono problem detekcji równowagi zmowy jawnej lub milczącej w kontekście wyboru właściwej metody ekonometrycznej, który determinowany jest ilością informacji posiadaną przez obserwatora. zaprezentowano jeden z markerów zmowy spójnych z równowagą właściwego modelu interakcji strategicznej – obecność zaburzeń strukturalnych w wariancji procesu ceny dla faz zmowy i konkurencji. jako poprawną teoretycznie metodę detekcji tego typu zmian bez wiedzy a-priori o momentach przełączania zaproponowano wykorzystanie przełącznikowego modelu markowa z przełączaniem reżimów wariancji. w celu weryfikacji skuteczności metody aplikowano ją dla szeregu cen rynkowych lysiny w czasie trwania i upadku zmowy jej producentów. s ł o w a k l u c z o w e: zmowa jawna i milcząca, równowaga, lysina, wariancja ceny, model przełącznikowy markowa microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 milda maria burzała poznań university of economics the synchronization of regional business cycles with nationwide cycles a b s t r a c t. this paper attempts to assess the level of synchronization between the business cycles of poland’s regions and those of the country as a whole. the measure of economic activity was an index of total industrial output sold, recorded monthly from january 1999 to december 2008, adjusted for seasonal and random fluctuations. the analysis of dominant business cycles was performed using spectral analysis. to assess the synchronization of cycles, characteristics of cospectral analysis were used: coefficient of coherence, amplitude intensification and phase difference. in the conclusion, an attempt is made to construct a synthetic indicator as a means of ranking the regions by degree of business cycle synchronization. k e y w o r d s: spectral analysis, cospectral analysis, business cycle, synthetic indicator. 1. introduction comparison of the level of economic activity in poland’s regions (also called provinces or voivodeships) with that of the nation as a whole provides a source of information for both the regional authorities and central government. from a macroeconomic point of view, the observation of changes in a region may be an important factor for evaluating the effectiveness of anticyclic policy and monitoring the channels by which fluctuations spread. the concept of synchronization of fluctuations is understood to relate to analysis of particular components of the business cycle in terms of similarity of shape, amplitude of fluctuations and relative time lags. to perform such an analysis, a measure of economic activity must be adopted. in the empirical investigation, the measure used was the monthly recorded indices of total sold industrial output from january 1999 to december 2008 (pr_irt = =100 ⋅ pr_irt / pr_irt-12), adjusted to eliminate seasonal and random fluctuations (statistica 8 program, census 2 method). hence the concept of economic activity is associated with changes resulting from the joint effect milda maria burzała 62 of growth factors and business cycle fluctuations. frequency domain analysis was carried out, including both spectral and cospectral analysis. spectral analysis, in relation to the polish economy, has been primarily used in macroeconomic research1, because its use depends on a sufficient length of time series2. the fact that the present regional division of poland has only been in existence since 1999 was a significant barrier in applying the spectral analysis. 2. tools of spectral and cospectral analysis spectral and cospectral analysis refer to stationary stochastic processes and to the frequency domain. transition from the time domain to the frequency domain is accomplished by means of fourier transformation. in the study, the stationarity assumption of the stochastic process was verified using the dickey– fuller (adf) test. spectral analysis leads to the determination of a power spectrum, namely a spectrum of the considered time series: ],,[for )( 2 1 cos)( 2 1 )( ππω τ π ωττ π ω τ ωτ τ −∈ == ∑∑ ∞ −∞= − ∞ −∞= kekf i (1) where k(τ) is the autocovariance function, τ = t–s is the distance between the two analysed points in time, and ω = 2π/n is the frequency of harmonic components. the power spectrum constitutes a distribution of variance of the analysed time series and makes it possible to identify the harmonic structure of the series and determine the contribution of individual components of the series to the variance of the process. joint spectral analysis enables testing the relations between particular frequencies of two time series. the basic magnitude in this case is the crossspectrum, which is a distribution of the joint covariance of the two processes kyx(τ): ].,[for )( 2 1 cos)( 2 1 )( ππω τ π ωττ π ω τ ωτ τ −∈ == ∑∑ ∞ −∞= − ∞ −∞= yx i yxyx kekf (2) 1 cf. for example p. skrzypczyński (2006, 2008), s. dudek, d. pachucki, k. walczyk (2008); include review of research with regard to synchronization of fluctuations, l. talaga, z. zieliński (1986). 2 in the literature it is stated that the minimum length of a time series should be such as to include approximately 100 observations. the synchronization of regional business cycles with nationwide cycles 63 using de moivre’s lemma, the cross-spectrum can be written in complex form: ),()( sin)( 2 1 cos)( 2 1 )( ωω ωττ π ωττ π ω τ τ yxyx yxyxyx iqc kikf −= =−= ∑ ∑ ∞ −∞= ∞ −∞= (3) where cyx(ω) is the co-spectrum (the real part of the cross-spectrum); qyx(ω) is the quadrature spectrum (the negative imaginary part of the cross-spectrum). in cospectral analysis three characteristics are introduced which form a basis for comparison of the course of two processes. in the study, the variable x was associated with a series of nationwide output indexes, while the variable y was associated with the corresponding series for a region. a) gain of the variable x with respect to y is interpreted as the modulus of the coefficient β in the regression of variable y with respect to x for a given frequency ω (gyx(ω) >1 means that the amplitude of process yt for frequency ω is associated with a lower amplitude of process xt): ].;[ 0)(, )( ])()([ )( 5,022 ππω ω ω ωω ω −∈ ≥ + = for g f qc g yx x yxyx yx (4) b) the phase difference identifies leads or lags of variable x with respect to y (a positive value represents a lag, a negative value represents a lead) for frequency ω: ].;[ )( )( )( ππω ω ω ωφ −∈⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = for c q arctg yx yx yx (5) c) coherence is a measure of the fit (r2) in the regression of y with respect to x for frequency ω: ].;[1)(0, )()( )()( 2 22 2 ππωω ωω ωω −∈≤≤ ⋅ + = fork ff qc k yx yx yxyx yx (6) in the empirical analyses we consider only such leads or lags at a given frequency which is associated with a high value of the coefficient of coherence (for strongly correlated frequency components). 3. study results in frequency domain theoretically, when adjusted for seasonal and random fluctuations, the annual indices of industrial output represent changes resulting not only from the business cycle, but also the underlying trend. the adf test applied to the milda maria burzała 64 annual indices of output adjusted for seasonal and random fluctuations, in the case of the series for each of the analysed regions, indicated a significant negativity of the parameter δ in the model (7): .___ 1 1 ∑ = −− +δ+=δ k k tititt irprirprirpr εδδ (7) the adf test statistics rejects the null hypothesis of unit root at the α = 0.001 significance level assuming first order autoregression (k=1). the stationarity of the time series means that any stochastic trend occurring in the annual indices of output can be regarded as a realization of low-frequency fluctuations. for example, figure 1 shows indices of total industrial output sold for poland. test for stationarity was carried out based on the model: ._95.0_032.0_ 11 pol t pol t pol t irprirprirpr −− δ+−=δ for parameter δ = -0,032 the statistic adf is equal to -6.128. sty-1999 paź-1999 lip-2000 kwi-2001 sty-2002 paź-2002 lip-2003 kwi-2004 sty-2005 paź-2005 lip-2006 kwi-2007 sty-2008 paź-2008 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 figure 1. indices of total industrial output sold for poland from january 1999 to december 2008 analysis of the dominant cycles was carried out based on the power spectrum. for a clear majority of regions it is possible to indicate one cycle which to a significant degree explains the fluctuations in the output indices (shown in bold type in table 1). the 40-month cycle turned out to be such a dominant cycle. in the case of the nationwide cycle it explains more than 52% of the variability of output indices. the effect of the remaining cycles is quite different. for the wielkopolskie region we can notice an equivalent effect from three cycles (120, 60 and 40 months). generally it is possible to identify groups of regions where economic activity is ruled by long cycles (120, 60 months), medium cycles (40, 30 months) and short cycles (24, 20 months). the regions with dominating long cycles are as follows: lubuskie (59.8%), wielkopolskie (59.05%) and mazowieckie (58.27%). medium cycles dominate in podkarpackie (70.86%), zachodniopomorskie (70.24%), świętokrzyskie (63.41%) and warmińskomazurskie (62.97). the synchronization of regional business cycles with nationwide cycles 65 table 1. dominant cycles: percentage of explained variability of the time series length of time series regions frequency (duration in months) 0.008 (120) 0.017 (60) 0.025 (40) 0.033 (30) 0.042 (24) 0,050 (20) 120 poland 9.46 6.19 52.10 0.92 15.61 4.42 120 dolnośląskie 3.46 8.94 42.43 6.97 5.17 4.51 120 kujawsko-pomorskie 3.99 13.75 40.01 4.70 16.71 8.90 120 lubelskie 0.33 6.94 40.48 14.52 11.04 1.35 120 lubuskie 15.33 44.46 6.73 16.65 0.44 0.20 120 łódzkie 41.14 5.33 20.46 5.79 7.51 1.50 120 małopolskie 6.96 3.14 33.43 13.37 3.19 2.82 118 mazowieckie* 17.88 40.39 18.58 0.28 7.54 7.69 60 opolskie*** xxx 29.66 xxx 37.53 xxx 2.26 120 podkarpackie 6.21 9.77 59.26 11.61 3.53 5.87 120 podlaskie 13.50 7.94 30.51 8.49 3.35 3.92 120 pomorskie 7.11 12.90 10.20 20.38 14.58 25.72 114 śląskie** 0.97 12.96 28.62 6.23 16.56 3.04 120 świętokrzyskie 5.17 6.90 62.12 1.28 4.67 0.61 120 warmińsko-mazurskie 15.81 2.89 59.74 3.23 14.57 0.11 120 wielkopolskie 28.03 31.01 30.41 0.21 3.19 6.73 120 zachodniopomorskie 10.02 7.32 67.58 2.66 1.47 1.77 note: with a number of observations less than 120, the length of the analysed cycles changes: * 118, 59, 39.3, 29.5, 23.6, 19.7; ** 114, 57, 38, 28.5, 22.8, 19; ***60, 30, 20 months. short cycles dominate mainly in pomorskie (40.30%). in the mentioned above regions, the variability of the series of output indices is explained to a much higher degree than for poland as a whole, where long cycles explain hardly 15.65%, medium cycles 53.02% and short cycles 20.02%. taking dominant cycles as a criterion, the greatest similarity to nationwide activity was displayed by those regions where medium-length cycles dominate. therefore the greatest synchronization is observed in the warmińsko-mazurskie and świętokrzyskie regions. cospectral analysis was performed for all cycles lasting from 20 to 120 months3. the cospectral analysis characteristics are presented in tables 2, 3 and 4. the measure of correlation in particular fluctuation bands is the coefficient of coherence. the highest values of k2 (which averages 0.933 across the regions) is observed for the dominant 40-month cycle and the long-term activity 3 for three regions, some data could not be obtained. in those cases the cycles analysed are accordingly shorter. these regions are asterisked in the tables, and the duration of the components is given at the foot of the table. the shortest series, for the opolskie region, allowed only 3 cycles to be analysed. milda maria burzała 66 represented by the 120-month cycle (average 0.830 across the regions). the most weakly correlated are the 30-month cycles (average k2 across the regions: 0.451). the results obtained confirm certain intuitive suppositions that the differentiation in economic activity may refer to shorter cycles. in the long term a higher coherence can be expected between regional and nationwide economic activity. analysing the various fluctuation bands, certain groups of regions can be identified where k2 ≥ 0.95 (shown in bold type in table 2). for short, 24and 20-month cycles there are far fewer such regions than for long cycles (120 and 60 months), which confirms the intuitive assumptions referred to above. taking as a criterion the average value of k2 for all frequencies, it is possible to identify the regions with high average coefficients of coherence (mazowieckie and wielkopolskie, with averages of 0.913 and 0.894 respectively) and with relatively low average coefficients of coherence (lubuskie and lubelskie, with averages of 0.448 and 0.491 respectively). the averages given are a measure of the mean correlation between the cycles distinguished for a given region and the nationwide cycles. table 2. values of coefficient of coherence length of time series regions frequency (duration in months) 0.008 (120) 0.017 (60) 0.025 (40) 0.033 (30) 0.042 (24) 0,050 (20) 120 dolnośląskie 0.817 0.809 0.970 0.071 0.862 0.891 120 kujawsko-pomorskie 0.579 0.511 0.964 0.639 0.966 0.921 120 lubelskie 0.219 0.691 0.948 0.038 0.822 0.227 120 lubuskie 0.733 0.616 0.671 0.253 0.267 0.145 120 łódzkie 0.982 0.361 0.940 0.461 0.938 0.667 120 małopolskie 0.975 0.938 0.963 0.203 0.749 0.930 118 mazowieckie* 0.960 0.737 0.897 0.972 0.977 0.934 60 opolskie*** xxx 0.967 xxx 0.912 xxx 0.662 120 podkarpackie 0.980 0.970 0.974 0.027 0.660 0.577 120 podlaskie 0.955 0.856 0.984 0.553 0.827 0.692 120 pomorskie 0.827 0.718 0.851 0.191 0.665 0.583 114 śląskie** 0.644 0.870 0.964 0.360 0.917 0.887 120 świętokrzyskie 0.923 0.909 0.993 0.526 0.912 0.100 120 warmińsko-mazurskie 0.964 0.060 0.975 0.746 0.967 0.437 120 wielkopolskie 0.991 0.827 0.963 0.740 0.930 0.912 120 zachodniopomorskie 0.993 0.986 0.995 0.531 0.707 0.573 note: with a number of observations less than 120, the length of the analysed cycles changes: * 118, 59, 39.3, 29.5, 23.6, 19.7; ** 114, 57, 38, 28.5, 22.8, 19; ***60, 30, 20 months. gain coefficients make it possible to compare the amplitudes of cycles observed in a region with the amplitude of nationwide cycles within particular bands of fluctuations. analysing the values given in table 3, it is noticed that the synchronization of regional business cycles with nationwide cycles 67 the average coefficients of intensification for 120and 60-month cycles (1.141 and 1.296 respectively) indicate that on average, the amplitude of long-term fluctuations within regions is higher than the amplitude of nationwide fluctuations. table 3. gain coefficients length of time series regions frequency (duration in months) 0.008 (120) 0.017 (60) 0.025 (40) 0.033 (30) 0.042 (24) 0.050 (20) 120 dolnośląskie 0.859 1.408 1.318 0.509 0.841 1.301 120 kujawsko-pomorskie 0.711 1.165 1.139 1.262 1.354 1.739 120 lubelskie 0.180 1.029 1.126 0.432 1.026 0.387 120 lubuskie 1.558 2.115 0.509 1.095 0.228 0.226 120 łódzkie 1.822 0.519 0.555 0.687 0.619 0.463 120 małopolskie 0.951 0.830 0.902 0.818 0.518 0.816 118 mazowieckie* 1.273 1.646 0.602 0.565 0.674 1.070 60 opolskie*** xxx 1.486 xxx 1.733 xxx 0.496 120 podkarpackie 0.974 1.344 1.246 0.314 0.529 0.925 120 podlaskie 0.792 0.639 0.524 0.672 0.322 0.524 120 pomorskie 1.147 1.339 0.635 1.144 1.213 2.304 114 śląskie** 0.473 1.609 1.312 1.073 1.701 1.282 120 świętokrzyskie 1.163 1.600 1.737 1.184 0.836 0.240 120 warmińsko-mazurskie 1.457 0.256 1.234 1.215 1.106 0.399 120 wielkopolskie 1.985 1.856 0.883 0.662 0.531 1.202 120 zachodniopomorskie 1.773 1.888 1.959 1.426 0.489 0.771 note: with a number of observations less than 120, the length of the analysed cycles changes: * 118, 59, 39.3, 29.5, 23.6, 19.7; ** 114, 57, 38, 28.5, 22.8, 19; 60, 30, 20 months. in turn, the short cycles (24 and 20 months, with average values 0.799 and 0.884 respectively) are characterized by a higher amplitude of nationwide fluctuations. closest to unity are the average gain coefficients for medium-term fluctuations (40 and 30 months, average values 1.045 and 0.924 respectively), which indicate the greatest degree of synchronization with nationwide fluctuations. for each band of fluctuations it is also possible to indicate the region displaying the greatest synchronization with the nationwide cycle: − podkarpackie and małopolskie for the 120-month cycle; − lubelskie for the 60-month cycle; − małopolskie for the 40-month cycle; − śląskie for the 30-month cycle; − lubelskie for the 24-month cycle; − mazowieckie for the 20-month cycle. milda maria burzała 68 table 4. phase difference in radians (in months) length of time series regions frequency (duration in months) 0.008 (120) 0.017 (60) 0.025 (40) 0.033 (30) 0.042 (24) 0,050 (20) 120 dolnośląskie -0.563 (-10.8) 0.485 (4.6) 0.017 (0.1) 0.527 (2.5) 0.652 (2.5) 1.058 (3.4) 120 kujawsko-pomorskie -1.490 (-28.5) 1.099 (10.5) 0.048 (0.3) 0.045 (0.2) 1.043 (4.0) 0.654 (2.1) 120 lubelskie -0.760 (-14.5) 1.099 (10.5) 0.421 (2.7) -0.502 (-2.4) 0.123 (0.5) 2.339 (7.4) 120 lubuskie -1.215 (-23.2) 0.497 (4.7) 0.722 (4.6) 0.278 (1.3) 1.299 (5.0) 1.762 (5.6) 120 łódzkie -1.766 (-33.7) -1.147 (-11.0) 0.556 (3.5) 1.194 (5.7) 1.428 (5.5) 0.323 (1.0) 120 małopolskie -0.212 (-4.0) 0.338 (3.2) 0.170 (1.1) -0.706 (-3.4) -0.132 (-0.5) -0.026 (-0.1) 118 mazowieckie* -0.635 (-11.9) -0.682 (-6.4) -0.081 (-0.5) -0.097 (-0.5) -0.087 (-0.3) -0.424 (-1.3) 60 opolskie*** xxx 0.145 (1.4) xxx -0.420 (-2.0) xxx -0.221 (-0.7) 120 podkarpackie -0.444 (-8.5) -0.087 (-0.8) 0.072 (0.5) 1.038 (5.0) 2.261 (8.6) -0.972 (-3.1) 120 podlaskie -0.624 (-11.9) 0.419 (4.0) 0.743 (4.7) 0.677 (3.2) 1.563 (6.0) 2.140 (6.8) 120 pomorskie -0.615 (-11.7) 0.797 (7.6) 0.531 (3.4) 0.740 (3.5) -1.431 (-5.5) 1.626 (5.2) 114 śląskie** -0.002 (-0.04) 0.449 (4.1) 0.018 (0.1) 0.881 (4.0) -0.404 (-1.5) 0.143 (0.4) 120 świętokrzyskie -0.268 (-5.1) 0.520 (5.0) 0.280 (1.8) 0.526 (2.5) 0.693 (2.6) -1.892 (-6.0) 120 warmińsko-mazurskie -2.058 (-39.3) -1.711 (-16.3) -0.266 (-1.7) -0.068 (-0.3) 1.070 (4.1) 1.186 (3.8) 120 wielkopolskie 0.438 (8.4) 0.139 (1.3) -0.163 (1.0) -0.129 (0,6) -0.676 (2.6) -0.594 (1.9) 120 zachodniopomorskie 0.302 (5.8) -0.070 (-0.7) -0.170 (-1.1) -0.429 (-2.0) 0.379 (1.4) -2.003 (-6.4) note: with a number of observations less than 120, the length of the analysed cycles changes: * 118, 59, 39.3, 29.5, 23.6, 19.7; ** 114, 57, 38, 28.5, 22.8, 19; 60, 30, 20 months. the phase differences presented in table 4 enable analysis of the leads and lags of nationwide components of the business cycle relative to the corresponding components for a region. it should be noted that for most regions (apart from wielkopolskie and zachodniopomorskie), the 120-month cycle, connected with long-term activity, displays a lag with respect to nationwide activity. the longest lags are found in this fluctuation band (more than three years for the warmińsko-mazurskie region). for each component of the cycle it is possible to indicate the regions where economic activity runs in parallel with nationwide activity (without leading or lagging). however these results are different from those obtained using amplitude of fluctuations as a criterion. the greatest the synchronization of regional business cycles with nationwide cycles 69 number of regions, for which the phase difference in radians is smaller in absolute value than 0.1, can be indicated as having a dominant 40-month cycle. such a phase difference is associated with lags (leads) in economic activity by up to one month. these are the regions dolnośląskie, kujawsko-pomorskie, mazowieckie, podkarpackie and śląskie. 4. indicator of synchronization of fluctuations to summarise the study, an attempt was made to build a synthetic indicator as a basis for assigning ranks to each region depending on the degree of synchronization of fluctuations. for this purpose three characterictics of cospectral analysis were used, together with a cycle pattern perfectly synchronized with the national cycle. such a cycle would be a cycle observed in a region having k2 = 1, g(ω) = 1 and φ(ω) = 0 (maximum correlation, no intensification of fluctuations and no phase difference). table 5. synthetic indicators and ranks for individual regions regions partial synthetic indicators final synthetic indicator ranks frequency (duration in months) 0.008 (120) 0.017 (60) 0.025 (40) 0.033 (30) 0.042 (24) 0,050 (20) dolnośląskie 0.179 0.239 0.094 0.519 0.181 0.265 0.175 2 kujawsko-pomorskie 0.431 0.367 0.055 0.197 0.251 0.309 0.191 4 lubelskie 0.586 0.270 0.110 0.546 0.085 0.755 0.220 7 lubuskie 0.407 0.487 0.341 0.320 0.633 0.742 0.429 15 łódzkie 0.468 0.505 0.213 0.435 0.322 0.298 0.388 14 małopolskie 0.051 0.113 0.062 0.420 0.228 0.075 0.148 1 mazowieckie* 0.174 0.353 0.149 0.135 0.103 0.101 0.235 11 opolskie*** xxx 0.156 xxx 0.277 xxx 0.276 0.226 9 podkarpackie 0.077 0.111 0.082 0.657 0.559 0.303 0.185 3 podlaskie 0.158 0.201 0.233 0.334 0.455 0.532 0.255 12 pomorskie 0.184 0.297 0.220 0.420 0.373 0.708 0.436 16 śląskie** 0.257 0.264 0.095 0.364 0.265 0.131 0.196 6 świętokrzyskie 0.106 0.259 0.231 0.284 0.171 0.772 0.227 10 warmińsko-mazurskie 0.422 0.756 0.106 0.152 0.191 0.515 0.192 5 wielkopolskie 0.317 0.298 0.66 0.194 0.240 0.166 0.222 8 zachodniopomorskie 0.243 0.242 0.271 0.331 0.285 0.490 0.272 13 note: with a number of observations less than 120, the length of the analysed cycles changes: * 118, 59, 39.3, 29.5, 23.6, 19.7; ** 114, 57, 38, 28.5, 22.8, 19; 60, 30, 20 months. for each region in each band of fluctuations, the modulus of the distance from the pattern was analysed. these distances were normalized according to the simple formula: modulus of observed distance divided by maximum distance. this approach meant that the range of variability could be normalized and the milda maria burzała 70 distances made independent of the units used. the partial synthetic indicator in a given band is the arithmetic mean of the normalized distances. the final synthetic indicator for a given region, encapsulating information from all bands, was based on a weighted average, where the weights reflected the degree of variability explained by each component of the cycle. the values of the synthetic indicators and the ranks assigned to each region are presented in table 5. the smaller the value of the synthetic indicator, the smaller the distance from the model. hence a rank of 1 denotes the highest degree of synchronization. 5. summary summing up the results of the spectral and cospectral analysis, it is possible to identify one cycle which is dominant both for nationwide activity and for the majority of the regions. this is the 40-month cycle, which also displays the greatest synchronization when nationwide changes are compared with the changes observed in the regions. however, the classification of the regions varies depending on the criterion adopted for comparison. the indicator constructed in subsection 4 can take values from 0 to 1. the final values of the indicator appearing in the table lie within the interval [0.148, 0.436]. the relative narrowness of this interval and the values of the synthetic indicator (< 0.5) indicate a high level of synchronization between the business cycles recorded in the regions and those of the country as a whole. references dudek, s., pachucki, d., walczyk, k. (2008), synchronizacja cyklu koniunkturalnego polskiej gospodarki z krajami strefy euro w kontekście struktury tych gospodarek, http://www.nbpnews.pl/r/nbpnews/pliki_pdf/nbp/publikacje/analityczne/irg_sghp.pdf skrzypczyński, p. (2006), analiza synchronizacji cykli koniunkturalnych w strefie euro, materiały i studia nbp, vol. 210, national bank of poland department of macroeconomic and structural analysis, warsaw. skrzypczyński, p. (2008), wahania aktywności gospodarczej w polsce i strefie euro, materiały i studia nbp, vol. 227, national bank of poland department of macroeconomic and structural analysis, warsaw. talaga, l., zieliński, z. (1986), analiza spektralna w modelowaniu ekonometrycznym, pwn, warsaw. zeliaś, a. (1988), metody statystyki międzynarodowej, pwe, warsaw. synchronizacja cykli koniunkturalnych województw z cyklami ogólnokrajowymi z a r y s t r e ś c i. w artykule podjęto próbę oceny stopnia synchronizacji cykli koniunkturalnych województw polski z cyklami ogólnokrajowymi. miernikiem aktywności gospodarczej były rejestrowane miesięcznie od stycznia 1999 do grudnia 2008 indeksy produkcji sprzedanej przemysłu ogółem oczyszczone z wahań sezonowych i przypadkowych. analizę dominujących cykli koniunkturalnych przeprowadzono z wykorzystaniem analizy spektralnej. do oceny synchronizacji cykli wykorzystano charakterystyki analizy kospektralnej: współczynnik koherencji, wzmocthe synchronization of regional business cycles with nationwide cycles 71 nienie amplitudy oraz przesunięcie fazowe. w podsumowaniu artykułu podjęto próbę budowy miernika syntetycznego, który był podstawą przypisania rang poszczególnym województwom ze względu na stopień synchronizacji cyklu koniunkturalnego. s ł o w a k l u c z o w e: analiza spektralna, analiza kospektralna, cykl koniunkturalny, miernik syntetyczny. dem_2019_57to85 © 2019 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2019.004 vol. 19 (2019) 57−84 submitted october 24, 2019 issn (online) 2450-7067 accepted december 22, 2019 issn (print) 1234-3862 wondatir atinafu * energy consumption and economic growth in ethiopia: evidence from ardl bound test approach a b s t r a c t. the present study aims to investigate the dynamic relationship between economic growth and energy consumption. specifically, the study tries to answer the questions whether energy consumption has any significance effect on economic growth of the country and it also determined the magnitude of the effect. in doing this, the study used an ardl bound test approach to analyze ethiopian data from 1970 to 2017 with real gdp as a function of energy consumption, human capital., physical capital., trade openness and policy change dummy. to do so, secondary data were obtained from wdi, unctad stat and nbe. co-integration test approves the existence of long-run relationship among the variables. moreover, the estimation result reveals that, energy consumption found statistically insignificant in affecting economic growth in the long-run. however, it was positive and statistically significant in short-run. likewise, the dummy variable incorporated to capture the policy change found insignificant in longrun and with positive significant result in short-run. also, we applied the granger causality test in linear multivariate models to evaluate how important is the causal impact of energy consumption on economic growth. the results give the evidence of causality running from economic growth to energy consumption supporting “conservation hypothesis”, implying that reducing energy consumption may be implemented with little or no adverse effect on economic growth. hence, this study recommended the policy makers to improve the existing policies on energy consumption so as to enhance the level of efficiency in the energy sector i.e. energy regulation policies supporting the shift from lower-quality to higher-quality energy services. k e y w o r d s: economic-growth, energy-consumption, ardl, ethiopia, causality. * correspondence to: wondatir atinafu, department of economics, jimma university, e-mail: wondatiratinafu@gmail.com. wondatir atinafu dynamic econometric models 19 (2019) 57–84 58 introduction arguably, energy plays a vital role in economic and social development. the role of energy in economic growth has long been a controversial topic in economics literature. as a result, the ongoing debate among energy economists about the relationship between energy use and output growth led to the emergence of two opposite views. one point of view suggests that energy is the prime source of value because other factors of production such as labor and capital cannot do without energy. according to this argument, energy use is expected to be a limiting factor to economic growth. the other point of view suggests that energy is neutral to growth. this is what became to be known in the literature as the ‘neutrality hypothesis’. the main reason for the neutral impact of energy on growth is that the cost of energy is very small as a proportion of gdp and, thus, it is not likely to have a significant impact on output growth. it has also been argued that the possible impact of energy use on growth will depend on the structure of the economy and the stage of economic growth of the country concerned (ghali and el-sakka, 2009). theoretical disagreement on the role of energy is matched by mixed empirical evidence. that is, whether economic growth leads to energy consumption or that energy consumption is the engine of economic growth. the direction of causality has significant policy implications. empirically it has been tried to find the direction of causality between energy consumption and economic activities for the developing as well as for the developed countries employing the granger or sims techniques. like other developing countries ethiopia is energy using growing economy, with energy production of 14.1 and 30.9 total million metric tons of oil equivalent in 1990 and 2016 respectively. the biomass energy use is predominant which accounts 93.9% and 90.2% for the year 1990 and 2016 respectively and the balance goes to the modern energy. this shows that there is a gradual shift from traditional to modern energy sources (wdi, 2017). moreover, it is believed that the modern energy penetration rate has increased as of 2017 because of the commissioning of the three hydro power plants in the country. ethiopia is non-oil producing countries and its fossil oil energy needs are met by large quantities of imports. the preceding facts show the power sector in ethiopia is underdeveloped and hence energy consumption is very low. as a result, ethiopia is far from having satisfied the current energy demand of its people. cognizant of this problem and in line with the millennium development goals, ethiopia is trying to provide energy to its citizens by investing in major modern energy infrastructures in the country. this show that ethiopia has recognized that energy consumption and economic growth in ethiopia… dynamic econometric models 19 (2019) 57–84 59 accessibility to affordable energy services is a prerequisite to poverty alleviation, and necessary condition for sustainable economic growth. this policy goal implies that increased energy consumption can help achieve social development and enhance economic growth. thus, to meet its growing needs of energy, ethiopia faces both energy constraints from the supply side and demand management policies (eea, 2009). the current concerns about global warming also poses a question about how can economic growths in ethiopia, will be reconciled with stabilization in the use of both traditional and fossil fuels. however, for any such policy making it is essential to determine the causal relationship between energy consumption and general economic activities. it is important, therefore, to ascertain empirically whether there is a causal link between energy consumption and economic growth in ethiopia. this is particularly true given the current debate about global warming and the need to reduce greenhouse gas emissions by conserving energy consumption, since any constraints put on energy consumption to help reduce emissions will have an effect on growth and development if causality from energy to gdp exists. moreover, ethiopia has huge potential of modern energy resources; however, availability of modern energy per se is not enough for the economic and social problems facing the country. the power investment that is currently taking place in ethiopia is part of the process of the recognition that the quality and quantity of modern power supply can play a pivotal role in the country’s social and economic development. this investment process is implicitly based on the assumption that investment in modern energy and the drive towards making the modern energy sector more efficient can promote economic growth. although energy use is a reflection of climatic, geographic and economic factors (such as the relative prices of energy), it is closely related to the growth in the modern sectors (industry, motorized transport and urban areas). “there is a strong connection between the energy sector and a national economy. on the one hand, energy demand, supply and pricing have significant impact on socio-economic development and the overall quality of life of the population. on the other hand, the nature of economic structure and the change in that structure, the prevailing macro-economic conditions are key factors of energy demand and supply” (eea, 2009). the data compiled by energy information administration for the periods 1980 to 2014 shows that gdp per capita have strong correlation coefficient of 0.6 with energy consumption (eea, 2016). although the existence of correlation between the two implies the existence of causality, on the other hand it wondatir atinafu dynamic econometric models 19 (2019) 57–84 60 is source of doubt, on the part of many growth theorists. the fact that economic growth tends to be very closely correlated with energy consumption, does not a priori mean that energy consumption is the cause of the growth. indeed, most economic models assume the opposite: that economic growth is responsible for increasing energy consumption. it is also conceivable that both consumption and growth are simultaneously caused by some third factor. with this background, (that is growing of different school of thought with regard to resource consumption in general and energy consumption in particular) there are numerous researches which have tried to figure out the casual relationship between energy use growth and economic growth. the answers to questions pose in the hypothesis, which are recognized in many previous studies, have important implications for policy makers. as noted by wolde-rufael (2005), amongst others, if causality runs from energy consumption to gdp then it implies that an economy is energy dependent and hence energy is a stimulus to growth implying that a shortage of energy may negatively affect economic growth or may cause poor economic performance, leading to a fall in income and employment. in other words, energy is a limiting factor in economic growth (stern, 2010). whereas if causality only runs from gdp to energy consumption this implies that an economy is not energy dependent hence, as noted by masih and masih (1997) amongst others, energy conservation policies may be implemented with no adverse effect on growth and employment. if, on the other hand, there is no causality in either direction (referred to as the ‘neutrality hypotheses), it implies that energy consumption is not correlated with gdp, so that energy conservation policies may be pursued without adversely affecting the economy. the non-existence of such research work in the country, at least to the knowledge of the research worker, shows there is a gap to be filled, so that energy policy lesson can be drawn. and the inconclusive empirical results which make it difficult to draw a conclusion about ethiopia and the important role energy plays in economic development in country, the purpose of this paper is therefore, to fill this gap by attempting to undertake the energy economic growth nexus employing multivariate model consisting of gdp, physical capital., human capital & energy consumption growth. moreover, previous studies are that most of them are used johansen cointegration method of vector autoregressive method as their method of analysis. even though the johansen’s co-integration technique is one of the widely used methods of time series analysis, its outcome could not be reliable for small sample size; that is observations less than 80 years for the time series data (narayan, 2005; udoh et al., 2012). relatively, the autoregressive distributed lag (ardl) method has some advantage over the johansen’s method energy consumption and economic growth in ethiopia… dynamic econometric models 19 (2019) 57–84 61 (pesaran et al., 1999). these advantages are it can be applied irrespective of whether the regressors are i(1) and i(0). it can also provide valid and statistically significant result or avoids the problem of biasness in small sample sizes (pesaran et al., 1999; narayan, 2005; chaudhry et al., 2006; udoh et al., 2012). this ardl procedure can provide unbiased and valid estimates of the long run model even when some of the regressors are endogenous (harris et al., 2003, pesaran et al., 1999; ang, 2007). furthermore, in using this approach, a dummy variable can be included in the co-integration test process, which is not permitted in johansen’s method (rahimi et al., 2011). hence in this paper, ardl model is adopted so as to provide valid empirical evidence on the main target of this study which is assessing the nexus between energy consumption and economic growth in ethiopia. the overall objective of this paper is to empirically investigate the nexus between energy consumption and economic growth in ethiopia. more specifically: to empirically examine the effect of energy consumption on the aggregate economic growth of ethiopia in both short run and long run and to investigate the possible causal relationship between economic growth and energy consumption in ethiopia, the ardl bound test approach and granger's causality test were used. the reminder of the paper organized as follows. section 2 includes review literature, section 3 presents methodology applied in the study, section 4 reports the findings and discussion of our analysis and conclusion follows in section 5. 1. review of related literature 1.1. theoretical literature the laws of thermodynamics and the conservation of matter describe the immutable constraints within which the economic system must operate. the mass-balance principle means that, in order to obtain a given material output, greater or equal quantities of matter must be used as inputs with the residual a pollutant or waste product. therefore, there are minimal material input requirements for any production process producing material outputs. the second law of thermodynamics (the efficiency law) implies that a minimum quantity of energy is required to carry out the transformation or movement of matter or, more generally, perform physical work. carrying out transformations in finite time requires more energy than these minima. all production involves work. therefore, all economic activities must require energy, and there must wondatir atinafu dynamic econometric models 19 (2019) 57–84 62 be limits to the substitution of other factors of production for energy so that energy is always an essential factor of production. primary factors of production are defined as inputs that exist at the beginning of the period under consideration and are not directly used up in production (though they can be degraded or accumulated from period to period), while intermediate inputs are those created during the production period under consideration and are used up entirely in production. mainstream economists usually think of capital., labor and land as the primary factors of production, and goods (such as fuels and materials) as intermediate inputs. the prices paid for the various intermediate inputs are seen as eventually being payments to the owners of the primary inputs for the services provided directly or embodied in the produced intermediate inputs. this approach has led to a focus in mainstream growth theory on the primary inputs, and in particular, capital and labor. the classical factor of land, including all-natural resource inputs, gradually diminished in importance in economic theory as its value share of gdp fell steadily and is usually subsumed as a subcategory of capital. growth models with resources and no technical change adding non-renewable natural resources that are essential in production to the basic mainstream growth models means that capital also needs to be accumulated to compensate for resource depletion. when there is more than one input – both capital and natural resources – there are many alternative paths that economic growth can take, determined by both the nature of technology and institutional arrangements. solow showed that sustainability is achievable in a model with a non-renewable natural resource with no extraction costs and non-depreciating capital when the elasticity of substitution between the two inputs is unity, and when certain other technical conditions are met. sustainability, and even indefinite growth in consumption, can occur when the utility of individuals is given equal weight without regard to when they happen to live. however, under competition the same model economy results in exhaustion of the resource and consumption and social welfare eventually fall to zero. with any constant discount rate, the efficient growth path also leads to eventual depletion of the natural resource and the collapse of the economy. the hartwick rule shows that if sustainability is technically feasible, a constant level of consumption can be achieved by reinvesting the resource rents in other forms of capital., which in turn can substitute for resources. a common interpretation of this body of work is that substitution and technical change can effectively decouple economic growth from the use of energy energy consumption and economic growth in ethiopia… dynamic econometric models 19 (2019) 57–84 63 and other resources. depleted resources can be replaced by more abundant substitutes, or by ‘equivalent’ forms of human-made capital (people, machines, factories, etc.). growth models with resources and technical change in addition to substitution of capital for resources, technological change might permit growth or at least constant consumption in the face of a finite resource base. when the elasticity of substitution between capital and resources is unity, exogenous technical progress will allow consumption to grow over time if the rate of technological change divided by the discount rate is greater than the output elasticity of resources. technological change might enable sustainability even with an elasticity of substitution of less than one. once again, technical feasibility does not guarantee sustainability. depending on preferences for current versus future consumption, technological change might instead result in faster depletion of the resource. therefore, mainstream economic growth theory assumes that resource consumption is a consequence, not a cause, of growth. synthesis: unified models of energy and growth the mainstream growth models ignore energy in the economic growth, by contrast, the ecological economics literature posits a central role for energy in driving growth but argues that limits to substitutability and/or technological change might limit or reverse growth in the future. but none of the models and theories reviewed so far really provides a satisfactory explanation of the long-run history of the economy. until the industrial revolution, output per capita was generally low and economic growth was not sustained. ecological economists point to the invention of methods to use fossil fuels as the cause of the industrial revolution. but the mainstream growth models that ignore energy resources can at least partly explain economic growth over the last half a century. there are currently two principal mainstream theories that explain the growth regimes of both the preindustrial and modern economies and the cause of the industrial revolution, which formed the transition between them. these are endogenous technical change approach, and the second approach is represented by two sectors – malthusian sector and solow sector. to integrate the different approaches, stern (2011) proposed to modify solow’s growth model. in the model stern added an energy input that has low substitutability with capital and labor, while allowing the elasticity of substitution between capital and labor to remain at unity. in this model, depending wondatir atinafu dynamic econometric models 19 (2019) 57–84 64 on the availability of energy and the nature of technological change, energy can be either a constraint on growth or an enabler of growth. omitting time indexes for simplicity, the model consists of two equations: y = [(1-g ) (albbb k1–b + g(aee)f )]f (1) d k = s(y – pee ) dk (2) equation (1) embeds a cobb–douglas production function of capital (k) and labor (l) in a constant elasticity of substitution (ces) function of value added and energy (e) that produces gross output y. f = (d 1)/ d; where d is the elasticity of substitution between energy and the value-added aggregate; pe the price of energy; and g is a parameter reflecting the relative importance of energy and value added. al and ae are the augmentation indexes of labor and energy, which can be interpreted as reflecting both changes in technology that augment the effective supply of the factor in question and changes in the quality of the respective factors. equation (2) is the equation of motion for capital that assumes like solow that the proportion of gross output that is saved is fixed at s and that capital depreciates at a constant rate d. as d -> 1and g -> 0 we have the solow model as a special case, where in the steady state, k and y grows at the rate of labor augmentation. additionally, depending on the scarcity of energy, the model displays either solow-style or energy constrained behaviour. 1.2. empirical literature review over the past few years, the relationship between energy consumption and economic growth has been extensively researched. yet, there seems to be no consensus regarding the direction of causality between energy consumption and economic growth. in a study of over more than hundred countries, chontanawat et al. (2008) find that the causal relationship between energy consumption and economic growth is more pronounced in developed than in developing countries. causality running from energy consumption to economic growth. ethiopia was included in the study and the result shows there is granger causality running from economic growth to energy consumption. stern (1993) examined the causal relationship between energy use and gdp in the usa. he employed a multivariate vector autoregressive (var) analysis and used a weighting index of energy quality, where content of energy use shifts from lower quality energy such as coal to high quality energy such as energy consumption and economic growth in ethiopia… dynamic econometric models 19 (2019) 57–84 65 electricity, rather than using a measure of total energy use. also, found that total energy use does not granger cause gdp. masih and masih (1996) used cointegration analysis to study this relationship in a group of six asian countries and found cointegration between energy use and gdp in india, pakistan, and indonesia. no cointegration is found in the case of malaysia, singapore and the philippines. the flow of causality is found to be running from energy to gdp in india and from gdp to energy in pakistan and indonesia. nondo and mulugeta (2009) applied panel data techniques to investigate the long-run relationship between energy consumption and gdp for a panel of 19 african countries (comesa) based on annual data for the period 1980– –2005. they have estimated the long-run relationship and test for causality using panel-based error correction models. the results indicate that long-run and short-run causality is unidirectional., running from energy consumption to gdp. the paper did not elaborate county specific result, it simply indicated the result in its aggregate form, and the study did not include ethiopia. using a bivariate analysis ebohon (1996), examines the causal directions between energy consumption and economic growth for nigeria and tanzania. the results show a simultaneous causal relationship between energy and economic growth for both countries. in a bivariate relationship between energy consumption and economic growth in african countries, wolde-rufael (2005) also found conflicting evidence with the neutrality hypothesis supported in a substantial number of countries, with little support for the hypothesis that energy consumption causes economic growth. bi-directional causality was detected for two countries, gabon and zambia. for the remaining nine countries where there was no causality in any direction between economic growth and energy consumption, energy consumption seems neither to promote nor to retard economic growth. the most striking result of the empirical evidence is that the introduction of both gross capital formation and labor has altered the direction of causality in thirteen countries that were previously investigated by wolde-rufael (2005). in seven of the countries where wolde-rufael (2005) found no evidence of causality in any direction between energy consumption and economic growth, he now found evidence of granger causality for seven of these countries, benin, senegal., south africa, sudan, togo, tunisia and zimbabwe. in benin and south africa causality runs now from energy consumption to economic growth; in senegal., sudan and tunisia causality runs now from economic growth to energy consumption, and in togo and zimbabwe we find now that energy and economic growth were mutually causal. wondatir atinafu dynamic econometric models 19 (2019) 57–84 66 causality was also reversed in another six counties: algeria, cameroon, gambia, ghana, morocco and nigeria. in algeria causality was reversed from economic growth to energy consumption, to the opposite causality running from energy consumption to economic growth contrary to the no causality found by chontanawat et al. (2008). amirat and bouri (2010) undertook analyses of the causal relationship between the per capita energy consumption and the per capita gdp in algeria by using annual data from 1980 to 2007. they include capital and labor as additional variables to the energy growth nexus. they used granger causality test and the variance decomposition analysis. the results give the evidence of causality running from energy consumption to economic growth. similarly, using a multivariate causality test, akinlo (2008) found also conflicting results for eleven african countries. the result shows that energy consumption is co-integrated with economic growth in seven of the countries. in addition, in few of the countries, the result suggests that energy consumption has a significant long run impact on economic growth. olatunji adeniran (undated) tested for causal relationship between energy consumption and gdp in nigeria using systematic econometric techniques. the study found that there is a unidirectional causality that runs from gdp to electricity consumption. jumbe (2004) examined cointegration and causality between electricity consumption (kwh) and, respectively, overall gdp (gdp), agricultural gdp (agdp) and non-agricultural gdp (ngdp) using malawi data for 1970– –1999 periods. the results show that kwh is, respectively, cointegrated with gdp and ngdp, but not with agdp. the granger causality results show a bidirectional causality between kwh and gdp, but a unidirectional causality running from ngdp to kwh. yohannes (2010) has conducted causal relationship between economic growth and energy consumption in ethiopia and he found energy consumption granger cause economic growth. 2. methodology of the study autoregressive distributive lag (ardl) approach to co-integration so as to capture the nexus between energy consumption and economic growth, time series secondary data was employed. data for all variables was taken from only two sources so as to keep its consistency and avoid possible biases due to difference in measurement techniques. the data sources for this study energy consumption and economic growth in ethiopia… dynamic econometric models 19 (2019) 57–84 67 were world bank (wb) and unctad. the study considers annual data of ethiopia for the years from 1970 to 2017. most of the time series studies in this area previously conducted are used the engle-granger approach following engle and granger (1987) and the johansen’s co-integration technique following johansen (1988) and johansen and juselius (1990). but its outcome could not be reliable for small sample size (narayan, 2005; udoh et al., 2012). relatively, the autoregressive distributed lag method of co-integration (ardl) has more advantage over the johansen’s method (pesaran et al., 1999). johansen’s co-integration technique requires that all the variables in the system have equal order of integration, that is the application of the johansen technique will fail when the underlying regressors have different order of integration, especially when some of the variables are i(0) (pesaran et al., 2001). that means the trace and maximum eigen value tests may lead to erroneous co-integrating relations with other variables in the model when i(0) variables are present in the system (harris, 1999). fortunately, to overcome this problem a new autoregressive distributed lag (ardl) model is developed by pasaran, shin and smith (2001) which have more advantages than the johansen co-integration approach. first, the ardl approach can be applied irrespective of whether the regressors are i(1) and i(0) or have a mix of these integration orders. the only exception is that none of the variables in the model is integrated of order 2 or higher. second, while the johansen co-integration techniques require large data samples for validity, the ardl procedure provides statistically significant result in small samples (pesaran et al., 1999; narayan, 2005; udoh et al., 2012). that means, it avoids the problem of biasness that arise from small sample size (chaudhry et al., 2006). third, the ardl procedure provides unbiased and valid estimates of the long run model even when some of the regressors are endogenous (harris et al., 2003; pesaran et al., 1999; ang, 2007). moreover, the ardl procedure employs only a single reduced form equation, while the other co-integration procedures estimate the long-run relationships within a context of system equations. further, in using the ardl approach, a dummy variable can be included in the co-integration test process, which is not permitted in johansen’s method (rahimi et al., 2011). therefore, in order to achieve the targeted objectives of the study, the model of economic growth equation is estimated using ardl model of econometric technique. the above advantages of the ardl technique over other standard co-integration techniques justify the application of ardl approach in the present study to investigate the link between economic growth and energy consumption. wondatir atinafu dynamic econometric models 19 (2019) 57–84 68 the empirical model in ardl framework according to pesaran and pesaran (1997), the ardl approach requires the following two steps. in the first step, the existence of any long-term relationship among the variables of interest is determined using an f-test. the second step of the analysis is to estimate the coefficients of the long-run relationship and determine their values, followed by the estimation of the shortrun elasticity of the variables with the error correction representation of the ardl model. by applying the ecm version of ardl, the speed of adjustment to equilibrium will be determined. according to pesaran and pesaran (1997), the ardl model is represented by the following equation: after checking for the order of integration of all variables in the model, the autoregressive distributed lag (ardl) model involves two steps for estimating the long-run relationship (pesaran et al., 2001). in the first step the existence of long-run relationship among all variables in an equation should be examined and then in the second step the long-run and short-run coefficients of the variables can be estimated in the model. one can run the second step only if we find along run co-integration relationship among the variables in the first step. in order to examine the long-run relationship and dynamic interaction between economic growth and energy consumption, this study employs an ardl model. in general., there are three steps in estimating the model. the first step is to estimate the long-run relationship among the variables. this is done by testing the significance of the lagged levels of the variables in the error correction form of the underlying ardl model. our ardl model can be written as follows: ∆𝐿𝑁𝑅𝐺𝐷𝑃! = 𝛼" + 𝛽#𝐿𝑁𝑅𝐺𝐷𝑃!$# + 𝛽%𝐿𝑁𝐸𝑁𝐸𝑅𝐺𝑌!$# + 𝛽&𝐿𝑁𝑇𝑂!$# + 𝛽'𝐿𝑁𝑃𝐶!$# + 𝛽(𝐿𝑁𝐻𝐶!$# + 2𝛿#∆𝐿𝑁𝑅𝐺𝐷𝑃!$) * )+# + 2𝛿%∆𝐿𝑁𝐸𝑁𝐸𝑅𝐺𝑌!$) * )+# + 2𝛿&∆𝐿𝑁𝑇𝑂!$) * )+# + 2𝛿'∆𝐿𝑁𝑃𝐶!$) * )+# + 2𝛿(∆𝐿𝑁𝐻𝐶!$) * )+# + 𝛾𝐷_𝑒𝑛𝑒𝑟𝑔𝑦 + 𝜀! where, lnrgdp is log of real gdp, lnenergy is log of energy consumption, lnto is log of trade openness, lnpc is log of physical capital., 𝐿𝑁𝐻𝐶 is log of human capital. the selection of the optimum lagged orders of the ardl models is based on akaike information criteria (aic). in order to test energy consumption and economic growth in ethiopia… dynamic econometric models 19 (2019) 57–84 69 co-integration among the variables, the wald f-statistics for testing the joint hypotheses has to be compared with the critical values as tabulated by pesaran et al. (2001). the joint hypotheses to be tested are: 𝐻": 𝛽# = 𝛽% = 𝛽& = 𝛽' = 𝛽( = 𝛽, = 𝛽= 0 𝐻#: 𝛽) ≠ 0 , 𝑖 = 1,2….7 if the f-statistics is higher than the upper bound critical value, the null hypothesis (𝐻") is rejected, indicating that there is a long run relationship between the lagged level variables in the model. in contrast, if the f-statistic falls below the lower bound, then the 𝐻" cannot be rejected and no long run relationship exists. however, if the f-statistics falls in between the upper bound and lower bound critical values, the inference is inconclusive. at this condition, the order of integration of each variable should be determined before any inference can be made. in the second step, once the co-integration is established, the conditional ardl (p,q,r,s,t,) long-run model of the economic growth and energy consumption can be estimated as below: 𝐿𝑁𝑅𝐺𝐷𝑃! = 𝛼" + 2𝛽#𝐿𝑁𝑅𝐺𝐷𝑃!$# * )+# + 2𝛽%𝐿𝑁𝐸𝑁𝐸𝑅𝐺𝑌!$# . )+" + 2𝛽&𝐿𝑁𝑇𝑂!$# / )+" + 2𝛽'𝐿𝑁𝑃𝐶!$# 0 )+" + 2𝛽(𝐿𝑁𝐻𝐶!$# ! )+" + 𝐷 + 𝜀! in the final step, we obtain the short-run dynamic parameters by estimating an error correction model (ecm) associated with the long-run estimates. this is specified as follows: ∆𝐿𝑁𝑅𝐺𝐷𝑃! = 𝛼" + ∑ 𝛽#∆𝐿𝑁𝑅𝐺𝐷𝑃!$) 1 2+" + ∑ 𝛽%∆𝐿𝑁𝐸𝑁𝐸𝑅𝐺𝑌!$) 1 2+" + ∑ 𝛽&∆𝐿𝑁𝑇𝑂!$) 1 2+" + ∑ 𝛽'∆𝐿𝑁𝑃𝐶!$) 1 2+" + ∑ 𝛽(∆𝐿𝑁𝐻𝐶!$) 1 2+" + 𝛾𝐷 + 𝛿𝐸𝐶𝑇𝑡 − 1 + 𝜀! where, 𝛿#, 𝛿%, 𝛿&, 𝛿', 𝛿(, 𝛿, , 𝛿are the short-run dynamic coefficients of the model’s convergence to equilibrium, and 𝛿 is the speed of adjustment. the theoretical foundation of the study is based on the augmented solow model and endogenous growth model for economic growth equation which aims to show the impact of energy consumption on economic growth of ethiopia. it is constructed based on the theoretical framework of the augmented solow model and endogenous growth model with a modification that extends wondatir atinafu dynamic econometric models 19 (2019) 57–84 70 the basic production function framework to permit human capital as an additional input in to the production function following romer (1996) and energy following stern and cleveland (2004). as implied by solow’s formulation, economic growth is a function of capital accumulation, an expansion of labor force and exogenous factor, technological progress which makes physical capital and labor more productive. the presence of co-integration alone does not indicate the direction of causality. hence, we need to test whether the relationship between the variables is unidirectional or bidirectional. since the underlying series (lnrgdp and lnenergy) are integrated of the same order, the ordinary granger causality test can be applied to perform causality tests. the test proceeds in estimating the following two equations. 𝐿𝑁𝑅𝐺𝐷𝑃! = 𝛼" + ∑ 𝛼#∆𝐿𝑁𝑅𝐺𝐷𝑃!$) 3 2+" + ∑ 𝛼%∆𝐿𝑁𝐸𝑁𝐸𝑅𝐺𝑌!$) 3 2+" + 𝜀#! 𝐿𝑁𝐸𝑁𝐸𝑅𝐺𝑌! = 𝛽" + ∑ 𝛽#∆𝐿𝑁𝐸𝑁𝐸𝑅𝐺𝑌!$) 3 2+" + ∑ 𝛽%∆𝐿𝑁𝑅𝐺𝐷𝑃!$) 3 2+" + 𝜀%! the null hypothesis is that: h0: 𝛽11 = β12 =.... = β1j = 0 implying lenergy does not granger cause lrgdp h1: β11 ≠ 𝛽12 ≠ .... ≠ β1j ≠ 0 implying lnenergy does granger cause lnrgdp the null hypothesis can be stated as: h0: 𝛼11 = α12 = .... = α1j = 0 implying lrgdp does not granger cause lnenergy h1: α11 ≠ 𝛼12 ≠ .... ≠ 𝛼1j ≠ 0 implying lnrgdp does granger cause lnenergy the decision is that there is causality from energy consumption (lnenergy) to economic growth (lnrgdp) if the null hypothesis h0: 𝛽11 = β12 = .... = β1j = 0 can be rejected at least at 10% level of significance. similarly, there is causality from economic growth to energy consumption if the null hypothesis h0: 𝛼11 = α12 = .... = α1j = 0 can be rejected at least at 5% level of significance. energy consumption and economic growth in ethiopia… dynamic econometric models 19 (2019) 57–84 71 description of the macroeconomic variables the descriptions of the dependent and the explanatory variables that are included in the study model are explained as follows: real gross domestic product (rgdp): it is the total market value of all final domestically produced products at constant price. it is a dependent variable of the model. here rgdp has been transformed into log so as to keep the linearity of the variable vis-á-vis the other variables. energy consumption (ec): energy consumption is proxied by (gdp per unit of energy use) which measured by the ppp gdp per kilogram of oil equivalent of energy use. physical capital (pc): capital stock is defined as the value of the existing supply of physical goods that are used in the production process at a given point of time and includes buildings, machinery, equipment and inventory. there are points of view that capital stock is generally believed to be of critical importance, not only as a component of final aggregate demand, but also in terms of the impact of capital stock on the economy’s growth and employment opportunities (ghali, 1999). hence, we expect that gross capital formation should have a positive coefficient in explaining economic growth. human capital (hc): in this study human capital is proxied by secondary school enrolments (% gross). romer (1996) and gungor (1997) notes that human capital which describes the knowledge and skills embodied in individuals are an important source of economic growth. human capital accumulation that is the ability of individuals to solve problems and to think critically is believed to promote higher growth by improving labor force which will be more productive. therefore, human capital variable is expected to have positive impact on the production and economic growth of the country. trade openness (to): trade openness is the sum of export and import divided by two divided by gdp and expected to affect economic growth positively. romer, (1993) claimed that the countries have higher possibility to implement leading technologies from other countries if countries are more open to trade. in addition, chang et.al (2005) emphasized trade openness promotes the efficient comparative advantage which allows the dissemination of knowledge and technological progress and encourages competition in the international market. policy dummy (d): changes in political and economic policies (the dummy variable d in the model) can influence the performance of the economy through investment on human capital and infrastructure, improvement in political and legal institutions and so on (easterly, 1993). wondatir atinafu dynamic econometric models 19 (2019) 57–84 72 3. results and discussion 3.1. empirical results for unit root testing it is vital and must to test the nature of stationarity of the variables before running ardl model, a model used to determine the existence of long run relationship among the variables. doing so avoids the possibility of running a spurious regression, which makes the result to be unreliable and inconsistent. the null hypothesis of no stationarity cannot be rejected for all variables in level. however, every variable become stationary with trend once they are first differenced. this indicates that none of the above variables are integrated of order two i(2), which is a precondition to use ardl model (see appendix 1) as a result, autoregressive distributed lag approach to co-integration is the right technique to apply in this scenario. therefore, ardl or bound testing approach to co-integration is the preferred and appropriate method of regression in this case. 3.2. bounds test for long run relationship in the ardl approach to co-integration, the first step is to test the presence of co-integration or long run relationship among the variables. this test for the long run relationship is done using the f-statistic. given the annual nature of the data; it is recommended that the optimal lag length for the ardl model is maximum two lags. moreover, aic is used to determine the optimal lag because of small sample size at hand. the f statistic will then be compared with the lower and upper bounds of kripfganz and schneider (2018) critical values, based on the rational mentioned in chapter three. the calculated f-statistics is 5.416 and this value is higher than the upper bound critical values at 5% level of significance. the results indicate that there is strong evidence of long-run relationship or cointegration between log of rgdp and the remaining variables. this represents a co-integrated rgdp equation in ethiopia. thus, the null hypothesis of no co-integration between log of rgdp and its fundamentals is rejected (see appendix 2). a. dynamic long-run ardl estimates based on the confirmation obtained from the unit root test about the absence of a variable which is i(2) and given the f statistic result which indicated the existence of long run cointegration among the variables, it is now possible energy consumption and economic growth in ethiopia… dynamic econometric models 19 (2019) 57–84 73 to proceed to the estimation of the long run coefficients of the model. the following table presents the results found after running the appropriate ardl model to find out the long run coefficients. table 1. estimated long run coefficients using the ardl approach (dependent variable is lnrgdp; 44 observations used for estimation from 1974 to 2017) regressors coefficient st. error t-ratio lnenergy 3.84 9.27 0.41 lnto 0.544* 0.287 1.90 lnhc 0.099 0.101 0.98 lnpc 0.455*** 0.138 3.3 d –0.274 0.176 –1.56 constant –5.299 15.675 –0.34 note: the signs ***, ** and * indicate the significance of the coefficients at 1%, 5% and 10% level of significance respectively; ardl (4, 1, 1, 0, 4, 4) selected based on akaike information criterion (aic). the real gdp equation or growth model is specified in a log-linear form; hence, the coefficient of the dependent variable can be interpreted as elasticity with respect to economic growth. as we observe from the long-run ardl regression result, log of energy consumption has an insignificant impact on log of real gdp. additionally, human capital found to be statistically insignificant in the long-run. the result is inconsistent with the outcome found by driffield and jones (2013), and fayissa and nsiah (2008) where human capital is found to positively and significantly affecting output. moreover, the dummy for policy change found statistically insignificant to affect economic growth (i.e. other things remain constant, policy change from derg regime to post derg regime of the country doesn’t significantly affect the performance of the economy in the long-run). apart from these, both trade openness and physical capital found to be positively and statistically significant to affect economic growth in ethiopia (see appendix 3). b. short-run error correction model the short run model results are different from the long run. for instance, energy consumption is significantly and positively affecting output which is dissimilar to the long run result. also, even though trade openness has statistically significant in both long run and short run estimate, it has negative sign in short run, however. the result also suggests that, openness can be pain for an economy and invoke a call for protectionism. this may arise in line with poor quality of institutions and weak exporting capacity of the country or large share of import content of the countries international trade participation. wondatir atinafu dynamic econometric models 19 (2019) 57–84 74 table 2. error correction representation for the selected ardl model (dependent variable is dlnrgdp; 44 observations used for estimation from 1974 to 2017) regressors coefficient st. error t-ratio dlnrgdp 0.3527* 0.194 1.82 dlnenergy 6.035** 2.486 2.43 dlnto –0.00971* 0.0506 –1.92 dlnpc –0.0321 0.0593 –0.54 dd 0.206*** 0.0708 2.92 ecm (–1) –0.2779** 0.12564 -2.21 r-squared = 0.8575 r-adjusted = 0.7448 note: the signs ***, ** and * indicate the significance of the coefficients at 1%, 5% and 10% level of significance respectively. more interestingly, the dummy of policy change found positive and statistically significant. that means the policy transition during 1991 (departure from the previous socialist system) had significant effect on economic growth of ethiopia in the short run. the speed of adjustment of any disequilibrium towards long-run equilibrium or the equilibrium error correction coefficient (ecm), estimated (–0.2779) is highly significant and has the correct sign. it implies a high speed of adjustment to equilibrium after a shock. approximately 27.79 % of the disequilibrium from the previous year’s shock converges back to the long-run equilibrium in the current year and such significant error correction term is another proof for the existence of a stable a long-run equilibrium relationship among the variables. regarding the short run model’s goodness of fit, the regression result imply that real gross domestic product is moderately explained by the explanatory variables incorporated in the model. the adjusted r-squared reveals that 74.48% of the short-run variation in real gross domestic product is explained by the explanatory variable (see appendix 2). diagnostic testing and model stability in this study akaike information criterion is used to determine the optimal lag length of each variable automatically because it is a better choice for small sample size data. moreover, according to pesaran and shin (1999), for the annual data a maximum of two lag length is recommended to choose the optimal lag for each variable. therefore, in this paper a maximum lag length of 2 was chosen for the conditional ardl model. finally, in this model, aic selects the optimal lag length of each variable (lnrgdp, lnenergy, lnto, lnhc, lnpc, d), respectively and it is ardl(4, 1, 1, 0, 4, 4). this energy consumption and economic growth in ethiopia… dynamic econometric models 19 (2019) 57–84 75 automatically determination of the lag length is to get the valid result and inferences (see appendix 4) to check the reliability and verifiability of the estimated long-run and short-run models, diagnostic tests are undertaken. these tests include serial correlation (breusch and godfrey lm test), functional form (ramsey’s reset test), normality (jarque-bera test), hetroscedasticity (breusch-pagangodfrey test) and also cumsum recursive residuals and cumsum square recursive residuals tests are applied to check the overall stability of the longrun and short-run coefficients which are recommended by pesaran et al. (2001). the results indicate that both the lm version and the f version of the statistics are unable to reject the null hypothesis specified for each test. hence, there is no serial correlation problem and the ramsey functional form test confirms that the model is specified well. likewise, the errors are normally distributed and the model doesn’t suffer from heteroskedasticity problem (see appendix 6). a. the null hypothesis of no serial correlation (bruesch-godfrey lm test) is failed to reject for the reason that that the p-value associated with test statistic is greater than the standard significant level (0.234> 0.05). since the lagged dependent variable appear as a regressor in the model, lm test avoid the use of the traditional durbin watson test statistic. b. for ramsey’s reset test, which tests whether the model suffers from omitted variable bias or not we failed to reject the null hypothesis of this test which says that the model is correctly specified, because the p-value is larger than the conventional significance value (0.716> 0.05). c. similarly, we could not reject the null hypothesis for the jarque-bera normality test which says that the residuals are normally distributed, for the reason that the p-value associated is larger than the standard significance level (0.627>0.05). therefore, the error term is normally distributed. d. the last diagnostic test is hetroscedasticity test and as we can understand from the result, the null hypothesis of no heteroscedasticity is failed to be rejected at 5% significant level due to its p-value associated is greater than the standard significance level (0.301> 0.05). pesaran and shin (1997) further suggested that structural stability or presence of structural break of the long run and short run relationships for the sample period can be better examined by cumulative sum (cumsum) and the cumulative sum of squares (cumsumsq) of the recursive residual test. the test is based the first set of n observations and is updated recursively which will then be plotted against the break points to assess the given parameter consistency. in this study the plot of cumsum and cumsumsq starts wondatir atinafu dynamic econometric models 19 (2019) 57–84 76 from 1994/95, implying that the test is based on the recursive residuals from observations before 1994/95. the test chooses the first n observation by itself. for the stability test the graph plots both the cumulative sum and the 5% critical lines. and, if the cumulative sum remains inside between the two critical lines or bounds back after it is out of the boundary lines, the null hypothesis of correct specification of the model cannot be rejected. but, if the cusum goes outside (never returns back) between the two critical bounds there exists series parameter instability problem. figure 1. cumulative sum of recursive residuals figure 2. cumulative sum of square of recursive residuals (ardl(4, 1, 1, 0, 4, 4) result). as the two plots above clearly reveal the plots of cumsum and cumsumsq stay within the lines, and, therefore, this confirms the equation is correctly specified and the model is stable. furthermore, the result shows that there is no structural instability in the model during the sample period. from this, the model appears to be robust in estimating short run and long run relationship between real gross domestic product and the included regressor. energy consumption and economic growth in ethiopia… dynamic econometric models 19 (2019) 57–84 77 granger causality test results granger causality test provides important information of the causal direction between the variables and knowing the direction of causality between the variables. in this study, granger causality wald test after var model was employed to look at the causal linkages between economic growth and energy consumption in ethiopia. c. pairwise granger causality test this section is concerned with tests of granger causality between gdp and energy. the estimated f-statistics of the causality test are reported in the tables below. from the result we fail to accept the null hypothesis that lngdp does not granger causes lnenergy, but we fail to reject the null hypothesis that lnenergy does not granger cause lngdp. therefore, it appears that granger causality runs one-way from gdp to energy and not the other way (see appendix 5). from the above pairwise granger causality we fail to reject the null hypothesis for lnenergy does not granger cause lngdp because the p-value is 0.93326 which much higher than 0.05. however, in the second case we can reject the ho and accept the alternative which states lnrgdp can granger cause lnenergy. d. vector error correction granger causality (wald test) after undertaking pair wise granger causality test, error correction model is also used and the result are shown in below. table 3. granger causality test results for lnrgdp equation dependent variable: lnrgdp (log of real gdp) excluded chi 2 p-value lnenergy 0.02524 0.874 lnto 2.8082 0.094* lnhc 0.5262 0.468 lnpc 8.2672 0.004*** d 3.7145 0.054* all 53.656 0.000*** in table 3 where gdp is dependent variable the null hypothesis energy consumption does not granger cause economic growth and the alterative hypothesis is energy consumption granger cause economic growth. from the table 3 it shown that the p-value is 0.874 and based on the ‘p-value’ we tend to accept ho. that is, energy does not granger cause economic growth. wondatir atinafu dynamic econometric models 19 (2019) 57–84 78 table 4. granger causality test results for lnenergy equation dependent variable: lnenergy (log of energy consumption) excluded chi 2 p-value lnrgdp 8.344 0.004*** lnto 5.5232 0.019** lnhc 1.1691 0.28 lnpc 4.413 0.036** d 16.982 0.000*** all 38.881 0.000*** note: the signs ***, ** and * indicate the significance of the coefficients at 1%, 5% and 10% level of significance respectively. in the table 4 where energy is dependent variable and with hypothesis gdp does not granger cause energy consumption and the alternative hypothesis that gdp does granger cause energy consumption. the ‘p-value’ is 0.004 and accordingly we have to reject the null hypothesis and hence we tend to accept the alterative hypothesis. therefore, the evidence of multi-variate analysis is in line with the growth-led energy consumption hypothesis where causality running from economic growth to energy consumption. the above two results that is the pair wise granger causality (which is bivariate analysis) and the vector error correction model granger causality test (which is multivariate analysis including physical & human capital., trade openness and policy dummy) are consistent with each other. both evidences are in line with the growth-led energy consumption hypothesis where causality running from economic growth to energy consumption, implying that economic development seems to take precedence over energy consumption and that economic growth caused greater demand for energy. the economy of ethiopia is heavily dominated by the agricultural sector. however, the energy use of the sector is insignificant. and the results show that shortage of energy may not adversely affect gdp growth or cause a fall in the gdp in the short run. this is because the agricultural sector does not depend on energy. the above result of granger causality running from economic growth to energy consumption in ethiopia goes in line with the finding of chontanawat et al. (2008) who found economic growth granger cause energy consumption using bivariate analysis for ethiopia. gdp is generally less in the developing world than the developed world (or alternatively causality from energy to gdp generally increases at higher stages of development). hence the results support the view that energy is generally neutral with respect to its effect on economic growth in the developing world, implying that the effect of energy conservation policies to help combat global warning would have a greater detrimental effect on the overall growth energy consumption and economic growth in ethiopia… dynamic econometric models 19 (2019) 57–84 79 of oecd/developed countries than that of the non-oecd/developing countries”. and it also supports the finding of wolde-rufael (2005) for five countries (algeria, democratic republic of congo, egypt, ghana and ivory coast) who found economic growth granger cause energy consumption using bivariate analysis. and similarly, it goes in line with what found by wolde-rufael (2009) for sudan and zimbabwe which granger causality test shows that economic growth granger cause energy consumption using multi-variate analysis consisting gdp, capital., energy and labor. however, it contradicts with the results for cameroon, gambia, ghana, morocco and nigeria. and the result goes in line with the result of akinlo (2008) who found for sudan and zimbabwe granger causality running from economic growth to energy consumption. the result is also consistent with masih and masih (1996) for pakistan and indonesia, olatunji adeniran(undated) for nigeria, jumbe (2004) for malawi. this finding is contrary with the result of yohannes (2010) in ethiopia and amirat and bouri (2010) who undertook analyses of the causal relationship between the per capita energy consumption and the per capita gdp in algeria adding capital and labor to the economic growth and energy consumption nexus and found granger causality running from energy consumption to economic growth which reverse the result of chontanawat et al. (2008) for algeria. it is also inconsistent with nondo et.al (2009) for 19 cemesa member countries. the implication of the uni-directional causality running from economic development to energy consumption result is that, the result may statistically suggest that energy conservation measures may be taken without jeopardizing economic development. in practice however, to suggest measures that can lead to the reduction of energy consumption to the end-user in order to halt any conservation problem arising out energy consumption may not be a viable option for ethiopia particularly given the magnitude of the energy problems and the fact that the current energy infrastructure of the country is still inadequate to support the quest for rapid economic growth that is required to eradicate poverty and to raise the living standards of the people. reducing energy consumption while the overwhelmingly majority of the population is still denied access to the use of modern form of energy may not be a viable option. ethiopia has not yet reached the energy ladder that may warrant such a suggestion but it can still substantially improve the detrimental consequences of energy consumption (example the loss of natural resource for energy and the subsequent loss of soil fertility and erosion) without reducing its use. by making its energy sector more efficient and by making it available to a larger part wondatir atinafu dynamic econometric models 19 (2019) 57–84 80 of the population (especially electricity) energy used per unit of output can be raised. conclusions and policy implications this study aimed to examine the dynamic relationship between economic growth and energy consumption in ethiopia. in order to achieve objectives, data from different relevant source were collected over the years 1970–2017 and the parameters of the model were estimated using ardl system of data estimation technique. the estimation result reveals that, energy consumption found statistically insignificant in affecting economic growth in the long-run. however, it was positive and statistically significant in short-run. similarly, the dummy variable incorporated to capture the policy change effect found insignificant in long-run and with positive significant result in short-run. regarding the causality, the evidence is in line with the growth-led energy consumption hypothesis where causality running from economic growth to energy consumption, implying that reducing energy consumption may be implemented with little or no adverse effect on economic growth. in practice however any conservation measures taken to reduce energy consumption may not be a viable option for ethiopia particularly given the magnitude of its energy problems and the fact that the current energy infrastructure of the country is still inadequate to support its quest for rapid economic growth and for eradicating poverty. the option therefore might be for ethiopia to enhance the level of efficiency in the energy sector. increasing energy efficiency can cut down growth of energy demand that can mitigate conservation and health problem. as noted by iea (2002), finding ways of expanding the quality and quantity of energy services while simultaneously addressing the environmental impacts associated with energy use represents one of the critical challenges africa is facing. this means that energy regulation policies supporting the shift from lowerquality (typically less efficient and more polluting) to higher-quality energy services could provide impulse to economic growth rather than be detrimental to the development process (costantini and martini, 2010). since short run energy shortages may have significant impacts on the long run economic performances, the country needs to attract new capital for its energy industries. however, expanding energy production is not the one and only solution to the growth problems of the country. promoting energy efficiency and focusing on decreasing energy intensity may also have positive impacts on economic growth rates without putting considerable pressure on energy consumption and economic growth in ethiopia… dynamic econometric models 19 (2019) 57–84 81 the environment. developing energy sources that are renewable and that have low or no carbon content seem to be essential for this purpose. irrespective of the strength of the causal relationship between energy consumption and economic growth, the energy challenge facing ethiopia is daunting. unfortunately, in africa, it is not energy lack that is the basic problem but the lack of institutions, rules, financing mechanisms, and regulations needed to make markets work in support of energy for sustainable development .until these elementary limitations that are restraining the development of an efficient and accessible energy sector are fully solved, energy supply will still persist to be a major obstacle for the economic and social development ethiopia. reference akinlo, a.e. (2008). energy consumption and economic growth: evidence from 11 african countries. energy economics, 30, 2391–2400. http://dx.doi.org/10.1016/j.eneco.2008.01.008 amirat, a. & bouri, a. (2010). energy and economic growth: the algerian case, typewritten. chontanawat, j., hunt, l. & pierce, r. (2008). does energy consumption cause economic growth? evidence from systematic study of over 100 countries. journal of policy modelling, 30, 209–220. http://dx.doi.org/10.1016/j.jpolmod.2006.10.003 easterly, w. (1993). how much do distortions affect growth? journal of monetary economics, 32(2), 187–212. ebohon, o. j. (1996). energy, economic growth and causality in developing countries: a case study of tanzania and nigeria. energy policy, 24, 447–453. eea (2009). problems and prospects of energy sector in ethiopia. bulletin of the ethiopian economic association, 3(5), 7–11. enders,w. (1996), applied econometric time series, lowa state university: john wiley & sons inc. engle, r.f. & granger, c.w.j. (1987). cointegration, error correction representation, estimation and testing. economertica, 55, 251–276. ghali, k.h. & el-sakka (2004). energy use and output growth in canada: a multivariate cointegration analysis. energy economics, 26, 225–238. http://dx.doi.org/10.1016/s0140-9883(03)00056-2 granger, c.w.j. (1988). causality, co-integration, and control. journal of economic dynamics and control, 12, 551–559. harris, r. (1999), using co-integration analysis in econometric modeling, london: prentice hall. jumbe, c.b.l. (2004). co-integration and causality between electricity consumption and gdp: empirical evidence from malawi. journal of energy economics, 26(1), 26–68. johansen, s. and k. juselius (1990), maximum likelihood estimation and inference of cointegration: with applications to the demand for money. oxford bulletin of economics and statistics, 52, 169–210. masih, a. m. m., masih, r. (1998). a multivariate co-integrated modeling approach in testing temporal causality between energy consumption, real income and prices with an application to two asian ldcs. applied economics, 30, 1287–1298. wondatir atinafu dynamic econometric models 19 (2019) 57–84 82 nondo and mulugeta. (2009). energy consumption and economic growth: evidence from comesa countries, annual meeting, january 31-february 3, atlanta, georgia. pesaran, m.h. & shin, y. (1998). generalised impulse response analysis in linear multivariate models. economics letters, 58, 17–29. pesaran, h. and y. shin (1999), an autoregressive distributed lag modeling approach to cointegration analysis, in: econometrics and economic theory in the 20th century: the ragnar frisch centennial symposium, strom, s. (ed.) cambridge university press. pesaran m.h., shin, y. and smith, r. (2001), bound testing approach to the analysis of level relationship. journal of applied econometrics, 16(3), 289–326. https://doi.org/10.1002/jae.616 rahimi, m. and a. shahabadi (2011), trade liberalization and economic growth in iranian economy, bu-ali sina university, hamedan, iran. http://dx.doi.org/10.2139/ssrn.1976299 romer, d. (1996), advanced macroeconomics. new-york: mcgraw-hill. stern, d.i. (1993). energy use and economic growth in the usa, a multivariate approach. energy economics, 15, 137–150. stern, d.i. (2010). energy quality. ecological economics, 69(7), 1471–1478. http://dx.doi.org/10.1016/j.ecolecon.2010.02.005 stern, d. i. (2011). the role of energy in economic growth. ecological economics reviews, 1219, 26–51. wolde-rufael, y. (2004). electricity consumption and economic growth: a time series experience for 17 african countries. energy policy, 34, 1106–1114. wolde-rufael, y. (2005). energy demand and economic growth: the african experience. journal of policy modelling, 27(8), 891–903. http://dx.doi.org/10.1016/j.jpolmod.2005.06.003 world bank (2017). world development indicators. yohannes, h. (2010). energy, growth, and environmental interaction in the ethiopian economy. journal of economic & financial modelling, 2(2), 35–47. websites accessed world bank, world development indicator accessed at http//www.world bank.org/data unctad, unctadstat, accessed at http//unctadstat.unctad.org/en/ appendix appendix 1. results of augmented dickey fuller test variables at level i(0) at 1st difference i (1) order of integration ln real gdp intercept 3.264 -4.643** i (1) trend 0.565 -5.633** ln energy cons. intercept 0.783 -6.427** i (1) trend -1.534 -6.991** ln trade openness intercept -1.209 -5.448** i (1) trend -2.001 -5.406** ln physical capital intercept 1.821 -6.852** i (1) trend -0.570 -7.883** ln human capital intercept -1.298 -4.026* i (1) trend -1.188 -4.067** energy consumption and economic growth in ethiopia… dynamic econometric models 19 (2019) 57–84 83 appendix 2. bound test result appendix 3. ardl regression result (using aic lag selection criteria) (if p-values < desired level for i(1) variables) both f and t are more extreme than critical values for i(1) variables reject h0 if (if p-values > desired level for i(0) variables) both f and t are closer to zero than critical values for i(0) variables do not reject h0 if t -2.518 -3.826 -2.867 -4.238 -3.572 -5.065 0.037 0.299 f 2.428 3.691 2.911 4.327 4.043 5.806 0.002 0.015 i(0) i(1) i(0) i(1) i(0) i(1) i(0) i(1) 10% 5% 1% p-value kripfganz and schneider (2018) critical values and approximate p-values finite sample (5 variables, 46 observations, 4 short-run coefficients) case 3 t = -3.009 h0: no level relationship f = 5.416 pesaran, shin, and smith (2001) bounds test . estat ectest _cons -5.29906 15.67556 -0.34 0.738 -37.65184 27.05371 l3d. .0888911 .0534729 1.66 0.109 -.0214716 .1992538 l2d. .1624917 .0663116 2.45 0.022 .0256313 .2993522 ld. -.0235592 .0707011 -0.33 0.742 -.1694792 .1223607 d1. .2069438 .0708412 2.92 0.007 .0607347 .3531528 d l3d. -.0716564 .0520453 -1.38 0.181 -.1790727 .0357599 l2d. -.1246635 .0604452 -2.06 0.050 -.2494161 .0000892 ld. -.0298997 .0594317 -0.50 0.619 -.1525608 .0927614 d1. -.0321597 .0593711 -0.54 0.593 -.1546955 .0903762 lnpc d1. -.0971763 .0506137 -1.92 0.067 -.2016378 .0072852 lnto d1. 6.035592 2.48643 2.43 0.023 .9038526 11.16733 lnenergy l3d. .334598 .1949743 1.72 0.099 -.0678092 .7370053 l2d. -.0992182 .1940987 -0.51 0.614 -.4998183 .3013818 ld. .3527601 .1940001 1.82 0.082 -.0476366 .7531567 lnrgdp sr d -.2748932 .1761806 -1.56 0.132 -.6385121 .0887256 lnpc .4555515 .1382543 3.30 0.003 .1702086 .7408945 lnhc .0992804 .1013946 0.98 0.337 -.1099879 .3085486 lnto .5444754 .2870811 1.90 0.070 -.0480309 1.136982 lnenergy 3.843454 9.277611 0.41 0.682 -15.30459 22.9915 lr l1. -.2779594 .1256454 -2.21 0.037 -.5372788 -.01864 lnrgdp adj d.lnrgdp coef. std. err. t p>|t| [95% conf. interval] log likelihood = 98.496424 root mse = 0.0349 adj r-squared = 0.7448 r-squared = 0.8575 sample: 1974 2017 number of obs = 44 ardl(4,1,1,0,4,4) regression . ardl lnrgdp lnenergy lnto lnhc lnpc d, aic ec wondatir atinafu dynamic econometric models 19 (2019) 57–84 84 appendix 4. optimal lag length for each variable (akaike information criterion appendix 5. pair wise granger causality test appendix 6: diagnostic tests of the model test statistics lm version f version serial correlation chsq(1)= 1.2024[.234]** f(4, 41)= .62469[.423]** functional form chsq(1)= .011370[.716]** f(4, 39)= .0053317[.943]** normality chsq(2)= 1.5745[.627]** not applicable heteroscedasticity chsq(1)= 1.3321[.301]** f(4, 38)= 1.3031[.263]** a: lagrange multiplier test of residual serial correlation b: ramsey's reset test using the square of the fitted values c: based on a test of skewness and kurtosis of residuals d: based on the regression of squared residuals on squared fitted values r1 4 1 1 0 4 4 lnrgdp lnenergy lnto lnhc lnpc d e(lags)[1,6] . matrix list e(lags) pairwise granger causality tests date: 04/06/19 time: 22:25 sample: 1970 2017 lags: 2 null hypothesis: obs f-statistic prob. lnenergy does not granger cause lnrgdp 46 0.069179... 0.93326... lnrgdp does not granger cause lnenergy 7.031050... 0.00236... microsoft word 00_tresc.docx dynamic econometric models vol. 10 – nicolaus copernicus university – toruń – 2010 joanna górka nicolaus copernicus university in toruń the sign rca models: comparing predictive accuracy of var measures† a b s t r a c t. evaluating value at risk (var) methods of predictive accuracy in an objective and effective framework is important for both efficient capital allocation and loss prediction. from this reasons, finding an adequate method of estimating and backtesting is crucial for both the regulators and the risk managers’. the sign rca models may be useful to obtain the accurate forecasts of var. in this research one briefly describes the sign rca models, the value at risk and backtesting. we compare the predictive accuracy of alternative var forecasts obtained from different models. empirical example is mainly related to the pbg capital group shares on the warsaw stock exchange. k e y w o r d s: family of sign rca models, value at risk, backtesting, loss function. 1. introduction nowadays, accurate modelling of risk is very important in risk management. this is a result of the globalisation of financial market, the evolution of the derivative markets and the technological development. value at risk (var) has become the standard measure to quantify market risk1. this measure can be used by the financial institutions to assess their risks or by a regulatory committee to set margin requirements. in the literature, many parametric var models and many forecasting accuracy assessments for var methods exist. the important representation of the parametric var models are the generalized autoregressive conditional heteroskedasticity models (garch) (bollerslev, 1986; engle, 1982). these models describe non-linear dynamics of financial time series. a different, alternative approach to the description of financial time series represent the † this work was financed from the polish science budget resources in the years 2008-2010 as the research project n n111 434034. 1 it was introduced by jp morgan in 1996. joanna górka 62 random coefficient autoregressive models (rca) (which were proposed by nicholls, quinn, 1982). thavaneswaran et al. proposed a number of expansions of the random coefficient autoregressive model order one. the new models, such as sign rca(1), rcama(1,1), sign rcama(1,1), rca(1)-garch(1,1) and sign rca(1)-garch(1,1) can be used to obtain value-at-risk measure. the aim of this paper is to use the family of sign rca models to obtain the var forecasts and compare the results obtained from sign rca models with other selected var models. 2. the family of sign rca models random coefficient autoregressive models (rca) are straightforward generalization of the constant coefficient autoregressive models. a full description of this class of models including their properties, estimation methods and some applications can be found in nicholls and quinn (1982). thavaneswaran, appadoo and bector (2006) proposed a first order random coefficient autoregressive model with a first order moving average component, i.e. rcama(1,1). in another paper thavaneswaran and appadoo (2006) proposed to add the sign function to rca(1) and rcama(1,1) models. the last modification is based on assumption that residuals from the rca model or the sign rca model can be described by a garch model. in this way, the rca(1)-garch(1,1) model and sign rca(1)-garch(1,1) model were created. all these modifications influence the increase of variance and kurtosis of processes2. in table 1 equations of individual models from the family of sign rca models and their names are presented. to ensure the existence of the i-vi models (table 1) the following assumptions must be satisfied: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 2 2 0 0 , 0 0 ~ ε δ σ σ ε δ iid t t , (1) 122 <+ δσφ . (2) the sign function, described by the following formula 1 for 0, 0 for 0, 1 for 0, t t t t y s y y >⎧ ⎪ = =⎨ ⎪− <⎩ (3) 2 theoretical properties of the family of sign rca models can be found in articles, i.e.: appadoo, thavaneswaran, singh (2006), aue (2004), górka, (2008), thavaneswaran, appadoo, bector (2006), thavaneswaran, appadoo (2006), thavaneswaran, appadoo, ghahramani, (2009), thavaneswaran, peiris, appadoo (2008). the sign rca models: comparing predictive accuracy of var measures 63 has the interpretation: if φ>+ tδφ , the negative value of φ means that the negative (positive) observation values at time 1−t correspond to a decrease (increase) of observation values at time t . in the case of stock returns it would suggest (for returns) that after a decrease of stock returns, the higher decrease of stock returns occurs than expected, and in the case of the increase of stock returns the lower increase in stock returns occurs than expected. table 1. the family of sign rca models (without conditions) model model equations no. rca(1) ( ) tttt yy εδφ ++= −1 i sign rca(1) ( ) ttttt ysy εδφ +φ++= −− 11 ii rcama(1,1) ( ) 11 −− +++= ttttt yy θεεδφ iii sign rcama(1,1) ( ) 111 −−− ++φ++= tttttt ysy θεεδφ iv rca(1)-garch(1,1) ( ) tttt yy εδφ ++= −1 , ttt zh=ε 11 2 110 −− ++= ttt hh βεαα v sign rca(1)-garch(1,1) ( ) ttttt ysy εδφ +φ++= −− 11 , ttt zh=ε 11 2 110 −− ++= ttt hh βεαα vi note: ts – sign function is described by equation (3); φ , θ , φ , iα , 1β – model parameters. condition (2) is necessary and sufficient for the second-order stationarity of process described by equation i, however conditions (1)-(2) ensure strict stationarity of this process. if conditions (1)-(2) are satisfied, then processes described by equations ii-iv are stationary in mean. if residuals from the rca model are described by a garch model, then the rca(1)-garch(p,q) model described by equation v, where ( )2,0~ zt nz σ , 00 >α , 0≥iα and 0≥jβ , is obtained. if the sign function is added to the rca-garch model, then the process described by equation vi is obtained. the conditions ensuring the positive value of conditional variance of this process are the following: ( )2,0~ zt nz σ , 00 >α , 0≥iα , 0≥jβ , 0α≤φ . predictors of the conditional mean and conditional variance of sign rca models are presented in table 2 and 3 respectively. joanna górka 64 table 2. conditional mean predictors models conditional mean rca(1), rca(1)-garch(1,1) ( )11p t t tt ty e y f yϕ++ = = sign rca(1), sign rca(1)-garch(1,1) ( ) ( )11 p t t t tt ty e y f s yϕ++ = = +φ rcama(1,1) ( )11p t t t tt ty e y f yϕ θε++ = = + sign rcama(1,1) ( ) ( )11p t t t tt ty e y f s yϕ θε++ = = +φ + table 3. conditional variance predictors models conditional variance rca(1), sign rca(1), rcama(1,1), sign rcama(1,1) ( ) 2 2 2 2 2 11 t t tt t e u f yε δσ σ σ++ = = + rca(1)-garch(1,1), sign rca(1)-garch(1,1) ( ) 2 2 2 2 2 11 ( )t t z t tt t e u f e h yδσ σ σ++ = = + 3. value-at-risk value-at-risk (var) is used as a tool for measuring market risk. it is defined as „the maximum potential loss that a portfolio can suffer within a fixed confidence level during a holding period”. formal definition of var is following (artzner, delbaen, eber, heath, 1999): ( ) ( ){ } ( ){ }var inf : inf : 1xx x f x x p x xα α α= ≥ = > ≤ − , (4) where ( )0,1α ∈ is a particular confidence level, xf – the cumulative density function. consider a time series of daily ex post returns ( ( )1100 ln lnt t tr p p−= − where tp is the share price at time t) and corresponding time series of ex ante var forecasts ( varα ), the formula (4) takes the form: ( )1 vartp r α α+ ≤ − = . (5) the negative sign arises from the convention of reporting var as a positive number. one-step-ahead conditional forecasts of value-at-risk are calculated by the formula: ( )1 1 1var , l t t t t t zαα μ σ+ + += + (6) where tt |1+μ , tt |1+σ are one-step-ahead conditional forecasts of mean and volatility respectively. the sign rca models: comparing predictive accuracy of var measures 65 3.1. estimation methods for var this section briefly describes the alternative models that we use for estimating var forecasts in this paper. the following models are used in the research to obtain var forecasts: − the historical simulation (hs)3. the var is estimated as the α-th quantile of the empirical distribution of returns. hs is based on the assumption that returns are iid time series of an unknown distribution. − the equally weighted moving average (ewma) model, i.e. 2 2 1 1 1 t it t i t k r k σ + = − + = ∑ , (7) where k – size of window, 2ir – returns. the returns are assumed to be normally distributed. − the riskmetrics (rm) model, i.e. ( ) ( )2 2 2 21 1 1 1 t t i i t tt t i t k r rσ λ λ λσ λ−+ = − + = − = + −∑ , (8) where ( )0,1λ ∈ is known as the decay factor, 2tλσ is the previous volatility forecast weighted by the decay factor, and ( ) 21 trλ− is the latest squared returns weighted by ( )1 λ− . the var is estimated under the assumption that returns are normally distributed (as in the case of ewma). − the ar(1)-garch(1,1) model, i.e. 1t t tr rφ ε−= + , (9) where t t tzε σ= , )1,0(~ nzt , 2 2 2 1 1t t tσ ω αε βσ− −= + + . (10) in this case, returns series is assumed to be conditionally normally distributed. − models from the family of sign rca models4. 3 hs is the oldest and still very popular estimator of the var. 4 they were presented in previous section. joanna górka 66 3.2. backtesting var estimates backtesting is based on testing whether the var estimates are statistically accurate. the ,,failure process” is defined as: ( )1 var , 1, ...,lt t ti r t t t n= < − = + + , (11) where ( )*1 denotes the indicator function returning a unit if the argument is true, and zero otherwise; t is the size of the sample used to estimate parameters of the model; n is the number of one-step-ahead var forecasts computed. the var forecasts are accurate if the { }ti series is iid with mean α , i.e. | 1t te i α−⎡ ⎤ =⎣ ⎦ . to test the statistical accuracy we used the standard likelihood ratio tests: 1. the proportion of failures test – lrpof (kupiec, 1995) 5: [ ] [ ]0 1: . :t th e i vs h e iα α= ≠ , 1 2 ln ˆ ˆ1 n n n poflr α α α α −⎡ ⎤−⎛ ⎞ ⎛ ⎞ = − ⎢ ⎥⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ ~ 21χ , (12) where n is the number of failures var, α̂ is the mle of α , i. e. nn . 2. the christoffersen independence test – lrind (christoffersen, 1998): 0 01 11:h α α= , ( ) ( ) ( ) 00 10 01 11 00 1001 11 01 01 11 11 1 2 ln 1 1 t t t t ind t tt t lr α α α α α α + +− = − − − ~ 21χ , (13) where: 0 1 ˆ ijij i i t t t α = + , 11011000 1101 tttt tt +++ + =α , ijt – number of i values followed by a j value in the ti series ( ), 0,1i j = . 3. the time between failures test – lrtbf (haas, 2001) 6: 1 1 1 2 ln 1 ivn tbf i i i lr α α α α − = ⎡ ⎤⎛ ⎞− ⎢ ⎥= − ⎜ ⎟ −⎢ ⎥⎝ ⎠⎣ ⎦ ∑ ~ 2nχ , (14) 5 similar, the lr test of unconditional coverage by christoffersen (1998) was proposed. other symbol of this test is the lruc. 6 haas extended the kupiec’s time until first failure test (tuff test) by adding test for every exception (second and next). the sign rca models: comparing predictive accuracy of var measures 67 where i i v 1 =α , 1v – time until first failure, iv – time between exception ( )1−i and exception i for 2, ...,i n= . if, in above tests the null hypothesis is not rejected, then a particular model gives accurate forecasts of var. however, if more than one model is deemed adequate, we cannot conclude which of var model should be selected. lopez (1998) suggested measuring the accuracy of var forecasts on the basis of distance between observed returns and forecasted var values. this approach does not give any formal statistical selection of model adequacy but it allows to rank the models. let 1 n t t f f = = ∑ means a total loss function. a model which minimizes the total loss function is preferred over the other models. in the literature, different loss functions were proposed (see lopez, 1998, 1999; blanco and ihle, 1998; sarma, thomas and shah, 2003, caporin, 2003; angelidis, benos and degiannakis, 2004). in this paper, the loss functions used to compare the accurate var forecasts are as follows: − the regulatory loss function – rl (lopez, 1999)7: ( ) 1 , 21 1 , 1 , 0 var , 1 var var . t r t t t r t t r t r f r r + + + + > −⎧⎪ = ⎨ + + ≤ −⎪⎩ (15) − the firm’s loss function – fl (sarma, thomas, shah, 2003): ( ) , 1 , 21 1 , 1 , var var , 1 var var . r t t r t t t r t t r t c r f r r + + + + > −⎧⎪ = ⎨ + + ≤ −⎪⎩ (16) where c is a measure of cost of capital opportunity. sarma, thomas and shah (2003) proposed testing for superiority of a model vis-á-vis another in terms of the loss function. they suggested a two-stage var model selection procedure. the first stage consists in testing the statistical accuracy for the competing var models. in the second stage of the var model selection procedure, the firm’s loss function is used to evaluate statistically var models8. 7 this name comes from sarma, thomas and shah (2003) who explain that (16) is able to express the regulatory concerns in model evaluation. however, no score is attached in case if exception does not occur. 8 only that var model for which the average number of failures was equal to the expected and these failures are independently distributed is included in the second stage. joanna górka 68 consider two var models, i and j. the hypotheses are: 0 1: 0 . : 0h vs hθ θ= < , where θ is the median of the distribution of , ,t i t j tz f f= − , where ,i tf and ,j tf are the values of loss function generated by model i and model j respectively. negative values of tz indicate a superiority of model i over j. the testing procedure is as follows: 1. define an indicator variable ( )0t tzψ = ≥1 and the number of non-negative tz ’s, as 1 t n ij t t t s ψ + = + = ∑ . 2. calculate the statistics as: ( ) 0.5 ~ 0,1 0.25 ij ij s n sts n n − = asymptotically, (17) ijsts is based on assuming that the tz is iid 9. alternatively, we can compare competing var models using the predictive quantile loss function (see giacomini and komunjer, 2005; bao et al., 2006). the expected loss function is given by: ( ) ( ) 1 1 1 var var n i i i i i q r r nα α = ⎡ ⎤= − < − +⎣ ⎦∑ . (18) the selected model is the var model which has the minimum of qα . 4. empirical application the data used in the empirical application are daily prices of twenty polish firms’ shares from the wig20 portfolio on the warsaw stock exchange (wse). the data were obtained from bossa.pl for the period from 23-rd september 2005 to 18-th february 2009, which yields 852 observations. however, one of shares was excluded because it was not quoted on 23 september 2005. to analyze daily percentage log returns of each share were used. this empirical study was composed of two parts. the first part (analysis i) was carried out with regard to all of twenty shares from wse. the research procedure was the following: 1. for the first 500 observations of each returns series the descriptive statistics and some tests were calculated. next, returns series with 9 for details on the sign test see diebold and mariano (1995). the sign rca models: comparing predictive accuracy of var measures 69 autocorrelation and kurtosis bigger than for normal distribution were chosen10. 2. parameters of six models from the family of sign rca were estimated for the first 500 observations of time series selected in step one. next, only models with statistically significant parameters were used. 3. the estimation of parameters for models selected in step 2 was performed for rolling window of 100, 150, 200, 250, 300, 400, 500 observations. in the same way the estimation of ar(1)-garch(1,1) models was obtained. 4. for all models from step 3 and for the historical simulation (hs), the equally weighted moving average (ewma) model, the riskmetrics (rm) models (with 0.95λ = and 0.99λ = ) var measures were calculated11. one-step-ahead forecasts of var (that is 751, 701, 651, 601, 551, 451, 351 forecasts, respectively) were calculated on the basis of these models. 5. the traditional var tests and loss functions for each model and window were calculated. 6. the obtained results in above step were compared. in the second part (analysis ii) only the pbg shares (pbg capital group) was chosen. all presented models of var for the last 250 observations were calculated12. for obtained var forecasts the two-stage var model selection procedure was applied. all model parameters (analysis i and ii) were estimated using maximum likelihood (mle) with the bfgs algorithm. calculations were carried out in the gauss program. 4.1. results of the analysis i selected results of the descriptive statistics and some tests are given in table 4. all series have a mean between -0.052 and 0.561, kurtosis bigger than for normal distribution. the standard deviations are different, ranging from 1.955 for pgnig to 5.354 for bioton. the skewness and kurtosis differ among all series. only 8 of 19 returns series have autocorrelation. the lbi test rejects the null hypothesis of non random coefficient to four stock returns. 10 this method of the elimination of initially selected companies can impact on the results. it would be worth to check out which results might be obtained for the whole set of companies. however, such analysis was omitted in this paper. 11 the returns series were assumed either to be normally distributed or conditionally normally distributed, respectively. 12 the set of 250 observations corresponds to roughly one year of trading days and according to the basel ii accord requirement the minimum of 250 var forecasts should be used to the backtesting approach. therefore, one-step-ahead forecasts of var at the same period (250 observations) were calculated. parameters were estimated for rolling windows of 125, 250, 375 observations each. the returns series were assumed either to be conditionally normally distributed or normally distributed respectively. joanna górka 70 next, the 7 different models were estimated for 8 returns series. further, only models with statistically significant parameters were chosen. in this way models like rca and sign rca were chosen. to present backtesting results for var forecasts of the pbg shares was chosen because for that share the autoregressive parameter in the rca models for all returns series has been the biggest. it is very important because we can expect the sign rca model to be better than other models. the traditional var tests and loss functions for the pbg for all models are presented in table 5 and the 5% at significance level. one can see that the accuracy test rejects the null hypothesis for windows size of 500, 400 observations for hs, ewma model, ar(1)-garch(1,1) model, rca model and sign rca model. for example, for window size 250 the regulatory loss function is the smallest for rm ( 0.95λ = ). next position in this ranking have ar(1)-garch(1,1) model, ewma model, rca model, sign rca model and the last position has rm ( 0.99λ = ). the hs method is not taken into consideration because the accuracy test rejects the null hypothesis for windows size of 250 observations. on the other hand, the firm’s loss function is the smallest for rm ( 0.99λ = ) and the next positions in ranking have sign rca model, rca model, ewma model, ar(1)-garch(1,1) model and rm ( 0.95λ = ). the differences between values of the firm’s loss function are small for estimated models. to compare these results, the tests for superiority of a model vis-á-vis another were used only for models included into the second stage at sarma, thomas and shah procedure. the results are presented in table 6. for the window size 300 we can see that the sign rca model is significantly better than other models, i.e. the null hypothesis is rejected in the test of superiority between the sign rca model and the other models presented in subsection 3.1. however, as the size of windows decreases the rm model ( 0.99λ = ) outperforms the sign rca model. rca and sign rca models are statistically the same for the window size 100. in cases when results with hs are compared one can see that hs is almost everywhere significantly better than others. the table 7 includes the results of the var tests and the loss function at the 2,5% significance level which are similar to the results obtained at the 5% significance level. only for hs with the window size 250 and for rca model with the window size of 300 observations, some differences can be noticed, i. e. in the case of hs the accuracy at the 2.5% is better than at the 5% significance level (except rca model). for the loss function conclusions are the same with one exception, i. e. the hs has the last rank for regulatory loss function and the first rank for firm’s loss function. at the 1% significance level we obtained more differences (see table 8). firstly, risk metrics models are accurate only for windows size 500 and 500, 400 for 0.95λ = , 0.99λ = , respectively. the rca, sign rca and ewma the sign rca models: comparing predictive accuracy of var measures 71 models are accurate for small windows (size 200, 150, 100). the regulatory loss function is the smallest for hs. the firm’s loss function has the lowest values for sign rca models for the window size 200. very strange results were obtained for hs and therefore we are not able to find any rules for accuracy and value of the regulatory loss function. 4.2. results of the analysis ii firstly, we calculated the 250 one-step-ahead forecasts of var of the pbg share using all models of var (presented in 3.1)13. the var forecasts were received from different models estimated for the different window sizes, i.e, t =125, 250 and 375. secondly, the competing var models were testing for statistical accuracy. for the established period of forecasting, only sign rcama(1,1) models (for t = 375 and all significance level, for t = 250 and α = 2.5%, 1%), sign rca(1)-garch(1,1) models (for α = 1% and rolling window size t = 375, 125) and risk metrics models (for λ= 0,99 and α = 1% and t = 125) did not fulfill the conditions used at first stage of sarma, thomas and shah procedure (the null hypothesis was rejected at least for one test, see (12)-(14)). for other models, the firm’s loss function (see the table 9), the sts test and the predictive quantile loss function (see the table 10) were calculated. lower values of the firm’s loss function for var forecasts were received from rcama(1,1), rca(1) and sign rcama(1,1) (if it was included at second stage) models (with the exception of the hs for α = 5% and t = 375, 250 and with the exception of the rm(λ= 0,99) for t = 125 and α = 5%, 2.5%). the test for superiority of a model vis-á-vis another indicates that: 1. at the 5% significance level, for different rolling window sizes, each of models having first rank is superior over other models. 2. at the 2.5% significance level, for rolling windows size of 125 observations, the rm (λ= 0,99) is superior over other models. the rcama(1,1) model is better than almost all other models (with the exception of hs method and rca(1) model for t = 375 and with the exception of the rca(1)-garch(1,1) model for t = 250, for which the predictive ability is equal). 3. for the α = 1%, for different rolling window sizes, each of models having first rank is superior over other models (with the exception of rcama(1,1) and rca(1) models for t = 375 that have equal predictive ability). other conclusions are formulated based on the predictive quantile loss function (table 10), which yields different position in the ranking. for var forecasts of the pbg share, for established forecasting period, the choice of the 13 one-step ahead forecasts on the period 19.02.2008-18.02.2009 were computed. joanna górka 72 best model from the competing models depends on the significance level and rolling window sizes. for the sign rca models the rolling window size of 125 observations seemed too small. this conclusion is similar to one from analysis i. 5. conclusions evaluating forecasts based solely on one criterion yield the limited information regarding the accuracy method. thus, in the literature is commonly accepted that results of each evaluation criterion are presented separately and then best performing method is selected. however, it can be noticed that the different evaluation criteria give the different choice of the best estimation method of var. therefore, it is difficult to make general remarks, nevertheless the empirical results showed that: 1. none of the presented methods gave a satisfactory var estimates. 2. the results showed no domination of either forecasting methods of var. 3. bigger sample did not lead to the better results. 4. it seems that the family of sign rca models should be used for the sample size of 150 to 300 observations. 5. in terms of the firm’s loss function the sign rca model was significantly better than the ar-garch model, rm ( 0.95λ = ) model and ewma model. the sign rca model was not worse than the standard rca model. 6. one should treat every share individually and use different methods and models for obtaining a good forecast of var. 7. the historical simulation gave better results (in terms of accuracy) at the 1% significance level than for other significance levels. it seems that the minimum window size should be 250 observations but smaller than 500 observations. 8. the rcama(1,1) model can be competitive to other var measures from the firm’s loss function point of view. 9. the sign rca models with garch errors did not give better forecasts of var for the pbg share. references angelidis, t., benos, a., degiannakis, s. (2004), the use of garch models in var estimation, statistical methodology, 1, 105–128. appadoo, s. s., thavaneswaran, a., singh, j. (2006), rca models with correlated errors, applied mathematics letters, 19, 824–829. artzner, p., delbaen, f., eber, j.-m., heath, d. (1999), coherent measures of risk, mathematical finance, 9, 203–228. aue, a. (2004), strong approximation for rca(1) time series with applications, statistics & probability letters, 68, 369–382. the sign rca models: comparing predictive accuracy of var measures 73 bao, y., lee, t.-h., saltoglu, b. (2006), evaluating predictive performance of value-at-risk models in emerging markets: a reality check, journal of forecasting, 25,101–128. blanco, c., ihle, g. (1998), how good is your var? using backtesting to assess system performance, financial engineering news, august, 1–2. bollerslev, t. (1986), generalized autoregressive conditional heteroscedasticity, journal of econometrics, 31, 307–327. caporin, m. (2003), evaluating value-at-risk measures in presence of long memory conditional volatility, working paper 05.03, greta. christoffersen, p. f. (1998), evaluating interval forecasts, international economic review, 39, 841–862. diebold, f. x., mariano, r. s. (1995), comparing predictive accuracy, journal of business & economic statistics, 13, 253–263. engle, r. f. (1982), autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation, econometrica, 50, 987–1006. giacomini, r., komunjer, i. (2005), evaluation and combination of conditional quantile forecasts, journal of business and economic statistics, 23, 416–431. górka, j. (2008), description the kurtosis of distributions by selected models with sing function, dynamic econometric models, 8, 39–49. haas, m. (2001), new methods in backtesting, financial engineering, working paper, bonn. lopez, j. (1998), methods for evaluating value-at-risk estimates, frbny economic policy review. lopez, j. (1999), regulatory evaluation of value-at-risk models, frbny economic policy review, 4, 119–124. nicholls, d., quinn, b. (1982), random coefficient autoregressive models: an introduction, springer, new york. sarma, m., thomas, s., shah, a. (2003), selection of value-at-risk models, journal of forecasting, 22, 337–358. thavaneswaran, a., appadoo, s. s. (2006), properties of a new family of volatility sing models, computers & mathematics with applications, 52, 809–818. thavaneswaran, a., appadoo, s. s., bector, c. r. (2006), recent developments in volatility modeling and application, journal of applied mathematics and decision sciences, 2006, 1–23. thavaneswaran, a., appadoo, s. s., ghahramani, m. (2009), rca models with garch innovations, applied mathematics letters, 22, 110–114. thavaneswaran, a., peiris, s., appadoo, s. (2008), random coefficien volatility models, statistics & probability letters, 78, 582–593. modele sign rca: porównanie trafności prognoz var z a r y s t r e ś c i. obiektywna i skuteczna ocena trafności prognozowania wartości narażonej na ryzyko (value at risk – var) jest bardzo ważna zarówno dla efektywnego zarządzania kapitałem jak i do prognozowania strat. z tego powodu znalezienie odpowiednich metod estymacji i weryfikacji var jest kluczowe zarówno dla instytucji nadzorujących jak i dla menadżerów. modele sign rca mogą być użyteczne do otrzymywania trafnych prognoz var. w artykule, pokrótce przedstawione są modele sign rca, wartość narażona na ryzyko i weryfikacja prognoz var. porównana jest trafność prognoz var otrzymanym z różnych alternatywnych modeli. przykład empiryczny skoncentrowany jest głównie na cenach akcji spółki pbg notowanej na giełdzie papierów wartościowych w warszawie. s ł o w a k l u c z o w e: modele klasy sign rca, value at risk, testowanie wsteczne, funkcja strat. table 4. results of the descriptive statistics, box-ljung tests and locally best invariant test company mean std. dev. skewness kurtosis b-l (1) b-l (2) lbi agora -0,052 2,451 -0,204 4,853 8,925*** 9,029*** 1,672 assecopol 0,169 2,493 -0,582 13,015 6,953*** 8,357*** 2,848** bioton -0,036 5,354 -8,286 138,621 1,673 2,111 -0,028 bre 0,256 1,972 0,263 4,055 3,915** 4,025 2,378** bzwbk 0,175 2,442 -0,135 3,472 1,478 2,738 1,034 cersanit 0,247 2,361 0,567 6,312 0,156 1,887 1,639 getin 0,231 2,646 0,523 11,370 0,008 0,837 0,954 gtc 0,255 2,737 0,461 5,383 1,510 8,046*** 0,461 kghm 0,212 3,011 -0,591 5,303 0,001 4,766 1,156 lotos 0,036 2,174 -0,329 4,835 1,596 2,249 -0,078 pbg 0,363 2,094 0,095 5,344 3,466* 3,468 1,909 pekao 0,071 2,160 0,219 3,616 0,005 0,044 0,273 pgnig 0,068 1,955 0,192 4,413 0,284 4,870* 2,929** pknorlen -0,021 2,170 -0,069 3,853 0,017 3,680 0,508 pkobp 0,117 2,055 0,324 3,912 3,625* 3,647 0,002 polimexms 0,366 2,420 -0,172 6,835 2,402 3,945 1,449 polnord 0,561 5,290 -1,387 28,269 2,085 2,489 -0,047 tpsa -0,022 1,978 -0,161 3,775 0,310 1,757 1,109 tvn 0,145 2,242 -0,083 3,716 3,004* 3,250 3,218** note: *, **, *** indicate rejection of h0 at the 10% ,5% and 1% significant level, respectively. b-l (1) – estimates of the box-ljung test statistics of order 1. b-l (2) – estimates of the box-ljung test statistics of order 2. lbi – estimates of the locally best invariant test statistics. table 5. results of the var tests (95% var for pbg) and the loss function model α̂ lrpof lrind lrtbf rl fl sh 500 10,54% 17,451*** 1,370 41,329 205,55 1366,59 400 9,09% 12,929*** 1,139 33,525 240,15 1722,11 300 7,62% 6,923*** 0,603 34,955 279,97 2106,98 250 7,15% 5,213** 0,494 34,272 281,66 2246,19 200 5,84% 0,914 0,026 32,805 261,26 2462,33 150 5,56% 0,453 0,016 40,548 263,62 2668,63 100 4,79% 0,068 0,394 35,815 224,13 2913,60 ewma 500 9,12% 10,179*** 1,967 32,075 186,89 1427,20 400 7,98% 7,207*** 0,350 28,098 208,58 1818,34 300 6,35% 1,961 0,027 26,143 236,04 2235,86 250 5,82% 0,817 0,723 28,900 229,42 2428,40 200 5,07% 0,007 0,348 31,243 223,04 2628,37 150 4,42% 0,512 0,121 32,300 222,40 2856,85 100 4,26% 0,907 0,117 35,007 215,48 3079,17 table 5. continued model α̂ lrpof lrind lrtbf rl fl rm (λ= 0,95) 500 7,12% 2,959* 3,850* 25,133 117,94 1629,97 400 6,43% 1,788 0,544 25,727 148,73 2036,69 300 5,99% 1,070 0,657 27,728 206,95 2394,60 250 5,49% 0,296 0,481 29,504 206,95 2546,11 200 5,07% 0,007 0,348 33,443 206,96 2686,53 150 4,85% 0,033 0,326 36,837 210,83 2856,07 100 4,66% 0,186 0,310 37,665 213,93 3052,66 rm (λ= 0,99) 500 6,55% 1,630 3,238* 24,787 137,97 1576,26 400 5,99% 0,872 0,306 25,381 172,62 1948,88 300 6,17% 1,484 0,796 28,060 230,84 2272,02 250 6,16% 1,581 0,041 28,334 240,02 2397,50 200 6,14% 1,678 0,104 35,573 255,05 2502,31 150 6,56% 3,293 0,000 46,305 287,75 2571,93 100 7,19% 6,720*** 0,253 55,251 346,13 2559,78 ar(1)-garch(1,1) 500 8,26% 6,628** 5,247** 25,450 168,05 1497,28 400 7,54% 5,331** 1,411 29,023 185,14 1884,31 300 6,72% 3,095* 0,117 31,296 233,15 2267,60 250 5,66% 0,525 0,596 27,342 219,27 2460,78 200 5,38% 0,190 0,550 31,366 214,92 2629,52 150 5,14% 0,027 0,514 36,663 215,14 2868,68 100 4,26% 0,907 0,117 36,932 221,00 3138,36 rca 500 8,83% 8,924*** 1,693 26,278 187,36 1442,89 400 7,76% 6,238** 1,630 28,136 204,92 1828,93 300 6,53% 2,498 0,065 27,313 237,79 2223,04 250 5,82% 0,817 0,723 26,199 230,09 2410,01 200 5,07% 0,007 0,348 25,277 221,17 2594,08 150 4,99% 0,000 0,415 37,995 227,54 2812,48 100 4,39% 0,604 0,171 34,631 220,41 3025,38 sign rca 500 8,83% 8,924*** 1,693 26,278 186,61 1438,09 400 7,54% 5,331** 5,564** 27,559 204,61 1838,01 300 6,53% 2,498 0,065 27,312 239,14 2219,78 250 5,82% 0,817 0,723 26,199 230,74 2404,58 200 5,07% 0,007 0,348 25,277 221,40 2586,80 150 4,99% 0,000 0,038 39,013 227,35 2801,30 100 4,79% 0,068 0,864 37,365 228,79 3003,25 note: *, **, *** indicate rejection of h0 at the 10% ,5% and 1% significant level, respectively, lrpof – the values of the proportion of failures test statistics, lrind – the values of the independence test statistics, lrtbf – the values of the time between failures test statistics, rl – regulatory loss function, fl – firm’s loss function. table 6. the test for superiority of a model vis-á-vis another sample: 300 ↓ better → sign rca rca ar-garch rm(0.99) rm(0.95) ewma hs sign rca x -7,455* -8,052* -10,863* -9,330* -12,397* rca 7,455 x -3,962* -10,182* -9,159* -3,451* ar-garch 8,052 3,962 x -2,343* -9,245* 0,724 rm(0.99) 10,863 10,182 2,343 x -8,989* 8,904 rm(0.95) 9,330 9,159 9,245 8,989 x 8,563 ewma 12,397 3,451 -0,724 -8,904* -8,563* x hs x sample: 250 ↓ better → sign rca rca ar-garch rm(0.99) rm(0.95) ewma hs sign rca x -5,262* -4,691* 1,020 -8,199* -6,323 rca 5,262 x -4,691* 1,999 -7,954* -5,099 ar-garch 4,691 4,691 x 4,854 -7,954* -0,612 rm(0.99) -1,020 -1,999 -4,854* x -9,994* -6,159 rm(0.95) 8,199 7,954 7,954 9,994 x 6,078 ewma 6,323 5,099 0,612 6,159 -6,078* x hs x sample: 200 ↓ better → sign rca rca ar-garch rm(0.99) rm(0.95) ewma hs sign rca x -5,369* -5,056* 11,640 -4,194* -8,662* 12,895 rca 5,369 x -3,253* 12,581 -3,880* -5,683* 13,992 ar-garch 5,056 3,253 x 14,619 -1,842 -2,234* 13,522 rm(0.99) -11,640* -12,581* -14,619* x -10,308* -16,971* 4,586 rm(0.95) 4,194 3,880 1,842 10,308 x 1,999 11,092 ewma 8,662 5,683 2,234 16,971 -1,999 x 14,619 hs -12,895* -13,992* -13,522* -4,586* -11,092* -14,619* x sample: 150 ↓ better → sign rca rca ar-garch rm(0.99) rm(0.95) ewma hs sign rca x -0,567 -2,984* 19,905 -2,002* -8,120* 10,462 rca 0,567 x -3,059* 21,113 -1,775 -7,063* 12,502 ar-garch 2,984 3,059 x 20,887 2,379 -2,757* 13,257 rm(0.99) -19,905* -21,113* -20,887* x -14,315* -24,135* -6,761* rm(0.95) 2,002 1,775 -2,379* 14,315 x -0,944 10,613 ewma 8,120 7,063 2,757 24,135 0,944 x 15,448 hs -10,462* -12,502* -13,257* 6,761 -10,613* -15,448* x sample: 100 ↓ better → sign rca rca ar-garch rm(0.99) rm(0.95) ewma hs sign rca x -1,715 -5,218* -2,153* -9,159* 4,634 rca 1,861 x -5,729* -1,861 -6,386* 6,313 ar-garch 5,218 5,729 x 4,415 2,007 9,086 rm(0.99) x rm(0.95) 2,153 1,861 -4,415* x 0,401 6,240 ewma 9,159 6,386 -2,007* -0,401 x 9,597 hs -4,634* -6,313* -9,086* -6,240* -9,597* x note: * indicate rejection of h0 at the 10% and 5% significant level. table 7. results of the var tests (97.5% var for pbg) and the loss functions model α̂ lrpof lrind lrtbf rl fl sh 500 5,98% 12,642*** 2,683 35,191** 123,33 1628,86 400 5,10% 9,659*** 0,031 30,168 135,69 2074,56 300 4,54% 7,587*** 0,019 27,865 171,95 2467,40 250 3,66% 2,912* 0,047 17,641 155,85 2764,23 200 3,69% 3,289* 0,015 24,976 154,39 2950,27 150 3,14% 1,086 0,130 20,965 158,94 3208,25 100 3,06% 0,911 0,117 23,765 157,49 3400,58 ewma 500 4,84% 6,234** 1,736 28,034** 114,91 1661,45 400 4,88% 8,226*** 0,006 29,047 130,49 2111,82 300 3,81% 3,358* 0,049 25,024 151,85 2597,32 250 3,33% 1,533 0,156 20,585 145,23 2833,12 200 3,07% 0,816 0,217 21,926 144,02 3070,24 150 2,85% 0,343 0,281 23,676 144,58 3332,11 100 2,53% 0,003 0,455 25,225 138,45 3610,65 rm (λ = 0,95) 500 3,13% 0,536 0,714 12,178 60,26 1932,02 400 3,10% 0,628 0,582 10,837 80,65 2404,45 300 3,27% 1,214 0,256 13,397 129,14 2810,04 250 3,00% 0,569 0,337 12,575 129,14 2990,57 200 2,76% 0,181 0,419 13,916 129,15 3157,89 150 2,85% 0,343 0,281 21,614 132,36 3352,30 100 2,80% 0,261 0,255 21,282 134,38 3584,68 rm (λ = 0,99) 500 3,99% 2,710 1,167 23,185* 80,16 1836,63 400 3,77% 2,586 0,186 21,844 104,95 2268,16 300 3,63% 2,537 0,099 21,407 148,64 2644,88 250 3,33% 1,533 0,156 20,585 153,45 2799,71 200 3,23% 1,291 0,143 25,544 164,68 2925,99 150 3,71% 3,668* 0,001 31,614 189,66 3000,01 100 4,93% 14,208*** 0,018 54,397** 240,05 2947,31 ar(1)-garch(1,1) 500 5,41% 9,215*** 2,182 26,878 103,79 1732,20 400 4,21% 4,517*** 1,676 22,917 109,42 2199,89 300 3,63% 2,537 0,099 21,407 148,91 2650,81 250 3,33% 1,533 0,156 20,585 139,01 2877,29 200 3,07% 0,816 0,217 21,926 136,56 3081,41 150 3,14% 1,086 0,130 27,158 134,82 3359,94 100 3,06% 0,911 0,117 22,612 142,15 3662,46 table 7. continued model α̂ lrpof lrind lrtbf rl fl rca 500 4,84% 6,234** 1,736 24,499* 115,61 1674,19 400 4,66% 6,888*** 2,057 29,051 127,91 2125,85 300 3,99% 4,277** 0,017 25,205 154,07 2583,65 250 3,33% 1,533 0,156 20,585 146,76 2814,50 200 3,07% 0,816 0,217 21,926 142,87 3032,33 150 2,85% 0,343 0,281 23,676 144,83 3292,28 100 2,80% 0,261 0,255 24,688 141,09 3540,01 sign rca 500 5,13% 7,666*** 1,953 26,390 115,69 1665,44 400 3,99% 3,494* 1,500 23,079 126,41 2142,75 300 3,81% 3,358* 0,049 21,490 154,02 2582,48 250 3,33% 1,533 0,156 20,585 147,33 2807,82 200 3,23% 1,291 0,143 24,923 143,88 3020,79 150 3,00% 0,665 1,299 27,904 145,29 3276,84 100 2,93% 0,539 0,179 27,440 145,11 3517,04 note: *, **, *** indicate rejection of h0 at the 10% ,5% and 1% significant level, respectively, lrpof – the values of the proportion of failures test statistics, lrind – the values of the independence test statistics, lrtbf – the values of the time between failures test statistics, rl – regulatory loss function, fl – firm’s loss function. table 8. results of the var tests (99% var for pbg) and the loss functions model α̂ lrpof lrind lrtbf rl fl sh 500 2,56% 6,056** 0,475 17,577** 42,00 2163,96 400 1,77% 2,218 0,290 13,426* 56,23 2775,29 300 1,27% 0,375 0,180 4,422 77,63 3452,54 250 2,16% 6,162** 0,576 17,518 91,70 3464,14 200 1,08% 0,036 0,152 1,888 67,93 4456,03 150 2,00% 5,459** 0,571 24,181** 105,19 4229,94 100 0,93% 0,036 0,132 3,400 60,19 5611,84 ewma 500 3,13% 10,313*** 0,714 23,320** 66,52 1933,39 400 2,66% 8,633*** 0,658 25,503** 74,00 2476,85 300 2,00% 4,285** 0,449 13,755 90,20 3046,29 250 2,00% 4,676** 1,436 11,420 86,18 3320,07 200 1,54% 1,624 0,313 7,374 86,15 3617,66 150 1,14% 0,135 0,185 3,069 84,75 3936,79 100 1,07% 0,032 0,173 3,535 81,86 4268,12 table 8. continued model α̂ lrpof lrind lrtbf rl fl rm (λ = 0,95) 500 1,99% 2,719* 0,286 9,927 26,51 2271,53 400 2,22% 5,014** 1,581 12,666 39,05 2820,60 300 2,36% 7,441*** 1,053 14,764 77,73 3286,66 250 2,16% 6,162** 1,181 12,415 77,73 3500,93 200 2,00% 5,067** 1,303 12,230 77,73 3699,53 150 2,00% 5,459** 1,184 14,802 78,92 3935,00 100 1,86% 4,516** 1,288 12,587 79,58 4213,58 rm (λ = 0,99) 500 1,71% 1,472 0,209 5,627 38,83 2171,52 400 1,55% 1,188 0,221 7,810 54,51 2679,38 300 2,00% 4,285** 0,449 14,208 89,53 3103,32 250 2,00% 4,676** 0,490 12,402 93,77 3280,42 200 2,15% 6,547** 1,074 15,053 104,02 3421,39 150 2,85% 16,200*** 0,281 39,564 124,04 3485,92 100 3,06% 20,831*** 0,117 48,907 157,58 3431,58 ar(1)-garch(1,1) 500 2,28% 4,259** 0,374 11,647 55,22 2048,00 400 2,22% 5,014** 0,455 15,475 58,34 2590,51 300 1,81% 2,977* 0,370 11,299 89,91 3118,01 250 1,83% 3,360* 0,411 9,867 81,74 3386,59 200 1,69% 2,592 0,379 7,917 81,97 3635,11 150 1,71% 2,958 0,419 14,533 79,46 3965,85 100 1,33% 0,755 0,270 9,869 79,37 4329,54 rca 500 3,13% 10,313*** 0,714 23,320** 67,06 1947,01 400 2,88% 10,707*** 0,774 25,957** 73,93 2487,60 300 2,18% 5,778** 0,535 18,055 92,98 3031,22 250 2,00% 4,676** 0,490 13,800 88,61 3302,35 200 1,69% 2,592 0,379 11,307 86,68 3571,25 150 1,43% 1,138 0,290 9,825 86,71 3881,87 100 1,20% 0,281 0,219 6,357 82,19 4187,14 sign rca 500 3,13% 10,313*** 0,714 23,320** 65,98 1940,98 400 2,88% 10,707*** 0,774 25,957** 75,11 2494,86 300 2,00% 4,285** 0,449 13,755 92,42 3029,72 250 2,00% 4,676** 0,490 13,800 88,94 3293,79 200 1,69% 2,592 0,379 11,307 86,81 3560,67 150 1,43% 1,138 0,290 11,433 85,97 3867,81 100 1,33% 0,755 0,270 11,177 84,15 4156,81 note: *, **, *** indicate rejection of h0 at the 10% ,5% and 1% significant level, respectively, lrpof – the values of the proportion of failures test statistics, lrind – the values of the independence test statistics, lrtbf – the values of the time between failures test statistics, rl – regulatory loss function, fl – firm’s loss function. table 9. results of the firm’s loss function model t = 375 t = 250 t = 125 fl rank fl rank fl rank α = 5% sym, hist, 1075,956 1 1141,764 2 1263,7779 10 ewma 1109,013 5 1192,913 9 1257,8114 8 rm (λ= 0,95) 1229,387 9 1251,9575 10 1263,0397 9 rm (λ= 0,99) 1189,228 8 1191,2823 8 1101,7305 1 ar(1)-garch(1,1) 1154,158 7 1169,4068 6 1207,2458 4 rca 1101,740 3 1153,1165 4 1212,1169 6 sign rca 1103,836 4 1172,3837 7 1239,823 7 rcama 1098,295 2 1149,6521 3 1204,0552 3 sign rcama 1102,4462 1 1180,7179 2 rca garch 1132,959 6 1158,4801 5 1209,4749 5 sign rca garch 1296,929 10 1339,8815 11 1404,0839 11 α = 2,5% sym, hist, 1284,699 5 1364,1283 5 1521,6798 10 ewma 1282,948 4 1377,3036 7 1454,5161 8 rm (λ= 0,95) 1447,782 9 1473,5854 9 1481,442 9 rm (λ= 0,99) 1379,883 8 1378,9041 8 1283,0513 1 ar(1)-garch(1,1) 1334,391 7 1366,5462 6 1426,3841 6 rca 1272,254 2 1341,8047 2 1406,9397 5 sign rca 1274,395 3 1358,9357 4 1433,1749 7 rcama 1270,903 1 1337,7536 1 1397,165 3 sign rcama 1361,558 2 rca garch 1305,188 6 1349,0442 3 1405,9143 4 sign rca garch 1549,821 10 1577,6011 10 1630,4145 11 α = 1% sym, hist, 1754,0863 9 1754,7449 9 2234,0469 9 ewma 1506,0564 4 1623,2342 7 1723,7757 7 rm (λ= 0,95) 1697,2481 8 1719,5178 8 1727,7362 8 rm (λ= 0,99) 1625,3806 7 1619,3369 6 ar(1)-garch(1,1) 1569,7932 6 1611,8139 5 1688,9632 5 rca 1494,4265 2 1573,8864 2 1663,8987 3 sign rca 1496,858 3 1593,0414 4 1698,2509 6 rcama 1492,2845 1 1568,9729 1 1650,9211 2 sign rcama 1600,3206 1 rca garch 1535,3552 5 1589,0235 3 1664,72 4 sign rca garch 1849,3472 10 note: t denotes the rolling window size, fl– the firm’s loss function. table 10. results of the the predictive quantile loss function model t = 375 t = 250 t = 125 qα rank qα rank qα rank α = 5% sym. hist 0,3183 7 0,3159 7 0,3202 7 ewma 0,3169 5 0,3141 5 0,3181 5 rm (λ= 0,95) 0,3191 8 0,3182 9 0,3189 6 rm (λ= 0,99) 0,3163 4 0,3153 6 0,3213 9 ar(1)-garch(1,1) 0,3215 9 0,3167 8 0,3122 1 rca 0,3156 1 0,3131 3 0,3173 3 sign rca 0,3161 3 0,3122 1 0,3169 2 rcama 0,3174 6 0,3130 2 0,3206 8 sign rcama 0,3363 11 0,3253 10 rca garch 0,3158 2 0,3137 4 0,3179 4 sign rca garch 0,3421 10 0,3283 10 0,3657 11 α = 2,5% sym. hist 0,1945 5 0,1908 2 0,1938 6 ewma 0,1943 4 0,1910 3 0,1914 2 rm (λ= 0,95) 0,1916 1 0,1914 5 0,1915 3 rm (λ= 0,99) 0,1923 2 0,1931 9 0,1995 9 ar(1)-garch(1,1) 0,1985 9 0,1913 4 0,1845 1 rca 0,1947 6 0,1925 6 0,1956 7 sign rca 0,1943 3 0,1896 1 0,1927 4 rcama 0,1957 8 0,1931 8 0,1982 8 sign rcama 0,2016 10 rca garch 0,1953 7 0,1926 7 0,1928 5 sign rca garch 0,2108 10 0,2048 10 0,2358 11 α = 1% sym. hist 0,1015 9 0,0992 9 0,0957 6 ewma 0,0978 3 0,0947 1 0,0941 4 rm (λ= 0,95) 0,0973 2 0,0962 5 0,0958 7 rm (λ= 0,99) 0,0957 1 0,0960 3 ar(1)-garch(1,1) 0,1000 7 0,0961 4 0,0909 1 rca 0,0997 6 0,0963 6 0,0950 5 sign rca 0,0988 5 0,0950 2 0,0934 2 rcama 0,1004 8 0,0967 7 0,0969 8 sign rcama 0,1022 9 rca garch 0,0980 4 0,0967 8 0,0938 3 sign rca garch 0,1143 10 note: t denotes the rolling window size, qα – the predictive quantile loss function. microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 jarosław krajewski nicolaus copernicus university in toruń estimating and forecasting gdp in poland with dynamic factor model† a b s t r a c t. presented paper concerns the dynamic factor models theory and application in the econometric analysis of gdp in poland. dfms are used for construction of the economic indicators and in forecasting, in analyses of the monetary policy and international business cycles. in the article we compare the forecast accuracy of dfms with the forecast accuracy of 2 competitive models: ar model and symptomatic model. we have used 41 quarterly time series from the polish economy. the results are encouraging. the dfm outperforms other models. the best fitted to empirical data was model with 3 factors. k e y w o r d s: dynamic factor models, principal components analysis, gdp. 1. introduction in recent years, dynamic factor models have become popular in empirical macroeconomics. they are believed to have been pioneered by geweke (1977) and sims & sargent (1977) who applied this type of models to the analysis of small sets of variables. dfms have a very wide field of applications. such models are widely used for forecasting, constructing leading indicators of business climate, monetary policy analysis or the analysis of international business cycles. the purpose of the article is to estimate a dynamic factor model of gdp in poland in 1997 – 2008. the second part of the article presents a concept of a dynamic factor model. in the third part the approach to estimating model parameters as well as common factors are discussed. the methods of specifying the number of factors in † scientific work is financing by european social fund and national budget of poland in the integrated regional operational programme framework, measure 2.6 "regional innovation strategies and transfer of knowledge" kujawsko-pomorskie voievodship own project "candidate for doctor's degree grants 2008/2009 – irop". jarosław krajewski 140 the model are also presented. the data used in the study and the empirical results have been described in the fourth part. final part summarises the whole study. 2. dynamic factor model the concept of factor models bases on the assumption that the behavior of most macroeconomic variables may be well described using a small number of unobserved common factors. these factors are often interpreted as the driving forces in the economy. the particular variables may be then expressed as linear combination of up-to-twenty common factors which usually make it possible to explain a major part of variability of those variables (kotłowski, 2008). let ty stand for a variable and let tx express the vector of n variables containing information that can be useful in modeling and forecasting the future values of ty . in the dynamic factor model we assume that all variables itx contained in vector tx may be expressed as a linear combination of current and lagged unobserved factors itf . ,,...,1for ,)( nielx itiit =+= tfλ (1) where ]',...,,[ 21 trtt fff=tf stands for vector r of unobserved common factors at moment t, qiqiiii lll λλλλλ ++++= ...)( 2 210 represent a lag polynomials and ite express an idiosyncratic errors for variable itx (see stock, watson, 1998). in turn, ty may be noted as the function of current and lagged common factors contained in vector tf and the past values of variable ty , with the following formula .)()( tttt eylly ++= γβ f (2) the model described with equations (1) and (2) is a dynamic factor model. 3. model estimation and selection of the factor’s number one of the most widely used methods of parameters and factors estimation in a factor models is the method of principal components. let us emphasise that both: the factor matrix and the coefficient matrix are unknown. model (1) is thus equivalent to the model in the matrix form of ,'1 eλfhhx += − (3) where matrix h is any non-singular matrix of dimension rr × . it is necessary to carry out the appropriate normalisation of matrix h . stock & watson (1998) estimating and forecasting gdp in poland … 141 suggest that for this purpose condition in the form of riλλ =)/( ' n may be imposed on the parameters of model which would render matrix h orthonormal. the estimation of matricies f and λ using the method of principal component consist in finding such estimates of matrices f̂ and λ̂ that would minimise the residual sum of squares in equation (3) as expressed with the following formula .)( 1 )( 1 1 2'∑∑ = = −= n i t t itxnt v ti fλλf, (4) it is necessary, in the first step, to perform a minimisation of function (4) in respect to factor matrix f with the assumption that matrix λ is known and fixed. then we obtain estimate f̂ , as function λ, which is subsequently substituted in equation (4) for the true value of f. in the second step, we minimise function (4) in respect to matrix λ with a normalisation condition ,)/( ' riλλ =n thus directly obtaining estimate .λ̂ it should be emphasised that it is equivalent to maximisation of expression ].)([ '' λxxλtr matrix λ̂ is a matrix whose subsequent columns are eigenvectors of matrix xx' multiplied by n corresponding to highest eigenvalues of the same matrix. in turn, the estimate of matrix f is expressed by the formula ./)ˆ(ˆ nλxf = (5) stock & watson (1998) emphasise that if the number of variables is higher than the number of observations, i. e. n > t, then from the computational point of view it is easier to apply a procedure which determine estimate f ~ by minimizing concentrated function (4) in respect to matrix f with the condition ./' riff =t matrix f ~ will then contain eigenvectors of matrix xx' corresponding to r highest eigenvalues of this matrix and multiplied by t . in turn, the estimate of matrix λ ~ will assume the following form ./) ~ ( ~ '' txfλ = (6) in practice, the number of factors necessary to represent the correlation among the variables is usually unknown. to determine the number of factors empirically a number of criteria were suggested. bai and ng (2002) have suggested information criteria to be used to estimate the number of factors. ,ln))(ln()(1 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + += tn nt nt tn kkvkic ) (7) jarosław krajewski 142 ,ln))(ln()( 22 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + += ntcnt tn kkvkic ) (8) , ln ))(ˆln()( 2 2 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ += nt nt c c kkvkic (9) where )(ˆ kv is residual sum of squares from k – factors model and { }.min tncnt = 4. description of data and empirical results the data used in the study are macroeconomic quarterly data describing polish economy and encompass the period from first quarter of 1997 to third quarter of 2008 (47 observations). as explained variable we used polish gdp. all of data were taken from polish central statistical office1. before embarking on the work on factor model specification, the date had to be appropriately modified. in the first step variables were adjusted for the impact of seasonal fluctuations. next, the series were transformed by taking logarithms and/or differencing so that the transformed series were stationary (green, 2003). in the final step, all variables were standardized. in total, 41 time series were considering, representing of following macroeconomic categories: output & sales, construction, domestic and foreign trade, prices and labour market, budgetary and monetary policy. after preliminaries principal components analysis were used to estimate factors. next bai and ng informational criteria were calculated to specify number of factors. table 1 includes eigenvalues and values of information criteria. the first two criteria reach minimum for the number of factors equal to 3, while the third criteria assumes its lowest value for 10 factors. due to the fact that two out of three criteria display the same value, we arbitrarily assume that the number of factors in the model is 3. the first three factors explain almost 82% of total variance of gdp. some econometricians maintain that factors estimated using the principal component method do not have an economic interpretation. however, in this paper try-out were done. for this reason, it is possible to carry out a regression of particular variables against each of the estimated factors and check which factor explains the behavior of a given variable to the greatest extent. the r-squared values on the regression of particular variables suggest that the first factor primarily affects the variability of labour market and foreign trade. the second factor determines the prices and incomes. the third factor to the greatest extent influences the values of sales. 1 www.stat.gov.pl estimating and forecasting gdp in poland … 143 the further stage of study bic criterion was used to determine the number of gdp and factors delays. the bic criterion indicated model only with current values of the first three factors. model estimation results are presented in table 2. all coefficients are significant under 5% level. factor model of gdp in poland estimated in this way has r-squared over 70%. real values of gdp and values based on the model are shown on the figure 1. table 1. selection of the number of factors in the model number of factors eigenvalues contribution to variance cumulative contribution to variance ic1 ic2 ic3 1 76.949 0.721 0.721 -3.694 -3.491 -3.833 2 6.249 0.059 0.779 -3.584 -3.177 -3.861 3 4.278 0.040 0.819 -4.555 -3.945 -4.971 4 2.921 0.027 0.846 -4.490 -3.677 -5.044 5 2.460 0.023 0.870 -4.467 -3.451 -5.160 6 1.976 0.019 0.888 -4.335 -3.115 -5.166 7 1.551 0.015 0.903 -4.192 -2.769 -5.162 8 1.429 0.013 0.916 -4.109 -2.482 -5.216 9 1.282 0.012 0.928 -4.246 -2.417 -5.493 10 1.066 0.010 0.938 -4.126 -2.093 -5.511 figure 1. actual and fitted values of gdp in poland in 1997 – 2008 next stage of analysis was to check if lagging some of variables will influence the final result of estimation. this caused increasing the number of factors to four. the resulting model with four factors is presented in table 3. it is not -3 -2 -1 0 1 2 3 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 actual fitted jarosław krajewski 144 hard to see that in four factors model r-squared is higher and true significance level of coefficients is lower. table 2. dynamic factor model of gdp in poland in 1997-2008 dependent variable: gdp coefficient std. error t statistic p value f1 -0.0362 0.0095 -3.7985 0.0005 f2 0.0683 0.0335 2.0404 0.0478 f3 0.3707 0.0405 9.1610 0.0000 r-squared 0.7142 akaike criterion 1.7264 adjusted r-squared 0.7003 schwarz criterion 1.8481 durbin-watson 2.1454 hannan-quinn criterion 1.7715 table 3. dynamic factor model of gdp in poland in 1997-2008 – after changes in data set dependent variable: gdp coefficient std. error. t statistic p value f21 0.1722 0.0273 6.3160 0.0000 f22 0.1686 0.0329 5.1320 0.0000 f23 -0.2548 0.0361 -7.0669 0.0000 f24 0.2241 0.0400 5.6090 0.0000 r-squared 0.7909 akaike criterion 1.4858 adjusted r-squared 0.7748 schwarz criterion 1.6497 durbin-watson 2.2511 hannan-quinn criterion 1.5463 table 4. forecast errors mape rmse r-squared ar 90.8002 0.8116 0.1246 dfm 1.7898 0.0160 0.7003 dfm2 12.8691 0.1150 0.7748 causal model 4.9267 0.0440 0.8694 in the last stage of our study we generated forecasts and forecast errors. the forecasting performance of the factor models was evaluated by comparing the accuracy of gdp forecasts obtained on the basis of the factor models with the accuracy of gdp forecasts derived from other competitive models. two competitive models were taken into consideration: an univariate autoregressive model and causal model with two variables. an univariate autoregressive model was adopted as the main benchmark model for evaluating the forecasting performance of the factor models (see marcellino, stock, watson, 2001). the bic criterion indicated model ar(1). the causal model included industrial production sales and average employment as explanatory variables. forecasting models were estimated on a shorter sample (up to 4 quarter 2007). the forecast was estimating and forecasting gdp in poland … 145 produced on the one period ahead. first factor model had the best forecast accuracy. table 4 presents results. 5. summary the principal component analysis reduced the number of explanatory variables from 41 to 3 factors. the resulting dynamic factor model of gdp in poland is satisfactory from the statistical point of view. changes in data set influenced the final result of model estimation. in this study it brought out increasing number of factors and improvement estimation performance. unfortunately, it did not improve forecasting performance. first dynamic factor model of gdp in poland in 1997 – 2008 gave the best forecasting performance in comparison with three competitive models described above. references bai, j., ng, s. (2002), determining the number of factors in approximate factor models, econometrica, 70, 191–221. geweke, j. (1977), the dynamic factor analysis of economic time series, [in:] aigner d. j., goldberger a. s. (ed.), latent variables in socio–economic models, amsterdam, north holland. greene, w. h. (2003), econometric analysis, pearson education, new jersey. kotłowski, j. (2008), forecasting inflation with dynamic factor model – the case of poland, working papers, 2-08, sgh, warszawa. marcellino, m., stock, j. h., watson, m. w. (2001), macroeconomic forecasting in the euro area: country specific versus area–wide information, working paper, 201, innocenzo gasparini institute for economic research. sargent, t., sims, c. (1977), business cycle modelling without pretending to have too much a-priori economic theory, in sims c. (ed.), new methods in business cycle research, minneapolis, federal reserve bank of minneapolis. sims, c. a. (1980), macroeconomics and reality, econometrica, 48, 1–48. stock j., watson, m. w. (1998), diffusion indexes, working paper, 6702, national bureau of economic research. zastosowanie dynamicznego modelu czynnikowego do modelowania i prognozowania pkb w polsce z a r y s t r e ś c i. referat traktuje o podstawach konstrukcji dynamicznych modeli czynnikowych i ich zastosowaniu empirycznym. dfm stosuje się do prognozowania, konstruowania głównych wskaźników koniunktury, analiz polityki monetarnej i badania międzynarodowych cykli koniunkturalnych. w referacie oszacowano dynamiczny model czynnikowy pkb w polsce w latach 1997–2008, a także oceniono trafność uzyskanych na jego podstawie prognoz w porównaniu do modelu ar i modelu symptomatycznego. zbiór danych wykorzystanych do badania zawiera 41 zmiennych makroekonomicznych. najlepszym ze statystycznego punktu widzenia okazał się model z 3 czynnikami. s ł o w a k l u c z o w e: dynamiczny model czynnikowy, metoda głównych składowych, pkb. microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 marcin fałdziński nicolaus copernicus university in toruń application of modified pot method with volatility model for estimation of risk measures a b s t r a c t. the main aim of this paper is the presentation and empirical analysis of the new approach which combines volatility models with peaks over threshold method that comes from extreme value theory. the new approach is applied for estimation of risk measures (var and es) in financial time series. for the empirical analysis the financial risk model evaluation was conducted. in this paper the pot method was compared with standard volatility models (garch and sv) in case of the conditional modeling. k e y w o r d s: extreme value theory, peaks over threshold, value-at-risk, expected shortfall. 1. introduction current literature in the area of risk management is very extensive (szegö, 2004) and results are frequently ambiguous. the existing approaches for estimating the profit/loss distribution of a portfolio of financial instruments can be schematically divided into three groups: nonparametric historical simulation methods, parametric methods based on volatility models (garch type models) and methods based on the extreme value theory. this work focuses on methods based on volatility models and peaks over threshold (pot) method. previous results from papers which applied the pot method in risk management were the main motivation to deal with a problem of extremes in financial time series. the main aim of this paper is to propose and analyse more complex approach taking into account extremes and non-extremes in risk management of financial time series. most of the attention is focused on the new approach proposed in this paper. 2. the pot method and volatility models in the peaks over threshold method we are interested in excesses over a high threshold value u . detailed description of pot method can be found in: marcin fałdziński 120 embrechts, klüppelberg, mikosch (2003) or osińska, fałdziński (2008). here will be presented mcneil and frey's approach, which joint volatility models and pot method. we assume that tx is a time series representing daily observations of log return on a financial asset price. we assume that dynamics of x process is given as: t t t tx zμ σ= + , (1) where innovations tz are the white noise process with zero mean and unit variance. we assume that tμ is the expected return and tσ is the volatility of the return, where both are measurable to information set 1t −f at time 1t − . to implement an estimation procedure for the process (1), we need to choose a dynamic conditional mean as well as a conditional variance model. many volatility models were proposed in econometric literature, from arch/garch models, and their different modifications and generalization, to sv models. mcneil and frey defined simple risk measures forms for one day horizon with relation to process (1) as: 1 1 ( ) t q t t qvar var zμ σ+ += + , (2) 1 1 ( ) t q t t qes es zμ σ+ += + , (3) where ( )tqvar z is the value-at-risk of tz process, and ( ) t qes z is the corresponding expected shortfall. the method proposed by them demands minimal assumptions for innovations distribution and focus on modeling distribution tails using extreme value theory. generally we can say, that we use two stage approach, which can be presented in the following steps: 1. fit a garch-type model (generally volatility model) to return series. estimate 1tμ + and 1tσ + using fitted model and calculate standardized residuals. it means, that fitted model is used to estimate one-day ahead predictions of 1tμ + and 1tσ + . 2. evt is used to estimate ( )qvar z and ( )qes z with application of pot method for mentioned residuals. a value-at-risk in the evt for the peaks over threshold method is equal: ˆ ˆ ( ) 1 ˆ u n var u n γ σ α α γ −⎛ ⎞⎛ ⎞ ⎜ ⎟= + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ , (4) where α is a tolerance level, u is a threshold, ˆ ˆ,γ σ are estimated parameters from generalised pareto distribution (gpd), n is the total number of realizations and un is number of extremes. because of var drawbacks the alternative risk measure was developed, which is called expected shortfall (es), and was application of modified pot method with volatility model … 121 proposed by artzner et al. (1997; 1999). the expected shortfall for the peaks over threshold method is given by: ˆˆ( ) ( ) ˆ ˆ1 1 var u es α σ γ α γ γ − = + − − , (5) in the literature we can find comparisons of models which estimate var and es where extreme value theory is applied (i.e. brooks, clare, dalle molle, persand, 2005; harmantzis, miao, chien, 2006; kuester, mittnik, paolella, 2006; osińska, fałdziński, 2008; fałdziński, 2008). in all of these papers authors find, that evt is good or very good approach to estimate risk measures. empirical results presented in these papers show that volatility models with application of extreme value theory more accurately estimate expected future values of asset returns, particularly in case of extremely rare events (i.e. extremes). standard volatility models better fit to mean values in financial time series in contrast to models with evt. therefore the new approach is based on such combination, that extremes are estimated using pot method, and non-extremes are estimated using standard volatility models. this combination is an attempt to identify extremes in financial time series. the new approach can be written in the following: 1 1 1 1 t t t t t t garch pot u n garch pot garch u μ σ μ σ + + + + + ≥⎧ = ⎨ + <⎩ (6) this new hybrid of volatility model and pot method is based on conditional volatility forecast 1tσ + , conditional mean 1tμ + and on threshold tu . if sum of 1tμ + and 1tσ + forecasts are higher than threshold then we apply mcneil and frey's approach, in other case we have standard garch model. threshold tu could be constant or time-varying. switching mechanism is formulated to identify whether the forecast of time series return is an extreme value, or not. forecasts of asset returns are very important, the more precise forecast of time series is, the extremes are better identified, which is connected to proper switching mechanism (6). the idea of new approach is completely based on the forecast capability from volatility model for time series (in this case it is garch model but we also used sv model). 3. backtesting a key part for risk measures is necessity to check correctness of estimation and simultaneously choose the most precise method for their estimates. risk models need to be validated and backtesting is the class of quantitative methods used to e.g. rank a group of models against each other (dowd, 2005; alexander, 2008). for backtesting we used three binominal tests: the failure test uclr , the mixed kupiec-test indlr k (haas, 2001) and the test of independence indlr ch marcin fałdziński 122 (christoffersen, 1998). unfortunately presented tests have weak power, and additionally these methods do not give the opportunity to create a ranking of the models. angelidis and degiannakis (2006) presented modified lopez's (lopez, 1999) loss function: 1 1| 1, 1 if violation occurs 0 else t t t t y es+ + + ⎧ −⎪ ψ = ⎨ ⎪⎩ (7) ( ) 2 1 1| 2, 1 if violation occurs 0 else t t t t y es+ + + ⎧ −⎪ ψ = ⎨ ⎪⎩ (8) to judge which model is the best, we compute the mean absolute error 1, 1 / t t t mae t = = ψ∑ % % , and the mean squared error 2, 1 / t t t mse t = = ψ∑ % % , where t% is the number of the forecasts, and total loss ( )tl is the sum of these errors (angelidis, degiannakis, 2006). the loss function approach is based on calculating magnitude of violations (or exceedances), i.e. cases when the risk measure underestimates a future value of asset. as we can see the presented loss functions measure only the underestimation of risk measure. a perfect situation is when an estimated risk measure does not underestimate and overestimate1 too much a future value of asset. for example, if we would have two estimated risk measures and both of them have almost equal value of standard loss functions, then the better risk measure is that which has the lower overestimation. an overestimation of risk measures was proposed to measure (fałdziński, 2009). the overestimation of loss functions are given in the following: 1 1 1 1 1, 1 1 if 0 0 if 0 t t t t t t es y y es y + + + + + + ⎧ − < <⎪ φ = ⎨ ≤⎪⎩ (17) ( )21 1 1 1 2, 1 1 if 0 0 if 0 t t t t t t es y y es y + + + + + + ⎧ − < <⎪ φ = ⎨ ≤⎪⎩ (18) we also compute the mean absolute error of overestimation 1, 1 / d over t t mae d = = φ∑ , and the mean squared error 2, 1 / d over t t mse d = = φ∑ , where 1 t i d = = ∑1 , 1 1 1 1 if 0 0 if 0 t t t y es y + + + < <⎧ = ⎨ ≤⎩ 1 is the number of the positive fore 1 cases when estimated risk measure is higher than value of asset application of modified pot method with volatility model … 123 casts (i.e. larger then 0, but smaller than the given risk measure). similarly we could construct the total loss of overestimation ( )over overotl mae mse= + . the otl could also be computed for var and srm, but then we have to change 1tes + for another estimated risk measure in formula (17), (18) and d . 4. empirical analysis the subject of the empirical analysis is the comparison of the estimated value-at-risk and expected shortfall measures for the new approach with volatility models. the comparison is based on selection of the best model for total loss tl proposed by angelidis and degiannakis and total loss of overestimation (otl). in the analysis the sv model with gaussian distribution and the garch model with gaussian and t-student error distributions were used. we have chosen sv and garch models because they represent the most standard volatility models. the parameters were estimated with the maximum likelihood method in case of garch models and the quasi-maximum likelihood method in the case of the sv models. the time series used in the analysis comprise 3000 observations of log returns (daily data: 07.11.1994 – 31.10.2006). we used 5 financial time series (wig, sp500, dax, ftse100, nikkei225) that represent the stock market returns. for each time series a thousand vars and ess were estimated for backtesting purposes. to compute the es for the volatility models we used dowd’s approach (dowd, 2005). also we used the timevarying threshold u , as a result of defining the number of extremes on 10% level for all observations in time series. this 10% level is a common standard in similar analyses. results for the binominal tests were computed besides their drawbacks. in short we can say that findings were very diverse and it was very difficult to make general conclusions that is why we decided no to show them. they are available upon request. based on findings contained in table 1 we can see that sv-pot model has the lowest value for total loss tl (next are n-sv-pot and sv models). it should not be a surprise because sv models more take into account the extremes than standard garch model. after the class of sv models (i.e. sv, sv-pot and n-sv-pot) we have four variant of garch-pot models with relatively higher values of tl. the end of the total loss ranking contains n-garch-pot and garch models alternately, but we should point out that n-garch-pot models are relatively better. on the other hand the class of sv models has the highest value of the total loss of overestimation (otl) for value-at-risk and this is the consequence of the same property mentioned before. the lowest value of otl for var is obtained for garch td model. we have this kind of result because the standard garch model do not take into consideration the extremes like the other models in the analysis and that is why the overestimation is the lowest. the next in the ranking are garch and ngarch-pot models alternately. marcin fałdziński 124 table 1. backtesting results for wig20 and sp500 wig20 α=0.05 model tl rank tl otl var rank otl var otl es rank otl es garch 0.0998 11 2.788 3 7.446 7 garch td 0.2019 15 2.680 1 10.111 14 ar-garch 0.0927 10 2.980 7 7.778 9 ar-garch td 0.1874 14 2.812 4 10.416 15 sv 0.0281 3 4.420 13 7.428 6 garch-pot 0.0498 4 3.219 12 5.913 3 garch-pot td 0.0525 5 3.115 9 5.733 2 ar-garch-pot 0.0734 7 3.118 10 6.817 4 ar-garch-pot td 0.0530 6 3.101 8 5.634 1 sv-pot 0.0069 1 7.631 15 9.425 12 n-garch-pot 0.0823 8 2.909 5 6.905 5 n-garch-pot td 0.1622 13 2.775 2 9.031 11 n-ar-garch-pot 0.0857 9 3.126 11 7.679 8 n-ar-garch-pot td 0.1609 12 2.922 6 9.957 13 n-sv-pot 0.0162 2 6.486 14 8.754 10 sp500 α=0.05 model tl rank lf otl var rank otl var otl es rank otl es garch 0.0554 11 1.890 3 5.025 8 garch td 0.1261 15 1.807 1 6.827 13 ar-garch 0.0495 10 2.080 11 5.379 11 ar-garch td 0.0997 14 2.003 5 7.168 15 sv 0.0102 3 2.344 13 3.850 5 garch-pot 0.0194 7 2.038 7 3.450 1 garch-pot td 0.0191 6 2.038 8 3.461 3 ar-garch-pot 0.0191 5 2.050 9 3.453 2 ar-garch-pot td 0.0175 4 2.051 10 3.489 4 sv-pot 0.0015 1 4.048 15 5.108 9 n-garch-pot 0.0435 8 1.890 4 4.762 7 n-garch-pot td 0.0854 12 1.822 2 6.222 12 n-ar-garch-pot 0.0440 9 2.084 12 5.245 10 n-ar-garch-pot td 0.0880 13 2.006 6 6.873 14 n-sv-pot 0.0098 2 2.959 14 4.315 6 note: n-ar-garch-pot td means the new approach proposed in this paper with ar(1)-garch(1,1) model and peak over threshold method where t-distribution was applied. respectively other abbreviation are constructed. it means that the switching mechanism takes into account non-extremes rather than extremes. in case of otl for var, garch-pot models are better than application of modified pot method with volatility model … 125 the class of the sv models, but relatively worse than the other models. the lowest values of otl for expected shortfall have been obtained for garchpot models. generally we can say that n-garch-pot models are relatively better than garch models in case of otl for es. table 2. ranking according to total loss tl for indices model wig wig20 sp500 dax ftse100 nikkei225 garch 10 11 11 9 11 10 garch td 15 15 15 15 15 15 ar-garch 11 10 10 8 8 11 ar-garch td 14 14 14 12 14 14 sv 3 3 3 3 3 3 garch-pot 6 4 7 5 4 7 garch-pot td 7 5 6 4 5 4 ar-garch-pot 4 7 5 7 6 5 ar-garch-pot td 5 6 4 6 7 6 sv-pot 1 1 1 1 1 1 n-garch-pot 9 8 8 11 10 9 n-garch-pot td 13 13 12 14 12 13 n-ar-garch-pot 8 9 9 10 9 8 n-ar-garch-pot td 12 12 13 13 13 12 n-sv-pot 2 2 2 2 2 2 table 3. ranking according to total loss of overestimation otl for var model wig wig20 sp500 dax ftse100 nikkei225 garch 3 3 3 4 8 11 garch td 1 1 1 2 2 8 ar-garch 6 7 11 12 12 12 ar-garch td 4 4 5 11 10 7 sv 13 13 13 13 13 13 garch-pot 10 12 7 7 4 4 garch-pot td 8 9 8 8 3 1 ar-garch-pot 12 10 9 5 5 3 ar-garch-pot td 9 8 10 6 6 2 sv-pot 15 15 15 15 15 15 n-garch-pot 5 5 4 3 7 9 n-garch-pot td 2 2 2 1 1 5 n-ar-garch-pot 11 11 12 10 11 10 n-ar-garch-pot td 7 6 6 9 9 6 n-sv-pot 14 14 14 14 14 14 marcin fałdziński 126 if we compare models at the same class based on total loss tl (table 2) we can state that new approach is placed between mcneil and frey's method and standard volatility models. similar conclusion can be deduced from the analysis of total loss of overestimation otl for expected shortfall (table 4). it means that the new approach is as good as the specific volatility model. to be accurate, if the forecast of conditional mean and conditional volatility for volatility model is more precise, then the new approach is better than two other methods. in case of total loss of overestimation otl for value-at-risk (table 3), we can see that volatility models are the best. it should not be a surprise, because var better fits to small and mean values of financial time series in comparison with es. the new approach which connects volatility models and pot method, like before is placed between two other methods, but sometimes this method is the best. table 4. ranking according to total loss of overestimation otl for es model wig wig20 sp500 dax ftse100 nikkei225 garch 8 7 8 8 10 8 garch td 13 14 13 11 14 14 ar-garch 9 9 11 9 11 9 ar-garch td 15 15 15 13 15 15 sv 5 6 5 12 5 5 garch-pot 3 3 1 4 2 4 garch-pot td 1 2 3 3 1 2 ar-garch-pot 4 4 2 2 3 3 ar-garch-pot td 2 1 4 1 4 1 sv-pot 10 12 9 15 7 11 n-garch-pot 7 5 7 5 8 6 n-garch-pot td 12 11 12 7 12 13 n-ar-garch-pot 11 8 10 6 9 7 n-ar-garch-pot td 14 13 14 10 13 12 n-sv-pot 6 10 6 14 6 10 generally we can say, that the new approach, which is a hybrid of standard volatility models and mcneil and frey's method, is as precise as the specific volatility model and is able to forecast the financial time series. references alexander, c. (2008), market risk analysis vol. iv: value-at-risk models, john wiley & sons ltd., new york. angelidis, t., degiannakis, s. (2006), backtesting var models: an expected shortfall approach, working papers, university of crete, athens university of economics and business. artzner, p., delbaen, f., eber, j. m., heath, d. (1997), thinking coherently, risk, 10, 68–71. application of modified pot method with volatility model … 127 artzner, p., delbaen f., eber j.m., heath, d. (1999), coherent measures of risk, mathematical finance, 9, 203–228. brooks, c., clare, a.d., dalle molle, j.w, persand, g. (2006), a comparison of extreme value theory approaches for determining value at risk, journal of empirical finance, 12, 339–352. christoffersen, p.f. (1998), evaluating interval forecasts, international economic review, 3. dowd, k. (2005), measuring market risk second edition, john wiley & sons ltd., new york. embrechts, p., klüppelberg, c., mikosch, t. (2003), modelling extremal events for insurance and finance, springer, berlin. fałdziński, m. (2008), model warunkowej zmienności wartości ekstremalnych (conditional extreme value volatility model), in zieliński z. (ed.), współczesne trendy w ekonometrii (contemporary trends in econometrics), wydawnictwo wyższej szkoły informatyki i ekonomii, olsztyn. fałdziński, m. (2009), on the empirical importance of the spectral risk measure with extreme value theory approach, forecasting financial markets and economic decisionmaking findecon, łódź, submitted. haas, m. (2001), new methods in backtesting, financial engineering research center, bonn harmantzis, f.c., miao, l., chien, y. (2006), empirical study of value-at-risk and expected shortfall models with heavy tails, journal of risk finance, 7, no.2, 117–126. kuester, k., mittnik, s., paolella, m.s. (2006), value-at-risk prediction: a comparison of alternative strategies, journal of financial econometrics, 1, 53–89. mcneil, j.a., frey, f. (2000), estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach, journal of empirical finance, 7, 271–300. osińska, m., fałdziński, m. (2008), garch and sv models with application of extreme value theory, in zieliński z. (ed.), dynamic econometric models, volume 8, umk, toruń. szegö, g. (2004), risk measures for the 21st century, john wiley & sons ltd., west sussex, uk. zastosowanie zmodyfikowanej metody pot z modelami zmienności do szacowania miar ryzyka z a r y s t r e ś c i. celem artykułu jest prezentacja nowego podejścia mającego na celu połączenie modeli zmienności z metodą peaks over threshold (pot), wywodzącą się z teorii wartości ekstremalnych. podejście to opiera się na możliwości szacowania ekstremów na podstawie metody pot, natomiast wartości średnich na podstawie modeli zmienności. nowe podejście jest zastosowane do estymacji miar ryzyka (var i es) dla finansowych szeregów czasowych. do oceny nowego podejścia wykorzystano procedury testowania wstecznego. w pracy zastosowano metodę pot dla stóp zwrotu indeksów rynków finansowych przefiltrowanych za pomocą modeli garch oraz sv, które porównano z wynikami otrzymanymi tylko za pomocą modeli garch i sv. s ł o w a k l u c z o w e: teoria wartości ekstremalnych, peaks over threshold, miary ryzyka microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 anna pajor cracow university of economics bayesian analysis of the box-cox transformation in stochastic volatility models† a b s t r a c t. in the paper, we consider the box-cox transformation of financial time series in stochastic volatility models. bayesian approach is applied to make inference about the box-cox transformation parameter (λ). using daily data (quotations of stock indices), we show that in the stochastic volatility models with fat tails and correlated errors (fcsv), the posterior distribution of parameter λ strongly depends on the prior assumption about this parameter. in the majority of cases the values of λ close to 0 are more probable a posteriori than the ones close to 1. k e y w o r d s: box-cox transformation, sv model, bayesian inference. 1. introduction the continuously compounded rates of return (or logarithmic returns) as well as the simple rates of return are commonly used in econometric analyses of financial data. these two types of data transformation are applied arbitrarily. in the derivatives pricing literature there is the tradition of using logarithmic returns, but when the logarithmic return is modelled as a conditionally student-t distributed random variable, the conditional expected simple rate of return is infinite. it violates the finite second moment condition for the asset payoff in call option pricing (see duan, 1999). duan (1999) uses the generalized error distribution (ged) for the logarithmic returns that also exhibits fat tails and includes the normal distribution as a special case. other researchers build model with sample returns instead of log-returns and with the student-t distribution (see e.g. hafner, harwartz, 1999; härdle, hafner, 2000; bauwens, lubrano, 2002). however, both the logarithmic return and simple one are variants of the well-known box-cox transformation of the xt/xt-1 ratio (where xt denotes the asset price at time t) with parameter 0 and 1, respectively. in the paper, we con † research supported by a grant from cracow university of economics. the author would like to thank janusz jaworski for language verification of the manuscript. anna pajor 82 sider the box-cox transformation of financial time series in stochastic volatility (sv) models. bayesian approach is applied to make inference about the boxcox transformation parameter (λ). as parameter λ is estimated along with other unknown parameters, information in the data is used to determine which transformation is appropriate for the data. the structure of the article is as follows: section 2 consists of a short presentation of the bayesian sv model with fat-tails correlated errors for the transformed data, section 3 focuses on the empirical results, and finally, section 4 incorporates the conclusions. 2. bayesian ar(1)-fcsv model for the transformed data let xt denote the price of an asset at time t, t = 0, 1, ..., t. the box-cox transformation of the xt/xt-1 ratio is defined as: ⎪ ⎩ ⎪ ⎨ ⎧ = > − = − − − 0)/ln( 0 1)/( ),/( 1 1 1 λ λ λλ λ tt tt tt xx xx xxb , t = 1, ..., t. for ),/( 1 λ−tt xxb we use an autoregressive structure 1: ,]),/([),/( 121111 ttttt xxbxxb εδλρδλ +−=− −−− t = 1, ..., t, (1) where {εt} is the stochastic volatility process with fat-tails and correlated errors (fcsv), introduced by jacquier et. al., (2004). the discrete-time fcsv process can be written as: ,/ tttt hu ωε = (2) ,lnln 1 thtt hh ησφγ ++= − (3) ,/)(~ 2 ννχωt ωt ⊥ (ul , ηl), t, l ∈ {1, …, t}, (ut , ηt)′ ~ , 1 1 ,0 ⎟⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ρ ρ in t = 1, …, t. where the abbreviation ″in″ denotes that the random vectors concerned are independent and normally distributed, ⊥ denotes stochastic independence. in the fcsv process, when ρ is equal to zero, ht is the inverse precision in the conditional distribution, p(εt|ht), that is, (v/v-2)ht (for v > 2) is the conditional variance. thus, the fcsv model specifies a log-normal autoregressive process for the conditional variance factor (ht) with correlated innovations in the conditional mean and conditional variance equations, i.e. in (2) and (3), respectively. 1 we use the autoregressive structure, because financial time series such as stock market indices often present positive autocorrelation of order one of the returns (see campbell et al., 1997). bayesian analysis of the box-cox transformation in stochastic volatility models 83 one interpretation for the latent variable ht is that it represents the random, uneven and autocorrelated flow of new information into financial markets (see clark, 1973). the parameter φ is related to the volatility persistence, and σh is the volatility of the log-volatility. the above model captures the leverage effect when the correlation ρ is negative. in fact, if ρ is negative, then a negative innovation ut is associated with higher contemporaneous and subsequent volatilities. on the other hand, a positive innovation ut is connected with a decrease in volatility (see jacquier et al., 2004). the bayesian model is characterized by the joint probability density function of the untransformed xt/xt-1 ratios (i.e. y = (y1, ..., yt)′, where yt = xt/xt-1), the latent variables (i.e. h = (h1, ..., ht)′, ω = (ω1, ..., ωt)′), and of the parameter vector θ : )(),|,,()|,,,( )0()0( θyθωhyyθωhy ppp = , (4) where ,)2(|| |),(|' 2 1 exp)|(),|,,( 5.1 1 5.05.0* 1 1* )0( − = −− = − ∏ ∑ × × ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −= t t t t tt t t tt h jtrvpp ωπ λ σ yrrσωyθωhy ),( 2 exp 22 )|( ),0( 1 2 1 1 2 ttt t t ip ωω ν ω νν ν ν ν +∞ − = − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −⎟ ⎠ ⎞ ⎜ ⎝ ⎛ γ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∏ω , 1 2 * ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = hh h σρσ ρσ σ ,)',( thtt u ησ=r ,)',,,,,,,( 2 11 λνρσφγρδ h=θ y(0) denotes initial values. the jacobian j(λ, y) is .),( 1 1∏ = −= t t tyj λλ y our model specification gets completed by assuming the following prior structure: ),()(),()()()()(),,,,,,,( 211 2 11 λνρσφγρδλνρσφγρδ pppppppp hh = where we use proper prior densities of the following distributions: δ1 ~ n(0, 1), ρ1 ~ u(-1,1) γ ~ n(0, 100), φ ~ n(0, 100) i(-1,1)(φ), v ~ exp(0.1), τ ~ ig(1, 0.005), ψ|τ ~ n(0, τ /2), )1(, 22 ρστρσψ −== hh . the prior distribution for δ1 is standardized normal, u(-1,1) denotes the uniform distribution over (-1,1). the prior distribution for φ is normal, truncated by the restriction that the absolute value of φ is less than one (i(-1, 1)(.) denotes the indicator function of the interval (-1, 1), which is the region of stationarity of lnht). the symbol ig(v0, s0) denotes the inverse gamma distribution with mean anna pajor 84 s0/(v0-1) and variance )]2()1/[( 0 2 0 2 0 −− vvs (thus, when ρ = 0, the prior mean for 2 hσ does not exist, but the precision, 2− hσ , has a gamma prior with mean 200 and standard deviation 200). the symbol exp(a) denotes the exponential distribution with mean 1/a (thus the prior mean for v is equal to 10 with the standard deviation equals 10). the prior distribution for (ψ, τ) induces a prior distribution for ),( 2hσρ , which has the following form: ,)1()()2()(),( 22 0 2 0 0 22 0 00 )1(2 )( 5.12)1(125.05.0 0 1 00 2 hh p v s v h v eepvsp σρ ψρσ σρ ρσπρσ − − − −−− − +−−− −γ= ν0 = 1, s0 = 0.005, ψ0 = 0, p0 = 2 (similar to jacquier et al., 2004). as far as the prior distribution for λ, we assume that our prior information regarding this parameter can be represented by the following: a) a non-standard distribution on the interval [0, 1]: )1()( xxep −−∝ βλ , where β = 30. this prior distribution is symmetrical and u-shaped, as shown in figure 1. b) the beta distribution with parameters 0.5 and 0.5; c) the uniform distribution on the interval [0, 1]; d) the exponential distribution with mean 1; figure 1. prior distributions for the box-cox transformation parameter (λ) as regards the initial condition for ht, i.e. h0, we assume that it is equal to 1. the joint posterior distribution is then .|||),(|' 2 1 exp )|()(),|,,( 5.1 1 5.05.0* 1 1* )0( − = − = − ∏∑ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −× ×∝ t t t t t t t tt hjtr vppp ωλ σyrrς ωθyyθωh 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0. 05 0. 1 0. 15 0. 2 0. 25 0. 3 0. 35 0. 4 0. 45 0. 5 0. 55 0. 6 0. 65 0. 7 0. 75 0. 8 0. 85 0. 9 0. 95 1 a non-standard b beta c uniform d exponential bayesian analysis of the box-cox transformation in stochastic volatility models 85 the posterior probability density function is used to make inference about the parameters and latent variables. 3. empirical results we consider ten international stock market indices, namely the s&p 500, nasdaq 100, djia (for the us), nikkei (for japan), the cac 40 (for france), the dax (for germany), the ftse 100 (for the uk), wig 20 (for poland), hang seng (for china), sptse 60 (for canada). the data set consists of the daily closing quotations of the stock market indices from january 2001 (or 2002) until february (or march) 2009 (see table 1). basic descriptive characteristics of the daily price ratios are presented in table 1. all series of xt/xt-1 ratio exhibit strong kurtosis, and they have highly nonnormal (truncated by zero) empirical distributions. table 1. sample characteristics for the data sets used time series (xt/xt-1 ratio of:) average std. dev. kurtosis period from: to: # obs. t wig 20 1.0000 0.0162 4.9800 02.01.2001 – 13.02.2009 2035 s&p 500 0.9998 0.0139 13.3286 03.01.2002 – 06.03.2009 1805 nikkei 225 0.9999 0.0163 11.3553 07.01.2002 – 06.03.2009 1760 ftse 100 0.9999 0.0137 10.9021 03.01.2002 – 06.03.2009 1813 dax 1.0000 0.0169 8.6642 03.01.2002 – 06.03.2009 1825 nasdaq 100 1.0000 0.0178 7.8639 03.01.2002 – 06.03.2009 1808 cac 40 0.9998 0.0159 9.7031 03.01.2002 – 06.03.2009 1838 sptse 60 1.0001 0.0132 14.2830 03.01.2002 – 06.03.2009 1798 hang seng 1.0002 0.0164 15.0382 03.01.2002 – 06.03.2009 1789 djia 0.9999 0.0130 12.5726 02.01.2001 – 13.02.2009 2039 note: the data were downloaded from the website http://finance.yahoo.com. in table 2 we present the posterior means and standard deviations (in parenthesis) of the parameters, in the case of the ar(1)-fcsv model with the uniform prior for λ on [0, 1]. our posterior results are obtained in gauss 9.0 using mcmc methods: metropolis-hastings within the gibbs sampler (see, e.g. pajor 2003 and jacquier et al., 2004 for detail).2 first, for more series the autoregressive parameters seem to be insignificantly different from zero. the posterior distributions of δ and ρ1 are located close to zero. second, all indices have persistent volatility as shown by φ the lowest posterior mean is 0.927 (for the wig20 index), the highest one is 0.97 (for nasdaq). it means that the halflife of shock to volatility, hl = ln(0.5)/ln(φ), is equal to about 9 days for the wig20 index and 20 days for the nasdaq index. we observed that the nasdaq index exhibits a lower variability of volatility as shown by the precision, σh-2. as regards the leverage effect parameter, ρ, the posterior means of ρ are negative, from -0.15 for the wig20 index to -0.62 for the cac40 index. 2 the results are obtained using 100 000 burnt-in and 1000 000 final gibbs passes. anna pajor 86 the parameter ρ is estimated precisely with a standard deviation around 0.068. almost all the posterior mass of ρ is in the negative region. thus, the leverage effect is strong for all indices excluding the wig20 index, for which it is significantly lower. the posterior means of the degrees of freedom are between 16 (for the hang seng index) and 39 (for the ftse 100 index). the hang seng index has the lowest posterior mean of degrees of freedom of the student-t distribution. for the remaining indices the posterior mean of v is above 23, indicating that the normal conditional distribution would not be strongly rejected by the data. table 2. posterior means and standard deviations (in parenthesis) of the parameters of the ar(1)-fcsv model, in the case of λ ~ u[0, 1] parameter wig 20 s&p 500 nikkei 225 ftse 100 dax nasdaq 100 cac 40 sptse 60 hang seng djia δ1∗10 4 4.74 4.74 6.52 5.43 9.70 5.77 6.59 8.68 7.20 4.14 (3.15) (1.65) (2.56) (1.63) (2.15) (2.63) (1.99) (1.78) (2.37) (1.60) ρ1 0.027 -0.095 -0.034 -0.092 -0.059 -0.070 -0.080 -0.060 0.005 -0.074 (0.023) (0.023) (0.024) (0.024) (0.023) (0.024) (0.023) (0.024) (0.023) (0.022) γ -0.622 -0.332 -0.429 -0.357 -0.328 -0.297 -0.335 -0.493 -0.408 -0.348 (0.119) (0.048) (0.068) (0.052) (0.049) (0.048) (0.047) (0.074) (0.069) (0.051) φ 0.928 0.965 0.951 0.962 0.963 0.966 0.963 0.948 0.955 0.963 (0.014) (0.005) (0.008) (0.006) (0.006) (0.006) (0.005) (0.008) (0.008) (0.005) σh -2 22.233 16.972 17.711 13.728 15.277 25.790 15.973 13.672 15.958 17.871 (5.675) (2.809) (3.395) (2.123) (2.592) (5.279) (2.491) (2.447) (3.156) (2.996) ρ -0.153 -0.607 -0.55 -0.578 -0.612 -0.492 -0.62 -0.484 -0.372 -0.55 (0.081) (0.063) (0.065) (0.061) (0.059) (0.081) (0.063) (0.067) (0.075) (0.064) ν 23.04 27.20 38.47 39.75 31.77 30.34 31.33 37.85 16.04 26.99 (9.96) (11.45) (14.52) (14.61) (12.76) (12.06) (12.55) (14.55) (7.22) (11.17) λ 0.397 0.472 0.504 0.448 0.405 0.387 0.399 0.497 0.401 0.433 (0.255) (0.265) (0.266) (0.263) (0.256) (0.253) (0.255) (0.266) (0.256) (0.262) table 3. posterior means and standard deviations (in parenthesis) of λ, in the case of the exponential prior distribution for λ (d) parameter wig 20 s&p 500 nikkei 225 ftse 100 dax nas-daq 100 cac 40 sptse 60 hang seng djia λ 0.431 0.666 0.801 0.579 0.45 0.412 0.44 0.798 0.437 0.531 (0.371) (0.556) (0.624) (0.497) (0.382) (0.35) (0.377) (0.652) (0.376) (0.453) finally, we consider the posterior evidence regarding the box-cox transformation parameter. figure 2 shows the prior and posterior distributions for λ in the case of the wig20 index. we see from the graphs that the prior distribution for λ strongly affects the posterior distribution for this parameter, e.g., a u-shaped prior distribution implies the u-shaped posterior distribution. in the case of the uniform prior for λ on the interval [0; 1], for most stock indices (considered here) the posterior mean is smaller than the prior mean, but the dispersion of posterior distribution is close to that of the prior distribution (in the case of c, bayesian analysis of the box-cox transformation in stochastic volatility models 87 the prior mean is equal to 0.5, the prior standard deviation is equal to 0.288). even though the prior distribution is symmetrical, in the majority of cases the posterior distributions are asymmetrical. the values of λ from the interval [0, 0.5] are more probable a posterior than those from [0.5, 1] (see the quantiles of the posterior distributions of the box-cox transformation parameter in table 4). a – non-standard b – beta distribution c – uniform distribution on the interval [0, 1] d – exponential sistribution figure 2. prior (solid line) and posterior (bars) distributions for λ (the wig20 index) in the case of the non-standard prior distribution for λ considered in (a), except for the nikkei and sptse 60 indices, the posterior medians are below 0.1, but the probability that λ is less than 0.9 is not zero. in table 5 we present the posterior probabilities that λ is in the interval [0, 0.01] and in the interval [0.99, 1]. except for the nikkei index, the values of λ from the interval [0, 0.01] are more probable a posterior than those from the interval [0.99, 1]. thus the data transformations which are close to the log-return are more probable a posterior than those which lead to the simple return. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0. 09 0. 18 0. 27 0. 36 0. 45 0. 54 0. 63 0. 72 0. 81 0. 9 0. 99 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0. 09 0. 18 0. 27 0. 36 0. 45 0. 54 0. 63 0. 72 0. 81 0. 9 0. 99 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0. 09 0. 18 0. 27 0. 36 0. 45 0. 54 0. 63 0. 72 0. 81 0. 9 0. 99 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0. 18 0. 36 0. 54 0. 72 0. 9 1. 08 1. 26 1. 44 1. 62 1. 8 1. 98 anna pajor 88 table 4. posterior quantiles for λ quantile of order wig 20 s&p 500 nikkei 225 ftse 100 dax nasdaq 100 cac 40 sptse 60 hang seng djia 0.05 0.003 0.004 0.005 0.004 0.003 0.003 0.003 0.005 0.003 0.004 0.25 0.016 0.023 0.033 0.021 0.016 0.015 0.016 0.031 0.016 0.019 a 0.5 0.042 0.082 0.824 0.062 0.044 0.039 0.042 0.665 0.042 0.055 0.75 0.135 0.957 0.972 0.937 0.161 0.111 0.144 0.97 0.133 0.914 0.95 0.988 0.995 0.996 0.993 0.989 0.985 0.988 0.996 0.988 0.992 0.05 0.005 0.011 0.016 0.008 0.005 0.005 0.004 0.013 0.004 0.007 0.25 0.088 0.163 0.217 0.131 0.095 0.081 0.088 0.194 0.087 0.116 b 0.5 0.277 0.433 0.516 0.375 0.294 0.26 0.28 0.489 0.278 0.343 0.75 0.573 0.746 0.806 0.693 0.597 0.545 0.575 0.787 0.576 0.658 0.95 0.939 0.981 0.987 0.973 0.945 0.926 0.939 0.985 0.938 0.963 0.05 0.042 0.063 0.076 0.055 0.044 0.04 0.042 0.072 0.043 0.051 0.25 0.185 0.252 0.286 0.229 0.192 0.178 0.186 0.278 0.188 0.215 c 0.5 0.362 0.462 0.506 0.429 0.374 0.35 0.366 0.496 0.369 0.409 0.75 0.585 0.686 0.723 0.656 0.597 0.571 0.589 0.716 0.591 0.636 0.95 0.864 0.914 0.929 0.901 0.871 0.855 0.865 0.926 0.867 0.891 0.05 0.034 0.058 0.077 0.047 0.036 0.032 0.035 0.071 0.034 0.044 0.25 0.158 0.252 0.322 0.213 0.167 0.153 0.161 0.307 0.16 0.196 d 0.5 0.332 0.52 0.652 0.445 0.351 0.32 0.341 0.63 0.337 0.411 0.75 0.598 0.932 1.13 0.805 0.629 0.574 0.615 1.122 0.61 0.74 0.95 1.172 1.771 2.037 1.577 1.208 1.11 1.186 2.095 1.186 1.435 note: prior distributions for λ: a – non-standard, u-shaped on the interval [0, 1], b – beta distribution, c – uniform distribution on [0, 1], d – exponential distribution. table 5. posterior results for λ case index: wig 20 s&p 500 nikkei 225 ftse 100 dax nasdaq 100 cac 40 sptse 60 hang seng djia u=pr(λ<0.01|y) 0.173 0.127 0.102 0.142 0.172 0.178 0.172 0.105 0.174 0.149 a v=pr(λ>0.99|y) 0.041 0.083 0.109 0.068 0.044 0.038 0.042 0.100 0.040 0.060 u/v 4.238 1.531 0.934 2.088 3.942 4.693 4.111 1.052 4.370 2.492 u=pr(λ<0.01|y) 0.077 0.050 0.040 0.060 0.073 0.081 0.079 0.045 0.081 0.064 b v=pr(λ>0.99|y) 0.018 0.035 0.044 0.030 0.019 0.016 0.018 0.039 0.017 0.024 u/v 4.294 1.418 0.913 1.999 3.829 4.924 4.402 1.137 4.769 2.656 u=pr(λ<0.01|y) 0.011 0.007 0.006 0.008 0.011 0.012 0.011 0.006 0.011 0.009 c v=pr(λ>0.99|y) 0.003 0.005 0.006 0.004 0.003 0.002 0.003 0.006 0.003 0.004 u/v 4.485 1.546 0.933 2.084 3.733 5.293 4.222 1.027 4.184 2.604 note: prior distributions for λ: a – non-standard, u-shaped on the interval [0, 1], b – beta distribution, c – uniform distribution on [0, 1]. it is important to stress that even though the prior distribution of λ has a strong effect on the posterior distribution of λ, it does not affect the posterior distribution of the remaining parameters. thus in table 3 we present the posterior characteristics only of λ, obtained in the ar(1)-fcsv model with the exponential bayesian analysis of the box-cox transformation in stochastic volatility models 89 distribution for the box-cox transformation parameter. although the prior mean is equal to 1, for all series the posterior mean is less than 1. finally, in table 6 we present the results of the formal bayesian model comparison. we consider three ar(1)-fcsv models: with, respectively, λ = 0 (m1), λ = 1 (m2), and λ ~ u(0, 1) (m3). if λ = 1, the relation (1) is linear in the simple returns. if λ = 0, it is linear in the logarithmic returns. to obtain the marginal data densities we use the newton and raftery method (see newton and raftery 1994). the newton and raftery estimator is quite stable for all our models. the drawback of this method in the fcsv models is that the models differ from one another by quite a few orders of magnitude. for all series, assuming equal prior model probabilities, the ar(1)-fcsv model with λ = 0 (log-returns) is more probable a posterior than with λ = 1 (simple returns). only for the djia index, the ar(1)-fcsv model with the uniform prior distribution of λ is quite a few orders of magnitude better than that with λ = 0. table 6. posterior probabilities (under equal prior model probabilities) and marginal data densities of the observation vector y in mi model (based on the newton – raftery method) index m1: λ = 0 m2: λ = 1 m3: 0 < λ < 1* p(y|m1) p(y|m2) p(y|m3)* wig 20 0.9754 0.0000 0.0246 2.4⋅10-170 1.8⋅10-176 6.0⋅10-172 s&p 500 0.9995 0.0000 0.0005 2.8⋅10-176 2.0⋅10-186 1.4⋅10-179 nikkei 225 0.9997 0.0000 0.0003 1.5⋅10-188 7.5⋅10-196 4.2⋅10-192 ftst 100 0.9931 0.0069 0.0000 4.1⋅10-161 2.8⋅10-163 4.3⋅10-177 dax 1.0000 0.0000 0.0000 3.4⋅10-122 2.4⋅10-141 1.3⋅10-129 nasdaq 100 1.0000 0.0000 0.0000 9.3⋅10-103 1.6⋅10-108 1.7⋅10-113 cac 40 1.0000 0.0000 0.0000 5.5⋅10-192 2.4⋅10-202 3.3⋅10-197 sptse 60 1.0000 0.0000 0.0000 3.3⋅10-45 2.5⋅10-53 1.1⋅10-55 hang seng 1.0000 0.0000 0.0000 3.3⋅10-50 5.2⋅10-55 2.9⋅10-55 djia 0.0000 0.0000 1.0000 2.3⋅10-202 7.8⋅10-204 2.6⋅10-195 note: *the results are obtained in the ar(1)-fcsv model with the uniform prior for λ on the interval (0, 1). 4. conclusions the paper presents the stochastic volatility models with the box-cox transformation of financial time series. the widely used logarithmic and simple returns are nested into the box-cox transformation by setting λ = 0 and λ = 1, respectively. using daily data, we show that in the stochastic volatility model with fat tails and correlated errors, the posterior distribution of the box-cox transformation parameter strongly depends on the prior assumption about this parameter. our empirical results show that in the majority of cases the values of anna pajor 90 λ close to 0 are more probable a posteriori than the ones close to 1. the formal bayesian model comparison indicates that the box-cox transformation with λ = 0 (log-return) is preferred by the data in the fcsv model. however, the posterior distributions of λ show that the simple returns are not completely inappropriate. references bauwens, l., lubrano, m. (2002), bayesian option pricing using asymmetric garch models, journal of empirical finance, 9, 321–342. campbell, j.y., lo, a.w., mackinlay, a.c. (1997), the econometrics of financial markets, princeton university press, chichester 1997. clark, p.k. (1973) a subordinated stochastic process model with finite variance for speculative prices, econometrica, 41, 135–155. duan, j.-c. (1999), conditionally fat-tailed distributions and the volatility smile in options, working paper, http://www.bm.ust.hk/~jeduan. hafner, c.m., harwartz, h. (1999), option pricing under linear autoregressive dynamics, heteroskedasticity, and conditional leptokurtosis, journal of empirical finance, 8(1), 1–34. härdle, w., hafner, c.m. (2000), discrete time option pricing with flexible volatility estimation, finance and stochastics, 4(2), 189-207. jacquier, e., polson, n., rossi, p. (2004), bayesian analysis of stochastic volatility models with fat-tails and correlated errors, journal of econometrics, 122, 185–212. newton, m.a., raftery, a.e. (1994), approximate bayesian inference by the weighted likelihood bootstrap (with discussion), journal of the royal statistical society b 56, 3–48. pajor, a. (2003), procesy zmienności stochastycznej sv w bayesowskiej analizie finansowych szeregów czasowych, czasowych (stochastic volatility processes in bayesian analysis of financial time series), doctoral dissertation published by cracow university of economics, kraków. zellner, a. (1971), an introduction to bayesian inference in econometrics, j. wiley, new york. bayesowska analiza transformacji boxa i coxa dla w modelach o zmienności stochastycznej z a r y s t r e ś c i. celem artykułu jest statystyczna analiza transformacji boxa i coxa ilorazu cen instrumentów finansowych w modelach fcsv. stosowane jest podejście bayesowskie, które pozwala zbadać, w jakim stopniu dane modyfikują wstępne przekonanie o parametrze transformacji. wyniki empiryczne pokazują, że założenia o rozkładzie a priori parametru transformacji ma istotny wpływ na kształt brzegowego rozkładu a posteriori tego parametru. jednak w większości rozważanych przypadków rozkłady te, w porównaniu z rozkładami a priori, są przesunięte w kierunku zera. zatem transformacje ilorazu cen dające wartości bliskie logarytmicznej stopie zwrotu są bardziej prawdopodobne a posteriori niż transformacje prowadzące do prostej stopy zwrotu. s ł o w a k l u c z o w e: transformacja boxa i coxa, model sv, wnioskowanie bayesowskie. microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 aneta włodarczyk, marcin zawada politechnika częstochowska the use of weather variables in the modeling of demand for electricity in one of the regions in the southern poland a b s t r a c t. the main objective of the paper is the verification of usefulness of the arfimafigarch class models in the description of tendencies in the energy consumption in a selected region of the southern poland taking into consideration weather variables. k e y w o r d s: weather variables, the arfima-figarch class model, weather risks. 1. introduction the companies specializing in the production or distribution of power are particularly exposed to the weather risk, understood as the possibility of change in the financial result of a company caused by the variability of daily weather conditions: air temperature, rainfall and snowfall, sun light exposure, wind speed and humidity. furthermore, the inability to store the power leads to the necessity of a precise measurement of the future demand for electricity by the companies specialising in its sale. therefore the search for statistical and econometrical tools enabling the modeling and forecasting of the demand for power in varying weather conditions has become such an important research problem. 2. review of research in the scope of the impact of the climatic factors on the electrical energy consumption identification and measurement of the weather risk are connected with the necessity to isolate from the observable electrical energy consumption a part which is sensitive to the effects of climatic factors. while analyzing historical time series relating to the demand for electrical energy, containing daily, weekly or monthly data from a dozen years, one may notice a strong long-term tendency, whose occurrence has been affected by social, demographic and economic aneta włodarczyk, marcin zawada 100 factors. in order to isolate the demand for electrical energy which is sensitive to weather factors, various ways of data filtration can be used. in empirical research on modeling the above relation the following methods are used: 1. the method of the decomposition of time series into the trend component, the calendar component, the periodic component and the irregular component (moral-carcedo, vicéns – otero, 2005; bessec, fouquau, 2008): , 1 ,0 ∑ = ++++= m j tttaug j jt fewdite κδαα (1) or ,33 2 210 ttt feyttte +++++= δαααα (2) where et is the demand for electricity, iaug,t is a dummy variable taking the value 1 if the observation of the demand corresponds to the month of august, wdt is the variable describing working day effect, yt is the seasonal unadjusted production in total manufacturing at time t, fet is the electricity demand with the deterministic component filtered out. 2. the index-related equalization of the long term tendencies which do not result from weather conditions in terms of the demand for electrical energy (sailor, muñoz, 1997; valor, meneu, casellles, 2001): , j ij ij e e msvi = (3) , jk ijk ijk e e dsvi = (4) where msviij is the index value for month i in year j, eij is the monthly electricity consumption for month i in year j, je is the monthly average electricity load for year j, dsviijk is the index value for day i of week j of year k, eijk is the electricity consumption for this same day, jke is the daily average electricity load for week j in year k. after the estimation of the demand for electrical energy which is sensitive to climatic factors, the strength and nature of the relations between the weather variables and the electrical energy consumption should be assessed. different types of models were used in the previous research: 1. pardo, meneu, valor (2002) estimated the following model: , )()( 11 1 6 1 10 tt k ktk i itittt hm dcddlhddltle εϖϕ δγβαα +⋅++ ++++= ∑ ∑ = = (5) the use of weather variables in the modeling of demand for electricity … 101 ,)1( 33 9 9 2 21 ttlll ξεφφφ =−−−− k (6) where dit is dummy variable for daily data (d1t = 1 for monday, d1t = 0 for other days of the week), mit is dummy variable for monthly data (m1t = 1 for january, m1t = 0 for other months of the year), ht is dummy variable for holidays (ht = 1 for holidays, ht = 0 for other days of the year). 2. moral-carcedo, vicéns – otero (2005) have constructed the following models in order to describe the non-linear relation between the energy consumption and air temperature: a) switch regression model ,tsttstt tmpfe εβμ ++= (7) b) threshold regression model [ ] [ ] ( ) , )pr( pr )pr( 2 1 ∑ = == == st ttt ttt tt ssdff isisdff is ψ (8) ⎪ ⎩ ⎪ ⎨ ⎧ >++ >=>++ <++ = .2 ,12 ,1 3 2 1 thtmptmp thtmpthtmp thtmptmp df ttt ttt ttt t εημ εγμ εβμ (9) in a warmer climate the relation between air temperature and energy consumption has a non-linear character with the form resembling letter u (valor, meneu, casellles, 2001; sailor and muñoz, 1997); i.e. the maximum demand for energy is observed at the low and high temperatures. the introduction of the hdd and cdd indices, which separate the winter and summer seasons, enables better quantification of the analysed relation. furthermore, the research by other authors (bessec and fouquau, 2008; among others) proves that in the climate zone, which includes poland, the effect of a bigger demand for energy in the summer season connected with the use of air conditioning equipment is not significant. 3. statistical analysis of characteristics of analysed time series for the purposes of this paper the authors used information concerning: power consumption (in kwh), air temperature (in oc) and wind speed (in m/s) in one of the regions in the southern poland in the period from september 1, 2005 to june 30, 2008. in the analysis and further calculations daily data was used in the following way: − the hdd index (heating degree days) was calculated on the basis of the relation: hdd = max (0,180c – ti),where ti – average daily air temperature on day number i; 102 − the c lation the f weather v cycles of grouping figure 1. c c a in ord consumpt were dete one o (with var power. m 2009). w this paper (5, 10, 20 test statis mcleod 1 table tion of the thors (włod dd index (c n: cdd = ma figure 1 pres variables ind f varying leng of variances changes in th cdd index, a a region in the der to analys tion and par ermined (cf. t of the pheno rious lengths meteorologica while focusin r the authors 0, 30, 40, 50) stics were sig and li test 1 presents the tests used in th darczyk, zawad aneta wło cooling degr ax (0,ti 180 ents the dail dicating the gth) in the a s. he daily powe air temperatur e southern pol se the charac rticular weat table 1). omena chara s of the cycl al data have ng on the dem s analysed th ) order with t gnificant at t are, in turn results of the l he paper, whose da, 2006, p. 313 odarczyk, marc ee days) was c). ly power con existence o average of an er consumptio re (middle pan land from sep cteristics of ther variable acteristic for les: daily, w a similar ch mand for pow he significanc the use of the the 0.001 sig n, the basis f ljung-box test e results are sh 3-321). cin zawada s calculated nsumption an f similar tim nalysed proce on, the hdd i nel) and wind ptember 1, 200 the distribut es, the basic r the power weekly, annu haracteristic wer and the ce of autocor e ljung-box gnificance le for the conc only for the or hown in table on the basis nd values of me structure esses and the index (upper d speed (lowe 05 till june 30 tions of the c descriptive market is p al) in the de (cf. benth a variables pr rrelation to t x test1. all the vel. the resu clusion that rder 30. a detai 1 was presente of the ref particular (for time e effect of panel), the er panel) in 0, 2008 electricity e statistics periodicity emand for and benth, esented in the fiftieth e obtained ults of the there is a iled descripd by the au strong co arch ef table 1. d st m standar mi ma ske ku llk note: symbo air te consumpt nificantly figure 2. c t w the a a signific in the sha cification process o the use of wea orrelation of ffect. descriptive sta tatistic mean rd deviation nimum aximum ewness urtosis -b(30) b2(30) kpss ol ** indicated the emperature a tion, wherea y lower (figu correlative gr tion, the hdd wind speed an analysis of c ant stage of ape of analys n of equation of demand fo ather variables f squares of atistics for the consumption 4532400.0000 504420.0000 3204400.0000 5657800.0000 -0.2376 -0.4689 10188.3000** 10402.1000** 1.1764** significance of th and the hdd s the influen ure 2). raphs presentin d index and a nd the cdd in characteristic econometric sed variables ns of conditio or power. in s in the modelin f given serie analysed vari n temperat 0 7.5579 0 8.4172 0 -20.208 0 26.0420 -0.3741 -0.3992 * 19372.20 * 16759.90 1.0654* he result at the 0. d index have nce of wind f ng the dispers air temperatur ndex (lower pa cs of time se c modelling, s brings effec onal mean a other words ng of demand f es, which is iables ture w 9 3.4 2 1.5 80 0.4 0 11.9 1 0.9 2 1.3 00** 539.6 00** 498.9 ** 0.87 01 level. calcula e the biggest force and of sion of points e (upper pane anel) eries made i as the identi cts in the for nd the condi , it enables t for electricity … characterise ind 495 1 5052 583 9170 3 086 241 6100** 191 9290** 121 796** 1 ations made in g@ t impact on the cdd ind for the energy el), energy co n this part c fication of re rm of the rele itional varian the construct … 103 ed by the hdd 10.6700 8.0799 0.0000 38.2080 0.5307 -0.3965 103.6000** 120.5000** 1.0585** @rchtm. the power dex is sig y consumpnsumption, constitutes egularities evant spence of the tion of the aneta włodarczyk, marcin zawada 104 congruent econometric model according to the concept of z. zieliński (zieliński, 1984). 4. estimation and verification of models of the demand for electricity at the first stage of the research the authors identified a deterministic trend connected with the impact of demographic, economic and social factors on the demand for power in a region in the southern poland. from the estimated various models of trend for the daily power consumption the author selected a third degree polynomial trend, taking into account the value of determination coefficient and the significance of the estimates of structural parameters of the models. due to the object of the research, which was the description of the relation between the impact of weather factors on the energy consumption, in the equation of demand for power the author included also the analysed weather variablesgiving them a dynamic structure. additionally the equation includes also dummies, whose task is to describe a weekly periodocity, annual seasonality and holiday effect in the shaping of demand for energy. finally the authors proposed the following specification of the model of energy consumption, expressed in logarithms: , ln 6 1 11 1 13121 5 0 5 0 11 3 3 2 210 ∑ ∑ ∑ ∑ = = +− = = −+−+ ++++++ ++++++= i j ttttjtjiti k k ktkktkt usssmd windtempttte κκκϕδ βλαααα (10) where: dit – dummy variable equals one for the day i, and zero otherwise, mjt – dummy variable equals one for the month j , and zero otherwise, st – dummy variable is equal to one for the holiday, and zero otherwise, st-1 – dummy variable equals one for the day preceding the holiday, and zero otherwise, st+1 – dummy variable is equal to one for the day following the holiday, and zero otherwise. the models of power consumption without weather variables, including the impact of air temperature, wind, as well as the hdd index were estimated with the ols method. on the basis of information criteria, tests for model residuals and the parameter significance test the authors selected the following models with the weather variables (table 2). the results of parameter estimation of the model (10) indicate that the current and one period lagged air temperature as well as one day lagged wind force have the significant impact on the power consumption in a given day. moreothe use of weather variables in the modeling of demand for electricity … 105 ver, estimates of parameters which stand by dummy variables and model periodicity in the weekly cycle on demand on energy indicate that on mondays, saturdays and sundays energy consumption is lower than the average level and higher in the other days of the week. in the case of dummy variables associated with monthly seasonal effects all estimates of parameters are significant and negative for summer months (may, june, july, august, and september). it is connected with the impact of seasonal factors, such as, air temperature, length of the day, level of sun light exposure on the demand for energy. all parameters standing by dummy variables associated with holidays and neighbouring days are statistically significant and negative which indicates that the energy consumption on holidays and neighbouring days is significantly lower in comparison with regular working days (as indicated by results of the wald test for equality of parameters). table 2. estimates of the parameters of the model (10) parameter coefficient p-value parameter coefficient p-value α0 15.3739 0.0000*** φ1 0.0579 0.0000*** α1 -0.0002 0.0000*** φ2 0.0261 0.0000*** α2 4.336e-07 0.0000*** φ3 0.0262 0.0000*** α3 -1.729e-010 0.0089*** φ4 0.0073 0.0667** λ1 -0.0009 0.0358** φ5 -0.0568 0.0000*** λ2 -0.0036 0.0000*** φ6 -0.0830 0.0000*** β2 0.0024 0.0040*** φ7 -0.0491 0.0000*** δ1 -0.0067 0.0235** φ8 -0.0478 0.0000*** δ2 0.0293 0.0000*** φ9 -0.0185 0.0000*** δ3 0.0392 0.0000*** φ10 0.0199 0.0000*** δ4 0.0408 0.0000*** φ11 0.0500 0.0000*** δ5 0.0404 0.0000*** κ1 -0.1845 0.0000*** δ6 -0.0059 0.0442* κ2 -0.0369 0.0000*** adjusted r2 0.88326 κ3 -0.0859 0.0000*** aic -3785.4470 bic -3652.0610 hannan-quinn -3734.8310 note: the symbol *** indicates the significance of the result at the 0.001 level. calculations made in gretl. in order to identify the autocorrelation effect, box-pierce test (the lag level: 5, 10, 20, 50) has been used for residuals of model (10) – all test statistics indicate for the significant autocorrelation in residuals. to verified the arch effect, two different test have been used: engle test (for 1, 2, 5, 10, and 20 lags) for residuals and box-pierce test for squared residuals (level of lag: 5, 10, 20, 50). similarly, in this case all test statistics indicate for the significant autocorrelation in squared residuals. using the geweke-porter-hudak test, the long memory effect in residuals and squared residuals of the electricity demand aneta włodarczyk, marcin zawada 106 model has been captured2. with regard to the verification of residuals properties, the model of arfima (p, d, q)-figarch (p, d, q) class can be used for description of correlation between weather variables and energy consumption3: ,)()()( ttt d bub εθμφ =−δ (11) ),1,0(~z , t iidhz ttt ⋅=ε (12) ),)]((1[)( 2 1 , 2 tt r k tkkt d hbxb −−++=δ ∑ = εβωωεϕ (13) where: jj j dd b j d b )1()1( 0 −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =−=δ ∑ ∞ = filter difference of order d, ss s dd b s d b )1()1( 0 −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =−=δ ∑ ∞ = filter difference of order d, -1< d < 0,5, 0 < d < 1, 0 1 , >+∑ = r k tkk xωω , p p bbb φφφ −−−= ...1)( 1 , q q bbb θθθ +++= ...1)( 1 , q q bbb ϕϕϕ −−−= ...1)( 1 , p p bbb βββ ++= ...)( 1 . introduction to the equation of conditional variance of regressor, which is a variability of weather factors or dummy variables which model periodicity of variance enables to connect dynamics of variability of energy consumption with variability of weather conditions of different structure of energy consumers in working days and holidays. in the current framework the following descriptive variables have been introduced to the equation of conditional variance of the process:4 dummy variables which model the effect of week day, dummy variables which model the month effect in the year, dummy variables which model holidays, square of increment of daily average temperature in subsequent days, square of increment of wind power in subsequent days. orders of models arfima(p,d,q)-figarch(p,d,q) were chosen on the basis of information criteria and significance of the model parameters. the best models in this class are presented in table 3. 2 because of limited size of this framework, results of conducted tests for model residuals have not been presented. 3 in order to guarantee stationarity of analysed models of time series, it is assumed that roots of polynomial 0(b) ,0)( == ϕφ b lie outside the unit circle (preś, 2007, p. 206; laurent, 2007, p. 55–74). 4 because of large number of model parameters and problems associated with estimation, proposed variables were separately attached to the equation of conditional variance. table 3. parameter estimates of arfima(1,1)-garch(1,1) models parameter arma(1,1)-garch(1,1)+r arfima(1,1)-garch(1,1)+r arfima(1,1)-garch(1,1) cst(m) 0.001950 [0.4575] 0.0020 [0.5154] 0.0023 [0.4644] d-arfima 0.0426 [0.6560] 0.04460 [0.6341] ar(1) 0.7198 [0.0000] 0.6904 [0.0000] 0.6774 [0.0000] ma(1) -0.0929 [0.0925] -0.1104 [0.0694] -0.0983 [0.1168] cst(v) 0.0004 [0.0000] 0.0003 [0.0000] 0.0003 [0.0032] dif(temp) 0.87e-5 [0.0000] 0.55e-5 [0.0000] arch1 0.2611 [0.0000] 0.2091 [0.0000] 0.2314 [0.0049] garch1 0.2709 [0.0032] 0.4439 [0.0000] 0.3931 [0.0299] skewness -0.1157 [0.0173] -0.1217 [0.0130] -0.1202 [0.0165] df-student 5.2049 [0.0000] 5.0735 [0.0000] 5.1046 [0.0000] aic -4.3980 -4.3939 -4.3926 sc -4.3549 -4.3460 -4.3494 h-q -4.3816 -4.3757 -4.3762 shibata -4.3981 -4.3941 -4.3927 note: p-values have been presented in the brackets. calculations made in g@rchtm. table 4. summary statistics for model residuals of models arfima-garch statistic arma(1,1)garch(1,1)+r arfima(1,1)garch(1,1)+r arfima(1,1)garch(1,1) q (box-pierce) statistics on standardized residuals q(5) 4.76412 [0.1899] 4.9544 [0.1752] 4.7439 [0.1915] q(10) 15.8704 [0.0443] 15.9747 [0.0427] 15.5973 [0.0485] q(20) 29.4930 [0.0427] 30.0496 [0.0370] 30.0668 [0.0368] q(50) 57.5181 [0.1633] 57.7321 [0.1586] 58.5882 [0.1407] q (box-pierce) statistics on squared standardized residuals q(5) 0.9382 [0.8162] 1.4152 [0.7020] 1.4080 [0.7037] q(10) 3.9715 [0.8597] 4.8275 [0.7758] 5.0532 [0.7519] q(20) 6.1725 [0.9954] 7.2110 [0.9882] 7.2413 [0.9879] q(50) 27.6562 [0.9919] 27.9235 [0.9909] 27.3902 [0.9927] engle’s lm arch test arch(1-2) 0.2695 [0.7638] 0.4347 [0.6476] 0.4418 [0.6430] arch(1-5) 0.1866 [0.9677] 0.2834 [0.9224] 0.2819 [0.9231] arch(1-10) 0.3822 [0.9547] 0.4635 [0.9137] 0.4853 [0.9003] nyblom stability test nyblom statistic for parameter vector 1.4860 stability 1.6558 stability 1.2591 stability nonstability parameter by nyblom test nonstability parameter ma(1) nonstability parameter : d-arfima, ar(1), ma(1) nonstability parameters : d-arfima, ma(1) sign bias test sb 1.6800 [0.0930] 1.7303 [0.0836] 1.7104 [0.0872] nsb 1.2895 [0.1972] 1.0867 [0.2772] 1.1378 [0.2552] the joint test 4.3732 [0.2239] 5.3313 [0.1491] 5.1391 [0.1619] adjusted pearson goodness-of-fit test empirical distribution is congruent with theoretical distribution empirical distribution is congruent with theoretical distribution empirical distribution is congruent with theoretical distribution note: p-values have been presented in the brackets. calculations made in g@rchtm. aneta włodarczyk, marcin zawada 108 when model estimates are assessed with regard to its quality the following results of tests conducted on its standardized residuals should be analysed: verification of uncorrelated standardized residuals (box-pierce test), lack of arch effect (box-pierce test for squared residuals and engle’s test), testing parameters stability in the model (nyblom test), lack of diversity of influence made by negative and positive innovations on the level of variability (sb test), lack of diversity of influence made by large and small negative (positive) innovations on the variability (nsb test), fit of a distribution of empirical standardized residuals with assumed distribution (pearson’s chi-square goodness-of-fit test).5 each time, the introduction of garch structure with conditional skewed distribution of t-student has been made, the result was that the effect of grouping variances, which was present in residuals of arfima model has been eliminated. in the case of different estimated models of arfima-figarch class the estimate of fractional integration parameter d in conditional variance equation was statistically insignificant. even when dummy variables which model the effect of week day, month, and holidays in the equation of conditional variance of the process were considered, the characteristics of the model were not improved significantly. next, the authors introduce the variability of the weather factors as the regressor to the conditional variance equation of the electricity demand. the result is that the autocorelation effect, which is found in standardized residuals of arfima-garch model, has been decreased or eliminated. 5. summary demonopolization in energy industry in poland has forced companies from energy industry to work out and implement internal procedures of risk management, because the risk is present in energy trade. companies from this industry more and more often use weather derivatives to hedge against effects of weather risk, because this activity allows to make financial results independent of changing weather conditions. analysis of influence of particular weather factors on energy consumption conducted by the authors concerned only a particular region of southern poland. unfortunately, polish conditions does not allow straight-forward access to these type of data because of the high cost of data purchase, whereas in many countries, databases concerning weather variables are available for free on web pages of meteorological stations of national entities which collect this type of data. introduction to the equation of conditional variance of the regressor, which is a variability of average daily temperature increase in the coming days (dif(temp)) enables to connect the dynamics of volatility of energy consump 5 all above-mentioned methods have been described in the econometric literature (doman, doman, 2004, p. 295–308; laurent, 2007, p. 41–46). the use of weather variables in the modeling of demand for electricity … 109 tion with the volatility of weather conditions. moreover, the assessment of the arfima-garch models on the basis of the residuals of model (10) made it possible to assess the conditional volatility of the process of demand for electrical energy. with the use of conditional volatility one can measure the volatility of the demand for electrical energy, i.e. the risk related to unpredictable change in the energy consumption under the influence of e.g. changing weather conditions. while extending analysis of the impact of weather factors on the functioning of the power energy industry branch company, one should apply the value at risk methodology to measure the weather risk. such approach will make companies dealing with the energy production and sales aware of the potential losses they may suffer as a result of unexpected change of weather factors. references benth, f. e., benth, j. s. (2009), dynamic pricing of wind futures, energy economics, 31, 16–24. bessec, m., fouquau, j. (2008), the non-linear between electricity consumption and temperature in europe: a threshold approach, energy economics, 30, 2705–2721. doman, m., doman, r. (2004), ekonometryczne modelowanie dynamiki polskiego rynku finansowego (econometric modeling of the dynamics of the polish financial market), wydawnictwo ae w poznaniu, poznań. laurent, s. (2007), estimating and forecasting arch models using g@rch™5, timberlake consultants ltd, london. moral-carcedo, j., vicéns – otero, j. (2005), modelling the non-linear response of spanish electricity demand to temperature variations, energy economics, 27, 477–494. pardo, a., meneu, v., valor, e. (2002), temperature and seasonality influences on spanish electricity load, energy economics, 24, 55–70. preś, j. (2007), zarządzanie ryzykiem pogodowym (management of weather risk), wydawnictwo cedewu, warszawa. sailor, d. j., muñoz, j. r. (1997), sensitivity of electricity and natural gas consumption to climate in the u.s.a. – methodology and results for eight states, energy, 22, 987–998. valor, e., meneu, v., casellles, v. (2001), daily air temperature and electricity load in spain, journal of applied meteorology, 40,1413–1421. włodarczyk, a., zawada, m. (2006), behavior of prices in the polish power exchange and european power exchanges. statistical – econometric analysis, 3rd international conference: the european electricity market eem06. challenge of the unification, warsaw, 313–321. zieliński, z. (1984), zmienność w czasie strukturalnych parametrów modelu ekonometrycznego (time variability of structural parameters in econometric model), przegląd statystyczny (statistical survey), 1/2, 135–148. zastosowaniem zmiennych pogodowych w modelowaniu zapotrzebowania na energię elektryczną w jednym z regionów polski południowej z a r y s t r e ś c i. głównym celem opracowania jest zweryfikowanie przydatności modeli klasy arfima-figarch do opisu kształtowania się zużycia energii elektrycznej w wybranym regionie południowej polski z uwzględnieniem zmiennych pogodowych. s ł o w a k l u c z o w e: zmienne pogodowe, model arfima-figarch, ryzyko pogodowe. microsoft word dem_2018_5to34.docx © 2018 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2018.001 vol. 18 (2018) 5−34 submitted march 10, 2018 issn (online) 2450-7067 accepted june 15, 2018 issn (print) 1234-3862 shravani sharma and supran kumar* dynamics of financial development and economic growth: panel data analysis for selected indian states a b s t r a c t. with the help of standard refined panel analysis techniques the present study analysed the dynamics of causal relationship between financial development and economic growth for selected indian states. mainly focusing on banking level indicators the present attempt measured the extent of financial development in the selected indian states. three major econometric techniques including panel unit root tests, cointegration tests and finally the panel error correction model have been implemented for identifying the relationship between variables. firstly, the series was tested for cross-sectional independence and then checked for the presence of unit roots. the results of both first and second generation unit root indicated an integration of order one for all the variables and a long-run relationship between financial development and respective economic growth indicators was confirmed by the pedorni’s and westerlund’s cointegration tests. the results of the present study emphasized on the critical role of credit provided by banks in the process of long run economic growth across states. apart from this the results of the study highlighted a very relevant fact that the indian economy has a lot of scope in harvesting the less financially developed areas of the states which can run rapidly on the greeny path of dynamic and sharp long term sustainable economic growth. k e y w o r d s: causality; economic growth; financial development; panel data; unit root. j e l classification: g21; c23; o40. * correspondence to: shravani sharma, faculty of management, school of business, shri mata vaishno devi university, katra, jammu & kashmir (india), e-mail: shravanisharma06@gmail.com; supran kumar, faculty of management, school of business, faculty of management, shri mata vaishno devi university, katra, jammu & kashmir (india) e-mail: suparn329@yahoo.co.in. shravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 6 introduction the pace of provincial and national economic growth of an economy is primarily depicted by positive and continuous transformation in the level of production of goods and services. the continuous evolvement in the production of goods and services directly depends upon the capability of an economy in efficient utilization of its available physical and human resources. moreover, effective aggregation of physical and human capital is basically catalysed and promoted by productive financial intermediation (fitzgerald, 2006). this accumulation requires regular mobilization of foreign and domestic savings by discovering the productive projects, meanwhile also keeping proper checks on the working of these projects (goldsmith, 1969) and here, it is pertinent to mention that the monitoring of a project is facilitated by an effective and sound financial system. financial system is considered as the mainstay of an economy. effective and conducive financial system is instrumental in providing sound and progressive business environment. however, endowment of this environment as well as financial infrastructure is indeed a crucial challenge faced by a nation. it is suggested that countries which do well in terms of financial development for constant period of time are equally productive on the front of reduction of their poverty level and generation of better infrastructure as well (barro, 1996). thus, financial development entails for the establishment and development of financial institutions which consequently promote growth and investment process. available literature on financial development and economic growth specifies that apart from managing the savings of individuals as well as groups, credit allocation in financial system plays a very prominent role since it is considered as the key for rational and inclusive economic development. modern concept of economic growth suggests that rational allocation of capital investment increases the effectiveness of financial institutions which in turn promotes emergence and growth of an economy (yang and yi, 2007). financial development is considered as one of the major policy areas for enhancing the economic efficiency and simultaneously economic growth of an economy. in the present context, financial system is one of such inputs which propel the economic growth of a country with its slow accumulation. there are variety of ways through which the financial system can affect the economic growth of any economy. here, it is worth mentioning that a more developed financial system accesses and thus, in turn utilizes correct information about the borrowers of funds (boyd and prescott, 1986). moreover, a more developed and sound financial system will also provide more reliable database regarding the prospective clients to deal with and thus, will in turn help in channeling the resources to higher yielding projects. so, development dynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 7 of financial intermediaries helps in increasing economic growth (greenwood and jovanovic, 1990). different channels through which the financial intermediaries diligently ascertain the authentication of the borrowers includes the mortgage system, interests rates, collateral requirements and so on (tiwary and thampy, 2015). these above mentioned channels not only make the utilization of funds effective but also create an environment for proper mobilisation of funds within an economy as well as for attracting funds from other economies as well (boyd and smith, 1998). the other prime channel through which the financial system can affect the growth of an economy is by offering more competitive and effective products. hence, a financial system which provides a proper system to effectively manage the high return projects directly helps in boosting the economic growth. though, there is ample of indication that financial development plays a vital role in endorsing economic growth of the industrialized countries (beck and levine, 2004), but evidences are relatively mixed in case of developing or emerging economies. with uncertain market condition, it has become imperative to identify the channels of finance to economic growth as the aforesaid matters have important policy implications. assuming financial development as an engine to economic growth policy makers should focus on the creation and promotion of modern financial institutions including banks, non-banks, and stock markets in order to promote genuine and long-term economic growth. however, on the other hand, there are studies (patrick, 1966; stern, 1989; ram, 1999; akinboade, 2000; favara, 2003; majid and mahrizal, 2007; demetriades and james, 2011; nain and kamaiah, 2014 and kumar et al., 2015) which do not agree to this view and have rejected any causal role for financial development in the growth process. if this contrasting argument is the accurate depiction of reality, then all the policy work and efforts to encourage financial development would be premature and in fact will lead to uneconomical use of limited resources. moreover, unnecessary emphasis on financial development will also turn away attention from other, perhaps more, urgent policy options to spur economic growth such as labour training and skill development programs to improve productivity, legal reforms to induce investment, and export promotion schemes etc. apart from this, majority of the studies that deal with this concern consider economic growth at the aggregate level, where it is very simple to establish and explain a causal relationship between financial development and economic growth (misra, 2003, acharya et al., 2009, giri and mohapatra, 2012, nain and kamaiah, 2014 and sharma and bardhan, 2016). in indian context, there have been dearth of studies that deal with the impact of domestic financial development on economic growth at regional and empirical level. to put it briefly, there is no universal consensus on the relationship between financial development shravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 8 and economic growth. there are some mixed results regarding the nexus between financial development and economic growth. thus, focusing on the development of the aforementioned issue the present study is an attempt in the direction to uncover the relationship between the financial development and economic growth with the help indian context. 1. literature review significant research studies are available on the aspect of financial development and economic growth and other related issues contributed by academicians, researchers and institutions. the writings on the relationship between financial development and economic growth can be traced from the pioneering work of schumpeter (1911), robinson (1952), mckinnon (1973), shaw (1973) and lucas (1988) etc. goldsmith (1969) using data of 35 countries for the period 1860–1963 studied the relationship between financial development and economic growth and identified that financial development exerts a causal influence on economic growth. the study concluded with a positive correlation between economic growth and financial development. in a similar study gregorio and guidotti (1995) by highlighting the importance of the efficiency of the financial system more specially the credit component instead of volume of investments uncovered the long-run relationship between the growth in financial system and growth of the economy. the study revealed that efficient allocation of credit in the financial system plays a significant role in uplifting the economic condition of the latin american countries. the reason for this was attributed towards the introduction of financial liberalization. by examining the casual relationship between the financial development and economic growth in indian context, bhattacharya and sivasubramanian (2003) suggested that there exists a one-way causal relationship from financial development to economic growth. by focusing on egypt, abu-bader and abu-qarn (2007) identified a bidirectional relationship between financial development and economic growth. furthermore, the study also uncovered that the relationship between the two is catalysed by increasing resources for investment and enhancing efficiency. in another study, hassan et al. (2011) provided substantiated evidence on the link of financial development and economic growth in various low and middle-income countries. in context of developing countries the study indicated a positive relationship between financial development and economic growth. similarly, al-malkawi et al. (2012) in an empirical study uncovered a negative and statistically significant relationship between financial development and economic growth for united arab emirates. lacth and gurgul dynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 9 (2012) by considering differnt aspects of financial development identified unique causality from stock market development to economic growth and from economic growth to banking sector development in poland. furthermore, focusing on significant parameters of financial development various studies identified uni-directional causal relationship from credit provided by banks to the development of economic growth. osman (2014) examined impact of private sector credit on the economic growth of saudi arabia using ardl model and concluded that there is long-run as well as short-run relationship between private sector credit and economic growth of saudi arabia. the study suggested bank credit to private sector as an important contributor towards the economic growth of saudi arabia. similarly, emecheta and ibe (2014) concluded a positive and statistically significant relationship between bank credit to private sector, broad money and economic growth in nigeria. in a more recent study korkmaz (2015) found a significant role of domestic credit provided by banking sector on economic growth for selected europian economies. furthermore, there are some studies in the recent literature which support the uni-directional causality hypothesis from economic growth to bank credit. in a study onuorah and ozurumba (2013) suggested a uni-directional relationship running from economic growth to credit provided by banks. similarly, obradovic and grbic (2015) suggested that economic growth contributes to financial deepening. the study identified a uni-directional causality running from private enterprise credit to economic growth of serbia. in a more recent study aydi and aguir (2017) considering southern mediterrean countries indicated a strong positive relationship between financial development and economic growth. the study focused more on the advancement and more innovation of banking sector. in a somewhat similar study witkowska and kompa (2017) identified strong relationship between change in the political envionment and the development of the financial system in poland. bist (2018) considering a panel of 16 african countries highlighted the presence of long-run relationship between financial development and economic growth with a positive relationship from the indicators of financial development towards the economic growth. the study emphasized on extending of credit facility to the private sector in these countries for achieving more growth. in another study younis and bechtini (2018) uncovered one-way causality from financial development to inequality in income in brics countries. in another study nyasha and odhiambo (2018) illustrated the crucial role of financial development in accelarting economic growth in south africa. in indian context considering 25 states, mishra (2003) studied the relationship between financial credit provided by banks and the economic growth and indicated a positive relationship between the two. furthermore, shravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 10 using multivariate cointegration and error correction techniques for indian economy, kamat and kamat (2007) suggested strong evidence in favour of finance-led growth hypothesis and suggested finance as a strong and leading indicator for the economic growth. moreover, the results of the study also provided relevant evidence that improvement in the stock market enhances the infrastructural growth of the economy. on the similar lines, the study conducted by singh (2008) indicated long-run as well as short-run relationship between financial development and economic growth. in another attempt to identify the relationship between financial development and economic growth, arora (2012) presented a multi-purpose and multidimensional representation of the bank credit. the study proposed that the credit provided by banks acts as a base for growth, globalization, urbanization, removing the inequalities between rural and urban areas, small and large borrowers and finally economic growth. the study subsequently, advocated a broader role of credit, which is growth oriented as well as developmental in nature. in another study, acharya et al. (2009) investigated the relationship between the financial development and economic growth in different sets of indian states including bimaru and nine other states using the credit outstanding as an indicator of financial development and confirmed the relationship between financial development and growth in indian states. hye (2011) identified the relationship between financial development and economic growth by constructing an index of financial development including four financial variables. the findings of the study suggested that during most of the years the index negatively related with economic growth of the country. giri and mohapatra (2012) supported the supply leading hypothesis and highlighted the importance of financial development for better growth in indian context. in the similar context, ray (2013) also highlighted the positive role of financial development in the economic growth of india during 1990–91 to 2010–11. however, using aggregate data, nain and kamaiah (2014) found no evidence of causality between financial development and economic growth in india. similarly, kumar et al. (2015) employed toda and yamamoto granger-causality tests for south africa for data over the period 1971 to 2011 and confirmed the absence of causality between financial development and economic growth, thus indicating that these two variables evolve independently with each other. similarly, kumar and chauhan (2015) did not find any evidence of causality between saving deposits with commercial bank and economic growth of india. the existing literature on the subject matter centred on different variables, covering different countries/states, utilizing different econometric techniques, considering different time frames, and hence, presented varied results too. though, there are many studies in indian context uncovering the dynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 11 relationship between the two however, most of them either focused on few numbers of states or the time frame considered was inadequate. therefore, testing the same relationship with larger data set will provide better and reliable results. moreover, the literature focusing on the pooled data of states with empirical data is also very limited in indian context. thus, in this regard the present study contributes to the literature by providing evidence and hence, fulfilling the gap by undertaking panel data analysis of 23 indian states over a period of 17 years. hence, the present study is pursued in this direction to fill the existing gap. 2. research methodology 2.1. sources of data and variables used there is significant advantage of using panel data or pooled data over cross-sectional and /or time-series data as the panel data increases the degree of freedom for estimation of parameters and hence facilitates the use of multivariate analysis techniques (hassan et al., 2011). the present study utilized the state level data of union of india for the period of 1997–1998 to 2015– –2016 and the data have been compiled from the various reports of reserve bank of india including handbook of statistics on indian economy, basic statistical returns of scheduled commercial banks, branch banking statistics for the aforesaid duration. a total of 23 states comprising andhra pradesh, arunachal pradesh, assam, bihar, gujarat, haryana, himachal pradesh, jammu and kashmir karnataka, kerala, madhya pradesh, maharashtra, manipur, meghalaya, nagaland, orissa, punjab, rajasthan, sikkim, tamil nadu, tripura, uttar pradesh and west bengal have been selected for the estimation of the relationship between financial development and economic growth. although, the study has tried to capture the overall scenario of the financial development and economic growth nexus by including all the major states, yet the union territories of india are kept out of the preview of the present study. at present, india is a federation of 29 states and 6 union territories. for the purpose of the analysis, the present study has left out the state of delhi and the six union territories, as these are smaller geographical units, working and process are different. among the remaining 29 stat0.es, three states namely uttaranchal, jharkhand, and chhattisgarh, and telengana carved out of the states of uttar pradesh, bihar, madhya pradesh and the state of andhra pradesh respectively, were formed in the year 2000 and later. for analysing the data for these four states two major techniques were available: (i) merging the data of these newly formed states with their parent states and (ii) the other option to analyse these states would have been to split the data shravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 12 before 2000 between new states and their parent states. for this, data at district level would be needed, which is not available in public domain for most of the variables used in the study. thus, the present study has merged these newer states with their parent states for the purpose of research modelling estimation and consequent analysis. in recent years, the structure of employment and income generation in the indian economy has gone through some critical changes. india, a primarily agrarian and rural economy, is the eleventh largest economy in the world in terms of nominal gdp and the fourth largest in the world in terms of purchasing power parity (imf reports, 2011). india is currently the second fastest growing economy (after china), which registered 8.9 per cent growth during 2010 and after a slight slowdown, it is again reaching new heights. an attempt has been made to incorporate a comprehensive list of variables (refer table 1), which reflects the general and informative results about different indicators of financial development. by considering some of the relevant financial development indicators the present study attempted to identify the relationship of financial development and economic growth in indian context. the table 1 provides the summary of different studies alongwith the nature of data taken, the indicators used, etc. in india scheduled commercial banks accounted for around 70 per cent of the total assets of the financial system (inoue and hamori, 2010) so the present study has focused on the banking sector as a measure of financial development. banking sector development is defined as the development in the quality, quantity and effectiveness of the banking services (pradhan et al., 2011). this procedure involves the interaction of many activities and cannot be measured by a single indicator (levine and zervos, 1998; rousseau and wachtel, 2000; beck and levine, 2004; naceur and ghazouani, 2009). most of the studies given in table 1 focused on banking level indicators and considering the above mentioned literature the present study has focused mainly on credit, deposits, number of branches and their per capita deposits and credits etc. measure as financial development indicators. in the present endeavour, for carrying out the estimations, the per capita state domestic product (pcsdp) and state domestic product (sdp) have been used as the proxy for economic development. on the other hand, the financial variables (proxy of financial development) used in the study for state level data are – (1) the number of branches of banks (nof) (2) the outstanding credit of all the scheduled commercial banks of the state to the different sectors(oc) (3) per capita credit (pcc) (4) per capita deposit of all scheduled commercial banks of selected states(pcd) (5) population per office(ppo) (6) number of debit accounts (da) and (7) number of credit acdynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 13 counts (ca). the above mentioned indicators are primarily considered as the indicators of depth of the financial institution in a country thus, it is presumed that higher the depth higher will be the economic growth (inoue and hamori, 2010). these measures are widely accepted and frequently used in the finance-growth relationship literature (kendall, 2012; aghion et al., 2007; jayaratne and strahan, 1996; king and levine, 1993a). for making the data comparable all the values are taken in their respective logarithmic form. the use of multiple indicators for financial development helps in better understanding of different aspects and processes of financial development. three major econometric techniques including panel unit root tests, cointegration tests and finally the panel error correction model have been implemented for identifying the relationship between variables. table1. different indicators of financial development s.no studies period name/number of economies type of data financial development indicators 1 king and levine (1993a, 1993b) 1960– 1989 80 cross-section data i) liquid liabilities of financial system divided by gdp ii) ratio of bank credit divided by bank credit by central bank domestic asset, iii) ratio of credit allocated to private enterprise to total domestic credit, iv) credit to private sector to gdp. 2 gregorio and guidotti (1995) 1960– 1985 80 pooled crosssection data i) bank credit ii) gdp 3 berthelemy and varoudakis (1995) 1960– 1985 91 panel data i) money supply ii) gdp 4 demetriades and lunitel (1996) 1961– 1981 india time-series iii) bank deposit liabilities i) gdp 5 demetriades and lunitel (1996a) 1962– 1982 nepal time-series data i) bank deposit liabilities ii) gdp 6 rajan and zingales (1998) 1980– 1990 55 panel data i) the ratio of credit to private sector to gdp, ii) accounting standards 7 levine and zervos (1998) 1976– 1993 41 time-series data i) ratio of market capitalization to gdp, ii) ratio of total value of trades to gdp, iii)turnover ratio shravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 14 table1. continued s.no studies period name/number of economies type of data financial development indicators 8 rousseau and wachtel (2000) 1980– 1995 47 panel data i) liquid liabilities, ii) stock market capitalization, iii) stock market value traded 9 beck et al (2000) 1960– 1995 63 for crosssectional and 77 for panel cross-section and panel data i) private credit to gdp ii) gdp 10 safdari et al.(2011) 1975– 2008 iran panel i) trade to gdp ratio ii) the share of gross fixed capital formation to nominal gdp 11 sharma and ranga (2014) 2000– 2012 india time-series i) saving deposits with commercial bank ii) gdp 12 kumar and chauhan (2015) 1975 to 2013 india time-series i) saving deposits with commercial bank ii) gdp 13 sherawat and giri(2015) 1993– 2012 india panel data i) credit as a share of state output ii) deposit as a share of state output iii) number of scheduled commercial bank branches 14 sharma and bardhan (2016) 1980– 2011 india panel data i) per capita deposits ii) per capita credit iii) per capita gdp 2.2. panel unit root test to identify the nature and pattern of data in the dataset under consideration, a series of panel unit root test were employed in the present analysis. otherwise problem with spurious regression could be faced. the literature on stationarity measurement has divided the unit root tests into first generation unit root test (levin, lin and chu (llc), breitung, hadri, im pesaran shin test (ips)) (maddala and wu, 1999; breitung, 2000; hadri, 2000; levin et al, 2002 and im et al., 2003) and second generation unit root tests (o’connell; breitung and das; moon and perron, bai and ng and pesaran developed by o’connell, 1998; breitung and das moon and perron, 2004; bai and ng, 2004; breitung and das, 2005 and pesaran, 2007) (hurlin, 2004 and hu et al., 2006). the first generation unit root tests are in-turn divided into homogeneous model hypothesis tests (levin, et al., 2002; breitung, 2000 and hadri, 2000) and heterogeneous model hypothesis tests (im et al., 2003 and maddala and wu, 1999). although, cross-sectional units in the present study are the states of india which more or less comes under homodynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 15 geneous model hypothesis, yet to reinforce the results in the present attempt tests from heterogeneous hypothesis model have also been implemented. in case of ips test, for analysing the presence of unit root, the model is specified as follows: yit = βiyit-1+ ∑ ∅#$ % $&' ∆yi t-j + xitp+ϵit (1) where i = 1,2,...,n are the number of cross-sections over period t = 1,2,…,ti here, y¡t stands for each variable under consideration in our model, β is autoregressive coefficient and ϵ is the error term. the null and alternate hypothesis of the respective model: h0 : βi = 0, for i = 1, 2,…, n. (2) this indicates that a unit root is present and thus, the model would be nonstationary in this case. h1: βi< 0, for i = 1, 2,…, n. (3) ips test uses different unit root test for all the cross-sections and the final test statistics is represented as 𝑡 ̅is the average of individual adf statistics. 𝑡̅=1/n ∑ (𝑡)%# , $&' where tpi is the individual t-statistic for testing the null hypothesis. under the null hypothesis of non-stationarity ips show that the t-statistic follows asymptotically a standard normal distribution. ips provides simulated critical values for t for different number of cross-sections n, series length t. on the other hand, llc estimates the simple regression equation of differenced concerned variable as follows: ∆xit=∝+δxit-1∑ ∅ij ∆ s r=1 xit-j + uit (4) where, x is the variable under consideration, δ is the coefficient, u is the error term and i = 1, 2,…, n implying the number of cross-sections and t = 1,2,…, t. the llc unit root test has developed a procedure utilizing pooled crosssection time-series data to test the null hypothesis that each individual timeseries contains a unit root. under llc the unit roots are tested using adf regression the llc test involves that rejection of the null can occur even when only a small sub-set of series are stationary (divino et al., 2009). apart from this, the llc test assumes that errors are independent across all crosssections (banerjee, 1999). in the present study, comparison of the results from the above mentioned types of tests would enable overall comparison of cross-section results. shravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 16 literature reveals that not considering the cross-sectional dependence across cross-section units may affect the sample behaviour of the panel unit root test which subsequently results to the incorrect decision in a unit root test (o’connell, 1998, banerjee 1999; pesaran, 2007). in order to evade this under-performance of the unit root tests, the present study carried out second generation panel unit root test which takes care of cross-sectional dependence and also allows for parameter heterogeneity and serial correlation between cross-section units in the panel (pesaran, 2007). pesaran (2007) panel unit root test takes care of cross-sectional dependence by augmenting the df regression (dicky and fuller, 1979) with crosssectional mean and its lag. this test is known as cross-sectionally augmented dicky–fuller (cadf) test and is based on the following regression. ∆yi, t = αi + ᵖi yi,t-1+ βoyt-1 + β1∆yt+ ϵi,t (5) where yt is the average of y at time t for all n observations. the presence of lagged cross-sectional average and its first difference accounts for the crosssectional dependence through a factor structure. cadf regression is run for each cross-sectional unit and then average is taken over all the cross-sections (similar to im et al. 2003) and the resulting test statistic is calculated as follows: cips(n,t)=' , ∑ 𝑡#(𝑁,𝑇) , #&' (6) where, ti(n, t) is the statistic obtained from individual cadf regression of ith cross-sectional unit and where cips stands for cross sectional augmented ips (im et al., 2003) unit root test) 2.3. panel cointegration tests long-run relationship between the variables after unit root is demonstrated with panel cointegration tests. the present study implemented pedroni (1999) for testing panel cointegration for the selected states of union of india. pedroni (1999) cointegration test is a combination of seven statistics including panel v-statistic, panel rho-statistic, panel pp-statistic, panel adf-statistic, group rho-statistic, group pp-statistic, group adf-statistic. these statistics are estimated on the basis of cointegrating regression equation yi, t = β0 + β 1i x 1i,t + … + β ni xni,t + ei,t , (7) t= 1,…,t; i = 1,…, s where, t is the number of observations over time, s denotes the number of individual members in the panel, and n is the number of independent variables. it is assumed here that the slope coefficients, β1i ,.., βni and the member dynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 17 specific intercept β0 can vary across each cross-section. to compute the appropriate panel cointegration test statistics, the cointegration regression in (7) is estimated by applying ols for each cross-section (orsal, 2007). secondly, the present study also utilised the four panel cointegration tests of westerlund (2007) which having sufficient sample properties and high power relative to popular panel cointegration tests (e.g. pedroni, 2004) and is also applicable in the situation of cross-section dependencies between the data. 2.4. dynamic panel causality and long-term relationships pedronis’ heterogeneous panel cointegration tests are only able to specify whether the variables are cointegrated and if a long-run relationship exists between them. since, the mentioned model does not indicate the direction of causality, the present study estimated the two-step panel-based vector error correction model (vecm) proposed by engle and granger (1987) and uses it to conduct causality test on the financial development and economic growth relationship for the specified data of the study. to identify the cointegration relationship, consider a two variable system equation: y it =β xi t where, y and x are two variables of interest, t is the time period and i is the number of cross sections. the corresponding panel vec model is: ∆yit = β1j +∑ 𝛼'#3 % 3&' ∆𝑌#673+∑ 𝜂'#3 9 3&' ∆𝑋#673 + λect1it-1 (8) ∆xit = β2j +∑ 𝛼;#3 % 3&' ∆𝑋#673+∑ 𝜂;#3 9 3&' ∆𝑌#673 + λect2it-1 (9) where, δ is a first-difference operator applied to the variables; p and q are the lag lengths; i represents state i in the panel (i = 1, 2…., n); t denotes the year in the panel (t = 1, 2, …., t); y and x are the variable of interest; ect is error correction term which is derived from the cointegration equation. the present study estimated the long-run equilibrium relationship given by the lagged error correction term (ect), which is a measure of the extent by which the observed values in time t-1 deviate from the long-run equilibrium relationship and in case the variables are cointegrated any such deviation at time t-1 should make alterations in the values of the variables in the next time point in an effort to force the variables back to the long-run equilibrium relationship (ageliki et al., 2013). shravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 18 3. results and discussion to study the casual relationship among different indicators of financial development and economic growth, panel vecm is implemented in the present study. however, before testing the extent and pattern of causality, the data-set is tested for cross-sectional dependence and stationarity. the present study examined the cross-sectional independence using friedman test (pala, 2016) having null hypothesis of no cross-section independence. the test is beneficial for the panel having small time period and large crosssection. the test is a non-parametric in nature where spearman’s rank correlation coefficient is used to estimates the cross-sectional dependency for the estimates. table 2 reveals that friedman test-statistic is highly significant for economic growth and measures of financial development. hence, evidence of cross-sectional dependence in the data-series indicates that indian states for these relevant indicators are cross-sectionally dependent or correlated due to various observed and unobserved common factors (sharma and bardhan, 2016; mishra and mishra, 2014). table 2. cross-sectional dependence test for panel data test statistics probability friedman test (chi-square) 260.96* 0.00 note: *: statistically significant at 1 per cent level of significance friedman’s test statistic showed an asymptotically χ2 distribution with t-1 degrees of freedom. further, to test the order of integration of the variables under consideration literature provides two types of unit root tests. the first generation panel unit root tests are mainly based on the restrictive assumption of crosssectional independence (levin et al., 2002 (llc); im et al., 2003 (ips); maddala and wu, 1999; breitung, 2000). literature reveals that not considering the cross-sectional dependence across cross-section units may affect the sample behaviour of the panel unit root test which subsequently results to the incorrect decision in a unit root test (o’connell, 1998, banerjee 1999; pesaran, 2007). in order to evade this under-performance of the unit root tests, the present study carried out second generation panel unit root test which takes care of cross-sectional dependence and also allows for parameter heterogeneity and serial correlation between cross-section units in the panel (pesaran, 2007). pesaran (2007) panel unit root test takes care of cross-sectional dependence by augmenting the dicky fuller regression (dicky and fuller, 1979). here, pesaran (2007) panel unit root test in the presence of crosssectional dependence fails to reject the null hypothesis of unit root at level for all the variables (table 3). however, first differenced series of state level dynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 19 per capita gdp, sdp and indicators of financial development are stationary except for population per office and outstanding credit. table 3. second generation panel unit root test with cross-sectional dependence variables level first difference order of integration sdp 1.95 –4.13* i(1) pcsdp 0.12 –4.24* i(1) pop 18.64 17.52 – ca 0.21 –2.80** i(1) da 6.13 –3.17* i(1) nof 2.92 –3.63** i(1) oc –0.05 –1.23 – pcc 3.50 –3.47* i(1) pcd 2.72 –4.27* i(1) note: *: statistically significant at 1 per cent level of significance; **: statistically significant at 5 per cent level of significance. in order to check the robustness of the model, the present study also estimated the unit root using the first generation individual panel unit root test like im, pesaran, sin; fisher-adf; fisher-pp tests and common root test like levin, lin, chu which propose the null hypothesis in favour of unit root too. the assumptions regarding the common unit root indicates that the tests are estimated using common autoregressive parameters for all the series included in the panel data-set, while the individual unit root tests provide different autoregressive coefficients in individual series in the panel data-set. the most widely used panel unit root tests in the literature involve the levin et al. (2002) and im et al. (2003). thus, for studying the robustness of the model, the present study utilised both type including common and individual unit root tests for analysis. further, the lag length criterion for group or pool unit root test is the automatic lag length selection that entails the information matrix criterion based for the number of lag difference terms and the neweywest automatic bandwidth selection. the lag values used for the computation of the model were on the basis of the default values. the null hypothesis for the panel unit root test is that the variables involve the presence of unit root. the test statistics for all the variables used in the data-set are shown in table 4. all the variables accepted unit root hypothesis of non-stationary at levels except for llc test in case of outstanding credit. the statistically insignificant values for rest all the tests statistics (refer table 4) in the level form suggest the presence of unit root and hence, confirm the data to be non-stationary. shravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 20 table 4. first generation panel unit root tests variables llc** ips** adf** pp** pcc level 3.29 6.70 10.54 12.23 first difference –6.55* –10.95* 202.37* 2530.66* pcd level 2.63 6.47 121.60* 366.04* first difference –7.39* –16.37* 297.91* 3339.01* ps level 7.33 6.21 15.66 14.91 first difference –5.91* –3.52* 80.48* 158.15* nof level 17.75 13.99 9.52 1.44 first difference 0.80 1.60 45.43 64.33 oc level –2.51* –0.60 38.42 35.52 first difference 19.01 3.44 16.51 223.74* da level 13.02 10.08 15.51 42.33 first difference –4.18* –6.03* 142.47* 1186.30* ca level 4.34 7.63 5.48 5.18 first difference –2.67* –3.76* 92.39* 203.65* sdp level –5.10* 1.78 32.13 218.95* first difference –9.71* –8.12* 156.84* 523.58* pcsdp level 2.96 3.19 19.78 99.67 first difference –11.98* –12.18* 222.09* 352.34* note: *: statistically significant at 1 per cent level of significance. probabilities for fisher tests are computed using an asymptotic chi-square distribution. all other tests assume asymptotic normality. automatic lag length selection based on sic: 0 to 3. newey-west automatic bandwidth selection and bartlett kernel. again by analysing the data in the first difference, the statistically significant values of all the statistics for all variables (except for number of bank branches and outstanding credit) rejects the null of unit root and thus, substantiate the data to be stationary at the first difference. in other words, authentication of levin, lin and chu t stat; im, pesaran and shin, w-stat; adf-fisher, chi-square; pp-fisher, chi-square statistics less than the critical value of 1 per cent level of significance authenticate the rejection of null hypothesis, thus, validating the absence of unit root in the first differenced variables except for number of bank branches and outstanding credit. these results imply that for the respective states the variables are integrated of order one i(1). comparison of the results of both the first generation unit root test with that of the second generation unit root provides consistent results for most of the indicators of financial development as well as of the economic growth. however, the variables number of bank branches and population per office are i(1) in one type of test but not in the other, so the present study proceeds with both number of bank branches as well as population covered by the banks for further analysis as deleting a variable for analysis is not appropriate. however, outstanding credit is not taken into consideration for further analysis as the results of both first as well as the second generation unit root dynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 21 test substantiate that the variable is stationary and hence can’t be taken for further analysis. the results of the panel unit root test suggest that the variables under consideration are non-stationary in their level forms and thus, the application of ols/gls or other techniques would yield inconsistent and biased results (ramirez, 2006). thus, it becomes necessary to move forward towards panel cointegration analysis to test the long-run relationship, if exists, between the variables. the panel cointegration test examines the presence of a long-run relationship between the variables (ghali, 1998; and basu et al., 2003). the cointegration test is estimated using the pedroni (1999, 2004) as well as through westerlund (2007) test which takes the cross-section dependence aspect also. the pedroni (1999, 2004) test provides seven statistics and which are further grouped into two sub-dimensions: the “panel statistics” or “within dimension,” which corresponds to the unit root statistic against homogenous alternatives (breitung and pesaran, 2008); and the “group mean statistics” or “between dimension,” which involve the averaging of the individually estimated autoregressive coefficients for each country, individually. the null hypothesis for all the seven pedroni tests is that there is no cointegration between variables involved in the model. another important aspect of the pedroni (1999, 2004) statistics is analysis of the test using critical value of −1.64. this means that a test statistic of less than −1.64 implies rejection of the null for all other tests except the v-statistic. for the panel-v, the critical value is 1.64. thus, a test statistic greater than 1.64 indicates that the null hypothesis of no cointegration is rejected. according to lund (2010) if the results indicate inconsistency, the panel adf and group adf should be implemented. the result of pedroni’s cointegration between number of credit accounts and sdp of the respective states suggests that all the four within dimensions panel cointegration test for the respective states reject the null hypothesis of no cointegration between the variables and overall all the seven statistics too rejected the null hypothesis of no cointegration at 1 per cent level of significance (refer table 5). similarly, for the second situation of number of debit accounts and sdp of the selected states suggests that all the seven statistics are statistically significant and thus, rejected the null hypothesis of no cointegration. similar results are obtained both in case of population per office and number of bank branches as well. in a similar way, the relationship of per capita credit and per capita sdp also reject the null hypothesis for all the pedroni (2004) major statistics. likewise, the relationship between per capita deposits and per capita sdp too reveals that all the seven statistics are significantly rejecting the null of no cointegration. the panel adf and group adf show consistent results. the pedroni (2000) test rejected the null hypothesis of no cointegrashravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 22 tion in the important tests like panel variance test, panel adf and group adf thus, on the whole, the results of the panel cointegration explicate that cointegration lies in all the specified empirical models and the results clearly point to a statistically significant long-run relationship between sdp and the discussed financial variables in different models. table 5. first generation panel cointegration test pedroni’s cointegration test method sdp and ca test statistic(p-value) sdp – da test statistic(p-value) sdp – pop test statistic(p-value) pcsdp – pcc test statistic(pvalue) pcsdp – pcda test statistic(pvalue) sdp – nof test statistic(p-value within dimension/panel statistic panel vstatistic (+) 20.98* 20.97* 24.74* –1.52 1.35** 14.92* panel ρstatistic –11.89* –13.22* –10.38* –8.30* –9.08* –12.71* panel ppstatistic –7.35* –8.60* –4.55* –13.05* –14.04* –8.00* panel adfstatistic –3.44* –3.93* –3.47* –8.85* –8.59* –9.86* between dimensions/group mean statistic group ρstatistic –3.29* –3.74* –2.96* –5.02* –5.35* –3.64* group ppstatistic –14.13* –14.92* –11.48* –16.05* –16.08* –15.26* group adfstatistic –8.72* –8.14* –8.82* –12.31* –11.79* –11.48* note: *: statistically significant at 1 per cent level of significance, **: statistically significant at 10 per cent level of significance. the pedroni (2004) statistics are one-sided tests with a critical value of −1.64 (ko−1.64 implies rejection of the null), except the v-statistic that has a critical value of 1.64 (1.64 suggests rejection of the null); selection of lags is based on schwarz information criterion (sic). newey-west automatic bandwidth selection and bartlett kernel. since, the variables in our study are cross-sectionally dependent (table 2) and pedroni’s test of cointegration assume cross-sectional independency so the study moved further with analysis of the so called second generation panel cointegration tests that allow for cross-sectional dependence. table 6 shows that two of the four westerlund’s panel cointegration tests; that are one each for the panel (pτ) and group mean statistics (gα), confirmed the presence of cointegration. hence, the above mentioned tests reinforce the results and highlight the presence of cointegration among the variables taken into consideration. dynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 23 3.1. panel vector error correction model (vecm) the causality test is conducted using vecm technique if the variables are integrated to the same order as well as cointegrated (moudatsou and kyrkilis, 2011; fowowe, 2011; emirmahmutoglu and kose, 2011). the question of long-run causal relationship between financial development and economic growth is now examined more thoroughly with the use of panel vector error correction models. defining the lagged residuals, the dynamic error correction models are estimated for the variable sets differently. the estimated results are presented in tables 1.7. the short-run causality is estimated by the statistical significance level of the coefficients of the first differenced variables while the long-run causality is determined by the statistical significance of the respective error correction term (ect) values using t-tests (minija, 2012). in the first case (sdp-ca) the statistical significant value of the t-statistics of sdp for the relation ca=f(sdp) suggests that in short run the economic growth helps in the advancement of credit in the financial system. however, the statistically insignificant value of the ect values for the same relation does not provide any evidence of the long run relationship from sdp to number of credit accounts. considering the other relation where sdp=f(ca) suggests that although, in short run credit do not have a significant impact on the economic growth, but the statistically significant ect value (kamal, 2015 and bhanumati and azhagaiah, 2014) illustrates that higher credit allocation in the long run in turn helps in boasting the growth of the economy. the estimated coefficient of the residual from the vecm test (ect) with statistically significant value and a negative sign confirms the long-run equilibrium relation between the independent and dependent variable at 1 per cent level of significance. here, the speed of adjustment implies that 2.5 per cent of the disturbance in the short-run will be corrected each year. the results are in line with the previous study of sharma and bardhan (2016) thus, this evidence emphasize on the critical role of bank credit in the process of long run economic growth across states. it has been suggested that in the absence of developed financial markets small and medium-sized industries which are considered the backbone of an economy, in particular, the economies very often fail to expand due to the dearth of funds. further, expansion of credit not only supports industrial activities but also helps in generation of more physical and human capital which are also essential elements of economic growth. since, small firms are generally considered labour intensive and are supposed to generate more employment opportunities so for a typical developing economy such as india, providing better credit facility has a positive implications on economic growth (sharma and bardhan, 2016). this also highlights that credit and shravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 24 economic growth affect each other directly thus, suggesting a key indicator for enhancing the regional as well as national economic growth. similarly, in case of sdp-da the statistical significant value of ect for the relation da=f(sdp) suggests that in long run, economic growth does provide a strong evidence of causality towards the financial development, though the statistically insignificant values for sdp for the same relation suggests that in the short run there is no impact of this variable on deposits as a whole. considering the other relationship sdp=f(da) the results indicate that accumulation of deposit in the short run enhances economic growth in the long run thus, reinforce the results obtained in table 5 and 6. the above findings verify that deposits are crucial indicator and are significantly beneficial to the economic growth of the indian states. higher deposits with the intermediaries means channelling of surplus funds to the deficit units of the economy hence, transforming deposits into loans and other advances (naira, 2016). the results are in line with the earlier studies of bhanumurty and singh (2013) and sehrawat and giri (2015) which suggest a bidirectional causality and emphasis that economic growth has a critical role in financial development and that financial development leads to further development of economic growth of an economy. from the results of this relation it can be inferred that policy makers should emphasis on the measures which enhance the economic growth but they also focus on the development of banking system by considering the long term perspective of the mutual causality between the two which will ultimately benefit the economy as a whole. table 6. second generation panel cointegration test westerlund cointegration test method sdp and ca test statistic (p-value) sdp – da test statistic (p-value) sdp – pop test statistic (p-value) pcsdp – pcca test statistic (pvalue) pcsdp – pcda test statistic (pvalue) sdp – nof test statistic (p-value) gτ –1.78* –1.94* –1.49* –1.72* –2.36* –1.74* gα –4.82 –4.97** –4.52 –4.65 –10.62* –4.51 pτ –8.18* –7.84* –8.84* –7.09* –9.17* –10.65* pα –16.74* –14.91* –14.86* –5.42* –8.53* –17.19* note: *: statistically significant at 1 per cent level of significance, **: statistically significant at 10 per cent level of significance, selection of lags is based on akie information criterion (aic) and schwarz information criterion (sic). overall the above two findings suggests that it is essential to expand the business and dealings of banks by augmenting the credit and deposits as it decides the degree of financial accessibility and in turn promotes the economic growth in the long run. dynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 25 further, the statistically significant value of coefficients of sdp in next scenario of ppo=f(sdp) points towards an important short-run relationship suggesting that with the increase in economic growth, the population served by the banks will also increases. with the increase in sdp of the state the circulation of the money will increase and hence, more people will utilize the services of the bank. moreover, considering the broader perspective higher population covered under the banking system will slowly and steadily enhance the economic growth in the long run. this is indicated by the statistically significant coefficient of ect in the sdp= f(pop). further, the statistically significant value of coefficient of sdp in next scenario of nof=f(sdp) points towards an important short-run relationship illustrating that with the increase in economic growth, the number of branches within a state will increase and hence, the population served by the banks. ardic and damar (2006) considered it as the indicator of financial depth and suggested that with the increase in number of branches the domestic competition increases and thus, results in the better financial services and in-turn helps in the long run growth. the short run dynamics of per capita credit and economic growth presented in table 7 suggest that a short-run increase of bank credit inturn induces an increase in economic growth in the selected indian states. however in the long-run, the magnitude of per capita credit coefficient is quite small and insignificant indicating that per capita credit partially determines the magnitude of real economic growth in the short-run only. this can be inferred that the growth of bank credit influences per capita sdp of the selected indian states, but the reverse is not necessarily true. in other words, it can be argued that economic growth in indian states is caused by the expansion and improvement of the banking system in the long-run. a significant relationship between the another proxy of financial development, the bank accounts (deposits) per person suggests that with 1 per cent increase in the deposits the economic growth will increase with the same amount. moreover, the estimated coefficient of the residual from the vecm test (ect) with statistically significant value and a negative sign confirms the presence of long-run equilibrium relation between the independent and dependent variable at 5 per cent level of significance. the speed of adjustment implies that 1.6 per cent of the disturbance in the shortrun will be corrected each year. however, the results of the study suggested that there is no significant relationship from economic growth to per capita sdp of the states in short run or in long run. shravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 26 table 7. panel vecm estimation dependent variable coeff t-stat dependent variable coeff t-stat direction of causality sdp sr ca 0.107 0.650 ca sr sdp –0.009*** –0.135 ca<–>sdp lr ect –0.025* –2.661 lr ect –0.001 –0.423 sdp sr da 0.002** 0.051 da sr sdp –0.429 –2.336 sdp<–>da lr ect –0.038 –0.878 lr ect –0.446* –6.784 pcsdp sr pcc 0.008* 1.826 pcc sr pcsdp 1.752 3.142 pcc–>pcsdp lr ect 0.014 3.252 lr ect –0.004 –0.378 pcsdp sr pcd 0.010* 0.951 pcd sr pcsdp 1.056 3.928 pcd–>pcsdp lr ect 0.016** 3.075 lr ect 0.006 0.437 sdp sr pop –0.132 –1.050 pop sr sdp 0.017** 0.410 pop<–>sdp lr ect –0.038* –3.240 lr ect –0.012 –0.190 sdp sr nof 0.560 0.345 nof sr sdp –0.26** –1.527 sdp<–>nof lr ect –0.162* –3.288 lr ect –0.031* –4.462 note: *: statistically significant at 1 per cent level of significance, **: statistically significant at 5 per cent level of significance, ***: statistically significant at 10 per cent level of significance; sr: short-run relationship, lr: long-run relationship the results in table 7 overall suggests a bi-directional causal relationship suggesting that credit is a crucial factor of production, and increasing proper credit allocation in the long-run will enhance the economic growth so there is a need to develop an advance current credit allocation process. so, focus increasing the number of business units of the banks in less developed areas will benefit the economy directly. thus, overall it is highlighted that policies which affect financial development in short-run or in long-run are likely to have an effect on economic growth and vice-versa. conclusions and policy implications with the help of standard refined panel analysis techniques including panel unit root, panel cointegration and panel error correction model for a sample of 23 states of india over the period of 1997–1998 to 2015–2016, the present study analysed the dynamics of causal relationship between financial development and economic growth in the sampled states of the country. the results of the study are robust as the present study have utilised both the first generation and well as the recently developed second generation unit root as well as cointegration tests. firstly, the series was tested for crosssectional dependence, and the statistically significant values of crosscorrelation data provided evidence in support of cross-sectional dependency. further, the results of both first and second generation unit root as well as cointegration test that account for cross-sectional dependence were used to examine the data, and the results indicated an integration of order one for all the variables (except for few as mentioned above). the results suggested the dynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 27 presence of cointegration and causality between financial development inputs and economic growth. the pedorni’s and westerlund’s cointegration test confirm the long-run relationship between financial development and respective economic growth indicators. policy implications of the present empirical results are presented as follows. in the present case, a cyclic relationship exits, financial market develops more in terms of credit allocation as a consequence of economic growth which in turn feedbacks a stimulant to real growth. hence, policies which emphasize more on the financial development and economic growth need to be developed. in the developing economy like india, the financial sector is mainly dominated by commercial banks. hence, to speed up the financial development of such economies, stringent efforts should be directed towards improvements in the banking sector with easy access to loans. the analysis reveals that there exists a positive relationship between the number of business units, the population served by the bank and economic growth suggesting that higher the number of bank branches higher will be the population served by the branches and hence, more will be the level of economic growth. these two significant indicators highlight a very relevant fact that the indian economy has a lot of scope in harvesting the less financially developed geographical areas of the state which can run rapidly on the greeny path of dynamic and sharp long term sustainable economic growth. so, the government needs to provide additional consideration to the banking development in less developed states so as to facilitate economic growth as well as social development and thus, minimize regional disparities in development and economic prosperity at regional and macro level. the banks need to support and assist opening of more bank branches in different states so that the commercial banks should increase the business potential by accelerating both deposits and advances to improve the business and the state government should improve the climate for investment through better governance so that the banking activities for developmental schemes are accelerated. the bi-directional relationship between credit and economic growth and in short-run the uni-directional causality from per capita credit suggests that the banking system in indian economy has a lot of scope of flourishing. any expansion of the domestic credit provided by the banking sector will promote economic growth of the indian economy and vice-versa. because the financial sectors in india is still developing, deeper and more efficient financial markets are needed to improve the levels of economic development. policies should aim at enhancing financial development and economic growth across states rather than concentrating on limited states which, in turn, possibly will help in growth convergence. moreover, this has a signifishravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 28 cant policy implication that a well-planned financial policy for promoting the development of domestic financial markets encompassing the banking sector is a crucial growth strategy for developing economy like india. identification of promising business ventures by keeping an eye on the upcoming market trends will not only help the banks to earn better and safe returns by reducing their non-performing assets but will also enhance the economic growth of our country. like, there are special credit schemes for small and medium enterprises but it is important to allocate more bank credit to such endeavours as experience of emerging economies like china shows that these enterprises contribute around 90 per cent to the gdp and are also important for employment generation and poverty alleviation. the two important aspect of the economic growth of indian economy are agriculture and the industrial sector. policy makers should make some important and strict decisions to reduce and rationally distribute loans to the non-performing agricultural and industrial sector. finally, on the basis of results obtained from the present study it should be recognized that economic growth itself may have the potential to promote further banking-sector development and hence, bring about additional economic prosperity through an interactive feedback effect. so, policy makers should pursue policies that attract lucrative investments in the country which will not only help in generating better avenues but also will create a more competitive environment which in-turn will benefit the economic growth of the country as well as the financial sector. further, it is pertinent to mention that the nature of data and the modelling method used in the present study uncovered some important aspect of the relationship between financial development and economic growth. however, extending the present study by capturing more aspect of financial development could provide some more interesting results on the aforementioned aspect. references abu-bader, s., abu-qarn, a. (2007), financial development and economic growth: the egyptian experience, journal of policy modeling, 30(5), 887–898, doi: https://doi.org/10.1016/j.jpolmod.2007.02.001. acharya, d., amanulla, s., joy, s. (2009), financial development and economic growth in indian states: an examination, international research journal of finance and economics, 24, 117–130. ageliki, a., dimitris, k., george, p. (2013), integrating the neighbours: a dynamic panel analysis of eu-enp trade relations, comparative economic studies, 58(1), 1–26, doi: https://doi.org/10.1057/ces.2015.23. aghion, p., fally, t., scarpetta, s. (2007), credit constraints as a barrier to the entry and post-entry growth of frm’s, economic policy, 22(52), 731–779. dynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 29 akinboade, o. a. (2000), the relationship between financial deepening and economic growth in tanzania, journal of international development, 12, 939–950, doi: https://doi.org/10.1002/1099-1328(200010)12:7<939::aid-jid668>3.0.co;2-i. al-malkawi, h. a. n., marashdeh, abdullah, n. (2012), financial development and economic growth in the uae: empirical assessment using ardl approach to cointegration, international journal of economics and finance, 4(5), 105–115. ardic, o. p., damar, h. e. (2006), financial sector deepening and economic growth: evidence from turkey accessed 26 march 2016, http://www.luc.edu/orgs/meea/volume9/pdfs/damar%20ardic%20-%20paper.pdf. arora, r. u. (2012), finance and inequality: a study of indian states, applied economics, 44(34), 4527–4538, doi: https://doi.org/10.1080/00036846.2011.597736. aydi, m., aguir, a. (2017), financial development and economic growth: the empirical evidence of the southern mediterranean countries, international journal of economics and financial issues, 7(3), 196–209. bai, j., ng, s. (2004), a panic attack on unit roots and cointegration, econometrica, 72(4), 1127–1177, doi: https://doi.org/10.1111/j.1468-0262.2004.00528.x. banerjee, a. (1999), panel data unit roots tests and cointegration: an overview, oxford bulletin of economics and statistics, 61(1), 607–29, doi: https://doi.org/10.1111/1468-0084.0610s1607. barro, r. j. (1996), determinants of economic growth: a cross-country empirical study, national bureau of economic research, working paper 5698, doi: https://doi.org/10.3386/w5698. basu, p., chakraborty. c., reagle. d. (2003), liberalization, fdi, and growth in developing countries: a panel cointegration approach, economic inquiry, 41(3), 510–516, doi: https://doi.org/10.1093/ei/cbg024. beck, t., levine. r. (2004), stock markets, banks and growth: panel evidence, journal of banking and finance, 28(3), 423–442, doi: https://doi.org/10.1016/s0378-4266(02)00408-9. beck, t., levine, r., loayza, n. (2000), finance and source of growth, journal of financial economics, 58(1), 261–300, doi: https://doi.org/10.1016/s0304-405x(00)00072-6. berthelemy, j-c., varoudakis, a. (1995), clubs de convergence et croissance. le rôle du développement financier et du capital humain, revue economique, 46(2), 217–235, doi : https://doi.org/10.3406/reco.1995.409640. bhanumati, k., azhagaiah, r. (2014), causal relationship between stock price and gold price in india: a granger causality test approach, international journal of research in management, science and technology, 2(2), 22–27. bhanumurty, n. r., singh, p. (2013), financial sector development and economic growth in indian states, international journal of economic policy in emerging economies, 6(1), 47–63, doi: https://doi.org/10.1504/ijepee.2013.054472. bhattacharya, p. c., sivasubramanian, m. n. (2003), financial development and economic growth in india: 1970–71 to 1998–1999, applied financial economics, 13(12), doi: https://doi.org/10.1080/0960310032000129590. bist, p. j. (2018), financial development and economic growth: evidence from a panel of 16 african and nonafrican low-income countries, cogent economics and finance, 6(2), 1–17, doi: https://doi.org/10.1080/23322039.2018.1449780. boyd, j., prescott, e. (1986), financial intermediary coalitions, journal of economic theory, 38(2), 211–232, doi: https://doi.org/10.1016/0022-0531(86)90115-8. boyd, j. h., smith, b. d. (1998), the evolution of debt and equity markets in economic development, economic theory, 12(3), 519–560, doi: https://doi.org/10.1007/s001990050234. shravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 30 breitung, j. (2000), the local power of some unit root tests for panel data, in: b. baltagi ed.., non-stationary panels, panel cointegration, and dynamic panels, advances in econometrics, 15, 161–178. breitung, j., pesaran, m. h. (2008), unit roots and cointegration in panels, in: l. matyas, and p. sevestre eds., the econometrics of panel data, third edition, 279–322, kluwer academic. breitung, j., das, s. (2005), panel unit root tests under cross-sectional dependence, statistica neerlandica, 59(4), 414–433, doi: https://doi.org/10.1111/j.1467-9574.2005.00299.x. demetriades, p. o., james, g. a. (2011), finance and growth in africa: the broken link, economic letters, 113 (3), 263–265, doi: https://doi.org/10.1080/1540496x.2016.1116282. demetriades, p. o., luintel, k. b. (1996), financial development, economic growth and banker sector controls: evidence from india, economic journal, royal economic society, 106(4), 359–374. demetriades, p. o., luintel, k. b. (1996a), banking sector policies and financial development in nepal. oxford bulletin of economics and statistics, 58(2), 355–372. dicky, d. a., fuller, w. a. (1979), distribution of estimators for autoregressive time series with a unit root, journal of the american statistical association, 74(366), 427–431, doi: https://doi.org/10.2307/2286348. divino, j. a., teles, v. k., andrade, j. p. (2009), on the purchasing power parity for latinamerican countries, journal of applied economics, 12(1), 33–54, doi: https://doi.org/10.1016/s1514-0326(09)60004-0. emecheta, b. c., and ibe, r. c, (2014), impact of bank credit on economic growth in nigeria: application of reduced vector autoregressive var technique, european journal of accounting auditing and finance research, 2(9), 11–21. emirmahmutoglu, f., kose, n. (2011), testing for granger causality in heterogeneous mixed panels, journal of economic modelling, 28(3), 870–876, doi: https://doi.org/10.1016/j.econmod.2010.10.018. engle, r. f., granger, c. w. j. (1987), co-integration and error-correction: representation, estimation and testing, econometrica, 55, 251–276. favara, g. (2003), an empirical reassessment of the relationship between finance and growth, imf working paper, no. 03/123. fitzgerald, v. (2006), financial development and economic growth: a critical view, accessed 26 january 2016, http://www.un.org/en/development/desa/policy/wess/wess_bg_papers/bp_wess2006_fi tzgerald.pdf. fowowe, b. (2011), financial liberalization in sub-saharan africa: what do we know? journal of economic surveys, 27(1), 1–37, doi: https://doi.org/10.1111/j.1467-6419.2011.00689.x. ghali, k. h. (1998), public investment and private capital formation in a vector errorcorrection model of growth, applied economics, 30(6), 837–844, doi: https://doi.org/10.1080/000368498325543. giri, a. k., mohapatra, g. (2012), financial development and economic growth: evidence from indian economy, international journal of applied research and studies, 12(1), 1–17, doi: https://doi.org/10.18488/journal.aefr/2015.5.10/102.10.1159.1173. goldsmith, r. w. (1969), financial structure and development, yale university press, new haven. greenwood, j., jovanovic, b. (1990), financial development and economic development, economic development and cultural change, 15(3), 257–268. dynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 31 gregorio, j. d., guidotti, e. p. (1995), financial development and economic growth, world development 23(3), 433–448. gurgul, h., lach, u. (2012), financial development and economic growth in poland in transition: causality analysis, czech journal of economics and finance, 62(4), 347–367. hadri, k. (2000), testing for stationarity in heterogeneous panel data, econometrics journal, 3(2), 148–61, doi: https://doi.org/10.1111/1368-423x.00043. hassan, k. m., sanchez, b., yu, j. k. (2011), financial development and economic growth: new evidence from panel data, the quarterly review of economics and finance, 51(1), 88–104, doi: https://doi.org/10.1016/j.qref.2010.09.001. hurlin, c. (2004), testing granger causality in heterogeneous panel data models with fixed coefficients, document de recherche leo. hye, q. m. a. (2011), financial development index and economic growth: empirical evidence from india, the journal of risk finance, 12(2), 98–111, doi: https://doi.org/10.1108/15265941111112820. im, kyung, pesaran, s., hashem, m., shin,y. (2003), testing for unit roots in heterogeneous panels, journal of econometrics, 115(1), 53–74, doi: https://doi.org/10.1016/s0304-4076(03)00092-7. inoue, t., hamori, s. (2010), how has financial deepening affected poverty reduction in india? empirical analysis using state level panel data, ide discussion paper no.249. jayaratne, j, philip, s. (1996), the finance-growth nexus: evidence from bank branch deregulation, quarterly journal of economics, 111(3), 639–670, doi: https://doi.org/10.2307/2946668. kamal, m. m. k. (2015), an ecm approach for long run relationship between real exchange rate and output growth: evidence from bangladesh, dhaka university journal of science, 63(2), 105–110. kamat, m., kamat m. (2007), does financial growth lead economic performance in india? causality-cointegration using unrestricted vector error correction models, mpra paper no. 6154. kendall, j. (2012), local financial development and growth, journal of banking and finance, 36(5), 1548–1562. king, r. g., levine, r. (1993a), finance and growth: schumpeter might be right, quarterly journal of economics, 108(3), 717–737, doi: https://doi.org/10.2307/2118406. king, r. g., levine, r. (1993b), finance, entrepreneurship, and growth: theory and evidence, journal of monetary economics, 32(3), 513–542, doi: https://doi.org/10.1016/0304-3932(93)90028-e. korkmaz, s. (2015), impact of bank credits on economic growth and inflation, journal of applied finance and banking, 5(1), 57–69. kumar, r. r., stauvermann, p. j., loganathan, n., kumar, r. d. (2015), exploring the role of energy, trade and financial development in explaining economic growth in south africa: a revisit, renewable and sustainable energy reviews, 52, 1300–1311, doi: https://doi.org/10.1016/j.rser.2015.07.188. kumar, s., chauhan, s. (2015), impact of commercial deposit in banks with gdp in context with pradhan mantri jan dhan yojna, bvimsr’s journal of management and research, 7(1), 53–59. levin, a., lin, c. f., chu, c. s. (2002), unit root tests in panel data: asymptotic and finite-sample properties, journal of economics metrics, 108(1), 1–24, doi: https://doi.org/10.1016/s0304-4076(01)00098-7. levine, r., zervos, s. (1998), stock markets, banks, and economic growth, american economic review, 88(3), 537–558. shravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 32 lucas, r. e. (1988), on the mechanics of economic development, journal of monetary economics, 101(3), 455–470, doi: https://doi.org/10.1016/0304-3932(88)90168-7. lund, m. t. (2010), foreign direct investment: catalyst of economic growth? phd thesis, university of utah, salt lake city. maddala, g. s., wu, s. (1999), a comparative study of unit root tests with panel data, oxford bulletin of economics and statistics, 61(4), 631–52, doi: https://doi.org/10.1111/1468-0084.0610s1631. majid, m. s. a., mahrizal (2007), does financial development cause economic growth in the asean-countries?, savings and development, 31(4), 369–398. mckinnon, r. i. (1973), money and capital in economic development brookings institution, washington, dc. minija (2012), financial development and economic growth in india. phd thesis, pondicherry university. mishra, a., mishra, v. (2014), examining the income convergence among indian states: time series evidence with structural breaks, accessed on 13 september 2016, http://www.isid.ac.in/~epu/acegd2014/papers/vinodmishra.pdf. mishra, b. s. (2003), allocative efficiency of the indian banking system in the post reform period: a state level analysis, rbi occasional papers, 24(3), 161–186. moon, h. r., perron, b. (2004), testing for a unit root in panel with dynamic factors, journal of econometrics, 122, 81–126, doi: https://doi.org/10.1016/j.jeconom.2003.10.020. moudatsou, a., kyrkilis, d. (2011), fdi and economic growth: causality for the eu and asean, journal of economic integration 26(3), 554–577. naceur, s. b., ghazouani, s. (2009), stock markets, banks and economic growth: empirical evidence from the mena region, research in international business and finance, 21(2), 297–315, doi: https://doi.org/10.1016/j.ribaf.2006.05.002. nain, m. z., kamaiahn, b. (2014). financial development and economic growth in india: some evidence from non-linear causality analysis, economic change restructuring, 47(4), 299–319, doi: https://doi.org/10.1007/s10644-014-9151-5. naira (2016), the impact of money deposit banks on the economic development of nigeria, accessed on 13 september 2016, http://nairaproject.com/projects/1333.html. nyasha, s., odhiambo, n. m. (2018), financial development and economic growth nexus: a revisionist approach (february 2018), economic notes, 47(1), 223–229, doi: http://dx.doi.org/10.1111/ecno.12101. o’connell, p. (1998), the overvaluation of purchasing power parity, journal of international economics, 44(1), 1–19, doi: https://doi.org/10.1016/s0022-1996(97)00017-2. obradovic, s., grbic, m. (2015), causality relationship between financial intermediation by banks and economic growth: evidence from serbia, prague economic paper, 24(1), 60–72, doi: https://doi.org/10.18267/j.pep.500. onuorah, a. c., ozurumba, b. a. (2013), bank credits: an aid to economic growth in nigeria, information and knowledge management, 33, 1125–1136. orsal, d. d. k. (2007), comparison of panel cointegration tests, humboldt-university berlin, germany sbf discussion paper no. 29. osman, e. g. a. (2014), the impact of private sector credit on saudi arabia economic growth gdp: an econometrics model using ardl. approach to cointegration, international journal of social science, 3(6), 109–117. pala, a. (2016), which energy-growth hypothesis is valid in oecd countries? evidence from panel granger causality, international journal of energy economics and policy, 6(1), 28–34. dynamics of financial development and economic growth: panel data analysis… dynamic econometric models 18 (2018) 5–34 33 patrick, h. (1966), financial development and economic growth in underdeveloped countries, economic development and cultural change, 20, 174–189, doi: http://www.jstor.org/stable/1152568. pedroni, p. (1999). critical values for cointegration tests in heterogeneous panels with multiple repressors, oxford bulletin of economics and statistics, 0305–9049, 653–70. pedroni, p. (2000), fully modified ols for heterogeneous cointegrated panels, advances in econometrics, 15, 93–130. pedroni, p. (2004), panel cointegration; asymptotic and finite sample properties of pooled time series tests, with an application to the ppp hypothesis, econometric theory, 20, 575–625. pesaran, m. h. (2007), a simple panel unit root test in the presence of cross-section dependence, journal of applied econometrics, 22(2), 265–312, doi: https://doi.org/10.1002/jae.951. pradhan, p. r. (2011), financial development, growth and stock market development: the trilateral analysis in india, journal of quantitative economics, 9(1), 134–145. rajan, r. g., zingales, l. (1998), financial dependence and growth, american economic review, 88(3), 559–586. ram, r. (1999), financial development and economic growth: additional evidence, journal of development studies, 35(4), 164–174, doi: https://doi.org/10.1080/00220389908422585. ramirez, m. d. (2006), economic and institutional determinants of fdi in chile: a time series analysis 1960–2001, contemporary economic policy, 24 (3), 459–71, doi: https://doi.org/10.1093/cep/byj027. ray, s. (2013), does financial development promote economic growth in india?, international journal of economic practices and theories, 3(3), 140–151. robinson, j. (1952), the generalization of the general theory, in: the rate of interest and other essays. macmillan, london. rousseau, p. l., wachtel, p. (2000), equity markets and growth: cross country evidence on timing and outcomes, 1980–95, journal of banking and finance, 24(12), 1933–1957. safdari, m., mahmoodi, m., mahmoodi, e. (2011), the causality relationship between export and economic growth in asian developing countries, american journal of scientific research, 25, 40–45. schumpeter, j. a. (1911), theory of economic development, munich duncker and humblot. sehrawat, m., giri, a. k. (2015), financial development, poverty and rural-urban income inequality: evidence from south asian countries, qual quant, 15, 1–14. sharma, d., ranga, m. (2014), impact of saving deposits of commercial banks on gdp, indian journal of applied research, 4(9), 95–96, doi: https://doi.org/10.15373/2249555x. sharma, r., bardhan, s. (2016), finance growth nexus across indian states: evidences from panel cointegration and causality tests, journal of economic change restructuring, 50(1), 1–20. shaw, e. (1973), financial deepening in economic development, new york: oxford university press. singh, t. (2008), financial development and economic growth nexus: a time-series evidence from india, applied economics, 40 (12), 1615–1627. stern, n. (1989), the economics of development: a survey, the economic journal, 99(397), 597–685, doi: https://doi.org/10.2307/2978421. tiwary, m. k., thampy, a. (2015), financial development and economic growth: evidence from states in india, journal of financial accounting and management, 6(1), 41–86. shravani sharma and supran kumar dynamic econometric models 18 (2018) 5–34 34 westerlund, j. (2007), testing for error correction in panel data, oxford bulletin of economics and statistics, 69,709–748, doi: https://doi.org/10.1111/j.1468-0084.2007.00477.x. witkowska, d., and kompa, k. (2017), how the change of governing party influences the efficiency of financial market in poland, dynamic econometric models, 7(1), 147–159. yang, y., yi, m. (2007), does financial development cause economic growth? implication for policy in korea, journal of policy modeling, 30(5), 827–840, doi: https://doi.org/10.1016/j.jpolmod.2007.09.006. younsi, m., bechtinib, m, (2018), economic growth, financial development and income inequality in brics countries: evidence from panel granger causality tests, munich personal repec archive, mpra paper no. 85384. microsoft word 00_tresc.docx dynamic econometric models vol. 9 – nicolaus copernicus university – toruń – 2009 roman huptas cracow university of economics intraday seasonality in analysis of uhf financial data: models and their empirical verification a b s t r a c t. the aim of this paper is to outline the typical characteristics of the ultra-highfrequency financial data and to present estimation methods of intraday seasonality of trading activity. ultra-high-frequency financial data (transactions data or tick-by-tick data) is defined to be a full record of transactions and their associated characteristics. we consider two nonparametric estimation methods: cubic splines and a nadaraya-watson kernel estimator of regression. both approaches are compared empirically and applied to financial data of stocks traded at the warsaw stock exchange. k e y w o r d s: financial uhf data, intraday seasonality, diurnal pattern, cubic splines, kernel estimation. 1. introduction the last dozen or so years has witnessed mounting global interest in analyses of the microstructure of financial markets. research into the microstructure of financial markets centres around explaining the process behind the shaping of the price of financial instruments and analyses of individual trade events. the impact of various transaction factors and mechanisms on the way in which instruments prices are shaped was captured by the so-called theoretical microstructure models. a review of those models and numerous issues collectively referred to as market microstructure effects was incorporated into o’hara (1995) and dacorogna et al. (2001) (cf. doman, doman, 2004; bień, 2006). analysis of financial market processes and empirical verification of hypotheses arising from theoretical microstructure models were made possible by the newly-gained access to transactional databases. these databases became the source of specific financial time series referred to as ultra-high-frequency or tick-by-tick data. new modelling tools for financial time series anticipate specific qualities of transaction data. they include, above all, asynchronous distributions of obserroman huptas 130 vations relative to time units and discrete price changes. additionally, the individual events of the transaction process manifest themselves with varying frequency from one time period to another. consequently, there may be – from one day to the next – certain repeat pattern of intensity with which transactions are concluded. in pertinent literature, this pattern is referred to as intraday seasonality of durations, with durations or waiting times standing for the time spans between trade events. however, before one can use econometric models in analyses of ultra-high-frequency time series, it is essential to eliminate intraday seasonality, which strongly manifests itself in the time series. in estimating intraday seasonality use is mostly made of selected nonparametric statistical methods. the aim of this paper is to outline the typical characteristics of uhf financial data and further to present modelling and estimation methods of intraday seasonality of transaction activities. within the framework of these methods, two nonparametric approaches will be presented: cubic splines interpolation and kernel estimations of regression functions. both approaches will be verified and compared empirically on the basis of data extracted from the polish share market. 2. characteristics of uhf financial data several years ago a new term became operative – „ultra-high frequency data”, also known as „tick-by-tick data” or „transaction data”. these are time series composed of trade event features to which the exact time of their appearance was assigned. thus observations are recorded asynchronously on time units. uhf financial data have a few characteristic qualities, which do not manifest themselves at lower frequencies. the characteristic features of time series of transaction data include: nonsynchronous distribution of observations over time units, discrete transaction price changes, appearance of a number of transactions in the same single second, bid-ask bounce of transaction prices and intraday seasonality i.e. transactions reveal a daily periodic pattern. the most important quality of uhf financial data is the nonsynchronous i.e. erratic distribution of observations over time units. those data can, for instance, be aggregated so that they correspond to equal sequential time units (multiples of minutes, hours, days) and then – in analyses – enable use of a whole range of the garch models. on the other hand, such aggregation of transaction data and their analysis as observations made and selected at equal intervals leads to a loss of information furnished by the transaction process itself. transactions or changes in the share price do not happen at equal intervals. consequently, the durations between transactions involving shares of a given company may provide relevant information as to the intensity of their trading. thus the assumption that changes in prices or transactions are equidistant in terms of time may intraday seasonality in analysis of uhf financial data: models … 131 cause us to draw false conclusions. the problem of nonsynchronous trading and relevant examples are dealt with more extensively in tsay (2002, p. 207) (cf. doman, doman, 2004; osińska, 2006). an alternative way of analysis of financial data distributed asynchronously over time units involves using the so-called transaction-time models (acd, uhf-garch models etc). with those models, raw data are used for analysis. owing to that, information inherent in the duration of the time between selected trade events can also come into focus. duration analyses may furnish information on the microstructure of the financial market, affording a more accurate insight into various market interdependencies. in pertinent literature, the most frequently modelled durations between trade events are trade durations, price durations, volume durations (cf. engle, russell, 1997, p. 1149). 3. intraday seasonality of durations under normal economic conditions, transactions reveal a daily seasonality factor. it appears that the number of transactions is higher immediately after the opening of business than prior to the close of the session (when the time gaps between transactions are the shortest), and markedly smaller during midday hours, i.e. in the middle of the session (socalled “lunchtime effect” when durations between transactions are also the longest). thus, there exists a certain repetitious pattern of transaction intensity for each day. this is termed „intraday seasonality of durations”. consequently, engle and russell (1997) recommend decomposition of duration into a deterministic component )( itφ depending on moment it of the commencement of a given duration, and a stochastic component ix̂ , which is free from the seasonality effect and which models process dynamics. pertinent literature (cf. engle, russell, 1997) recommends that data be transformed as follows: , )( i i i t x x φ = ) (1) where: 1−−= iii ttx duration between transactions at time it and 1−it , ix̂ duration purged of the seasonality effect, )( itφ multiplied factor of intraday seasonality at time it . the seasonal factor )( itφ is construed as the average duration of each time unit during which we made data observations (most often denoting the average duration for each second). the diagram which illustrates intraday seasonality pattern, also known as diurnal pattern or time-of-day function mostly has the shape of the letter u turned upside down. in numerous situations researchers lack adequate information to fully specify a parametric function of intraday seasonality. despite the fact that intraday roman huptas 132 cyclicality is not the key issue of investigation, it still cannot be ignored, but much rather needs to be included in analyses. thus, in order to estimate the time-of-day function use can be made of selected nonparametric statistical methods such as splines, fourier series, neural networks, wavelet analyses or kernel methods. in most works on duration modelling use is made of cubic splines or kernel estimations. in pertinent literature the time-of the day function is determined most commonly by means of splines. this is a method which allows smoothing of average durations between events in subsequent time periods. firstly, all durations during all the subsequent hours of the sessions on each day are averaged. then a cubic spline with knots on every full hour of the session is determined. knots correspond to previously determined average durations. this version of approximation of the daily period factor was presented in the paper (engle, russell, 1997). with a view to ensuring enhanced elasticity, the authors added a knot on the half-hour of the last hour of the session to capture the fast growing trade activity prior to the close of the stock market. a slightly different approach to cubic spline approximation can be observed in paper (bauwens, giot, 2000, p. 135). the authors note that the intraday seasonality factor may vary from one day of the week to the other, i.e. the shape of the periodic factor for monday can be distinct from that for tuesday etc. consequently, the estimation of the intraday seasonality function was conducted separately for each day of the week to allow for possible seasonality within the week. in the first step, durations for the subsequent half-hours of the session were averaged separately for each day of the week and then the parameters of cubic splines were estimated for knots at full hours and half hours. an alternative and second most commonly practised method of estimation of intraday seasonality function is the kernel estimation method. the intraday seasonality pattern is estimated as the nadaraya-watson kernel estimator of regression of raw durations on the time of the day (cf. bauwens, veredas, 2004, p. 398): ,)( 1 1 ∑ ∑ = = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = n i n i n i n i i h tt k h tt kx tφ where: t number of seconds since the midnight of each day (or since the start of a session), ix durations corresponding to moments it ( ix is a dependent variable), it number of seconds since the midnight of each day (or since the start of a session) until the moment of a given transaction, k kernel function, nh bandwidth, s standard deviation of sample it , n number of observations. intraday seasonality in analysis of uhf financial data: models … 133 as far as the kernel function is concerned, the paper (bauwens, veredas, 2004, p. 398) makes use of the quartic kernel (with optimal bandwidth of 5/178,2 −sn ) which has the following shape: , 1||,0 1||,)1( 16 15 )( 22 ⎪⎩ ⎪ ⎨ ⎧ > ≤−= xdla xdlaxxk and in paper (bauwens, giot, 2002, p. 13) use is made of the gamma kernel function. in the case of paper (bauwens, veredas, 2004, p. 398) the estimation of the intraday seasonality function is made separately for each day of the week to incorporate possible seasonality arising from the transaction repetition patterns also over a week-long period. 4. empirical example the empirical verification of the methods presented above was carried out on the basis of time series involving trades in the shares of three companies listed in the wig20 index: telekomunikacja polska s.a. (tpsa)/polish telecom/, agora s.a. (agora) and cez s.a. (cez) over a period between 22 march 2009 and 25 june 2009. the analysis covers transactions closed during the continuous quotation phase. on the basis of such time series, durations between each transaction were determined. additionally, the time lags between the close of the session and the opening of next day’s trading were removed. tabel 1. descriptive statistics of transaction durations cez agora tpsa number of observations 13919 19183 65166 mean 98.930 84.840 25.110 standard deviation (sd) 224.720 176.560 43.640 dispersion index ( =mean/sd) 2.270 2.080 1.740 minimum 1 1 1 maximum 4196 4003 833 acf(1) 0.220 0.225 0.212 acf(2) 0.157 0.176 0.168 q(5) 1805.430 2799.010 8948.820 q(10) 2569.120 4111.970 13294.120 q(15) 3056.570 5179.310 16546.610 q(20) 3298.350 5866.790 19483.450 note: acf(k) – the value of the k-th order autocorrelation coefficient, q(k) – the value of the ljung-box q-statistic of k-th order, descriptive statistics in seconds. roman huptas 134 the basic descriptive statistics of transaction durations for the shares in question are illustrated in table 1. in our example, we witness three companies experiencing different trading activity patterns. the majority of the transactions involved tpsa, for which the average duration between transactions is 25 seconds. cez reported the fewest transactions and the average transaction time is 99 seconds. agora, in turn, is a company of average liquidity and the average duration is around 85 seconds. in an analysis of the features of the distribution of durations our attention is momentarily attracted to marked overdispersion, i.e. the standard deviation exceeds the mean. the dispersion indexes (the ratio standard deviation to mean) are generally very high, which may imply great dynamics of the series in question. it is worth noting that the greater the frequency of trades, the lower the value of the dispersion index. the values of the q ljung-box statistics in table 1 formally test the null hypothesis whereby there is no autocorrelation of durations respectively for the fifth, the tenth, the fifteenth and the twentieth order. clearly, on the basis of the determined test statistics, the null hypothesis whereby there is no autocorrelation is definitely rejected for all three companies. duration autocorrelation is thus extremely strong. figure 1. autocorrelation functions of transaction durations for cez, agora and tpsa graphs representing the autocorrelation function of the durations for the companies in question are to be found in figure 1. regardless of the company, the first values of the autocorrelation function are surprisingly low and stand at around 0.22. for cez shares the acf function fairly soon shrinks to zero for the first several dozen delays, only to level off. as far as tpsa shares are concerned, the autocorrelation function takes much more time to decrease, approximately at hyperbolic speed (rate), which is typical of long memory processes. this evidences high „stability” (persistence) of the process. moreover, the high values of lower order autocorrelations indicate stronger clustering of transaction activities. the dynamics of duration processes and the effect of clustering of transaction activities may be noted in figure 2 containing graphs of the series of the first 5000 observations. acf cez -0,05 0,00 0,05 0,10 0,15 0,20 0,25 0 100 200 300 400 500 lags acf agora -0,05 0,00 0,05 0,10 0,15 0,20 0,25 0 100 200 300 400 500 lags acf t psa -0,05 0,00 0,05 0,10 0,15 0,20 0,25 0 100 200 300 400 500 lags intraday seasonality in analysis of uhf financial data: models … 135 figure 2. plots of trade durations for agora and tpsa – first 5000 observations the existence of very powerful autocorrelation of durations may result from the midday seasonality of transactional activity. with this in mind, patterns of intraday seasonality were estimated by means of four methods, and further verified empirically to determine whether the method selected translates into effective elimination of high autocorrelation of the series under research. the following approaches, described extensively in point 3, were applied: 1. nadaraya-watson estimator of regression of the duration on the time of the day, determined separately for each day of the week (nw_days); 2. nadaraya-watson estimator of regression of the duration on the time of the day, ignoring possible seasonality over a week-long period (nw); 3. cubic splines with knots on every full hour and every half-hour of the session, determined separately for each day of the week (cs_days); 4. cubic splines with knots on every full hour and every half-hour of the session, ignoring possible seasonality over a week-long period (cs). in the event of kernel estimators, use was made of the quartic kernel with the optimal bandwidth of 5/178,2 −sn . next, after the intraday seasonality factor was estimated, durations purged of the seasonality effect were determined pursuant to formula (1). all algorithms and procedures were implemented using the gauss software. figure 3. intraday seasonality patterns for cez, agora and tpsa figure 3 depicts the shape of intraday seasonality patterns (nw and cs methods) for the three companies under analysis, without regard for the effect that the different days of the week have. figure 4 shows plots of the estimated time-of-day functions for subsequent days of the week for cez, agora and 0 2000 0 2000 4000 s e c o observations trade durations agora 0 1000 -1000 1000 3000 5000 s e c o observations trade durations tpsa -20180 9s e c o hours intraday seasonality cez n w c s -20180 9s e c o hours intraday seasonality agora n w c s 0 9s e c o hours intraday seasonality tpsa n w c s roman huptas 136 tpsa respectively, obtained by means of the kernel estimator (nw) and cubic splines (cs). figure 4. intraday seasonality patterns for the days of the week for cez, agora, tpsa the graphs presenting the estimated time-of-day functions have the shape of the letter u turned upside down and reveal unequivocally that durations are subject to daily seasonality. the durations between transactions are markedly shorter after the opening and before the close of the session than at midday. the extent of trading activity between 12:00 a.m. and 2:00 p.m. is noticeably less, due, amongst others, to the lunchtime effect. it should be noted that in the case of companies listed on the american and west european stock markets, the effect is less manifest than in the case of their australian counterpart, where the effect is more pronounced, i.e. the hump in the graph is manifestly spikier (cf. (bauwens, giot, 2001; hautsch 2004)). similarly, trading activity at the opening of the session is more intense as traders begin to accommodate the information of the night before (macroeconomic data, etc). trading activity at the close of trade can be explained in terms of some investors’ attempt to close their open positions. it is worth noting that intraday seasonality will vary from one day of the week to the next (figure 4). it seems that regardless of the type of company, trading activity is most intense on tuesdays and wednesdays than on any of the other days. duration statistics purged of intraday seasonality by means of the four above-named methods were included in table 2. deseasonalisation of the data 20220 9s e c o hours intraday seasonality (nw) the days of the week agora mo n tue 20220 9 s e c o hours intraday seasonality (cs) the days of the week agora mo n tue -200 300 9 s e c o hours intraday seasonality (nw) the days of the week cez mo n tue 0500 9s e c o hours intraday seasonality (cs) the days of the week cez mo n tue -40 60 9 s e c o hours intraday seasonality (nw) the days of the week tpsa mo n tue 0100 9s e c hours intraday seasonality (cs) the days of the week tpsa mo n tue intraday seasonality in analysis of uhf financial data: models … 137 partly reduced the autocorrelation of transaction durations. from the point of view of effective elimination of seasonality impact on autocorrelation “measured” in terms of the values of the ljung-box test statistics, the nw_days (table 2) appears to be the most effective method. in the case of tpsa and agora it was definitely the most successful, i.e. the values of test statistics are the lowest of all four methods used. on the other hand, in the case of cez company, it ranked number two, with the cs_days approach ranked the highest. tabel 2. descriptive statistics of adjusted transaction durations after deseasonalisation by means of the four methods stock method mean sd disp. index acf(1) q(5) q(10) q(15) q(20) cez nw_days 0.990 2.320 2.340 0.205 1360.150 1897.080 2101.860 2240.300 nw 0.990 2.200 2.220 0.215 1537.830 2111.380 2475.990 2676.680 cs_days 0.980 2.280 2.320 0.190 1290.340 1782.540 1972.950 2121.000 cs 0.970 2.130 2.190 0.209 1535.590 2137.890 2530.380 2759.580 agora nw_days 0.990 1.980 2.000 0.215 2518.120 3521.420 4333.470 4886.400 nw 0.990 1.980 2.000 0.211 2675.970 3719.670 4600.870 5169.360 cs_days 0.990 1.950 1.970 0.209 2599.490 3632.710 4520.890 5097.310 cs 0.990 1.950 1.970 0.215 2553.890 3604.400 4480.420 5078.150 tpsa nw_days 0.990 1.660 1.670 0.195 7357.560 10568.660 12843.060 14864.480 nw 0.990 1.680 1.700 0.197 7614.770 10964.070 13401.270 15598.530 cs_days 0.990 1.670 1.690 0.197 7527.590 10874.810 13294.540 15400.190 cs 0.990 1.680 1.700 0.200 7793.400 11248.550 13802.000 16085.570 note: sd – standard deviation; acf(k) – the value of the k-th order autocorrelation coefficient, q(k) – the value of the ljung-box q-statistic of k-th order, descriptive statistics in seconds. an analysis of the values of ljung-box statistics implies that the nw_days and cs_days approaches should be used. consequently, in eliminating intraday seasonality, the “day of the week” effect i.e. possible seasonality arising from variations in trading activity over the entire week should be taken into account. regardless of the deseasonalisation method used, the values of ljung-box test statistics for the companies in question dropped by approximately 15%-25%, but still continued to be very high. thus, the null hypothesis implying lack of autocorrelation continues to be rejected on each reasonable level of significance. this bears witness to the fact that the dynamics of transaction durations are influenced by factors other than the purely deterministic seasonality effect, which in turn is due to the structure of the share market. 5. summary based on the results of empirical data, the application of the kernel estimator of regression separately for each day of the week appeared to be the most effective method of elimination of intraday seasonality impact on the autocorrelation of transaction durations. it is noteworthy, though, that the results for all analytical methods used are highly similar. in the case of splines, their prelimiroman huptas 138 nary averaging of durations between events for the subsequent full or half-hours may become something of a drawback. the extent of data aggregation and the elasticity of the estimated function can be reduced by increasing the number of knots used in the spline. so it appears that the inclusion of intraday seasonality models in the base models is a natural step, and wins over earlier data filtrations and testing if the use of a two-step or one-step approaches will have identical impact on the quality of the estimators determined. references bauwens, l., giot, p. (2000), the logarithmic acd model: an application to the bid-ask quote process of three nyse socks, annales d’économie et de statistique, 60, 117–149. bauwens, l., giot, p. (2001), econometric modelling of stock market intraday activity, kluwer academic publishers, boston. bauwens, l., giot, p. (2002), asymmetric acd models: introducing price information in acd models, core discussion paper 9844. bauwens, l., veredas, d. (2004), the stochastic conditional duration model: a latent variable model for the analysis of financial durations, journal of econometrics, 119, 381–412. bień, k. (2006), model acd – podstawowa specyfikacja i przykład zastosowania (acd model – basic specification and example of application), przegląd statystyczny (statistical survey), t.53, z. 3, 83-97. dacorogna, m. m., gençay, r., müller, u., olsen, r. b., pictet, o. v. (2001), an introduction to high-frequency finance, academic press, san diego. doman, m., doman, r. (2004), ekonometryczne modelowanie dynamiki polskiego rynku finansowego (econometric modelling of dynamics of polish financial market), wydawnictwo ae w poznaniu, poznań. engle, r. f., russell, j. r. (1997), autoregressive conditional duration: a new model for irregularly spaced transaction data, econometrica, 66, 1127–1162. hautsch n. (2004), modelling irregularly spaced financial data, springer-verlag, berlin, heidelberg. o’hara, m. (1995), market microstructure theory, blackwell inc., oxford. osińska, m. (2006), ekonometria finansowa (financial econometrics), pwe, warszawa. tsay, r.s. (2002), analysis of financial time series, wiley series in probability and statistics, john wiley& sons, new york. wewnątrzdzienna sezonowość w analizie danych finansowych uhf: modele i ich empiryczna weryfikacja z a r y s t r e ś c i. celem artykułu jest krótkie przedstawienie cech charakterystycznych dla danych finansowych uhf oraz prezentacja metod modelowania i szacowania wewnątrzdziennej sezonowości aktywności transakcyjnej. w ramach tych metod są przedstawione dwa podejścia nieparametryczne: interpolacja za pomocą kubicznych funkcji sklejanych oraz estymacja jądrowa funkcji regresji. oba prezentowane podejścia są zweryfikowane i porównane empirycznie na podstawie danych z polskiego rynku akcji. s ł o w a k l u c z o w e: dane finansowe uhf, wewnątrzdzienna sezonowość, funkcje sklejane, estymator nadaraya-watsona. microsoft word dem_2014_93to104.docx © 2014 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2014.005 vol. 14 (2014) 93−104 submitted october 20, 2014 issn accepted december 23, 2014 1234-3862 ewa m. syczewska* the eurpln, dax and wig20: the granger causality tests before and during the crisis a b s t r a c t. in this paper the possible interdependence between bilateral exchange rate behavior and the corresponding stock indices is checked, with application to the eurpln rate and the dax and wig20 stock indices. methods and results are similar to previous study of usdpln exchange rate, and sp500 and wig20 indices. the linear (including instantaneous) causality test and the diks-panchenko test are applied to logarithmic returns and to the daily measure of volatility ln , , . differences between beforeand during-crisis period results are less vivid than in case of the u.s. and the polish instruments. but there is a substantial difference between linear (and diks-panchenko) test results and the instantaneous granger-causality test results, on the other hand – between returns and daily volatility. k e y w o r d s: exchange rates, stock indices, financial crisis, risk, granger causality, instantaneous causality, diks-panchenko test. j e l classification: c4, g19. introduction the results presented here are part of the research aimed at detecting interdependencies between bilateral exchange rates and stock indices of the corresponding two countries, while possible influence of the crisis is taken into account. the idea of improvement of the exchange rate model by use of the stock indices from the corresponding two countries has been suggested * correspondence to: ewa m. syczewska, warsaw school of economics, collegium of economic analysis, institute of econometrics, 6/8 madalinskiego street, 02-513 warsaw, poland, e-mail: ewa.syczewska@sgh.waw.pl. ewa m. syczewska dynamic econometric models 14 (2014) 93–104 94 by the bauwens et al. (2008) model of the norwegian krona, and used with success in the previous research by the author. in syczewska (2010, 2013, 2014) this method and the granger linear causality tests were applied to the usd/pln daily data and two indices – one from the warsaw stock exchange, the other representing the u.s. stock indices. it was shown that forecast errors from the arima and garch models diminished when the stock indices were included in the model as additional explanatory variables (see also matuszewska and witkowska (2007) for comparison to other methods of forecasting). in syczewska (2013) research it was shown that it is possible to find a stationary relationship between the usdpln exchange rate and the u.s. and polish stock indices. as the u.s. stock exchange opens before the closing time of the warsaw stock exchange, it was decided to use opening values of the s&p500 stock index and the closing values both for the usdpln exchange rate and for the wig stock index. there was a linear combination of the three variables, for which hypothesis of nonstationarity had to be rejected. in this paper we first perform the granger causality tests for the eurpln exchange rate and two stock indices: dax and wig20. we check whether the stock-index returns granger-cause the relative changes of the exchange rate. to this aim we use both the granger test and the granger test of instantaneous causality1. to take into account possible effects of the crisis, we perform the causality tests and the estimation exercise for two subsamples: up to sept. 15th 2008 (“before crisis”) and since sept. 16th 2008 (“during crisis”). next we apply to the same variables the diks-panchenko causality test, which is assumed to detect also nonlinear causality. we use daily data provided by stooq.pl, choose the period since the beginning of may 2004 (corresponding to the eu accession) up to beginning of october 2013. in syczewska (2010) behavior of rates and quality of modeling was compared for two subperiods: before (up to september 2008) and during crisis (up to end of july 2009), showing that2: 1. volatility of returns, hence errors of forecasts from the arma and garch models of returns, hugely increased during the crisis. 2. introduction of corresponding stock indices returns slightly improved performance of the models and forecasts. 1 this was suggested by specification of models for daily returns of norwegian krona (bauwens et al., 2008). 2 the paper by syczewska (2010) was presented at the international conference “zagadnienia aktuarialne – teoria i praktyka” in warsaw, 2nd-4th september 2009. the eurpln, dax and wig20: the granger causality… dynamic econometric models 14 (2014) 93–104 95 the polish economy performance during the crisis was better than for the other european economies. the current condition of the german economy (the main trade partner of poland) and expectations of future development give hope for further improvement. the eurpln exchange rate is important for the economic analysis due to the proportion of transactions made in euro. 1. description of the data we use daily data since 4th may 2004 (the eu accession) until september 2008 and from september 2008 until the september 2013 for the indices and exchange rates, and for the sake of comparison we perform the tests to the data from the first period (“before the crisis”), and the second period (“during the crisis”) starting in mid-september 2008, with the same number of observations as the first one3. as a measure of returns we use the standard formula: )ln(ln*100 1 ttt yyz , (1) where ty – closing values of an instrument, and also logarithm of proportion of daily maximum and minimum: )/ln(*100 min,max, 2 ttt yy , (2) as a measure of variance/volatility (brooks, 2008). figure 1 shows a typical behavior of the dax index returns: there is a marked increase of volatility during the 2008-09 crisis, later slight stabilization and again increase of volatility due to problems of the euro zone. figure 2 illustrates measure of volatility of the same stock index defined by equation (2). there is a marked increase in volatility during the subprime crisis and subsequent global crisis on financial markets. next, in the year 2009, the situation seems to slightly stabilize, but later due to the problems in the euro area, volatility increases again. the summary statistics in the tables 1 and 2, corresponding to the “precrisis” and “crisis” periods, show that the distribution of the logarithmic returns and returns (2) are skewed, which has to be expected. the results of the doornik-hansen test show that the empirical distribution of all the variables is non-gaussian. note that the empirical distribution of volatility, rdax etc. defined by eqn. (2), is more skewed than the distribution of logarithmic returns for the same instrument. 3 http://stooq.pl database, opening, closing, minimum and maximum daily quotes. we use only the dates, for which all the quotes of variables of interest were available. ewa m. syczewska dynamic econometric models 14 (2014) 93–104 96 figure 1. logarithmic returns of the dax index figure 2. volatility as in (2) of the dax index during the subprime, global and euro area crisis the volatility substantially increases. the last column of table 2 gives a measure of this increase, computed as a relative change of standard deviations for the second as comthe eurpln, dax and wig20: the granger causality… dynamic econometric models 14 (2014) 93–104 97 pared to the first subsample. the increase is close to 25% for the wig20 returns, and around 70% for the eurpln exchange rate and the dax index. the volatility (2) increases are even higher, by 60% for wig20, almost 100% for the dax index and almost 130% for the eurpln exchange rate. table 1. summary statistics, 1098 daily observations from 30th apr. 2004 to 15th sept. 2008 (“before crisis”) variable mean median std. dev. skewness kurtosis doornikhansen test ld_eurpln –0.0323 –0.0361 0.46162 0.1670 1.4535 67.946 ld_dax 0.0377 0.1084 1.0459 –0.5392 3.8284 238.571 ld_wig20 0.0274 0.0696 1.4138 –0.2944 1.4172 61.228 rdax 1.2353 1.0810 0.69478 2.7689 16.812 696.955 rwig20 1.7240 1.5224 0.84969 1.9266 6.7395 568.593 reurpln 0.6831 0.6207 0.31109 1.5026 4.3757 329.235 2. the granger causality test the granger test of the granger causality, applied to the returns of the exchange rate and the corresponding stock indices, can be treated as a tool for choice of explanatory variables in the model for exchange rates. according to the well-known definition, a variable is called a granger-cause for a variable , if lagged values of used as additional regressors in a model describing can improve quality of model and/or forecasts4. there are several tests of this property. let denote observation of the variable for period , , 1,2,…, – lagged observations for the same variable, , 0,1,2,…, – current and lagged values of , , , 1,2,…, , 0,1,2,…, – parameters of the model, – error terms. the granger test of granger causality is based on var – type regressions (regression of y on its lagged values and the same lags of the x variable): tktkttktktt xbxbxbyayay   12121111111 ...... . (3) the null h0: 0... 11211  kbbb means that the does not granger-cause the y variable. another variant is so-called instantaneous causality: the regression includes current value of the variable: 4 but see also the detailed discussion of granger causality concepts and their interpretation in osińska (2008, 2011). ewa m. syczewska dynamic econometric models 14 (2014) 93–104 98 tktktttktktt xbxbxbxbyayay   1212111101111 ...... . (4) the null h0: 0... 1121110  kbbbb in (4) means that is not an instantaneous cause for . table 2. summary statistics, 1098 daily observations from the period 16 sept. 2008 to 1rd feb. 2013 (“during crisis”) v ar ia bl e m ea n m ed ia n s td . d ev s ke w ne ss k ur to si s d oo rn ik -h an se n te st in cr ea se o f s td . d ev . ldeurpln 0.0197 –0.019 0.812 0.040 2.963 216.86 75.97% lddax 0.0233 0.089 1.754 0.123 5.389 507.16 67.68% ldwig20 0.0051 0.057 1.756 –0.289 3.150 217.24 24.19% rdax 2.0195 1.638 1.384 2.212 7.333 1074.24 99.16% rwig20 1.9622 1.552 1.359 2.544 10.458 1224.63 59.95% reurpln 1.0716 0.850 0.715 2.283 7.116 1382.02 129.80% the results of the granger causality tests for the eurpln exchange rate returns and the corresponding german and polish stock indexes returns are the following. (we use ls estimates with robust standard errors; p-values are given in brackets.) 2.1. comparison of results for periods before and during the crisis first we perform the test for subperiod between 2004–2008, i.e. before the crisis (more precisely: since 29 april 2004 until 15th sept. 2008, 1098 observations), and compare the results for a similar subsample (with the same number of observations) starting on 16th sept. 2008, and ending at 2nd feb. 2013. for each pair of the variables the var model was estimated. the ∙,∙ test statistic presented in the second and third column of the table 3 corresponds to the null hypothesis that all lags of the explanatory variable are insignificant in the equation for the dependent variable. next, in order to test instantaneous granger causality, we estimate the regression (4) using the ols with heteroskedasticity and autocorrelation consistent standard errors, and test the restriction h0: 0... 1121110  kbbbb . the results of this test for both periods are given in table 4. in case of pre-crisis period, the null of no causality was not rejected, for data covering the crisis period, only in one case – namely the causality from the eurpln exchange rate returns to the wig20 stock index returns, the null of causality has not been rejected (see table 3). the results of the linear the eurpln, dax and wig20: the granger causality… dynamic econometric models 14 (2014) 93–104 99 granger instantaneous causality test are different. the null hypothesis of lack of causality has been rejected in all the cases (see table 4). table 3. linear granger causality before and during the crisis causality the test statistic [p-value] pair of variables before the crisis during the crisis lddax to ldeurpln 0.550 [0.731] 0.217 [0.955] ldeurpln to lddax 1.521 [0.180] 1.032 [0.397] ldwig20 to ldeurpln 0.880 [0.494] 0.771 [0.571] ldeurpln to ldwig20 1.648 [0.145] 3.660 [0.0027]*** lddax to ldwig20 0.984 [0.427] 1.5016 [0.187] ldwig20 to lddax 0.469 [0.800] 0.283 [0.923] table 4. instantaneous granger causality before and during the crisis instantaneous causality the test f-statistic [p-value] pair of variables before the crisis during the crisis lddax to ldeurpln 10.875 [8.798e-012]*** 46.066 [2.090e-050]*** ldeurpln to lddax 9.652 [2.279-010]*** 27.905 [4.170e-031]*** ldwig20 to ldeurpln 15.021 [1.439e-016]*** 31.744 [2.636e-035]*** ldeurpln to lnwig20 11.026 [5.888-012]*** 47.805 [3.519e-052]*** ldwig20 to lddax 36.004 [6.862-040]*** 75.458 [8.618e-079]*** lddax to ldwig20 54.246 [1.215-058]*** 116.092 [3.142e-113]*** linear causality tests for the volatility measure (2) the similar computations of the linear granger test have been repeated using the volatility measure (2) instead of the log returns. before the crisis the linear test indicates causal relation only from volatility of wig20 to one of the dax stock index (and we would expect the influence to work the other way round). during the crisis the null hypothesis of lack of causality is rejected for all pairs of instruments with exception of rdax to reurpln and to rwig20 (see table 5). table 5. the granger causality test for volatility measure (2) causality the test statistic [p-value] pair of variables before the crisis during crisis rdax to reurpln 1.1059 [0.3554] 1.7194 [0.1273] reurpln to rdax 0.8221 [0.5339] 2.5659 [0.0256]** rwig20 to reurpln 1.2961 [0.2631] 3.0783 [0.0091]*** reurpln to rwig20 0.1288 [0.9859] 2.4572 [0.0317]** rdax to rwig20 0.5797 [0.7156] 1.4111 [0.2175] rwig20 to rdax 2.4223 [0.0340]** 2.4853 [0.0300]** ewa m. syczewska dynamic econometric models 14 (2014) 93–104 100 table 6. the instantaneous causality linear test instantaneous causality the test statistic [p-value] before the crisis during the crisis rdax to reurpln 3.934 [0.0006]*** 11.447 [1.92e-12]*** reurpln to rdax 2.251 [0.0365]** 9.099 [9.91e-10]*** rwig20 to reurpln 1.886 [0.0801]* 8.247 [9.50e-09]*** reurpln to rwig20 0.734 [0.6222] 5.999 [3.48e-06]*** rdax to rwig20 23.263 [6.06e-26]*** 24.122 [6.61e-27]*** rwig20 to rdax 9.934 [1.08e-10]*** 17.241 [4.07e-19]*** the results of the instantaneous causality test before the crisis are in better agreement with the intuition. there is a strong influence from rdax to the exchange rate, weaker from rwig20 to reurpln, and feedback between the two stock indices (see the second column of the table 6). last column of the same table shows results indicating strong feedback between all pairs of variables. 3. the diks-panchenko causality test the improved version of nonlinear hiemstra-jones test was introduced by diks and panchenko (2006). the idea of this nonparametric test is the following: the variable x granger-causes the y , if the current and lagged values of the kttt xxxx  ,...,,: 1 contain information concerning the future values ,..., 21  tt yy , additional to that contained in current and past values of this variable. in their paper diks and panchenko test conditional independence of finite number of lags: ),...,,(|~),...,,,,...,,(| 11111111  xyx lttttltttltttt xxxyyyyxxxy , where lx, ly denote numbers of lags of the variables and , respectively. the diks-panchenko test is an improved version of the hiemstra-jones test, based on comparison of the conditional distributions5. diks and panchenko use one lag for the variables and the 1ty forecast for the next period. let ),,(),,( 1 zyxyyx ttt  and assume that x and y are strictly stationary variables. the null hypothesis of lack of causality means that the conditional distribution of z with respect to y and x is the same as the conditional distribution of z with respect to y alone. the joint distribution and the marginal distributions are described by the formula 5 more detailed analysis of this test can be found in osińska (2008), pp. 226–229. the eurpln, dax and wig20: the granger causality… dynamic econometric models 14 (2014) 93–104 101 )(/),(),(/),,( ,,,, yfzyfyxfzyxf yzyyxzyx  . diks and panchenko (2006) show some deficiencies of the hiemstra and jones approximation of both sides of this formula, and propose their own versions: the null hypothesis of lack of causality implies that 0),,( )( ),( )( ),( )( ),,( ,,,,                 zyxg yf zyf yf yxf yf zyxf eq y zy y yx y zyx g , (5) where ),,( zyxg denotes a positive weights function, e.g. for )(),,( 2 yfzyxg y this simplifies to: )],(),()(),,([ ,,,, zyfyxfyfzyxfeq zyyxyzyxg  . under the null, the expression in parentheses is equal to zero, hence (5) is equal to zero. the null of no causality is rejected if the test statistic is high. diks and panchenko (2006) advocate use for (5) of the following estimator based on index function                   i ikk jj yz ij xy ik y ij zyx ik ddd n iiii nnn t zyx , 1, ,, )2)(1( )2( )(   , where: n – number of observations, xd – number of elements of the x vector etc., w jii , is the index function )(,  ji w ji wwii is equal to 1 if  ji ww and to 0 otherwise,  denotes the supremum norm. diks and panchenko give formula for choice of the  (bandwith) depending on number of observations (the default value is 0.5). they show that when 1 zyx ddd , the estimator is consistent and has asymptotic gaussian distribution. their code for application of the test both for linux and for windows is available at the panchenko’s webpage: http://research.economics.unsw.edu.au/vpanchenko/software/2006_gc_jed c_c_and_exe_code.zip. the results of the diks-panchenko test in our computations we assume 0.5 bandwidth, and we compute the nt statistic for one lag. the results are the following (see table 7). during the ewa m. syczewska dynamic econometric models 14 (2014) 93–104 102 crisis, the null of lack of causality is strongly rejected for all pairs of variables. for period 2004–2008, “before the crisis”, the test does not indicate causal relationship for logarithmic returns, with only exception of lddax  ldwig20 influence (the upper part of the table 7). in case of our volatility measure, the nt test statistic indicates feedback between rdax and rwig20, and some influence from rwig20 towards reurpln (the lower part of the table 7), all at 10%. table 7. the diks-panchenko causality test causality for variables the diks-panchenko test statistics [p-value] logarithmic returns before the crisis during the crisis ¬ lddax → ldeurpln 0.479 [0.3158] 2.697 [0.0035]*** ¬ ldeurpln → lddax –0.661 [0.7458] 3.506 [0.00023]*** ¬ lddax → ldwig20 1.541 [0.0617]* 3.589 [0.0017]*** ¬ ldwig20 → lddax –0.916 [0.8202] 3.118 [0.0009]*** ¬ ldeurpln → ldwig20 –0.760 [0.7763] 3.625 [0.0014]*** ¬ ldwig20 → ldeurpln –1.648 [0.9503] 2.525 [0.0056]*** ln(maxpt/minpt) before the crisis during the crisis ¬ rdax → reurpln –1.104 [0.8652] 2.922 [0.0017]*** ¬ reurpln → rdax –1.253 [0.8949] 3.612 [0.0002]*** ¬ rwig20 → reurpln 1.634 [0.0511]* 4.168 [0.0000]*** ¬ reurpln → rwig20 –1.087 [0.8615] 4.145 [0.0000]*** ¬ rdax → rwig20 1.505 [0.0662]* 2.824 [0.0024]*** ¬ rwig20 → rdax 1.595 [0.0554]* 4.559 [0.0000]*** conclusions the results of the two approaches for testing the granger causality, presented here for a daily closing values of the eurpln exchange rate and the two corresponding stock indices – one for germany as representing the euro zone (dax), the other for poland (wig20) – seem to be influenced by period and the method of testing. economic intuition and previous results concerning similar analysis for the usdpln exchange rate and the two representative indices for the u.s. and the polish stock exchange suggested that the global crisis have changed the direction and strength of the causal relationship for the logarithmic returns, as measured by the linear granger test of the granger causality. there were differences in lack or presence of g-causality between the analysed pairs of variables, and seeming increase of the u.s. data influence during the crisis (perhaps a reflection of temporal changes in capital flows in the global economy). here, for the eurpln exchange rate and the dax and wig20 stock index, we note much higher homogeneity of results – lack of causality before the eurpln, dax and wig20: the granger causality… dynamic econometric models 14 (2014) 93–104 103 crisis, overall presence of two-side causality during the crisis – when the tests are applied to the volatility measure (2). both the linear granger test and the nonlinear diks-panchenko test give similar results (see table 5 and the lower part of the table 7). results of the linear granger test for the logarithmic returns are different: this version of the test does not reject lack of linear causal relationship neither before the crisis nor during the crisis (see table 3) with one exception of granger-causality from the eurpln towards wig20 during the crisis. the diks-panchenko nonlinear test, on the other hand, does not reject lack of causality before the crisis, but strongly rejects it during the crisis (see upper part of the table 7). additional computations for the instantaneous causality in the linear framework show that there is a feedback between all pairs of logarithmic returns and indicate granger causality also for the volatility measure, even before the crisis (see table 4 and 6). the possible explanation of this difference between (lagged) causality test and instantaneous causality test merits further attention – perhaps partial explanation could be based on possible analysis of efficiency of the markets, as it seems that changes in returns and volatility seem to influence the other instrument at once, on the same day. the future research can also take into account indirect influence of other variables (e.g., the u.s. indices) on causal relationship between the two european instruments. references bauwens, l., pohlmeier, w., veredas, d. (2008), high frequency financial econometrics. recent developments, physica-verlag a springer company, heidelberg. bauwens, l., rime, d., succarat, g. (2008), exchange rate volatility and the mixture of distribution hypothesis, in bauwens, l., pohlmeier, w., veredas, d., high frequency financial econometrics. recent developments, physica-verlag a springer company, heidelberg, 7–29. brooks, ch. (2008), introductory econometrics for finance, 2nd ed., cambridge university press, new york. diks, c., panchenko, v. (2006), a new statistics and practical guidelines for nonparametric granger causality testing, journal of economic dynamics and control, 30, 1647– –1669, doi: http://dx.doi.org/10.1016/j.jedc.2005.08.008. hiemstra, c., jones, j. d. (1994), testing for linear and nonlinear granger causality in the stock price-volume relation, journal of finance, 49(5), 1639–1664, doi: http://dx.doi.org/10.2307/2329266. matuszewska, a., witkowska, d. (2007), wybrane aspekty analizy kursu euro/dolar: modele autoregresyjne z rozkładami opóźnień i sztuczne sieci neuronowe (some aspects of the eurousd exchange rate analysis: adl models and neuron network), metody ilościowe w badaniach ekonomicznych viii, modele ekonometryczne, 203–212. ewa m. syczewska dynamic econometric models 14 (2014) 93–104 104 osińska, m. (2008), ekonometryczna analiza zależności przyczynowych (econometric analysis of causal relationships), wydawnictwo naukowe uniwersytetu mikołaja kopernika, toruń. osińska, m. (2011), on the interpretation of causality in granger sense, dynamic econometric models, 11, 129–139. syczewska, e. m. (2010a), increase of exchange rate risk during current crisis, roczniki kolegium analiz ekonomicznych, 21, 99–122. syczewska, e. m. (2010b), changes of exchange rate behaviour during and after crisis, in metody ilościowe w badaniach ekonomicznych (quantitative methods in economics), 11, 145–157. syczewska, e. m. (2013), on exchange-rate model with stock indices as additional regressors (during and after crisis), presented at the international conference „cest’2013. current economic and social topic international colloquium, focused on gender disparities and financial market analysis”, may 23–24, 2013 r., warsaw. syczewska, e. m. (2014), przyczynowość w sensie grangera – wybrane metody (the granger causality – selected tools), metody ilościowe w badaniach ekonomicznych, 15(4), 169–180. eurpln, dax i wig20: testy przyczynowości grangera przed i podczas kryzysu z a r y s t r e ś c i. artykuł dotyczy przyczynowości w sensie grangera zastosowanej do dziennych notowań kursu eurpln oraz odpowiadających mu dwu indeksów – dax i wig20. przedstawia wyniki zastosowania dwu wariantów liniowego testu grangera – w tym wariantu dla tzw. natychmiastowej przyczynowości – oraz nieliniowego testu diksapanchenki do logarytmicznych zwrotów badanych instrumentów oraz do miary odzwierciedlającej zmienność dzienną ln , , jako proporcję minimalnych i maksymalnych notowań w dniu . różnice pomiędzy wynikami testu przyczynowości dla par zwrotów pomiędzy okresem poprzedzającym kryzys a okresem kryzysu nie są tak duże jak w badaniach dotyczących kursów i indeksów polskich i amerykańskich. widoczna jest natomiast różnica pomiędzy wynikami dla zwrotów i dla miary zmienności. s ł o w a k l u c z o w e: kurs walutowy, indeksy giełdowe, kryzys finansowy, ryzyko, przyczynowość w sensie grangera, natychmiastowa przyczynowość, test diksa-panchenki. microsoft word dem_2014_145to160.docx © 2014 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2014.008 vol. 14 (2014) 145−160 submitted october 20, 2014 issn accepted december 23, 2014 1234-3862 joanna górka* option pricing under sign rca-garch models a b s t r a c t. after black and scholes’s groundbreaking work, the literature concerning pricing options has become a very important area of research. numerous option valuation methods have been developed. this paper shows how one can compute option prices using sign rca-garch models for the dynamics of the volatility. option pricing obtained from sign rca-garch models, the black and scholes’s valuation and other selected garch option pricing models are compared with the market prices. this approach was illustrated by the valuation of the european call options on the wig20 index. the empirical results indicated that rca-garch and sign rca-garch models can be successfully used for pricing options. however none of the models can be indicated as the best one for the option valuations for every period and every time to maturity of the options. k e y w o r d s: sign rca-garch models, option pricing, garch models. j e l classification: g13. introduction following the seminal work of black and scholes (1973) and merton (1973), the option literature has developed into an important area of research. the black-scholes formula (henceforth bs) assumes that stock price varies according to the geometric brownian motion. the relationship between the geometric brownian motion and the bs formula presents the following equivalence (elliott and kopp, 1999)1: * correspondence to: joanna górka, nicolaus copernicus university, faculty of economic sciences and management, 13a gagarina street, 87-100 toruń, poland, e-mail: joanna.gorka@umk.pl. 1 this equivalence is obtained by applying itô lemma with function   tt ssf log . joanna górka dynamic econometric models 14 (2014) 145–160 146 , 2 2 0 twt ttttt essdwsdtsds           (1) where: ts – a stock price,  – the drift rate, annualized expected value of st, t – time, tw – the wiener process (brownian motion), 0 – the annualized volatility of st. the bs formula assumes that the returns of the underlying asset (stock price) follow a normal distribution with constant volatility. empirical evidence has shown, however, that the model is in conflict with facts, especially for short-run returns2. the financial markets research indicated that financial series, such as stock returns, foreign exchange rates and others, exhibit leptokurtosis and volatility varying in time. hence the assumption of constant volatility is often strongly violated. therefore, several option valuation models have been developed to incorporate stochastic volatility. one approach is to use continuous-time stochastic volatility models. another approach is to use discrete-time generalized autoregressive conditionally heteroskedastic (garch) models (amongst others engle,1982; bollerslev,1986). the choice of discrete-time garch models for this study was motivated by two facts, that:  the inclusion of linear autoregressive dynamics, ar(1), affects option prices (hafner and herwartz, 2001),  the random coefficient autoregressive models with the sign function (sign rca) are straightforward generalization of the constant coefficient autoregressive models (thavaneswaran et al., 2006a). the random coefficient and the sign function have influenced the unconditional kurtosis. the value of unconditional kurtosis in the rca-garch and the sign rca-garch models is bigger in comparison with ordinary argarch. in addition, the sign function allows the modelling of asymmetry, such as response of returns on various information from the market. the purpose of this work is to apply the sign rca-garch models to pricing european call options, and compare these results and results obtained from the black-scholes model and from other selected garch options pricing models with the market prices. such use of sign rca-garch 2 more frequent than monthly. option pricing under sign rca-garch models dynamic econometric models 14 (2014) 145–160 147 models as far as we know has not been applied in option pricing except the work by górka (2012). 1. theoretical framework 1.1. option pricing as a consequence of the equation (1), the price of a european call option is given by equation: ,)()(= 21 dnkednsc r tt  (2) where: , 2 = 2 1           r k s ln d t ,12  dd ts – the stock price at time t, k – the exercise price, r – the risk-free interest rate,  – the time to maturity of the option,  n – the cumulative normal density function,  – the volatility of rate of the return on the stock. the valuation of derivative is about moving to the world free of risk, in which risky assets have the same return as the risk-free. the general idea of the valuation of derivatives is based on the following theorem. theorem (elliott and kopp, 1999). if the process s satisfies the equation tttt dwsdtsds  = , (3) then also satisfies the equation tttt wdsdtrsds ~ =  , (4) where is r the risk-free interest rate, t r ww tt   = ~ is a wiener process. duan (1995) introduced the garch option pricing model by generalizing the traditional risk neutral valuation methodology to the case of conditional heteroskedasticity. letting the conditional mean t and conditional variance 2 t be measurable functions with respect to the information set (f), the general model joanna górka dynamic econometric models 14 (2014) 145–160 148 under the data generating probability measure p is given by (hafner and herwartz, 2001):    ,;;, ,1,0...~ ,= 22    tsf ndii y sst t tttt   (5) where f is a parametric function with parameter vector  . in garch models, it is not possible to find a risk-neutralization procedure that leaves unchanged the marginal variance of the process or the conditional variance beyond one period. therefore duan (1995) introduced the local risk-neutral valuation relationship (lrnvr; equivalent martingale measure q). the local risk-neutral valuation relationship is an essential feature of the equivalence of the conditional variances under the data generating probability measure p (historical measure) and the equivalent martingale q. under the measure q, the model is as follows (hafner and herwartz, 2001):       , , ,;;, ,1,0...~ ,= 22 t t t ttt sst t ttttt r tsf ndii y             (6) where: r is the risk-free interes rate, ry tt =]|[e 1f q ,    11 |var|var   tttt yy ff qp . this procedure leaves unchanged the one period ahead conditional variance and the conditional expected future return is equal to the risk-free interest rate at each time t. discounted asset price under the measure q is a martingale. 1.2. option pricing under sign rca-garch models the rca-garch models were proposed by thavaneswaran et al., (2006a). the rca(1)-garch(1,1) model has the following form:   ,1 tttt yy    (7) option pricing under sign rca-garch models dynamic econometric models 14 (2014) 145–160 149 ,ttt   (8) , 21-t1 2 110 2 t   t (9) where , ,0 ,1 1 are parameters of model,  ,,0...~ 2 diit  ,,0...~ 2 diit ,00  ,01  .01  theoretical properties of this model can be found among others: górka (2012), thavaneswaran et al., (2006a, 2006b, 2008, 2009). it is worth noting, that the value of unconditional variance and kurtosis increases in comparison with ordinary ar(1)-garch(1,1). we can define the rca(1)-garch(1,1) option pricing model under the historical measure p:     , ,1,0~ ,)(= 2 11 2 110 2 1     ttt t ttttt n yy    (10) and under the measure q:        , , ,1,0~ ,= 1 121 1 2 11 2 1110 2 1         t tt t tttt t ttt ry n ry       (11) where 1r is the one-day risk-free interest rate. for the rca(1)-garch(1,1) model, like for the ar(1)-garch(1,1) (hafner and herwartz, 2001), we can obtain the unconditional variance under the measure q. proposition 1 (górka 2012). under the measure q, the unconditional variance of yt under stationarity is finite if   11 1221    , and       1221 222 110 11 1 var        r yt q . the sign rca(1)-garch(1,1) models proposed by thavaneswaran, et al., (2006a) have the following form: joanna górka dynamic econometric models 14 (2014) 145–160 150 ,)(= 11 ttttt ysy    (12) ,= ttt  (13) ,= 2 11 2 110 2   ttt  (14) where  ,  , 0 , 1 , 1 are parameters of model,  ,,0...~ 2 diit  ,,0...~ 2 diit ,00  ,01  ,01        0<for1 0=for0 0>for1 = t t t t y y y s . the sign function  ts has the interpretation: if  t , the negative value of  means that the negative (positive) observation values at time 1t correspond to a decrease (increase) of observation values at time t . in the case of stock returns it would suggest (for returns) that after a decrease of stock returns, the higher decrease of stock returns occurs than expected, and in the case of the increase of stock returns the lower increase in stock returns occurs than expected. theoretical properties of this model can be found among others: górka (2012), thavaneswaran et al., (2006a, 2006b, 2008, 2009). it is worth noting, that the adding the sign function has influence the increase of unconditional variance and kurtosis in comparison with rca(1)-garch(1,1), and therefore also with ordinary ar(1)-garch(1,1). under the historical measure p the sign rca(1)-garch(1,1) option pricing model can be defined:     , ,1,0~ ,)(= 2 11 2 110 2 11     ttt t tttttt n ysy    (15) under the martingale measure q the sign rca(1)-garch(1,1) option pricing model takes the form:        , , ,1,0~ ,= 1 1221 1 2 11 2 1110 2 1         t ttt t tttt t ttt rys n ry       (16) option pricing under sign rca-garch models dynamic econometric models 14 (2014) 145–160 151 for the sign rca(1)-garch(1,1) model, like for the rca(1)garch(1,1), we can obtain unconditional variance under the measure q. proposition 2 (górka 2012). under the measure q, the unconditional variance of yt under stationarity is finite if   11 12221    , and       12221 2222 110 11 1 var        r yt q . 1.3. monte carlo simulations garch models are very popular and effective for modeling the volatility dynamics in many asset markets. unfortunately, existing garch models do not have closed-form solutions for option prices. these models are typically solved by simulation. the monte carlo simulation procedure for option pricing can be described in following steps (duan, 1995; hafner and herwartz, 2001; lehar et al., 2002; piontek, 2002, 2004): 1. parameter estimation under the empirical measure p. 2. simulation of sample paths for the underlying asset price under the equivalent martingale measure q (50000 paths), i.e.        n s stisti n s sti nr tni ess 1 0,0, 1 2 0,1 0 5.0 ,  , where: i – i-th path, sti 0,  – the current values of the innovation, 2 , 0 sti   – the current values of the variance. 3. correction to the standard monte carlo simulation procedure (empirical martingale simulation, duan and simonato, 1998), i.e. , 1 , 1 ,* , 1 0    m i nim ninr tni s s ess where: m – number of paths. 4. discounting the expected payoffs to yield of the monte carlo price of option, i.e.   .|0,maxe 0 1 0 * , q tni nr t ksec f joanna górka dynamic econometric models 14 (2014) 145–160 152 where: 0t c – the corresponding call price obtained by monte carlo simulation, qe – the risk-neutral conditional expectation operator. the risk-free interest rate ( 1r ) was approximated on the basis of the interest rate of the wibid3 and the wibor4. to quantify the deviation of theoretical option prices from the prices observed at the market the statistical error measures were applied (lehar et al., 2002):  the relative pricing error t tt c cc   ˆ rpe ,  the absolute relative pricing error t tt c cc   ˆ arpe , where tc and tĉ denote the observed price and the model price, respectively. the rpe is a measure of the bias of the pricing model. a non-zero rpe may therefore indicate the existence of systematic errors. the arpe measures both the bias and the efficiency of pricing (lehar et al., 2002). 2. an empirical analysis the data used in the empirical study were the wig20 index and prices of the european call options on wig20 index on the warsaw stock exchange (wse). the sample period for the wig20 runs from 19-th of november 2003 to 21-th of february 2011. evolution of the wig20 index was displayed in figure 1. the wig20 index went up during the first 4 years, after that it rapidly went down and since 2009 started to increase again. 3 warsaw interbank offered rate. 4 warsaw interbank bid rate. option pricing under sign rca-garch models dynamic econometric models 14 (2014) 145–160 153 figure 1. the wig20 index (november 19, 2003 – february 21, 2011) two periods were chosen to calculate option prices. the first one was at the turn of 2007–2008 (it is about 3 months long) and the second one was at the turn of 2010–2011. first day of the option pricing at the turn of year 2007–2008 was made on 22-th of november 2007 (valuation on the november 23, 2007) and 16-th of november 2010 at the turn of 2010–2011 (valuation on the november 17, 2010). option pricing were made using:  the standard black-scholes (bs),  monte carlo simulation (mcs; duan's method). for comparison, the valuation of options were also made using the ar(1)-garch(1,1) model and the ar(1)-gjr-garch(1,1) model. for all model specifications, parameter values were obtained from the mle using wig20 index daily logarithmic returns. the sample sizes on which models were estimated are as follows:  252 observations (~ year),  504 observations (~ 2 years),  1008 observations (~ 4 years). all computation were made using authors codes written in gauss 6.0. example results of the valuation of european call options for wig20 stock index on a particular day (november 17, 2010), on three month to maturity and different sample sizes, and market prices of these options (the closing price) are shown in table 1. joanna górka dynamic econometric models 14 (2014) 145–160 154 table 1. european call option prices for wig20 stock index on three months to maturity and market prices of these options strike (k) bs rca-garch sign rca-garch ar-garch ar-gjr-garch market price 17.12.2010 252 observations 2300 476.72 491.21 493.56 524.49 494.79 483.50 2400 377.64 399.56 403.62 433.41 405.20 388.50 2500 278.57 314.31 320.34 348.30 322.02 294.90 2600 179.50 238.20 246.04 271.50 247.50 221.05 2700 80.43 173.31 182.39 204.68 183.29 133.00 2800 1.31 120.84 130.42 149.05 130.64 78.00 2900 0.00 80.65 89.98 104.83 89.51 45.10 3000 0.00 51.66 59.92 71.13 58.89 23.00 3100 0.00 31.79 38.66 46.57 37.26 9.10 504 observations 2300 476.72 487.08 489.88 486.94 487.81 483.50 2400 377.64 390.36 392.63 389.99 391.45 388.50 2500 278.57 297.02 298.39 296.19 298.16 294.90 2600 179.50 211.25 211.15 209.70 211.28 221.05 2700 80.43 138.46 136.63 136.25 135.72 133.00 2800 1.31 82.89 79.92 80.40 76.51 78.00 2900 0.00 45.48 42.27 43.25 36.75 45.10 3000 0.00 23.27 20.53 21.61 14.78 23.00 3100 0.00 11.24 9.37 10.18 4.87 9.10 1008 observations 2300 476.72 495.32 495.34 495.31 519.97 483.50 2400 377.64 405.43 405.45 405.42 440.70 388.50 2500 278.57 322.00 322.00 321.97 368.18 294.90 2600 179.50 247.43 247.40 247.40 303.11 221.05 2700 80.43 183.66 183.60 183.63 245.94 133.00 2800 1.31 131.79 131.69 131.76 196.75 78.00 2900 0.00 91.65 91.51 91.62 155.33 45.10 3000 0.00 62.22 62.06 62.19 120.91 23.00 3100 0.00 41.44 41.29 41.41 93.08 9.10 note: the bold number denotes the theoretical option prices the closest to the market price. option prices calculated by the bs formula were underestimated, while option prices calculated by other models were overestimated for the sample size of 252 observations and 1008 observations. for the sample size of 504 observations the option prices were overestimated for some strikes, but for other – underestimated. it depends of the type of option and of the model on which theoretical option prices were calculated. for the out-of-the-money options differences between market prices and theoretical option prices calculated by models on the simple size of 504 observations were small, while for others simple sizes these differences were greater. in this study, for the option pricing under sign rca-garch models dynamic econometric models 14 (2014) 145–160 155 sample size of 504 observations the theoretical option prices were closest to the market prices. it holds for all model specifications. figure 2. the annualized implied volatility of the rca-garch option pricing model (504 observations; day of the option pricing – november 16, 2010) figure 3. the annualized implied volatility of the sign rca-garch option pricing model (504 observations; day of the option pricing – november 16, 2010) figures 2 and 3 show the relationship between implied volatility, the exercise price and the time to maturity of the option. this shape resembles a smile and is called the volatility smile. it is often observed in financial markets (lehar et al., 2002; piontek, 2002). when the time to maturity increases, the smile tends to become flatter. with the increase of the time to maturity of joanna górka dynamic econometric models 14 (2014) 145–160 156 the option the increase (figure 3) or decrease (figure 2) of volatility for options with the same moneyness is often observed. this result is similar to the result obtained from other garch models (piontek, 2002, 2004; hafner and herwartz, 2001; duan, 1995). on the basis of the valuation of options for a day, it is difficult to draw more general conclusions on the usefulness of rca-garch and sign rca-garch models. therefore, the valuation of options in the two periods were made using different models as a result of progressive estimation models. in the first period (november 23, 2007 – january 25, 2008) the values of ow20c85 and ow20f8 options with different exercise prices for the next 63 days were determined. in the second period (november 17, 2010 – february 21, 2011) the valuation of ow20c1 and ow20f1 options with different exercise prices for the next 67 days was made. table 2. the mean option pricing errors  180> error options rca-garch sign rca-garch ar-garch ar-gjr-garch 252 observations rpe itm 0.8079 0.1860 0.1988 0.2022 atm 1.8958 0.4736 0.5088 0.4358 otm 5.8016 1.1948 1.2816 1.0673 arpe itm 0.8079 0.1873 0.1988 0.2067 atm 1.8958 0.4736 0.5088 0.4408 otm 5.8016 1.1948 1.2816 1.2015 504 observations rpe itm 0.0764 0.0911 0.2228 0.2022 atm 0.1517 0.1830 0.5339 0.4358 otm 0.3341 0.4280 1.4065 1.0673 arpe itm 0.0812 0.0996 0.2228 0.2067 atm 0.1596 0.1979 0.5339 0.4408 otm 0.3589 0.4911 1.4198 1.2015 1008 observations rpe itm 0.3268 0.3114 0.3268 0.3628 atm 0.8260 0.7756 0.8260 0.8607 otm 2.2219 2.0916 2.2217 2.1698 arpe itm 0.3268 0.3176 0.3268 0.3628 atm 0.8260 0.7884 0.8260 0.8607 otm 2.2219 2.1179 2.2217 2.1698 note: atm – at-the-money, itm – in-the-money, otm – out-of-the-money. the bold number indicate minimum of absolute error. the obtained values of option prices were subsequently split according to the time to maturity of the options (in days), i.e. 5 type of call options. option pricing under sign rca-garch models dynamic econometric models 14 (2014) 145–160 157  short maturity ( 60 ),  medium maturity ( 180<60  ),  long maturity ( 180> ). then, in the first place the option at-the-money (atm) was determined, and then statistical measures of errors for the four options in-the-money (itm) and four options out-of-the-money (otm) were calculated6. the results for the second period were shown in tables 2–4. table 3. the mean option pricing errors  180<60  error options rca-garch sign rca-garch ar-garch ar-gjr-garch 252 observations rpe itm 0.2549 0.1153 0.1117 0.1435 atm 0.6812 0.3362 0.3339 0.4123 otm 4.5922 1.6938 1.6740 1.8805 arpe itm 0.2585 0.1209 0.1147 0.1441 atm 0.6966 0.3676 0.3392 0.4123 otm 4.6231 1.7395 1.6768 1.8805 504 observations rpe itm –0.0002 0.0426 0.1000 0.1532 atm –0.1188 0.0651 0.2646 0.4006 otm 0.1759 0.5314 1.5774 2.2177 arpe itm 0.0913 0.0685 0.1166 0.1676 atm 0.3081 0.1690 0.3375 0.4734 otm 0.7708 0.6223 1.6448 2.4192 1008 observations rpe itm 0.1702 0.1901 0.1702 0.2003 atm 0.5001 0.5692 0.5001 0.5282 otm 2.5682 3.2605 2.5682 2.2497 arpe itm 0.1715 0.2031 0.1715 0.2035 atm 0.5010 0.6349 0.5010 0.5448 otm 2.5682 3.3311 2.5682 2.2857 note: atm – at-the-money, itm – in-the-money, otm – out-of-the-money. the bold number indicate minimum of absolute error. obtained results depend mainly on the time to maturity and size of sample. however, the smallest absolute error values were received for the sample of 504 observations regardless of the choice of the model for the theoretical option prices (see table 2–4). for each time to maturity and size of 6 firstly, 4 options in-the-money and out-of-the-money were right next to the option atthe-money. second, for each day of option pricing the error measures had had 3 values, one of each type of option. for the whole of the period (for example, a option with medium maturity), the result was the average of the results for the option of this period (for example, for the option of 180<60  ). joanna górka dynamic econometric models 14 (2014) 145–160 158 sample different conclusions may be drawn. for example, for long time to maturity (table 2) and the sample of 504 observations the smallest values of the mean pricing error (absolute and relative) were obtained for rcagarch models, while for the short time to maturity (table 4) – for sign rca-garch models. this holds for each type of options. in this study, regardless of the sample size, the out-of-the-money options for the short time to maturity were substantially overestimated. the similar results for the long time to maturity for the first period (november 23, 2007 – january 25, 2008) were found. table 4. the mean option pricing errors  60 error options rca-garch sign rcagarch ar-garch ar-gjr-garch 252 observations rpe itm 0.0266 0.0287 0.0248 0.0517 atm 0.1834 0.1548 0.1891 0.2992 otm 1.7632 1.7607 1.8024 1.8375 arpe itm 0.0442 0.0452 0.0474 0.0622 atm 0.2039 0.2035 0.2277 0.3092 otm 1.7765 1.9044 1.8163 1.8760 504 observations rpe itm 0.0250 0.0211 0.0479 0.0520 atm 0.1704 0.0942 0.3498 0.2910 otm 2.0401 1.1652 4.3350 3.5110 arpe itm 0.0503 0.0471 0.0725 0.0751 atm 0.2180 0.1922 0.4135 0.3458 otm 2.1327 1.3513 4.4305 3.6669 1008 observations rpe itm 0.0413 0.0434 0.0413 0.0542 atm 0.2634 0.2738 0.2636 0.2903 otm 2.8569 2.9869 2.8585 1.6641 arpe itm 0.0544 0.0565 0.0544 0.0651 atm 0.2747 0.2851 0.2748 0.3048 otm 2.8630 2.9928 2.8645 1.7021 note: atm – at-the-money, itm – in-the-money, otm – out-of-the-money. the bold number indicate minimum of absolute error. comparing the rpes for the four different models (table 2–4), one can see systematic overpricing across all models (except the rca-garch model for the itm and atm options for 180<60  and the sample of 504 observations – table 3). in other words, the volatility of the underlying asset price was systematically overestimated. in some cases the differences between the mean option pricing errors were small (e.g. for the sample of 1008 observations in table 2, for the sample of 252 observations in table 4), while in other cases these differences option pricing under sign rca-garch models dynamic econometric models 14 (2014) 145–160 159 were substantial (e.g. for the sample of 504 observations in table 2 or 4). it is worth noting, that sign rca-garch models outperform the selected garch models, because the absolute error were substantially lower (e.g. for the sample of 504 or 1008 observations in table 2, for the sample of 504 or 1008 observations in table 4). however, the better performance of sign rca-garch models was not well established and depends on the time to maturity, size of sample or the period of the data. conclusions this paper has applied sign rca-garch models to compute theoretical option prices. this approach was illustrated by the valuation of the european call options on the wig20 index, together with a comparison of their values obtained on the selected garch models. it is difficult to make general remarks, nevertheless the empirical results showed that:  the black-scholes model cannot explain the prices of out-of-the-money options,  rca-garch and sign rca-garch models can be successfully applied in pricing options,  none of the models can be indicate as the best one for the option valuations for every period and every time to maturity of the options,  the choice of a sample size for estimating the option pricing model has a significant impact on the option pricing,  the choice of the volatility model is important for achieving a satisfying pricing performance. references black, f., scholes, m. (1973), the pricing of options and corporate liabilities, journal of political economy, 81(3), 637–654, doi: http://dx.doi.org/10.1086/260062. bollerslev, t. (1986), generalized autoregressive conditional heteroscedasticity, journal of econometrics, 31(3), 307–327, doi: http://dx.doi.org/10.1016/0304-4076(86)90063-1. duan, j.-c. (1995), the garch option pricing model, mathematical finance, 5(1), 13–32, doi: http://dx.doi.org/10.1111/j.1467-9965.1995.tb00099.x. duan, j.-c., simonato, j.-g. (1998), empirical martingale simulation for asset prices, management science, 44(9), 1218–1233, doi: http://dx.doi.org/10.1287/mnsc.44.9.1218. elliott, r. j., kopp, p. e. (1999), mathematics of financial markets, springer, new york, doi: http://dx.doi.org/10.1007/978-1-4757-7146-6. engle, r. f. (1982), autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation, econometrica, 50, 987–1006, doi: http://dx.doi.org/10.2307/1912773. joanna górka dynamic econometric models 14 (2014) 145–160 160 górka, j., (2012), modele klasy sign rca garch. własności i zastosowania w finansach (sign rca garch models. properties and application in finance), wydawnictwo umk, toruń. hafner, c. m., herwartz, h. (2001), option pricing under linear autoregressive dynamics, heteroskedasticity, and conditional leptokurtosis, journal of empirical finance, 8, 1–34, doi: http://dx.doi.org/10.1016/s0927-5398(00)00024-4. lehar, a., scheicher, m., schittenkopf, c. (2002), garch vs. stochastic volatility: option pricing and risk management, journal of banking & finance, 26, 323–345, doi: http://dx.doi.org/10.1016/s0378-4266(01)00225-4. piontek, k. (2002), modelowanie i prognozowanie zmienności instrumentow finasowych, praca doktorska pod kier. k. jajugi, ae we wrocławiu, maszynopis. piontek, k. (2004), wspołczynniki greckie w modelu wyceny opcji uwzględniającym efekt ar-garch, in tarczyński, w. (ed.), rynek kapitałowy – skuteczne inwestowanie, cześć ii, zeszyty naukowe uniwersytetu szczecińskiego nr 389, wydaw. nauk. uniwersytetu szczecińskiego, szczecin, 35–50. thavaneswaran, a., appadoo, s. s., bector, c. r. (2006a), recent developments in volatility modeling and application, journal of applied mathematics and decision sciences, 1–23, doi: http://dx.doi.org/10.1155/jamds/2006/86320. thavaneswaran, a., appadoo, s. s., ghahramani, m. (2009), rca models with garch innovations, applied mathematics letters, 22, 110–114, doi: http://dx.doi.org/10.1016/j.aml.2008.02.015. thavaneswaran, a., peiris, s., appadoo, s. s. (2008), random coefficien volatility models, statistics & probability letters, 78, 582–593. thavaneswaran, a., singh j., appadoo s. s. (2006b), option pricing for some stochastic volatility models, the journal of risk finance, 7(4), 425–445, doi: http://dx.doi.org/10.1108/15265940610688982. modele sign rca-garch w wycenie opcji z a r y s t r e ś c i. po ukazaniu się przełomowej pracy blacka i scholesa literatura dotycząca wyceny opcji stała się bardzo ważnym obszarem w badaniach. zostały opracowane liczne metody wyceny opcji. w artykule tym pokazano, jak można obliczyć ceny opcji wykorzystując model sign rca-garch do opisu dynamiki zmienności. wyceny opcji uzyskane przedstawioną metodą oraz wyceny opcji uzyskanych z wykorzystaniem modelu blacka-scholesa i wybranych modeli garch zostały porównane z ceną rynkową. podejście to zostało zilustrowane wyceną europejskich opcji kupna na indeks wig20. empiryczne wyniki wskazują, że modele rca garch i sign rca garch mogą być z powodzeniem stosowane do wyceny opcji. jednak żadnego z przedstawionych modeli nie można wskazać jako najlepszego do wyceny opcji dla dowolnej wielkości próby czy dowolnego czasu pozostającego do wygaśnięcia opcji. s ł o w a k l u c z o w e: model sign rca-garch, wycena opcji, modele garch. microsoft word dem_2014_125to144.docx © 2014 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2014.007 vol. 4 (2014) 125−144 submitted october 20, 2014 issn accepted december 23, 2014 1234-3862 elżbieta szulc*, dagna wleklińska, karolina górna, joanna górna the significance of distance between stock exchanges undergoing the process of convergence: an analysis of selected world stock exchanges during the period of 2004–2012 a b s t r a c t. the paper concerns the convergence of selected world stock exchanges from the point of view of their development in the context of geographical and economic distance between them. it presents the methodological approach which points up the necessity of taking into account spatial and economic connections among stock markets in convergence analyses. the research includes 46 largest trading floors analyzed in the period of 2004–2012. the empirical data refer to six diagnostic variables acknowledged as the important determinants of the development of stock markets. k e y w o r d s: stock exchanges, convergence, physical and economic distance, connectivity matrix, spatial panel models. j e l classification: c10, c12, c58, g15. introduction the paper concerns the convergence of selected world stock exchanges in the context of geographical and economic distance between them. the importance of the geographical and economic distance in revealing the linkages between stocks has already been considered in literature on the subject * correspondence to: elżbieta szulc, nicolaus copernicus university, department of econometrics and statistics, 13a gagarina street, 87-100 toruń, poland, e-mail: eszulc@umk.pl. elżbieta szulc, dagna wleklińska, karolina górna, joanna górna dynamic econometric models 14 (2014) 125–144 126 (e.g. suchecka and łaszkiewicz, 2011; wójcik, 2009; asgharianet et al., 2013). in particular, the premises of the spatial perspective on capital markets’ analyses in two areas: geography of finance and capital, and behavioral finance were indicated (see e.g. suchecka and łaszkiewicz, 2011). a spatial analysis of linkages between securities markets was carried among others by asgharian et al. (2013). on the basis of their research, they stated that the similarity with regard to the economies’ components is the strongest source of linkages between stock markets but the connections that result from geographical neighborhood, bilateral fdi and stability of the bilateral exchange rate are important as well. one of the directions of the analysis of the stock exchanges’ relationships is considering the convergence of these markets from the point of view of their specific characteristics. this process is strictly connected with the integration and liberalization of stock markets and their growing interdependence, which in turn is associated with the liberalization of capital flows and technological innovation. these processes are favorable for the development of stock markets, and thus the distinctions between them are becoming increasingly blurred over time. the literature on the convergence of stock markets includes e.g. fraser et al. (1994), koralun-bereznicka (2008) and caparale et al. (2009). in the works the problem of linkages between the markets in geographical and economic spaces is particularly interesting. in the recent literature the hypotheses of convergence vs. divergence are formulated mainly in the context of the contemporary financial crisis of 2007–2010 (see e.g. aspergis et al., 2014). the aim of the paper is to investigate whether, in the light of the current empirical analyses, one can observe the process of convergence of main stock markets in the world. in addition the importance of distance between the markets for the process is evaluated. particularly the role of economic distance is considered. in the research the hypothesis that in the convergence of stock markets the spatial and economic connections among them are important, and so that relative location of a stock exchange affects the growth rate of the exchange, is verified. the structure of the paper is as follows: in section 1 the subject and range of the investigation are defined. it qualifies the investigated stock exchanges and characterizes the specified diagnostic variables. section 2 presents the methodology. in this section a taxonomic measure of stock exchanges’ development is defined and the theoretical models of β-convergence in formulation of the regressions for the pooled time series and cross-sectional data are presented. moreover, in section 2 the diagnostic the significance of distance between stock exchanges … dynamic econometric models 14 (2014) 125–144 127 tests for verification of the empirical models are pointed out. section 3 contains preliminary data analysis. the results of the research are presented in section 4. conclusion formulates final remarks and indicates further investigations. 1. subject and range of the investigation the subject of the investigation contains the selected worldwide stock exchanges, characterized in terms of their level of development. the study included 46 largest trading floors in the period of 2004–2012. the specification of the exchanges with the assignment to the relevant country are presented in table 1. the level of stock exchange development was defined by a synthetic measure based on six diagnostic variables, i.e. x1  the capitalization of domestic shares, x2  the capitalization of newly listed domestic shares, x3  the total value of share trading, x4  gdp per capita, x5  the top 10 most heavily capitalized domestic companies, x6  the ratio of market capitalization to gdp. it was recognized that, in the light of theory and empirical analyses, the specified variables are important determinants of the development of stock exchanges. taking into account the connections of the capital market with the economy of the country of its location was also an important issue. the range of information provided by the world federation of exchanges played a significant role as well. the capitalization of domestic shares is one of the most important parameters reflecting the situation in the securities market. it is calculated as the total number of shares issued by domestic companies. a high value of this indicator encourages large investors to invest their capital in a given market and shows its attractiveness compared to others. from the point of view of the development of a given equity market, another important indicator is the capitalization of newly listed domestic shares. contemporary capital market is very often treated as a short term mechanism where one can earn or lose money suddenly. in many cases, the financial performance of companies is ignored. with increasing stock quotation and improving situation in the stock market, the interest of new companies investing in a given trading floor is also growing. in practice, this means that growing prices of shares allow investors to make a profit. however, the number of initial public offerings do not affect the conditions of a stock exchange. only on the basis of their capitalization, the development potential of a given market can be assessed (wiśniewski, 2003). elżbieta szulc, dagna wleklińska, karolina górna, joanna górna dynamic econometric models 14 (2014) 125–144 128 table 1. specification of the stock exchanges considered north/south america brazil bm&bovespa (bov) chile santiago se (sse) canada tmx group (tmx) colombia colombia se (cse) mexico mexican exchange (bmv) bermuda bermuda se (bsx) argentina buenos aires se (bcba) peru lima se (bvl) united states nasdaq omx (nasdaq) nyse euronext (us) (nyse) asia/pacific australia australian se (asx) philippines phillippine se (pse) china honk kong se (sehk) shanghai se (shse) shenzen se (szse) taiwan se corp. (tsec) japan osaca se (ose) tokyo se group (tse) india national se india (nse) indonesia indonesia se (ise) south korea korea exchange (krx) malaysia bursa malaysia (bm) sri lanka colombo se (clse) thailand thailand se (thse) singapore singapore se (sse) europe/middle east/africa austrian wiener borse (ag) cyprus cyprus se (cpse) egypt cairo&alexandria se (egx) greece athens exchange (athex) spain bme spanish exchange (bme) netherlands nyse euronext (europe) (nee) iran tehran se (thrse) ireland irish se (irse) israel tel aviv se (tase) luxemburg luxemburg se (lxse) malta malta se (mse) mauritius mauritius se (sem) germany deutsche borse (db) norway oslo bors (obe) poland warsaw se (wse) south africa johannesburg se (jse) switzerland six swiss exchange (six) sweden nasdaq omx nordic exchange (nomx) turkey istanbul se (isse) hungary budapest se (bdse) great britain london se (lse) other important parameter characterizing the capital market is the value of turnover. it is calculated as a total number of shares traded multiplied by their respective matching prices within a year. for a well-developed market the desired situation is that the annual turnover is higher than the total value of shares traded. the significance of distance between stock exchanges … dynamic econometric models 14 (2014) 125–144 129 in order to connect the capital market with the country’s economy the gdp per capita indicator was used. it is one of the most popular parameters which reflects the level of citizens’ wealth. it has to be emphasized that most of the highly developed stock exchanges are located in developed countries. it is legitimate to use this indicator due to the fact that almost all exchanges, described as mature, are located in countries that have achieved a high level of development. the top 10 most heavily capitalized domestic companies is one of the indicators reflecting market concentration. this parameter, as the only one out of the six variables taken into account, was treated as a destimulant. the concentration phenomenon takes place when a small number of large companies has a significant share in the capitalization of a given stock exchange. a high value of this indicator is undesirable since it characterizes poorly developed markets. the ratio of market capitalization to gdp reflects the relationship between economic development and maturity of an equity market. it was observed that for well-developed stock markets, the value of this indicator is higher than 60 percent. it is believed that such participation allows operators to gather national and foreign capital. at the same time, the results of empirical studies demonstrate that this ratio needs to be above 2 percent for the stock exchange to have significant influence on particular processes of a national economy. on the other hand, this variable should be treated with caution because of the fact that currently one company may be listed on more than one market (łuniewska and tarczyński, 2006, p. 45). 2. methodology the research was conducted in relation to the aggregate characteristic of the stock exchanges in the form of taxonomic measure of development. this indicator is understood as a synthetic normalized formula expressed by (see hellwig, 1968): , 2 1' q i i sq q q   (1) where: iq  the synthetic variable determining the level of development of the i-th exchange in relations to a development standard, q  the average value of the synthetic variable, qs  the standard deviation of the variable. elżbieta szulc, dagna wleklińska, karolina górna, joanna górna dynamic econometric models 14 (2014) 125–144 130 in this approach the values of the synthetic variable iq are calculated according to the formula:   , 1 2 0   m j jiji zzq (2) where: ijz  the value of j-th diagnostic variable for i-th exchange standardized to 0–1, jz 0  the value of j-th diagnostic variable for the standard of development standardized to 0–1. thus, iq means a distance between i-th exchange and the development standard. through the use of the taxonomic measure of stock exchanges’ development it is possible to present the rankings of exchanges and their changes in time, the evaluation of the correlation between stock exchanges in terms of development, the identification of the linkages between markets in economic space, and finally the analysis of the stock exchanges’ convergence, which is meant as equalizing their development levels. in the analyses of the stock exchanges' convergence the econometric models of β-convergence, in particular the spatial models for pooled time series and cross-sectional data (tscs) and spatial panel models, were used. the model tscs with spatial component takes the form of the spatial autoregressive model (sar_pooled), i.e.   ,lnlnln ' 1 ' ' 1' 1 ' it jt jt ij ijit it it q q wq q q                     (3) or of the model with spatial autoregressive residuals (se_pooled), i.e.   ,lnln ' 1' 1 ' itit it it q q q          .itjt ij ijit w     (4) the spatial panel models used in the investigation were as follows:   ,lnlnln ' 1 ' ' 1' 1 ' it jt jt ij ijiti it it q q wq q q                     (5) i.e. the spatial autoregressive panel model with individual fixed effects (the spatial autoregressive fixed-effect model) (sar_fe_ind) and the significance of distance between stock exchanges … dynamic econometric models 14 (2014) 125–144 131   ,lnln ' 1' 1 ' ititi it it q q q          ,itjt ij ijit w     (6) i.e. the spatial error panel model with individual fixed effects (se_fe_ind). elements wij in the formulas (3)–(6) come from connectivity matrix w which refers to the linkages between exchanges considered. assuming that there are n stock exchanges, the matrix has as many rows and columns as there are exchanges, i.e. n by n matrix w is considered. each row of the matrix contains non-zero elements in columns which correspond to the connected objects, according to the received criterion. furthermore, the given object cannot be connected to itself, so wij = 0 for all i = j. thus, the diagonal elements of w are zeros. in the majority of the spatial analyses the starting point in establishing the spatial connections is the binary matrix of neighborhood. the neighbors are usually established according to the common border criterion. then, the rows in the connectivity matrix are normalized, so that the row sums are equal to 1, as a result of dividing each entry on a row by the sum of the row values (the so-called row standardization to one). the weights wij that are established in this way signify that each j-th neighbour of the i-th spatial units is treated identically, and the greater the strength of its interactions with the neighbours is, the fewer neighbours it has. a different situation occurs when the weights wij are functions of some properties of the space, e.g. of the length of the common border, of the distance between the centers of the regions or of other measures of similarity between the regions, e.g. of the so-called economic distance between them. various types of weights wij may be pointed out according to the established criteria (see e.g. haining, 2005, p. 83–84 ). in this paper the linkages between stock exchanges will be defined with the use of two approaches. the first one uses a matrix of connections with weights established on the basis of the physical distance between the centers of the countries where the stock exchanges are located. the second one consists in that in the matrix of connections the economic distance (the essence of which is to establish similarity of the exchanges on the basis of the value of the taxonomic measure of exchanges’ development) is taken into consideration. the economic distance was expressed as:   , 1 2   m j kjijik zzd (7) elżbieta szulc, dagna wleklińska, karolina górna, joanna górna dynamic econometric models 14 (2014) 125–144 132 where: kjij zz ,  the values of standardized diagnostic variables for each i-th and k-th stock exchange, j = 1, 2, …, 6  the number of the diagnostic variable. in both approaches the elements of the linkages’ matrix are equal to:        ki ki dw ikik if0, if, 1 (8) then, as a result of row standardization to one, the matrixes of the connections based on the physical or economic distance are obtained. since the models (3)–(6) refer to the pooled time series and crosssectional data, the block matrixes of connections were used, i.e. , 9 2 1              w00 0w0 00w w     (9) where: 921 ... www  – matrixes of the spatial connections based on the physical distance, the same for all the considered years, and , 9 2 1                    w00 0 0w0 00w w     (10) where: 921 ...   www – matrixes of connections, taking into account the economic distance between exchanges, different for successive years. the convergence of the exchanges is confirmed by the data, if the parameter estimates β in models (3)–(6) are negative and statistically significant. in addition, if parameters  in models (3) and (5) and parameters  in models (4) and (6) are significantly different from zero, then in the convergence process the spatial connections among stock exchanges are important and the hypothesis that the rate of growth of any stock exchange is related to that of its neighbors is confirmed. the inclusion of the spatial elements in the analysis of the convergence of stock exchanges allows us to identify the relationships between them in the significance of distance between stock exchanges … dynamic econometric models 14 (2014) 125–144 133 a geographical and economic space. furthermore, the spatial models of convergence have better statistical properties, and thus allow for wider economic interpretation. in order to evaluate the quality of the empirical models in the investigation, the following tools were used: the moran test for verifying spatial independence of the residuals, the lagrange multiplier tests (lmlag, lmerr) and their robust versions (rlmlag, rlmerr) as spatial dependence diagnostics, the likelihood ratio test (lr) for testing the significance of the spatial dependence, the breusch-pagan heteroskedasticity test, the chow test for verifying the need for including fixed effects into the spatial panel models (on the tools see e.g. arbia, 2006; millo and piras, 2012; mutl and pfaffermayr, 2011; baltagi et al., 2003; suchecki (ed.), 2012). all calculations were performed with r (version 3.0.1) and the graphical illustrations – with the use of mapviever and corel. 3. preliminary data analysis figure 1 shows locations of investigated exchanges on the world map and bar charts of taxonomic measure of development (tmd) in the years 2004–2012. this presentation allows us to observe changes in the level of development of the individual stock exchanges and a comparison of the dynamics of change in the arrangement of their spatial location as well. it is interesting that some of the asian trading floors, e.g. ose, szse, krx, pse, did not record any decrease in the value of taxonomic measure of development after the beginning of the financial crisis in 2007. in the worst case, these stock exchanges reacted with slowdown or stagnation of growth. obviously, the beginning of a crisis has caused a decline in the us stock exchanges but also stimulated the process of making up the development imbalance between emerging and well-developed markets. interestingly, the same exchanges, for which synthetic variable already showed lower values in 2008 (nomx, se, shse, bm), recorded an increase in the value of this variable the following year (as opposed to all other stock exchanges). elżbieta szulc, dagna wleklińska, karolina górna, joanna górna dynamic econometric models 14 (2014) 125–144 134 figure 1. bar charts of tmd for the investigated stock exchanges in the years 2004–2012 figures 2 and 3 show the value of taxonomic measure of development (surface of the wheel) for each stock exchange included in the study for 2004 and 2012 respectively. this graphical presentation is useful for a preliminary assessment of changes in the global capital market over the considered period. in 2004, two dominant financial centers are clearly visible. in the west, it is nyse and nasdaq, while in central europe, the london stock exchange and nyse euronext europe stand out in particular. the reason for achieving such good results by the latter is certainly the fact that nyse euronext europe is an example of a trading platform created by the consolidation of the stock exchanges of paris, amsterdam, brussels and lisbon. against the background the tokyo stock exchange stands out of the asian stock exchanges in 2004. in 2012, by contrast, a slight strengthening of the position of the two largest us stock exchanges: nyse and nasdaq may be observed. however, the most spectacular changes can be seen in the case of the nomx central european stock exchange. comparing figures 2 and 3 allows to observe that the asian stock markets have strengthened themselves at the expense of the us and european stock exchanges within the nine years of the research period. asia is currently the largest region of emerging markets in the world and is a cradle of the fastest-growing economies. the advantage of these markets arises not only from the fact that this area is inhabited by more than 60 percent of the world's population, but also from the reasonable, in comparison to other countries, fiscal and monetary policy. economic liberalization and increasing competitiveness of these markets still attracts many foreign investors. the significance of distance between stock exchanges … dynamic econometric models 14 (2014) 125–144 135 figure 2. the taxonomic measure of stock exchanges’ development in 2004 figure 3. the taxonomic measure of stock exchanges’ development in 2012 for the purpose of a preliminary assessment of the relationship between analyzed stock markets in the context of physical distance, for all pairs of stock exchanges the values of pearson correlation coefficient for the tdm in the period of 2004–2012 were calculated. figure 4 is a graphic illustration of the relationships, for which the values of the correlation coefficient are greater than 0.9. it may be noticed that most of these connections are located on the old continent. on one hand, it demonstrates a high integration of european stock exchanges but also carries the risk of transmitting negative pulses occurring within a trading floor, for further linked to it causing the contagion effect. for comparison, the links identified on the basis of the value of the correlation coefficient for the period 2007–2010 crisis, are shown in figure 5. according to some beliefs (see e.g. login and solnik, 2001) the strength of the relationship between the stock exchanges during the downturn increases and decreases with the improvement of the general economic situation in the world. this hypothesis is elżbieta szulc, dagna wleklińska, karolina górna, joanna górna dynamic econometric models 14 (2014) 125–144 136 confirmed by the number of links between stock exchanges marked in figure 5, for which the value of the correlation coefficient exceeds 0.9. therefore, it seems that along with deteriorating sentiments in the global capital market, an increase of the correlation between securities markets might be expected, as long as the trend will not be reversed. figure 4. the significant linkages between investigated stock markets according to the pearson correlation coefficient in the period of 2004–2012 figure 5. the significant linkages between investigated stock markets according to the pearson correlation coefficient in the period of 2007–2010 4. results of the econometric analysis the successive tables presented below contain the information on the usefulness of various methodological conceptions expressed by the spatial models, presented in section 3, in comparison with the linear regression model, i.e. the traditional model without the spatial effects. the significance of distance between stock exchanges … dynamic econometric models 14 (2014) 125–144 137 table 2. results of the estimation and verification of β-convergence models for pooled time series and cross-sectional data – variant i linear regression spatial autoregressive model spatial error model parameters     –0.2276 (0.0000) –0.1252 (0.0000) – – –0.1704 (0.0000 ) –0.0965 (0.0000) 0.7082 (0.0000) – –0.1702 (0.0004) –0.0922 (0.0000) – 0.7255 (0.0000) goodness of fit adjusted r2 aic 0.0644 –161.0100 – –264.4000 – –261.9500 heteroskedasticity breuch-pagan test 1.8260 (0.1766) 1.1870 (0.2759) 1.5371 (0.2151) autocorrelation of residuals moran test 17.7228 (0.0000) –0.8701 (0.1923) –0.7309 (0.2312) spatial dependence lr lmlag lmerr rlmlag rlmerr – 316.6723 (0.0000) 291.9936 (0.0000) – – 105.3900 (0.0000) – – 26.0514 (0.0000) – 102.9300 (0.0000) – – – 1.3727 (0.2414) speed of convergence half-life 0.0167 41.47 0.0127 54.63 0.0121 57.31 note: numbers in brackets refer to the p-values. table 2 contains the results of estimation and verification of three models for pooled time series and cross-sectional data: the linear regression model (tscs), the spatial autoregressive model (sar_pooled) and the spatial error model (se_pooled). in the spatial models for the purpose of quantification of the connections among exchanges investigated the matrix w was used, taking into account the physical distance between them (variant i). table 3 presents the results for the three analogical models, but in the spatial elżbieta szulc, dagna wleklińska, karolina górna, joanna górna dynamic econometric models 14 (2014) 125–144 138 models the connectivity matrix w* of the economic distance between the exchanges was used (variant ii). table 3. results of the estimation and verification of β-convergence models for pooled time series and cross-sectional data – variant ii linear regression spatial autoregressive model spatial error model parameters     –0.2276 (0.0000) –0.1252 (0.0000) – – –0.1670 (0.0000) –0.0981 (0.0202) 0.8030 (0.0000) – –0.1477 (0.0110) –0.1221 (0.0000) – 0.8311 (0.0000) goodness of fit adjusted r2 aic 0.0644 –161.0100 – –279.9800 – –285.4900 heteroskedasticity breuch-pagan test 1.8260 (0.1766) 0.4405 (0.5069) 0.2427 (0.6223) autocorrelation of residuals moran test 25.8554 (0.0000) 3.0317 (0.0011) 3.3425 (0.0004) spatial dependence lr lmlag lmerr rlmlag rlmerr – 597.6765 (0.0000) 590.0261 (0.0000) 120.9600 (0.0000) – – 9.6901 (0.0019) – 126.4800 (0.0000) – – – 2.0397 (0.1532) speed of convergence half-life 0.0167 41.47 0.0129 53.69 0.0163 42.57 note: numbers in brackets refer to the p-values. the classical model estimated using the pooled time series and crosssectional data does not satisfy the fundamental criterions of statistical verification (see tables 2 and 3). this result is consistent with our prediction because the assumptions of the model, especially the same variance in the space and independence across residuals for all singled out objects, are usuthe significance of distance between stock exchanges … dynamic econometric models 14 (2014) 125–144 139 ally unrealistic in practice. though in this case the breusch-pagan statistic is insignificant, on the basis of the moran test the hypothesis of independence of the traditional model residuals should be rejected. as the moran test does not admit an explicit alternative hypothesis opposed to the null, the lagrange multiplier tests (lm) were used (see tables 2 and 3). the lm tests for the linear model for the pooled time series and cross-sectional data used consider the spatial lag model (spatial autoregressive) and the spatial error model as alternatives (lmlag and lmerr, respectively). tables 2 and 3 report the results of using the robust tests (rlmlag, in which h0:  = 0 under the assumption that   0 and rlmerr, where h0:  = 0 under the assumption that   0) as well. since the lmlag tests are more significant than the lmerr, and the rlmlag are significant while the rlmerr are insignificant, the spatial lag models should be preferred. subsequently, the significance of the spatial effects in sar and se models using the likelihood ratio test (lr) were confirmed. likewise, irrespective of which connectivity matrix (of physical or of economic distance) in the spatial models has been used, parameters ρ and λ are statistically significant. it is worth noting that the fact of including the connectivity matrixes in the considered models has crucial impact on convergence parameters (). absolute values of the parameters for the models sar and se are lower than for the traditional model which does not take into account the connections across investigated stock exchanges. in turn, comparing the  parameters in the spatial models which contain the matrix of physical distance with the parameters of the models which contain the matrix of economic distance one can see that the convergence parameters are higher in the second case. it can be supposed that geographical distance has less impact on the process of equalizing the differentiation of stock markets. evaluation of statistical properties of the received empirical models reveals that in the models constructed with the use of economic distance between the stock exchanges the problem of autocorrelation of the residuals has not been eliminated. it is a significant drawback of these models. solving the problem requires further investigation towards an appropriate modification of the connectivity matrix. tables 4 and 5 contain the results of the estimation and verification of exemplary panel models used in the investigation, i.e. the panel model with fixed effects without spatial component, the spatial autoregressive panel model with fixed effects, and the spatial error panel model with fixed effects. just as in the pooled time and cross-sectional data models also in the panel data models the connections among the stock exchanges in two variants (connections according to geographical/economic distance) were taken into elżbieta szulc, dagna wleklińska, karolina górna, joanna górna dynamic econometric models 14 (2014) 125–144 140 account. fixed effects are significant in the considered models. it means that individual characteristics of every exchange are valid for their convergence. table 4. results of the estimation and verification of panel models with fixed effects – variant i fe_ind sar_fe_ind se_fe_ind parameters     –1.5352 (0.0000) –0.8535 (0.0000) – – –1.2475 (0.1027) –0.6956 (0.0510) 0.5141 (0.0000) – –1.3614 (0.0000) –0.7559 (0.0000) – 0.6723 (0.0000) goodness of fit adjusted r2 aic 0.3629 –260.7000 – –327.3200 – –328.9800 heteroskedasticity breuch-pagan test 98.3157 (0.0000) 99.3100 (0.0000) 96.7010 (0.0000) autocorrelation of residuals moran test 13.5916 (0.0000) 13.5916 (0.0374) –0.4039 (0.3432) spatial dependence lr lmlag lmerr rlmlag rlmerr – 122.7834 (0.0000) 171.3005 (0.0000) 68.6220 (0.0000) – – 62.2343 (0.0000) – 70.2730 (0.0000) – – – 13.7172 (0.0002) chow test f – 87.4795 (0.0000) 482.3193 (0.0000) speed of convergence half-life 0.2400 2.89 0.1489 4.65 0.1763 3.93 note: numbers in brackets refer to the p-values. diagnostics for the considered models suggest that the classical panel model is the worst of them. in this case, the breusch-pagan statistic is significant, leading to rejecting the model assumption of homoskedasticity. in the significance of distance between stock exchanges … dynamic econometric models 14 (2014) 125–144 141 addition, on the basis of the moran test the hypothesis of independence of the model residuals should be rejected. table 5. results of the estimation and verification of panel models with fixed effects – variant ii fe_ind sar_fe_ind se_fe_ind parameters     –1.5352 (0.0000) –0.8535 (0.0000) – – –1.2113 (0.0000) –0.6740 (0.0000) 0.6129 (0.0000) – –1.3165 (0.0000) –0.7465 (0.0000) – 0.8174 (0.0000) goodness of fit adjusted r2 aic 0.3629 –260.7000 – –336.8200 – –350.8200 heteroskedasticity breuch-pagan test 98.3157 (0.0000) 98.5343 (0.0000) 95.8613 (0.0000) autocorrelation of residuals moran test 18.8266 (0.0000) 5.6920 (0.0000) 3.9209 (0.0000) spatial dependence lr lmlag lmerr rlmlag rlmerr – 184.0365 (0.0000) 311.6529 (0.0000) 78.1170 (0.0000) – – 17.7730 (0.0000) – 92.1220 (0.0000) – – – 145.3894 (0.0000) chow test f – 83.2527 (0.0000) 95.0155 (0.0000) speed of convergence half-life 0.2400 2.89 0.1401 4.95 0.1715 4.04 note: numbers in brackets refer to the p-values. the necessity of model re-specifications towards the spatial panel models was also confirmed by the lagrange multiplier tests. all the tests are statistically significant and unfortunately robust versions of the tests do not provide unambiguous conclusions on what kind of the spatial connections, elżbieta szulc, dagna wleklińska, karolina górna, joanna górna dynamic econometric models 14 (2014) 125–144 142 autoregressive or error, should be applied to the models. moreover, the significance of the spatial effects with the aid of the lr test has been confirmed. for investigating the reasonableness of including the fixed effects in the spatial models the chow test (the spatial model for pooled tscs data vs. the spatial panel model with fixed effects) was used. the results of the chow test have pointed out the statistical significance of the fixed effects in the spatial autoregressive panel model, as well as in the panel spatial error model (see tables 4 and 5). taking into account the geographical connections (variant i) among the investigated stock exchanges in the panel convergence models has removed the problem of autocorrelation of the residuals (at the level of significance =0.01). however, in the case of using the matrix of economic distance (variant ii) the autocorrelation of model residuals has not been eliminated. in turn, the problem of heteroskedasticity has remained in both cases. therefore, in further investigation searching for the spatial regimes will be performed. conclusions the paper's findings show that including the linkages that result from physical and/or economic distances between stock exchanges in the models of their convergence is justified and very important for the analyses of the phenomenon. in other words, the results of the investigation provide evidence of spatial effects in the empirical models of stock exchanges' convergence. as a result, it is possible to define the influence of the distance between exchanges on their economic development, the estimates of convergence parameter are more precise, and some statistical properties of the models are better. during the investigation it was observed that geographical distance has less impact on the process of equalizing differentiation of stock markets then the economic distance between them. due to the heteroskedasticity, the empirical panel models for the exchanges investigated as a whole were not entirely satisfactory. it means that there are differentials in relationships between objects considered and their speed of convergence. thus, in further investigation the spatial regimes will be searched for. for example, we will investigate the convergence of the european, asian and american stock markets, separately. in addition, we will continue the work on improving other properties of the empirical models, e.g. on removing the problem of autocorrelation of the residuals. the significance of distance between stock exchanges … dynamic econometric models 14 (2014) 125–144 143 references arbia, g. (2006), spatial econometrics. statistical foundations and applications to regional convergence, springer-verlag, berlin heidelberg, doi: http://dx.doi.org/10.1007/3-540-32305-8. asgharian, h., hess, w., liu, l. (2013), a spatial analysis of international stock market linkages, journal of banking & finance, 37(12), 4738–4754, doi: http://dx.doi.org/10.1016/j.jbankfin.2013.08.015. aspergis, n., christou, c., miller, s. m. (2014), country and industry convergence of equity markets: international evidence from club convergence and clustering, the north american journal of economics and finance, 29(c), 36–58. baltagi, b. h., song, s. h., koch, w. (2003), testing panel data regression models with spatial error correlation, journal of econometrics, 117(1), 123–150, doi: http://dx.doi.org/10.1016/s0304-4076(03)00120-9. caporale, g. m., erdogan, b., kuzin, v. (2009), testing for convergence in stock markets: a non-linear factor approach, cesifo working paper series no. 2845, doi: http://dx.doi.org/10.2139/ssrn.1496819. fraser, p., helliar, c. v., power, d. m. (1994), an empirical investigation of convergence among european equity markets, applied financial economics, 4(2), 149–57, doi: http://dx.doi.org/10.1080/758523959. haining r. (2005), spatial data analysis. theory and practice, cambridge university press, 3th ed., cambridge. hellwig, z. (1968), zastosowanie metody taksonomicznej do typologicznego podziału krajów ze względu na poziom ich rozwoju oraz zasoby i strukturę wykwalifikowanych kadr (the application of the taxonomic method to the typological division of a countries due to their level of development, resources and structure of qualified personnel), przegląd statystyczny (statistical survey), 4, 307–327. koralun-bereźnicka, j. (2008), zjawisko konwergencji rynków kapitałowych na rynkach europejskich (the convergence of the capital markets in the european markets), studia i prace kolegium zarządzania i finansów (working papers of collegium of finance and management), szkoła główna handlowa w warszawie, dom wydawniczy elipsa, 87, 87–98. login, b., solnik, f. (2001), extreme correlation of international equity markets, journal of finance, 56(2), 649–676, doi: http://dx.doi.org/10.1111/0022-1082.00340. łuniewska, m., tarczyński, w. (2006), metody wielowymiarowej analizy porównawczej na rynku kapitałowym (methods of multidimensional comparative analysis in the capital market), pwn, warszawa. millo, g., piras, g. (2012), splm: spatial panel data models in r, journal of statistical software, 47(1), 238. mutl, j., pfaffermayr, m. (2011), the hausman test in a cliff and ord panel model, econometrics journal, 14(1), 48–76, doi: http://dx.doi.org/10.1111/j.1368-423x.2010.00325.x. suchecka, j., łaszkiewicz, e. (2011), the influence of spatial and economic distance on changes in the relationships between european stock markets during the crisis of 2007–2009, acta universitatis lodziensis, folia oeconomica, 252, wydawnictwo uniwersytetu łódzkiego, 69–84. suchecki, b. (ed.) (2012), ekonometria przestrzenna ii. modele zaawansowane (spatial econometrics ii. advanced models), wydawnictwo c.h.beck, warszawa. elżbieta szulc, dagna wleklińska, karolina górna, joanna górna dynamic econometric models 14 (2014) 125–144 144 wiśniewski, t. (2003), do unii europejskiej – giełdy krajów kandydackich, (for the european union – stock exchanges of candidate countries), nasz rynek kapitałowy (our capital market), 6, 22–25. wójcik, d. (2009), the role of proximity in secondary equity markets, in clark, g. l., dixon, a. d., and monk, a. h. b. (eds.), managing financial risk. from global to local, oxford university press, oxford, 140–162, doi: http://dx.doi.org/10.1093/acprof:oso/9780199557431.003. znaczenie odległości między giełdami papierów wartościowych w procesie ich konwergencji. analiza wybranych giełd światowych w okresie 2004–2012 z a r y s t r e ś c i. artykuł dotyczy analizy konwergencji wybranych giełd światowych z punktu widzenia poziomu ich rozwoju, w kontekście geograficznej i ekonomicznej odległości między nimi. przedstawia podejście, które wskazuje na potrzebę uwzględniania przestrzennych i ekonomicznych powiązań między rynkami giełdowymi w analizach ich konwergencji. badanie obejmuje 46 największych parkietów, analizowanych w okresie 2004–2012. dane empiryczne odnoszą się do 6 zmiennych diagnostycznych, uznanych jako ważne determinanty rozwoju rynków giełdowych. s ł o w a k l u c z o w e: giełda papierów wartościowych, konwergencja, odległość fizyczna, odległość ekonomiczna, macierz sąsiedztwa, przestrzenne modele panelowe. microsoft word dem_2014_5to28.docx © 2014 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2014.001 vol. 14 (2014) 5−28 submitted may 5, 2014 issn accepted december 12, 2014 1234-3862 juliusz jabłecki*, ryszard kokoszczyński*, paweł sakowski, robert ślepaczuk*, piotr wójcik does historical vix term structure contain valuable information for predicting vix futures? a b s t r a c t. we suggest that the term structure of vix futures shows a clear pattern of dependence on the current level of vix index. at the low levels of vix (below 20), the term structure is highly upward sloping, while at the high vix levels (over 30) it is strongly downward sloping. we use these features to predict future vix futures prices more precisely. we begin by introducing some quantitative measures of volatility term structure (vts) and volatility risk premium (vrp). we use them further to estimate the distance between the actual value and the fair (model) value of the vts. we find that this distance has significant predictive power for volatility futures and index futures and we use this feature to design simple strategies to invest in vix futures. k e y w o r d s: volatility term structure, volatility risk premium, vix, vix futures, volatility futures, realized volatility, implied volatility, investment strategies, returns forecasting, efficient risk and return measures j e l classification: g11, g14, g15, g23, c61, c22 * the views presented in this text are those of the authors and do not necessarily represent those of the national bank of poland or union investment tfi s.a. correspondence to: robert ślepaczuk, university of warsaw, faculty of economic sciences, ul. długa 44/50, 00-241 warsaw, poland, tel: +48 22 55 49 174, fax: +48 22 831 28 46, e-mail: rslepaczuk@wne.uw.edu.pl.  we gratefully acknowledge government financial support via grant no. umo-2011/ 03/b/hs4/02298. j. jabłecki, r. kokoszczyński, p. sakowski, r. ślepaczuk, p. wójcik dynamic econometric models 14 (2014) 5–28 6 introduction we observe that term structure of vix futures shows a clear pattern of dependence on the current level of the vix1. figure 1 shows that the term structure is highly upward sloping when the level of vix is relatively low (below 20) and it is significantly downward sloping when the level of vix is high (over 30). we assume that this is mostly due to the market perception of risk in the short and the long-term. that perception reflects strong mean reversion effects and long memory – both visible in volatility time series. figure 1. vix and s&p500 index quotations from 01/01/2006 until 01/07/2013 in order to investigate the term structure of vix futures phenomenon more deeply, we introduce two quantitative characteristics of volatility term structure derived directly from the levels of the consecutive vix futures maturities. these measures are calculated independently for different quintile groups2 of vix and various times to expirations of vix futures. further, we use these measures in order to estimate the distance between the actual and the fair (or “theoretical”) value of volatility term structure for particular maturity of vix futures. subsequently, we include this information in the 1 vix – volatility index quoted on cboe, based on implied volatility of s&p500 index options. it reflects market’s expectation of stock market volatility in 30 days horizon. more information could be found in whaley (1993), cboe (2003) and cboe (2009). 2 as those dependence patterns are not homogenous in the whole sample, we divide our data into vix levels quintile groups. the rest of the study is conducted consistently on this basis. does historical volatility term structure contain valuable information… dynamic econometric models 14 (2014) 5–28 7 process of forecasting vix levels and vix futures and we find that this distance has significant predictive power. in the last section, we propose a simple investment strategy that uses vts in order to predict vix futures prices. volatility research and, in particular, volatility forecasting seems to be one of the most active and successful areas in financial econometrics in recent decades (andersen et al., 2005). the literature on the vix and its derivatives is growing very fast, but the number of studies testing the predictability of vix futures prices is – according to our best knowledge – still very low. konstantinidi and skiadopoulos (2011) show only weak evidence of statistically predictable patterns in the evolution of volatility futures prices. they also cannot find a trading strategy with economically significant profits. some papers suggest strongly that the use of the information content of the volatility term structure may improve this situation, but the number of studies attempting to examine directly the relationship between the term structure of vix futures and their future returns or the underlying equity returns is still very limited (fassas, 2012; asensio, 2013; huskaj and nossman, 2013 and the references therein). in our previous research (jabłecki et al. 2013a) we found significant relationship between lagged term structure of vix futures and current level of vix, yet this didn’t translate into better predictions of vix level. the paper is thus organized as follows. the next section describes the data. methodology of this research is presented in the second section. the third section presents the description of simple measures of volatility term structure (vts). next section describes the dependence between vix and s&p500 index and measures of vts. forecasting properties of vts are presented in the same section. the investment strategies based on these results are presented in the fifth section. the last section concludes and presents possible extensions of this research. 1. data description for each trading day we gathered close prices for vix, vix futures and s&p500 index3. initially, the data set included 24 expiration months (from january 2006 until july 2013), but we had to limit their number to 7 because 3 data are from the following sources:  vix http://cfe.cboe.com/products/historicalvix.aspx  vix futures http://cfe.cboe.com/products/historicalvix.aspx  s&p500 www.stooq.pl j. jabłecki, r. kokoszczyński, p. sakowski, r. ślepaczuk, p. wójcik dynamic econometric models 14 (2014) 5–28 8 of liquidity problems for longer maturities. data preparation for vix returns also included the process of gap correction in order to omit the problem of very high positive returns at the moment of series change. figure 1 and table 1 present time series for vix and s&p500. it is worth noticing that s&p500 returns are leptokurtic and negatively skewed while vix returns are also leptokurtic and positively skewed. additionally, both returns series are negatively correlated. table 1. the descriptive statistics for daily returns of vix and s&p500 index vix returns s&p500 returns number of obs. 1872 1872 minimum –0.3505 –0.0946 maximum 0.4960 0.1095 1st quartile –0.0401 –0.0049 3rd quartile 0.0325 0.0061 mean 0.0001 0.0001 median –0.0052 0.0008 se mean 0.0016 0.0003 lcl mean –0.0030 –0.0005 ucl mean 0.0034 0.0007 std. deviation 0.0719 0.0144 skewness 0.7022 –0.2994 kurtosis 4.1499 8.9920 correlation –0.7550 note: all calculations were made on the data from 01/01/2006 until 01/07/2013. 2. methodology to answer research questions and verify hypotheses we decided to undertake the following steps. as some earlier studies suggest (i.e. giot, 2005; simon and wiggins, 2001), we investigate the relationship we are interested in having classified daily vix closing prices into five quintile groups4. as we observe very low liquidity for longer vix futures maturities, we withdraw from the sample all futures prices for contracts with expiration longer than seven months5. then, we use ols to estimate quadratic regressions for closing prices of vix and vix futures, independently for each quintile group: , (1) 4 the selection of quintile groups is somewhat arbitrary, nevertheless it was dictated by very heterogeneous shapes of volatility term structure of vts in different market conditions. it is important to add that it really does not matter if we choose quintile or quartile groups. 5 this amounted to be less than 25% of initial sample size. does historical volatility term structure contain valuable information… dynamic econometric models 14 (2014) 5–28 9 where – daily closing price of vix or vix futures, – time to maturity of -th vix futures ( 1,…, , for vix 0), – size of the vix -th quintile group ( 1,…,5). as a result, we are able to define the shape of relationship between prices of vix contracts and their times to expiry. we refer to this relationship as volatility term structure (vts). on figure 2, we draw all observations and shapes of volatility term structure, separately for five vix quintile groups. it confirms our initial presumption that volatility term structure is dependent on vix level. it is upward sloping for initial four quintile groups where parameter is negative. on the other hand, it is downward sloping for the fifth vix quintile group where parameter has positive value. figure 2. quadratic regressions for term structure of vix futures in five different vix quintile groups. all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. vix futures are quoted on cfe based on this initial intuition that volatility term structure is dependent on vix level and that the slope of vts depends on the current level of vix, we propose two different measures of vts and three different measures of volatility risk premium (vrp) in order to quantify vts and risk associated with it. we use the results of regression (1) to construct a reference (theoretical) price of volatility futures as a function of the vix level (quintile) and time to expiration. the distance between the actual price and the reference price allows us to estimate the vrp. then, we show that future returns of vix and s&p500 index are dependent on the actual level of vix and the shape of volatility term structure. this is confirmed with simple regressions j. jabłecki, r. kokoszczyński, p. sakowski, r. ślepaczuk, p. wójcik dynamic econometric models 14 (2014) 5–28 10 trying to find some robust patterns which are used later on to construct investment strategy, where the investment algorithms are based on vts. 3. measures of volatility term structure we propose two measures of vts and two measures of vrp. the detailed formulas are presented below. slope1 is a sum of differences of vix futures prices with consecutive maturities divided by actual vix level, calculated separately within each vix quintile group: slope1 , (2) where – close price of 7-th vix futures within -th vix quintile group. actual values of slope1 confirm our observations from the previous section concerning volatility term structure. we see substantial positive differences between last and first contract levels for first four quintile groups and significant negative differences for the last quintile group (figure 3). figure 3. boxplot for slope1 with respect to vix quintile groups. all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. diamonds denote group mean values, while black dots denote outliers which are outside 150% of interquartile range the second vts measure, slope2, is the slope coefficient of the simple linear regression, estimated using ols separately for every quintile group of vix levels: does historical volatility term structure contain valuable information… dynamic econometric models 14 (2014) 5–28 11 β β , 3 where – daily closing price of vix and vix futures, – time to maturity of -th vix futures (for vix = 0, 1,…, ), – size of the vix -th quintile group ( 1,…,5). slope2 results give support to very similar conclusions as those inferred from slope1 values (table 2). table 2. the descriptive statistics of slope2 for vix quintile groups vix quintile group size parameters min max avg med sd vix levels 1 374 (0;13.14] 0.9 20.3 5.7 5.4 3.1 2 374 (13.14;16.03] –5.4 20.1 9.4 10.6 5.6 3 375 (16.03;19.55] –2.2 19.0 8.0 8.7 4.9 4 373 (19.55:25.41] –10.5 16.1 4.3 4.6 5.1 5 374 (25.41;80.86] –71.8 8.4 –9.9 –5.6 13.9 all 1870 (0:80.86] –71.8 20.3 3.5 4.8 10.3 note: all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. as the next step we calculate individual volatility risk premium (vrp) by comparing actual vix futures prices with their „theoretical” values given by formula (1), for each quintile group separately. , is defined as a percentage deviation of -th futures current price from its „theoretical” price: , , , , (4) where: – the number of the quintile group ( ∈ 1,2,3,4,5 , – reference to consecutive vix futures contracts with ascending time to maturity ( ∈ 1,2,3,4,5,6,7 ), – size of the vix -th quintile group ( 1,…,5), , – closing price of -th vix futures contract within -th quintile group, , – theoretical value of -th vix futures index future, calculated from eq. (1), within vix -th quintile group. shows that inside first four vix quintile groups we observe on average quite substantial departures from „theoretical” volatility term structure. they range between –20% and 20%. what is more, these deviations are neither skewed towards positive nor negative direction, which means that on average they are equal to zero. the fifth vix quintile group shows much different picture. deviations here are much more volatile and they are heavily skewed towards positive values (figure 4). at the same time, much more j. jabłecki, r. kokoszczyński, p. sakowski, r. ślepaczuk, p. wójcik dynamic econometric models 14 (2014) 5–28 12 than half of observations from this group have values below zero (median is on the level of 10%). figure 4. boxplot for with respect to vix quintile groups. all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. diamonds denote group mean values, while black dots denote outliers which are outside 150% of interquartile range in the next step, we estimate the aggregated volatility risk premium for all maturities and for each quintile group. this measure is proposed in two versions in order to correctly present the direction and the magnitude of deviations from the “theoretical” shape:  aggregated volatility risk premium is the sum of individual volatility risk premiums for all vix futures maturities, separately for each vix quintile group: , ∑ , , (5)  absolute aggregated volatility risk premium is the sum of absolute individual volatility risk premium for all vix futures maturities separately for each vix quintile group: , | | ∑ , , (6) where: – the number of the quintile group ( ∈ 1,2,3,4,5 ), – reference to consecutive vix futures contracts with ascending time to maturity ( ∈ 1,2,3,4,5,6,7 ). does historical volatility term structure contain valuable information… dynamic econometric models 14 (2014) 5–28 13 , contains information about the magnitude of departures for all maturities taken together. this could be an important factor for estimating the value of overall shift of vts. on the other hand, , | | contains information about the direction of departures for all maturities together. similarly, this could helpful to estimate the direction of overall shift of vts. figure 5 shows the direction of deviation ( , ), while figure 6 presents their aggregated values ( , | | ). figure 5. boxplot for , with respect to vix quintile groups. all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. diamonds denote group mean values, while black dots denote outliers which are outside 150% of interquartile range figure 5 allows for similar conclusions as figure 4 but the range of fluctuations inside each quintile group is now much wider – the reason for that being quite trivial because this is the sum of individual volatility risk premiums. moreover, we observe a large number of outliers inside the first quintile group. they signal quite substantial deviations of the vix futures from their theoretical shape, most often for maturities from the second one up – the result thereof is the highly upward sloping vts for days with very high values of , or , | | . furthermore, we observe a positive skewness for the fifth vix quintile group, as is the case for . j. jabłecki, r. kokoszczyński, p. sakowski, r. ślepaczuk, p. wójcik dynamic econometric models 14 (2014) 5–28 14 the situation is quite similar when we analyze figure 6. once again we observe much wider fluctuations in each quintile group than in the case of figure 4. moreover, we identify much more outliers in each quintile group than in figure 5. observations in each quintile group are more or less positively skewed with the highest skewness in the fifth quintile group. additionally, median and mean in fifth quintile group are much higher than in other four quintile groups what is partly the result of the highest fluctuations in case of this group on figure 5. figure 6. boxplot for , | | with respect to vix quintile groups. all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. diamonds denote group mean values, while black dots denote outliers which are outside 150% of interquartile range 4. forecasting properties of volatility term structure in order to check predictive power of proposed measures of vts and vrp we present below tables with median 1-month returns of s&p500 index and vix conditional on five quintile groups of vix and:  five quintile groups for slope1 (table 3 and table 4),  five quintile groups for slope2 (table 5 and table 6), does historical volatility term structure contain valuable information… dynamic econometric models 14 (2014) 5–28 15  four quintile groups6 for (table 7 and table 8),  four quintile groups for , (table 9 and table 10),  four quintile groups for , | | (table 11 and table 12). table 3. median 1-month s&p500 returns (in %) conditional on vix and slope1 quintile groups vix quintile group slope1 i quintile group ii quintile group iii quintile group iv quintile group v quintile group all vix level [–0.53: –0.02] (–0.02: 0.1] (0.1: 0.25] (0.25: 0.39] (0.39: 0.92] [–0.53: 0.92] 1 [9.89:14.18] na 3.03 1.22 1.74 1.14 1.27 2 (14.18:17.74] 5.46 0.81 2.17 –0.18 0.72 0.86 3 (17.74:21.68] 0.31 –3.13 0.11 2.34 3.14 1.19 4 (21.68:26.85] 1.98 1.39 2.84 3.2 4.12 2.48 5 (26.85:80.86] 0.52 2.94 8.00 na na 1.23 all [9.89:80.86] 0.91 0.87 1.55 1.74 1.4 1.33 note: all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. table 4. median 1-month vix returns (in %) conditional on vix and slope1 quintile groups vix quintile group slope1 i quintile group ii quintile group iii quintile group iv quintile group v quintile group all vix level [–0.53: –0.02] (–0.02: 0.1] (0.1: 0.25] (0.25: 0.39] (0.39: 0.92] [–0.53: 0.92] 1 [9.89:14.18] na –0.04 0.91 1.43 5.01 2.52 2 (14.18:17.74] –27.7 –2.88 –8.46 9.84 6.54 5.36 3 (17.74:21.68] 1.27 14.95 –0.92 –7.27 –8.53 –3.39 4 (21.68:26.85] –14.99 –1.34 –7.70 –8.50 –11.94 –7.62 5 (26.85:80.86] –11.44 –12.78 –16.01 na na –12.06 all [9.89:80.86] –11.95 –1.9 –3.07 –1.12 2.92 –2.68 note: all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. bold fonts denote values significantly different from zero. we try further to find with various vts and vrp measures any patterns of relationship between s&p500 index and vix returns. table 3. shows that we cannot find any clear dependence between s&p500 return and vix quintile groups and this observation does not change for all other tables describ 6 we divide vrp measures into quartile groups because their fluctuations are not homogenous. we choose quartile groups for vrp instead of quintile groups selected for vix to make results and then definitions of investment strategies more transparent. j. jabłecki, r. kokoszczyński, p. sakowski, r. ślepaczuk, p. wójcik dynamic econometric models 14 (2014) 5–28 16 ing s&p500 returns (table 3, table 5, table 7, table 9, table 11). additionally, we can see that s&p500 returns do not depend on slope1 levels. table 5. median 1-month s&p500 returns (in %) conditional on vix and slope2 quintile groups vix quintile group slope2 i quintile group ii quintile group iii quintile group iv quintile group v quintile group all vix level [–71.83: –0.72] (–0.72: 2.87] (2.87: 6.88] (6.88: 10.54] (10.54: 20.36] [–71.83: 20.36] 1 [9.89:14.18] na 1.07 1.34 1.23 2.61 1.27 2 (14.18:17.74] 5.46 0.81 1.76 1.67 0.07 0.86 3 (17.74:21.68] 1.04 –2.57 –2.29 1.63 2.99 1.19 4 (21.68:26.85] 1.98 –2.72 2.79 3.27 3.21 2.48 5 (26.85:80.86] 0.59 2.88 1.97 9.10 na 1.23 all [9.89:80.86] 0.98 0.55 1.44 1.75 1.83 1.33 note: all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. table 6. median 1-month vix returns (in %) conditional on vix and slope2 quintile groups vix quintile group slope2 i quintile group ii quintile group iii quintile group iv quintile group v quintile group all vix level [–71.83: –0.72] (–0.72: 2.87] (2.87: 6.88] (6.88: 10.54] (10.54: 20.36] [–71.83: 20.36] 1 [9.89:14.18] na 8.63 0.85 3.04 –0.84 2.52 2 (14.18:17.74] –27.7 –2.88 –11.78 2.53 10.79 5.36 3 (17.74:21.68] –14.21 13.89 13.99 –5.05 –8.47 –3.39 4 (21.68:26.85] –14.99 0.63 –3.38 –11.93 –8.33 –7.62 5 (26.85:80.86] –11.91 –11.24 –14.69 –16.10 na –12.06 all [9.89:80.86] –12.64 0.87 –1.71 –0.99 –0.57 –2.68 note: all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. bold fonts denote values significantly different from zero. on the other hand, we can observe – in all relevant tables describing vix returns (table 4, table 6, table 8, table 10 and table 12) – that vix returns decrease almost monotonically, while vix level moves from the first to the fifth quintile group. at the same table, we can see that vix returns increase almost monotonically while slope1 moves from the first to the fifth quintile group. generally, table 4 informs us about very high positive vix returns for the first and the second vix quintile groups and for the fourth and the fifth slope1 quintile groups. on the other hand, we observe very high negative vix returns for the fourth and the fifth vix quintile groups and for the first and for the second slope1 quintile groups. does historical volatility term structure contain valuable information… dynamic econometric models 14 (2014) 5–28 17 slope2 discloses additional information. table 5 shows that s&p500 index returns increase monotonically while slope2 moves from the first to the fifth quintile group. table 6, describing the dependence of vix returns on the slope2 values, allows for almost the same conclusions as in the case of table 4. the general conclusion concerning the part of the table with high positive and high negative returns is once again the same as in the case of table 4. table 7. median 1-month s&p500 returns (in %) conditional on vix quintile groups and quartile groups vix quintile group i quartile group ii quartile group iii quartile group iv quartile group all vix level [–0.34: –0.08] (–0.08: –0.01] (–0.01: 0.06] (0.06: 0.79] [–0.34: 0.79] 1 [9.89:14.18] 0.68 0.95 1.83 1.56 1.27 2 (14.18:17.74] 1.77 0.44 –1.74 2.18 0.86 3 (17.74:21.68] 1.84 1.19 0.19 1.78 1.19 4 (21.68:26.85] 0.56 1.23 2.6 4.78 2.48 5 (26.85:80.86] 1.08 2.99 5.68 –1.60 1.23 all [9.89:80.86] 1.15 1.1 1.28 1.92 1.33 note: all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. table 8. median 1-month vix returns (in %) conditional on vix quintile groups and quartile groups vix quintile group i quartile group ii quartile group iii quartile group iv quartile group all vix level [–0.34: –0.08] (–0.08: –0.01] (–0.01: 0.06] (0.06: 0.79] [–0.34: 0.79] 1 [9.89:14.18] 5.74 0.26 0.49 5.43 2.52 2 (14.18:17.74] –6.32 2.10 15.27 1.69 5.36 3 (17.74:21.68] –12.00 0.31 3.52 –6.55 –3.39 4 (21.68:26.85] –3.67 –8.01 –8.59 –6.31 –7.62 5 (26.85:80.86] –11.79 –15.9 –16.55 –9.76 –12.06 all [9.89:80.86] –5.37 –2.28 0.15 –3.03 –2.68 note: all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. bold fonts denote values significantly different from zero. table 7 shows that s&p500 returns increase monotonically with the increase of values. on the other hand, we cannot see any clear dependence between vix returns and values (table 8). contrary to the j. jabłecki, r. kokoszczyński, p. sakowski, r. ślepaczuk, p. wójcik dynamic econometric models 14 (2014) 5–28 18 latter, the general conclusion concerning the part of the table with high positive and high negative returns still holds. in table 9 we find the most ideal dependence of s&p500 returns. they increase monotonically with the increase of , values. there is no clear dependence between vix returns and , values (table 10). once again, the general conclusion concerning the part of the table with high positive and high negative returns still holds. table 9. median 1-month s&p500 returns (in %) conditional on vix quintile groups and , quartile groups vix quintile group . i quartile group ii quartile group iii quartile group iv quartile group all vix level [–2.19: –0.55] (–0.55: –0.11] (–0.11: 0.58] (0.58: 3.68] [–2.19: 3.68] 1 [9.89:14.18] 0.77 0.82 2.11 2.02 1.27 2 (14.18:17.74] 1.41 1.40 –1.93 2.16 0.86 3 (17.74:21.68] 0.28 –0.40 –0.17 2.75 1.19 4 (21.68:26.85] –0.56 1.88 2.60 4.21 2.48 5 (26.85:80.86] 0.42 2.90 5.50 0.28 1.23 all [9.89:80.86] 0.6 1.05 1.4 2.58 1.33 note: all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. table 10. median 1-month vix returns (in %) conditional on vix quintile groups and , quartile groups vix quintile group . i quartile group ii quartile group iii quartile group iv quartile group all vix level [–2.19: –0.55] (–0.55: –0.11] (–0.11: 0.58] (0.58: 3.68] [–2.19: 3.68] 1 [9.89:14.18] 7.59 3.01 –2.64 2.96 2.52 2 (14.18:17.74] –7.72 0.06 12.13 2.17 5.36 3 (17.74:21.68] –10.23 5.36 –1.48 –9.96 –3.39 4 (21.68:26.85] –5.87 –12.34 –5.21 –7.42 –7.62 5 (26.85:80.86] –8.72 –16.39 –18.52 –11.68 –12.06 all [9.89:80.86] –5.13 –0.59 –0.56 –4.63 –2.68 note: all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. red font denotes significantly negative returns while green font denotes significantly positive returns. table 11 and table 12 do not add any new information. there is no any clear dependence between , | | and s&p500 return. the same concludoes historical volatility term structure contain valuable information… dynamic econometric models 14 (2014) 5–28 19 sion we can drawn in the case of vix returns and , values (table 12). the general conclusion concerning the part of the table with high positive and high negative vix returns holds only in case of the negative returns. table 11. median 1-month s&p500 returns (in %) conditional on vix quintile groups and , | | quartile groups vix quintile group . | | i quartile group ii quartile group iii quartile group iv quartile group all vix level [0.03: 0.33] (0.33: 0.58] (0.58: 0.87] (0.87: 3.68] [0.03: 3.68] 1 [9.89:14.18] 1.69 0.75 1.31 2.24 1.27 2 (14.18:17.74] –2.28 0.87 1.63 2.16 0.86 3 (17.74:21.68] 0.95 –0.76 1.66 2.01 1.19 4 (21.68:26.85] 2.74 1.12 2.22 3.83 2.48 5 (26.85:80.86] 4.25 3.27 0.38 0.22 1.23 all [9.89:80.86] 1.47 0.86 1.36 1.82 1.33 note: all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. table 12. median 1-month vix returns (in %) conditional on vix quintile groups and , | | quartile groups vix quintile group . | | i quartile group ii quartile group iii quartile group iv quartile group all vix level [0.03: 0.33] (0.33: 0.58] (0.58: 0.87] (0.87: 3.68] [0.03: 3.68] 1 [9.89:14.18] –2.49 7.54 6.39 –0.44 2.52 2 (14.18:17.74] 13.68 5.89 –3.32 –0.57 5.36 3 (17.74:21.68] 0.35 1.56 –10.17 –8.34 –3.39 4 (21.68:26.85] –13.8 –3.5 –3.67 –9.24 –7.62 5 (26.85:80.86] –26.61 –15.39 –8.24 –10.27 –12.06 all [9.89:80.86] –3.24 1.45 –2.73 –6.42 –2.68 note: all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. bold fonts denote values significantly different from zero. having found strong dependence patterns we will try to use them to design a simple investment strategy which is supposed to beat the market (s&p500 buy&hold strategy). j. jabłecki, r. kokoszczyński, p. sakowski, r. ślepaczuk, p. wójcik dynamic econometric models 14 (2014) 5–28 20 5. the investment model the main objective of this section is to design investment strategies which will implement the dependence of vix and vix futures returns on various vts and vrp measures. this idea has been directly motivated by our previous study (jabłecki et. al. 2013a), where we found evidence of significant relationship between lagged slope of vix futures term structure and current level of vix. this finding however, couldn’t help us to get significantly lower average prediction errors of futures vix level, when compared with naïve forecasts. we decided therefore to try to translate apparent connection between term structure of vix futures and vix level into a profitable investment strategy. our investment algorithms use the idea of mean reversion characteristics of vix fluctuations and additionally, the information hidden in term structure of vix futures shape and vrp values. tables and figures presented earlier confirmed our initial intuition that vix returns (and partly s&p500 returns7) depend on the current level of vix, vts shape and vrp values. higher slope of vts (or high vrp value) together with lower vix quintile group generally implicates high vix returns, while lower slope of vts (or low vrp value) together with higher vix quintile group generally implicates very low vix returns. additionally, we observed that average s&p500 index returns rise almost monotonically with the increase of both vrp values and slope2, revealing strong dependence between index future returns and the current level of risk perceived by market participants. we propose five simple strategies which invests in vix futures contracts. for comparison purposes we use s&p500 buy&hold strategy results. the general assumptions for all strategies are as follows:  transaction costs = 0.1%,  data gathering window: minimum one year,  first signals: 2008-01-01, then the current closing price is used to generate a new signal for each consecutive day,  trade price: closing price of vix futures with nearest maturity,  switch to the 2nd contract on the last trading day (rolling yields included),  leverage: 100%,  margin: we do not receive any additional interests from cash above margin. 7 we admit that dependence in case of s&p500 returns is much weaker than in case of vix returns. does historical volatility term structure contain valuable information… dynamic econometric models 14 (2014) 5–28 21 the detailed assumptions for each strategy and theirs results are presented below. 5.1. strategy i positions are determined by following conditions:  buy: if is in 1st quartile group,  sell: if in 4th quartile group,  close: if in 2nd quartile group after sell signal or in 3rd after buy signal,  hold: if in 2nd quartile group after buy signal or in 3rd after sell signal. strategy i utilizes the assumption that vrp is mean reverting process and every departure from „theoretical” volatility term structure should be reverted and should go back to zero. the problem is that while it is true inside vix quintile groups, the logic of the signal can be weakened by switches between vix quintile groups. this could be the reason of rather poor results of this strategy, which are presented in detail on figure 7 and in table 13. 5.2. strategy ii positions are determined by following conditions:  buy: if `in 1st quartile group,  sell: if in 4th quartile group,  hold: if in 2nd or 3rd quartile group. strategy ii is very similar to strategy i but it has less strict rules concerning the moment when we close the position. the problem with switches between vix quintile groups still exists but due to less frequent close signals the results of this strategy are most striking among all strategies (figure 8 and table 13). 5.3. strategy iii positions are determined by following conditions:  buy: if in 1st or 2nd quartile group,  sell: if in 3rd or 4th quartile group. strategy iii is characterized by most frequent switches between short and long positions. nevertheless, this modification does not enhance the results (figure 9 and table 13). j. jabłecki, r. kokoszczyński, p. sakowski, r. ślepaczuk, p. wójcik dynamic econometric models 14 (2014) 5–28 22 figure 7. equity line and signal of strategy i for vix futures and s&p500 index. all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration figure 8. equity line and signal of strategy ii for vix futures and s&p500 index. all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration does historical volatility term structure contain valuable information… dynamic econometric models 14 (2014) 5–28 23 figure 9. equity line and signal of strategy iii for vix futures and s&p500 index. all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration figure 10. equity line and signal of strategy iv for vix futures and s&p500 index. all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration 5.4. strategy iv positions are determined by following conditions:  buy: if vix in 1st or 2nd quintile group, j. jabłecki, r. kokoszczyński, p. sakowski, r. ślepaczuk, p. wójcik dynamic econometric models 14 (2014) 5–28 24  sell: if vix in 4th or 5th quintile group,  hold: otherwise. strategy iv signals depend only on vix level. it uses extreme quintiles of vix in order to generate buy and sell signals for vix futures. the position is hold until the opposite signal is generated. the results of this approach are rather poor (figure 10 and table 13). figure 11. equity line and signal of strategy v for vix futures and s&p500 index. all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration 5.5. strategy v positions are determined by following conditions:  buy: if vix in 1st or 2nd quintile group,  sell: if vix in 4th or 5th quintile group,  close: otherwise. strategy v uses the same logic as strategy iv but closes its positions more often what in fact is once again not the best option (figure 11 and table 13). 5.6. comparison of all strategies results are summarized in table 13. to compare the strategies under consideration, we use several popular risk-return measures. after detailed analysis of presented results we can notice that only strategy ii can beat the market represented by s&p500 buy&hold (strategy vi). return (arc) and return-risk statistics (information ratio, sharpe and treynor coefficients) are much better for this strategy. on the other hand, risk does historical volatility term structure contain valuable information… dynamic econometric models 14 (2014) 5–28 25 statistics show that strategy ii is characterized by relatively high beta coefficient, and what is more, by much higher level of standard deviation (asd) and maximum drawdown (maxd) when compared with the benchmark strategy. table 13. return and risk statistics for all investment strategies strategy i strategy ii strategy iii strategy iv strategy v strategy vi buy&hold arc (%) 6.56 30.81 –36.84 –17.80 –29.24 2.28 asd (%) 71.48 81.02 81.17 54.97 81.09 25.43 ir 0.09 0.38 –0.45 –0.32 –0.36 0.09 sharpe 0.08 0.37 –0.46 –0.34 –0.37 0.06 treynor 0.03 0.18 –0.21 –0.14 –0.20 0.02 beta 1.67 1.67 1.80 1.32 1.47 1.00 maxd (%) –69.58 –71.57 –95.04 –89.55 –90.82 –53.25 note: all calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of vix futures with up to 7 months to expiration. arc – annualized return compounded, asd, annualized standard deviation, ir – information ratio, sharpe – sharpe ration, treynor – treynor ratio, beta – slope of regression of a given strategy on buy&hold strategy, maxd – maximum drawdown. conclusions based on the presented results we can conclude that: 1. volatility time structure (vts) shape and volatility risk premium (vrp) values are important in order to predict vix and partly s&p500 index futures. 2. we observe very high positive vix returns for the first and the second vix quintile groups and for the fourth and the fifth slope1 (slope2, , , , , | | quintile groups. on the other hand, we observe very high negative vix returns for the fourth and the fifth vix quintile groups. 3. it is possible to use information from term structure of vix futures to construct profitable strategies (strategy ii) which enhance our returnrisk ratio when compared with s&p500 buy&hold strategy vi. it seems that it would be important to extend these conclusions of this research on other volatility and equity index futures (e.g. vstoxx and eurostoxx50, vnky and nikkei 225 and other volatility futures quoted on cboe/cfe). additionally, we would like to test investment strategies where each characteristics will be calculated on rolling two years window instead of anchored window used in this study. further research on various definition of volatility term structure are even more important because more adej. jabłecki, r. kokoszczyński, p. sakowski, r. ślepaczuk, p. wójcik dynamic econometric models 14 (2014) 5–28 26 quate reference to so called “normal” or equilibrium level of vts is the crucial point in defining diverse volatility arbitrage strategies. references alexander, c., korovilas d. (2011), the hazards of volatility diversification, icma centre discussion papers in finance dp2011-04, icma, doi: http://dx.doi.org/10.2139/ssrn.1752389. andersen, t. g., bollerslev, t., christoffersen, p. f., diebold, f. x. (2005), volatility forecasting, nber working paper 11188, doi: http://dx.doi.org/10.2139/ssrn.673405. asensio, i. o. (2013), the vix-vix futures puzzle, mimeo. bakshi, g., kapadia, n. (2003), delta-hedged gains and the negative market volatility risk premium, review of financial studies, 16(2), 527–566. black, f., scholes, m. (1973), the pricing of options and corporate liabilities, journal of political economy, 81(3), 637–54, doi: http://dx.doi.org/10.1086/260062. bollen, n. p. b., whaley, r. e. (2004), does net buying pressure affect the shape of implied volatility functions?, journal of finance, 59(2), 711–753. doi: http://dx.doi.org/10.2139/ssrn.319261. bondarenko, o. (2004), market price of variance risk and performance of hedge funds, university of illinois chicago working paper. bossu, s. (2006), introduction to variance swaps, wilmott magazine, 50–55. brenner, m., galai, d. (1989), new financial instruments for hedging changes in volatility, financial analysts journal, 45(4), 61–65, doi: http://dx.doi.org/10.2469/faj.v45.n4.61. briere, m., fermanian, j.-d., malongo, h., signori, o. (2011), volatility strategies for global and country specific european investors, doi: http://dx.doi.org/10.2139/ssrn.1945703. carr, p., lee, r. (2007), realised volatility and variance: options via swaps, risk, 20, 76–83. carr, p., lee, r. (2009), volatility derivatives, annual review of financial economics, 1(1), 319–339, doi: http://dx.doi.org/10.1146/annurev.financial.050808.114304. carr, p., lewis, k. (2004), corridor variance swaps, risk, 17, 67–72. carr, p., madan, d. (2001), towards a theory of volatility trading, [in:] jarrow, r. (ed.), volatility. new estimation techniques for pricing derivatives, in jouini, e., cvitanic, j., musiela, m. (ed.), option pricing, interest rates and risk management, cambridge university press, cambridge, 458–476. carr, p., wu, l. (2009), variance risk premiums, review of financial studies, 22(3), 1311– –1341, doi: http://dx.doi.org/10.1093/rfs/hhn038. cboe (2003), the cboe volatility index – vix, http://www.cboe.com/micro/vix/vixwhite.pdf. cboe (2009), the cboe volatility index – vix, chicago board options exchange. chen, k., he, x., poon, s.-h. (2010), the art of volatility modelling. a case study based on dbs, mimeo. cox, j. c., rubinstein, m. (1985), options markets. prentice hall. daigler, r. t., rossi, l. (2006), a portfolio of stocks and volatility, the journal of investing, 15(2), 99–106, doi: http://dx.doi.org/10.3905/joi.2006.635636. dash, s., moran, m. t. (2005), vix as a companion for hedge fund portfolios, the journal of alternative investments, 8(3), 75–80, does historical volatility term structure contain valuable information… dynamic econometric models 14 (2014) 5–28 27 doi: http://dx.doi.org/10.3905/jai.2005.608034. demeterfi, k., derman, e., kamal, m., zou, j. (1999), a guide to volatility and variance swaps, the journal of derivatives, 6(4), 9–32, doi: http://dx.doi.org/10.3905/jod.1999.319129. derman, e., demeterfi, k., kamal, m., zou, j. (1999), more than you ever wanted to know about volatility swaps, quantitative strategies research notes, goldman sachs. derman, e., kamal, m., kani, i., mcclure, j., pirasteh, c., zou, j. z. (1998), investing in volatility, [in:] futures and options world, special supplement on the 25th anniversary of the publication of the black-scholes model. derman, e., taleb, n. n. (2005), the illusions of dynamic replication, quantitative finance, 5(4), 323–326, doi: http://dx.doi.org/10.1080/14697680500305105. dupire, b. (1993), model art, risk, 6, 118–120. dupire, b. (2004), a unified theory of volatility, working paper, paribas capital markets. egloff, d., leippold, m., wu, l. (2010), the term structure of variance swap rates and optimal variance swap investments, journal of financial and quantitative analysis, 45(5), doi: http://dx.doi.org/10.1017/s0022109010000463. fassas, a. p. (2012), the relationship between vix futures term structure and s&p500 returns, review of futures markets, 20, 293–313. fleming, j., ostdiek, b., whaley, r. e. (1995), predicting stock market volatility: a new measure, journal of futures markets, 15(3), 265–302. galai, d. (1979), a proposal for indexes for traded call options, the journal of finance, 34(5), 1157–1172, doi: http://dx.doi.org/10.2307/2327241. gastineau, g. l. (1977), an index of listed option premiums, financial analysts journal, 33(3), 70–75, doi: http://dx.doi.org/10.2469/faj.v33.n3.70. giot, p. (2005), relationship between implied volatility index and stock index returns, journal of portfolio management, 31, 92–100. guobuzaite, r., martellini, l. (2012), the benefits of volatility derivatives in equity portfolio management, edhec – risk institute. hafner, r., wallmeier, m. (2008), optimal investments in volatility, financial markets and portfolio management, 22(2), 147–167, doi: http://dx.doi.org/10.1007/s11408-008-0076-8. herrmann, r., luedecke, t. (2002), why the volax future has failed, [in:] 9thsymposium on finance, banking, and insurance, universitaet karlsruhe (th), germany. huskaj, b., nossman, m. (2012), a term structure model for vix futures, journal of futures markets, 33(5), 421–442, doi: http://dx.doi.org/10.1002/fut.21550. jabłecki, j., kokoszczyński, r., sakowski, p., ślepaczuk, r., wójcik, p. (2012), pomiar i modelowanie zmienności – przegląd literatury, ekonomia, 31, 22–55. jabłecki, j., kokoszczyński, r., sakowski, p., ślepaczuk, r., wójcik, p. (2013a), struktura czasowa zmienności kontraktów terminowych na vix – modelowanie i własności prognostyczne, zarządzanie i finanse, 11(2), 181–192. jabłecki, j., kokoszczyński, r., sakowski, p., ślepaczuk, r., wójcik, p. (2013b), instrumenty pochodne na zmienność – nowa klasa aktywów?, wne uw. kitces, m. e. (2012), what makes something an alternative asset class, anyway?, journal of financial planning, 22–23. kolanovic, m. (2012), the vix: rewards and risks of a rapidly growing market, discussion paper, columbia university financial engineering practitioners seminar, http://ieor.columbia.edu/financial-engineering-practitioners-seminar. konstantinidi, e., skiadopoulos, g. (2011), are vix futures prices predictable? an empirical investigation, international journal of forecasting, 27, 543–560, j. jabłecki, r. kokoszczyński, p. sakowski, r. ślepaczuk, p. wójcik dynamic econometric models 14 (2014) 5–28 28 doi: http://dx.doi.org/10.1016/j.ijforecast.2009.11.004. merton, r. c. (1973), theory of rational option pricing, bell journal of economics, 4(1), 141–183, doi: http://dx.doi.org/10.2307/3003143. moran, m. t., dash, s. (2007), vix futures and options: pricing and using volatility products to manage downside risk and improve efficiency in equity portfolios, the journal of trading, 2(3), 96–105, doi: http://dx.doi.org/10.3905/jot.2007.688954. neuberger, a. (1994), the log contract, the journal of portfolio management, 20(2), 74– –80, doi: http://dx.doi.org/10.3905/jpm.1994.409478. nieto, b., novales, a., rubio, g. (2012), variance swaps, non-normality and macroeconomic and financial risks, the quarterly review of economics and finance, 54(2), 257– –270, doi: http://dx.doi.org/10.1016/j.qref.2013.12.002. signori, o., briere, m., burgues, a. (2010), volatility exposure for strategic asset allocation, journal of portfolio management, 36(3), 105–116, doi: http://dx.doi.org/10.3905/jpm.2010.36.3.105. simon, d., wiggins, r. (2001), s&p futures returns and contrary sentiment indicators, journal of futures markets, 21, 447–462, doi: http://dx.doi.org/10.1002/fut.4. szado, e. (2009), vix futures and options: a case study of portfolio diversification during the 2008 financial crisis, the journal of alternative investments, 12(2), 68–85. whaley, r. e. (1993), derivatives on market volatility: hedging tools long overdue, journal of derivatives, 1, 71–84, doi: http://dx.doi.org/10.3905/jod.1993.407868. yamai, y., yoshiba, t. (2005), value-at-risk versus expected shortfall: a practical perspective, journal of banking & finance, 29(4), 997–1015, doi: http://dx.doi.org/10.1016/j.jbankfin.2004.08.010. czy struktura terminowa zmienności zawiera istotne informacje w celu prognozowania zachowania się kontraktów terminowych na zmienność i indeksy giełdowe? z a r y s t r e ś c i. badanie pozwoliło nam stwierdzić, że struktura terminowa kontraktów na zmienność (indeks vix), a konkretniej jej nachylenie, jest zależna od aktualnego poziomu indeksu vix. w momencie, w którym indeks vix jest na niskim poziomie (poniżej 20) struktura terminowa ma wysokie dodatnie nachylenie, natomiast w momencie, w którym indeks vix jest na wysokim poziomie (powyżej 30) to wtedy struktura ma ujemne nachylenie. wykorzystujemy te obserwacje, aby lepiej przewidywać zachowanie się stóp zwrotu kontraktów terminowych na zmienność. na początek wprowadzamy miary ilościowe struktury terminowej zmienności (vts) oraz premii za ryzyko zmienności (vrp). vrp pozwala nam określić stopień odchylenia obecnej struktury terminowej od tzw. modelowej dla danego poziomu indeksu vix. zauważamy, że wielkość odchylenia ma istotne własności predykcyjne i dlatego w końcowej części artykułu proponujemy strategie inwestycyjne wykorzystujące tę koncepcję przy budowie algorytmów inwestycyjnych generujących sygnały na rynku kontraktów terminowych na indeks vix. s ł o w a k l u c z o w e: struktura terminowa zmienności, premia za ryzyko zmienności, kontrakty terminowe na zmienność i indeksy giełdowe, zmienność zrealizowana, zmienność implikowana, strategia inwestycyjna, prognozowanie stóp zwrotu, efektywne miary ryzyka i stóp zwrotu microsoft word dem_2014_105to124.docx © 2014 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2014.006 vol. 14 (2014) 105−124 submitted october 20, 2014 issn accepted december 23, 2014 1234-3862 krzysztof kompa, dorota witkowska* pension funds in poland: efficiency analysis for years 1999–2013 a b s t r a c t. the reform of the pension system in poland took place in 1999, when the onepillar pay-as-you-go system (payg) was replaced by the three-pillars system consisting of two mandatory (payg and fully funded) pillars and voluntary (funded) one. however problems concerning budget deficit in poland caused that the polish government introduced significant changes in distribution of the pension contribution between both mandatory pillars and in the pension funds’ portfolio composition in 2011 and 2013. the aim of this study is to analyze the performance of the pension funds operating in poland in the years 1999–2013. applying sharpe and treynor ratios the study provides evidence that well diversified portfolio protects pensioners’ interest better than portfolios constructed due to the new rules. k e y w o r d s: pension system, pension funds, sharpe and treynor efficiency ratios. j e l classification: g11, g23, h55, j26, j32. introduction the main ideas of changes in the retirement system consist in heightening the pension age and introducing funded system instead of pay-as-yougo system (payg). the most frequent reason given in the public policy debate for a funded system is the apparently superior performance of the capital market (in terms of the rate of return on investment it can offer) in comparison to the returns on payg pension contributions (sinn, 2000; feld * correspondence to: krzysztof kompa, wuls in warsaw, department of econometrics and statistics, 166 nowoursynowska st., 02-787 warsaw, e-mail: krzysztof_kompa@sggw.pl; dorota witkowska, university of lodz, department of finance and strategic management, e-mail: dorota.witkowska@uni.lodz.pl. krzysztof kompa, dorota witkowska dynamic econometric models 14 (2014) 105–124 106 stein, 1997). however risk of equity instruments must be also taken into account. by now it is widely accepted in most oecd countries that pension systems and rules need to be changed over time, although these changes vary from country to country. ensuring coverage of employees through one or more pension plans is fundamental to fighting income poverty in old age. all oecd countries have set up mandatory or quasi-mandatory pension plans, either public or private, to achieve quasi-universal coverage. however, mostly in lowincome countries, there is still a significant share of population not covered by public or national plans. the age at which workers can retire is the most often discussed component of a pension system. people now live longer thus it is necessary to increase the pension age, and to adjust women’s retirement age upwards in line with men’s age. many oecd countries have recently done precisely that since it requires an administrative decision thus it is the easiest element of the pension scheme to change. in effect, the majority of oecd countries will have a retirement age of at least 67 years by the middle of this century. the high costs of administering private pension plans that are passed on to members have been a policy concern for many oecd countries in recent years, especially in the states where systems are mandatory or quasimandatory. however, administrative efficiency is also a policy priority in voluntary plans. high fees discourage workers from joining voluntary plans and make mandatory ones very costly. in fact, cost inefficiencies are a threat to the sustainability and suitability of plans themselves. changes in the demographic structure, caused by declining fertility rates and the rise in life expectancy, lead to significant increase of the old-agedependency ratios1 in poland, which required a transformation of the pension system. the main reform2 was implemented in 1999, introduced a new system consisted of three pillars: two mandatory ones: pay-as-you-go pillar (social insurance institution – zus) and fully funded pillar (open pension funds – ofe), and a voluntary, funded pillar. the aim of our research3 is to analyze efficiency of the private pension funds, which were operating in poland in the years 1999–2013 and compare 1 old-age-dependency ratio is the population age 65 and over divided by population age 15-64 (eurostat). 2 detailed description of the pension reform can be found in góra and rutkowski (2000), hausner (2002) among others. 3 research is founded by the grant “analysis of open pension funds market as compared to the open investment funds market functioning in poland” 2013/09/b/hs4/00493 financed by national science center. pension funds in poland: efficiency analysis for years 1999–2013 dynamic econometric models 14 (2014) 105–124 107 their performance to the efficiency of constructed benchmarks. we use sharpe and treynor ratios evaluated on the basis of monthly returns from the accounting units in the period from august 17, 1999 to october 17, 2013. the analysis is provided for the entire period and separately for the subperiods when certain market tendency is observed (i.e. bull, bear and neutral market). 1. changes concerning pension funds open pension funds started to operate in poland in 1999 creating the second mandatory pillar of the “new” pension system. each person could select only one fund and participants were able to change funds with no charge or penalty after a statutory minimum 12 months period of contribution to a fund. pension funds operate like other open-end mutual funds i.e. they collect retirement savings from employees and their employers, and invest this money in a wide range of assets. at the beginning, there were 21 ofes but at the end of 2013 only 13 open pension funds were operating on the polish market, and one more pension fund diapered from the market after last regulations introduced by the polish government in 2014. in the years 1999–2013 number of participants as well as value of assets were steadily growing, at the end of december, 2013 there were more than 16.3 millions of participants and value of ofes’ assets exceeded 299 billions pln4. the summary of evolution of ofes’ in poland is presented in table 1. retirement savings have been created by a contribution of 12.22% of earnings (or 19.52% for employees born between 1949 and 1968 who did not choose funded tier). the contribution was credited to individuals’ notional accounts, while 7.3% of earnings were to be transferred to the pension funds, which created the second mandatory pillar5. there were strict regulations concerning open pension funds investment in risky assets (in order to protect pension plans’ participants). thus during the worldwide financial crisis the losses of ofe were not as substantial as the ones reported by pension funds in developed countries. however the crisis of 2007–2009 caused some serious problems also in polish economy. firstly, open pension funds lost a major part of the profits earned for their members before the crisis. secondly, slower gdp growth caused the increase of the public deficit and the public debenture in relation to gdp. 4 http://www.igte.com.pl/files/notowania/dane_ofe_12_2013.pdf 5 the ceiling to contributions and pensionable earnings was set at 2.5 times average monthly earnings projected for a given year in the state budget law. krzysztof kompa, dorota witkowska dynamic econometric models 14 (2014) 105–124 108 as a result, poland has been no longer in line with the maastricht criteria. therefore, in the years 2011–2013 the polish government introduced several changes in the pension system. table 1. basic characteristics of ofe in years 1999–2014 year net assets [billion pln ] contributions [billion pln] members [million] accounting unit weighting average [pln] 1999 2.3 2.3 7.0 n.a. 2000 9.9 7.6 10.3 n.a. 2001 19.4 8.7 10.6 n.a. 2002 31.6 9.5 11.0 15.80 2003 44.8 10.3 11.5 17.58 2004 62.6 11.4 12.0 20.08 2005 86.1 14.0 11.7 23.09 2006 116.6 16.2 12.4 26.88 2007 140.0 17.7 13.1 28.55 2008 138.3 20.5 13.8 24.51 2009 178.6 21.0 14.3 27.88 2010 221.3 22.4 14.9 31.10 2011 224.7 15.1 15.5 29.56 2012 269.6 8.0 15.9 34.39 2013 299.3 10.5 16.4 36.88 2014 153.1 7.7 16.6 38.08 note: all data are registered in december but from 2014 concerns november; http://www.mpips.gov.pl/ ubezpieczenia-spoleczne/ubezpieczenie-emerytalne/skladka-na-ubezpieczenie-emerytalne/, http://www.zus.pl/default.asp?p=2&id=1319&name=of141105.xls. the first manipulation in the original pension system was made in 2011, when the contribution transferred to pension funds was diminished from 7.3% to 2.3%. the remaining 5% was placed in a special individual subaccount. the share of contributions allocated in the sub-accounts within the social security fund and in the funded scheme was to be changing until 2017, when it was to reach 3.8% and 3.5% (for zus and ofe accounts, respectively). however, regulation introduced in 2013 affected this scheme transferring 4.38% and 2.92% of earnings to zus and ofe accounts, respectively (affected from 2014). the new law, which went into affect in february 2014, shifted 51.5% of the assets, held by the ofes (about 150 billion pln) to the social insurance institution, including all debt securities issued and guaranteed by the state treasury. according to the new regulation, pension funds are no longer obligatory and each employee will have four months every four years to decide whether 2.92 percent of their income goes to a chosen private fund or to pension funds in poland: efficiency analysis for years 1999–2013 dynamic econometric models 14 (2014) 105–124 109 zus6. overhaul of the pension system also concerned changes in the ofes’ investment portfolio, since private pension funds are no longer allowed to invest in government bonds. that will leave the pension funds with most of their assets held in shares of companies listed on the warsaw stock exchange or abroad, and give them an increasingly peripheral role in the future retirement benefits of poles. the government considered the changes (which took place in the years 2011 and 2014) necessary to lower poland’s budget deficit. many specialists called these changes “significant step backward”7, “un-privatizing the pension system” (hagemejer, 2013) or even “the most drastic nationalization of private assets since soviet times”, although polish prime minister donald tusk asserted that “it is no more than a bookkeeping change in the way to handle the public’s retirement money” (bilefsky and zurawik, 2013). after new regulations, at the end of november 2014 the private pension funds equaled only 153.1 billion pln (in september 2014 it was 159 billion pln8) and it was reported that 2.5 millions of ofe members (only 15.2% of employees) decided to stay in pension funds9. due to polish financial supervision authority, the value of total contribution transferred to the pension funds in september 2013 was 1050.8 millions pln compared to 254.3 millions pln10 in september 2014. the new regulations introduced in 2014 lead11 to a change in the composition of assets’ portfolios managed by ofes 6 for the first time, employees had to decide till the end of july, 2014 if they stay in the private pension fund by filling and sending to zus special form, when the form was not sent the employee was automatically excluded from ofe. 7 david mcmillan, chief executive of aviva europe in london, which manages a private pension fund in poland with 17.5 billion euros in assets (bilefsky and zurawik, 2013). 8 http://www.analizy.pl/fundusze/wiadomosci/17222/aktywa-funduszy-emerytalnych-%28 wrzesien-2014%29.html 9 it is considered as a very good result for ofe since the previous forecasts evaluated that only 5% of employed stay in ofe. 10 http://www.knf.gov.pl/opracowania/rynek_emerytalny/dane_o_rynku/rynek_ofe/dane_ miesieczne/dane_miesieczne_ofe.html. 11 new regulations also include the following changes: (1) gradual transfer of each person’s retirement funds managed by ofe to zus, which will start ten years before reaching retirement age; (2) automatic transfer of retirement contributions to zus, instead of ofe, unless an individual ofe member files a declaration (first time-slot will be between 1 april and 31 july 2014) requesting his/her contributions be transferred to ofe; (3) decrease of the maximum fee ofe can charge from contributions from 3.5% to 1.75%; (4) value of certain categories of assets in ofes portfolio (i.e. investment certificates issued by closed-end funds, units of open-ended funds or specialized open-ended funds, or units issued by foreign collective investment undertakings of the closed or open-ended type) will not be included in the krzysztof kompa, dorota witkowska dynamic econometric models 14 (2014) 105–124 110 not only due to the forced transfer of assets to zus but also due to new rules applicable to ofe investment activities. according to the polish financial supervision authority, shares of treasury bonds and equity instruments in the ofes’ portfolios in 2013 were the biggest among all instruments and nearly equal (42% and 43%, respectively). in addition to the management of the fund, the market conditions determine the performance of the investment portfolio. in the analyzed period (i.e. the years 1999–2013) the situation of the polish economy (and the financial market) was changing from high rate of the polish gdp growth and bull market to recession and bear market in warsaw stock exchange (see figure 1, which contains comparison of rates of return generated by equity market, represented by the warsaw stock exchange index wig and ofe in investigated years). figure 1. cumulative annual returns from ofe and wig in years 1999–2007 note: to figure 1. own evaluation on the basis of http://www.nbp.pl/publikacje/operacje_or/2012/ raport2012.pdf and http://www.gpw.pl/analizy_i_statystyki_pelna_wersja. another important aspect of the introduced changes is how the pension fund reform will affect the polish capital market since the increasing capitalization of pension funds made them one of the most important institutional investors generated from 16% to 22% of the warsaw stock exchange turnover among all institutional investors in the years 2005–2010 (marcinkiewicz, overall value of total net assets managed by ofe, which means that ofes may not charge a management fee from these assets. 50 100 150 200 250 300 350 ofe wig pension funds in poland: efficiency analysis for years 1999–2013 dynamic econometric models 14 (2014) 105–124 111 2011). smaller contribution to the pension funds will cause the decrease of the investment level. there is also a danger associated with the changes that the risk of investing in poland has increased. the private funds hold assets worth about $92 billion, i.e. more than onefifth of poland’s gross domestic product, and are among the biggest investors on the warsaw stock exchange (bilefsky and zurawik, 2013). also, due to high market concentration, there is a lack of price and investment competition between open pension funds. in 2011, the ofes’ commission equaled 553 millions pln while wages for management totaled 981 millions pln12. such situation created broadly critique of pension funds in poland. as a result of the widespread resentment, polish government introduced the new pension law. it was estimated that the transfer of 51.5% of ofes‘ assets would lead to a decrease of public debt in poland from around 55% gdp to 47% gdp. this is the main short-term purpose of the reform, rather than providing improved financial security for retirees (mrowiec and mrukzawirski, 2014). 2. methodology and data applied in research the aim of the research is to analyze performance of the private pension funds. the investigation is provided for the accounting units of eleven private pension funds that have been operating in poland from august 17, 1999 to october 17, 2013. these open pension funds contain 85% of all pension funds’ members and 88% net assets (table 2). comparison of analyzed funds at the end of july in the years 2013 and 2014 shows that position of particular pension funds on the market did not change however the values of accounting units and 3-year returns increased in 2014 in comparison to the previous year. values of the accounting units are quite similar and the range is only 5.8 and 6.4 pln in 2013 and 2014 (i.e. 16.5% and 17.8% of average, respectively). the weighted average of accounting units increased by 2.3% in 2014 in comparison to 2013. rates of return exhibit more variation since the range is about 32% of the weighted average in both years and they increased in average by 23.8% in 2014 in comparison to returns obtained in the previous year. 12 see forbes (2012). krzysztof kompa, dorota witkowska dynamic econometric models 14 (2014) 105–124 112 table 2. main characteristics of the analyzed pension funds measured on july, 31, 2013 and 2014 pension funds names percentage share of the market accounting unit [pln] 3-year rates of return [%] total returns [%] members net assets 2013 2014 2013 2014 2013 2014 2013 2014 2015 aegon 5.81 5.62 4.24 4.26 33.56 34.56 15.27 18.68 260.1 allianz 3.43 4.01 3.04 3.08 33.18 33.99 19.13 21.22 254.9 aviva 16.49 16.05 22.38 22.29 34.25 35.16 17.23 19.95 266.6 axa 7.17 6.97 6.27 6.32 34.40 35.32 16.29 19.62 270.1 generali 6.23 6.05 5.03 5.02 35.90 36.39 14.76 19.45 279.9 ing 18.83 18.45 23.99 23.98 37.63 38.31 17.98 21.79 295.1 nordea 5.48 5.98 4.52 4.55 36.39 37.32 19.05 24.28 291.1 pekao 2.11 2.05 1.50 1.51 32.74 33.47 14.64 18.71 244.6 pocztylion 3.67 3.56 1.90 1.88 31.83 31.91 13.77 17.53 234.4 pzu 13.73 13.40 13.42 13.40 34.74 35.68 15.38 19.61 265.6 warta 1.92 2.50 1.34 1.35 35.33 36.04 16.82 20.34 292.3 total 84.87 84.64 87.63 87.64 35.15* 35.95* 16.64** 20.60** note: total returns denotes rate of return from the whole period i.e. from the first day of the pension fund operating till jan. 29, 2015; * denotes weighted averages in considered period, ** weighted averages evaluated for the period 31.03.2010–29.03.2013, and 31.03.2011–31.03.2014, respectively (http://www.igte.com.pl/files/notowania/dane_ofe_07_2013.pdf). the first stage of the research is statistical analysis of 6-months subperiods. in the second stage, we consider daily registered monthly logarithmic rates of return from the accounting units in the entire period of analysis and five sub-periods due to the situation on the warsaw stock exchange i.e. stock index wig daily quotations: 17.09.1999–20.11.2003 stagnation 1 (1055 observations), 21.11.2003–06.07.2007 bull market 1 (912 observations), 07.07.2007–17.02.2009 bear market (403 observations), 18.02.2009–18.07.2011 bull market 2 (609 observations), 19.07.2011–17.10.2013 stagnation 2 (564 observations). sharpe and treynor ratios are evaluated for all pension funds and their performance is compared to the efficiency of constructed benchmarks. treasury bonds represent the risk free instrument. portfolios performance usually is measured by comparing their rates of return and risk measures. the former seem to be the most important for the pension funds participants when they select the pension funds. our study tests several hypotheses in order to find out if the expected value of the analyzed rates of return and their variances significantly differ from the benchmarks. we start from verifying the hypothesis if expected returns differ from zero: pension funds in poland: efficiency analysis for years 1999–2013 dynamic econometric models 14 (2014) 105–124 113 0)(:0 preh (1) using well-known test statistics: .t s r u p p (2) to test the significance of differences between expected rates of return generating by pension funds and the benchmarks: bp rreh )(:0 (3) we employ the following test statistics: ,t s rr u p bp  (4) where: )( pre – expected rate of return of the analyzed open pension fund, ,pr br – average rates of return of the analyzed portfolio and the benchmark, ps – standard deviation of rates of return generated by the pension fund, t – number of observations, u is normally distributed statistics. in the next step we test equality of the pension fund’s and benchmark’s variances. the null hypothesis is: 22 0 )(: bp srdh  (5) and the test statistics is defined as: , 2 2 2 b p s st   (6) ,1)1(22 2  tu  (7) where: )(2 prd – variance of the portfolio, bs – standard deviation of rates of return generated by the benchmark, other symbols are described above. various researchers who have highlighted numerous factors influencing the portfolio performance have documented investment efficiency. the two krzysztof kompa, dorota witkowska dynamic econometric models 14 (2014) 105–124 114 traditional measures of the portfolio performance are the treynor and the sharpe indexes13: , p fp p s rr ws   (8) , p fp p rr wt    (9) where: pws and pwt – sharpe and treynor ratios, respectively, ,pr fr – average returns from the analyzed portfolio and the risk free instrument, respectively, p – beta coefficient from the single-index model (also called sharpe’s model14) estimated separately for each pension fund: ,tbtppt err   (10) where: ,ptr btr – returns from pension funds and benchmark observed in the period t. significance of beta coefficient, i.e. * 0 :  ph (11) can be tested using the test statistics: , * p s b t p    (12) where: t – t-student statistics, pb – parameter estimates of , ps – standard error, 0*  or 1. ratios (8) and (9) are compared to the efficiency measures evaluated for the constructed benchmarks, bws and .bwt application of these traditional efficiency measures requires selection of the representative market index and risk-free instrument. we assumed the significance level of 0.05 for all tests. 13 sharpe and treynor ratios are composite measure of portfolio performance that also included risk, for details see: treynor (1965), sharpe (1966). application of sharpe index to evaluate the private pension funds efficiency is presented in antolin (2008). 14 discussion about sharpe’s model estimated for polish capital market can be found in tarczyński et al. (2013) among others. pension funds in poland: efficiency analysis for years 1999–2013 dynamic econometric models 14 (2014) 105–124 115 3. empirical analysis in the first stage of research the logarithmic rates of return from of 6months sub-periods are analyzed. we distinguish 27 such time spans from january 1, 2000 to june 30, 2013 (denoted as p2–p28) and two shorter periods (p1: 17.09.1999–31.12.1999 and p29: 1.07.2013–17.10.2013). p30 denotes the entire period of investigation. for all analyzed pension funds and for hypothetical portfolios rates of return are evaluated. table 3a. rates of returns no. end of the period aegon all aviva axa gen ing norea poc p1 31.12.1999 0.0903 0.0764 0.1058 0.0591 0.0628 0.0777 0.0909 0.0796 p2 30.06.2000 0.0755 0.1026 0.0531 0.0981 0.0796 0.0874 0.0502 0.0888 p3 29.12.2000 0.0337 0.0194 0.0471 0.0281 0.0555 0.0591 0.0438 0.0483 p4 29.06.2001 –0.0452 –0.0350 –0.0170 –0.0187 –0.0433 –0.0336 –0.0177 –0.0476 p5 31.12.2001 0.0809 0.1064 0.1131 0.0951 0.1003 0.1072 0.1126 0.0722 p6 28.06.2002 0.0535 0.0643 0.0562 0.0473 0.0543 0.0786 0.0717 0.0388 p7 31.12.2002 0.0693 0.0662 0.0513 0.0485 0.0617 0.0829 0.0669 0.0539 p8 30.06.2003 0.0504 0.0510 0.0428 0.0456 0.0545 0.0547 0.0495 0.0446 p9 31.12.2003 0.0488 0.0545 0.0561 0.0509 0.0591 0.0552 0.0600 0.0578 p10 30.06.2004 0.0423 0.0452 0.0399 0.0553 0.0506 0.0351 0.0361 0.0455 p11 31.12.2004 0.0848 0.0608 0.0814 0.0865 0.0859 0.0863 0.0779 0.0816 p12 30.06.2005 0.0536 0.0487 0.0655 0.0572 0.0580 0.0702 0.0540 0.0520 p13 31.12.2005 0.0663 0.0525 0.0738 0.0701 0.0763 0.0764 0.0678 0.0789 p14 30.06.2006 0.0374 0.0580 0.0441 0.0410 0.0568 0.0453 0.0454 0.0599 p15 29.12.2006 0.0995 0.0900 0.0971 0.1049 0.1086 0.1066 0.0964 0.0979 p16 29.06.2007 0.1018 0.1102 0.1076 0.1062 0.0918 0.0951 0.1047 0.0967 p17 28.12.2007 –0.0484 –0.0479 –0.0429 –0.0480 –0.0381 –0.0475 –0.0548 –0.0579 p18 30.06.2008 –0.0904 –0.0841 –0.0936 –0.0823 –0.0855 –0.0932 –0.0807 –0.0810 p19 31.12.2008 –0.0449 –0.0363 –0.0642 –0.0445 –0.0510 –0.0565 –0.0498 –0.0468 p20 30.06.2009 0.0294 0.0226 0.0147 0.0283 0.0377 0.0213 0.0252 0.0292 p21 31.12.2009 0.0932 0.0953 0.0990 0.0922 0.0961 0.0994 0.0865 0.0879 p22 30.06.2010 0.0138 0.0174 0.0206 0.0201 0.0173 0.0221 0.0271 0.0210 p23 31.12.2010 0.0779 0.0843 0.0846 0.0750 0.0692 0.0877 0.0842 0.0822 p24 30.06.2011 0.0185 0.0266 0.0212 0.0205 0.0196 0.0261 0.0256 0.0183 p25 30.12.2011 –0.0689 –0.0742 –0.0704 –0.0560 –0.0618 –0.0734 –0.0671 –0.0824 p26 29.06.2012 0.0466 0.0547 0.0533 0.0439 0.0459 0.0550 0.0596 0.0471 p27 28.12.2012 0.1011 0.1059 0.0955 0.0906 0.0901 0.0934 0.1025 0.0912 p28 28.06.2013 –0.0206 –0.0123 –0.0110 –0.0083 –0.0123 –0.0052 0.0008 –0.0039 p29 17.10.2013 0.0760 0.0765 0.0803 0.0737 0.0789 0.0850 0.0796 0.0813 p30 17.10.2013 1.1671 1.2454 1.2501 1.2213 1.2681 1.3210 1.2669 1.1773 note: description of abbreviations: all means allianz, gen – generali, poc – pocztylion. considered returns generated by pension funds in analyzed intervals (tables 3a and 3b) we notice that in general all of them show similar perkrzysztof kompa, dorota witkowska dynamic econometric models 14 (2014) 105–124 116 formance. negative returns are observed only in the following periods: 02.01.2001–29.06.2001, 29.06.2007–31.12.2008, 01.07.2011–30.12.2011, and 02.01.2013–28.06.2013 (except nordea in the period p28). table 3b. rates of returns no. end of the period pekao pzu warta average bond wig wibor p1 31.12.1999 0.0669 0.0738 0.1101 0.0814 –0.0071 0.0921 0.4291 p2 30.06.2000 0.0611 0.0663 0.0648 0.0751 0.0112 0.0342 –0.1283 p3 29.12.2000 0.0187 0.0300 0.0793 0.0425 –0.0102 –0.0904 0.0793 p4 29.06.2001 –0.0228 –0.0291 –0.0575 –0.0336 –0.0196 –0.2369 –0.1942 p5 31.12.2001 0.1131 0.1223 0.0774 0.0999 –0.0265 0.0072 –0.2891 p6 28.06.2002 0.0312 0.0648 0.0536 0.0561 0.0000 0.0140 –0.2609 p7 31.12.2002 0.0379 0.0752 0.0433 0.0600 0.0324 0.0138 –0.2663 p8 30.06.2003 0.0471 0.0532 0.0524 0.0497 0.0359 0.1061 –0.2586 p9 31.12.2003 0.0538 0.0604 0.0664 0.0567 –0.1041 0.2790 0.0187 p10 30.06.2004 0.0673 0.0486 0.0635 0.0479 –0.0505 0.1173 0.0721 p11 31.12.2004 0.0836 0.0814 0.0788 0.0809 0.1059 0.1086 0.1144 p12 30.06.2005 0.0495 0.0570 0.0548 0.0566 0.0800 0.0590 –0.2648 p13 31.12.2005 0.0604 0.0670 0.0750 0.0697 –0.0267 0.2183 –0.0933 p14 30.06.2006 0.0708 0.0479 0.0598 0.0513 –0.0284 0.1201 –0.1102 p15 29.12.2006 0.1174 0.1092 0.0992 0.1025 0.0276 0.2125 0.0000 p16 29.06.2007 0.1423 0.1097 0.0966 0.1055 –0.0249 0.2550 0.1189 p17 28.12.2007 –0.0759 –0.0457 –0.0630 –0.0517 –0.0188 –0.1651 0.1805 p18 30.06.2008 –0.0963 –0.1033 –0.0904 –0.0893 –0.0267 –0.2996 0.1507 p19 31.12.2008 –0.0553 –0.0403 –0.0560 –0.0497 0.0608 –0.3968 –0.1176 p20 30.06.2009 0.0236 0.0136 0.0211 0.0243 –0.0183 0.0711 –0.3913 p21 31.12.2009 0.0997 0.1038 0.0960 0.0954 0.0061 0.2646 –0.0053 p22 30.06.2010 0.0201 0.0195 0.0253 0.0204 0.0192 –0.0345 –0.0190 p23 31.12.2010 0.0775 0.0854 0.0766 0.0804 0.0010 0.1876 0.0055 p24 30.06.2011 0.0219 0.0185 0.0235 0.0219 0.0000 0.0085 0.2357 p25 30.12.2011 –0.0788 –0.0748 –0.0685 –0.0704 0.0035 –0.2557 0.0320 p26 29.06.2012 0.0488 0.0474 0.0510 0.0503 0.0050 0.0630 0.0289 p27 28.12.2012 0.0953 0.0937 0.0950 0.0958 0.0082 0.1565 –0.1514 p28 28.06.2013 –0.0109 –0.0047 –0.0093 –0.0088 –0.0084 –0.0724 –0.4126 p29 17.10.2013 0.0845 0.0814 0.0824 0.0800 –0.0047 0.1607 –0.0561 p30 17.10.2013 1.1941 1.2483 1.2485 1.2379 0.0244 1.1593 –1.6368 similar, to some extend, tendencies can be observed in the bond, equity and money market (table 3b). negative returns appear in the mentioned above periods (except wibor in p17, p18, and p25 and bonds in p19 and p25). however negative rates of return are observed additionally in following periods: for bond market: p1, p3, p5, p9, p10, p13, p14, p16, p20, p29; for wibor: p2, p5–p8, p12–p14, p20–p22, p27, p29 and the entire period of analysis p30. wig generates negative returns in p22. to summarize, pension funds in poland: efficiency analysis for years 1999–2013 dynamic econometric models 14 (2014) 105–124 117 among 29 considered sub-periods, pension funds generated negative returns in 6 periods, wig in 8, bonds in 14 and wibor in 16 periods. table 4. the structure of the hypothetical portfolios asset representative structure of the portfolios ofe due to the regulation from kompa & wiśniewski (2014) 1997 dec., 6, 2013 portfolio 1 portfolio 2 portfolio 3 bond market: treasury bonds 42% – 30% equity market: wig 46% 79% 70% monetary market: wibor 12% 21% – to analyze efficiency of the pension funds we construct three hypothetical portfolios employing aggregate measures of equity, money and bond markets. they are represented by wig (warsaw stock exchange index), wibor (warsaw interbank offered rate) and treasury bonds, respectively. these portfolios, are treated as market benchmarks in evaluation of the pension funds performance. the idea of the portfolio structures (presented in table 4) is to illustrate changes concerning the structure of the pension fund portfolios due to regulations from 1997 and 2013. the first portfolio is constructed due to the regulation from 199715, the second one – due to the regulation from dec. 6, 201316 while the third one is the optimal portfolio structure, which was simulated by kompa, wiśniewski, (2014), kompa, (2014) assuming that the portfolio contains only two types of assets. next we test hypotheses that expected returns obtained in 6 months periods are significantly different from zero (table 5). the analysis is provided for selected pension funds and three benchmarks. the positive expected rates of return are denoted by “+”, negative by “–“, and blank cells denote the situation when the null hypothesis could not be rejected at the significance level 0.05. as one can notice most funds generated positive returns in the majority of periods. there are only 4 periods when all pension funds and created portfolios generated loses, i.e. from july 1, 2007 to december 31, 2008 and from july 1, to december 31, 2011. in fact, pension funds performed better than constructed benchmarks. 15 file:///d:/katalogi%20robocze/archiwum%20dorota/rok%202014/ofe/ustawa%20o %20ofe.pdf. 16 http://orka.sejm.gov.pl/opinie7.nsf/nazwa/1946_u/$file/1946_u.pdf. krzysztof kompa, dorota witkowska dynamic econometric models 14 (2014) 105–124 118 table 5. verification hypothesis of expected returns (1)–(2) no. port.1 port.2 port.3 aeg all avi axa gen ing nor pek poc pzu war p1 + + + + + + + + + + + + + p2 + + + + + + + + + + + + + + p3 – – – + – + + p4 – – – + + p5 – – + + + + + + + + + + + p6 + + + + + + + + + + + + p7 – – + + + + + + + + + + + p8 + + + + + + + + + + + + + p9 + + + + + + + + + + + + + + p10 + + + + + + + + + + + + + + p11 + + + + + + + + + + + + + + p12 + + + + + + + + + + + + p13 + + + + + + + + + + + + + + p14 + + + + + + + + + + + + + p15 + + + + + + + + + + + + + + p16 + + + + + + + + + + + + + + p17 – – – – – – – – – – – – – – p18 – – – – – – – – – – – – – – p19 – – – – – – – – – – – – – – p20 + + + + + + + + + + + + p21 + + + + + + + + + + + + + + p22 + + + + + + + + + + + + p23 + + + + + + + + + + + + + + p24 + + + + + + + + + + + + + + p25 – – – – – – – – – – – – – – p26 + + + + + + + + + + + p27 + + + + + + + + + + + + + + p28 – – + + + + + + + + + + + p29 + + + + + + + + + + + + + + note: description of abbreviations: port.1 – portfolio1, port.2 – portfolio2, port.3 – portfolio3, aeg – aegon, all – allianz, avi – aviva, gen – generali, nor – nordea, pek – pekao, poc – pocztylion, war – warta. the conclusion is supported by the tests (3)–(4) and (5)–(7) in the whole analyzed period (i.e. for p30). table 6 demonstrates that returns from the pension portfolios exceed benchmarks, and variances of all portfolios are smaller than the benchmarks’ ones. therefore, it can be concluded that the open pension funds investment policy was relatively well established. that type of analysis is also provided for the all sub-periods p1–p29. results presented in table 7 indicate “+” if value of expected returns or variance generated by pension funds is higher than the ones obtained from benchmarks; “–“ the opposite situation and “0” – when null hypotheses cannot be rejected. in general, majority of the ofes’ performance was not worse than constructed portfolios. expected returns of pension funds are significantly higher than pension funds in poland: efficiency analysis for years 1999–2013 dynamic econometric models 14 (2014) 105–124 119 returns generated by considered benchmarks in 59% of cases for the portfolio 1, in 47% of cases for the portfolio 2, and in 42% of cases in comparison to the last benchmark. for all pension funds, variance of returns (that describe risk) is significantly smaller than the one estimated for the portfolios 2 and 3 while in comparison with variability of returns of the portfolio 1 it is usually smaller. table 6. values of the test statistics hypotheses )()(:0 bp rereh  2 0 2 0 )(: prdh portfolio no. portfolio no. 1 2 3 1 2 3 aegon 13.58 10.06 5.87 –29.86 –51.17 –46.86 allianz 14.96 11.43 7.22 –30.01 –51.27 –46.97 aviva 13.73 10.50 6.65 –24.98 –48.21 –43.51 axa 14.60 11.10 6.87 –29.96 –51.24 –46.93 generali 14.65 11.30 7.30 –27.27 –49.60 –45.08 ing 13.82 10.81 7.22 –20.70 –45.6 –40.57 nordea 15.12 11.62 7.46 –29.47 –50.94 –46.59 pocztylion 12.78 9.52 5.62 –25.69 –48.65 –44.00 pekao 13.28 9.97 6.02 –26.51 –49.14 –44.56 pzu 14.07 10.76 6.81 –26.44 –49.10 –44.51 warta 13.57 10.38 6.57 –24.29 –47.79 –43.04 table 7. comparison of expected returns and risk in analyzed periods p1–p29 hypotheses )()(:0 bp rereh  20 2 0 )(: prdh portfolio no. portfolio no. 1 2 3 1 2 3 + – 0 + – 0 + – 0 + – 0 + – 0 + – 0 aegon 16 6 7 14 11 4 11 13 5 0 28 1 0 29 0 0 29 0 allianz 17 7 5 14 10 5 14 13 2 0 26 3 0 29 0 0 29 0 aviva 15 5 9 14 11 4 12 13 4 0 25 4 0 29 0 0 29 0 axa 16 7 6 14 11 4 11 13 5 0 28 1 0 29 0 0 29 0 generali 17 6 6 15 11 3 11 12 6 0 26 3 0 29 0 0 29 0 ing 17 5 7 13 11 5 14 13 2 2 23 4 0 29 0 0 29 0 nordea 17 4 8 13 10 6 13 14 2 0 28 1 0 29 0 0 29 0 pocztylion 18 6 5 14 11 4 11 12 6 1 26 2 0 29 0 0 29 0 pekao 20 7 2 13 11 5 12 13 4 0 26 3 0 29 0 0 29 0 pzu 17 3 9 13 11 5 13 13 3 0 28 1 0 29 0 0 29 0 warta 19 5 5 13 11 5 13 12 4 1 25 3 0 29 0 0 29 0 percentage share 59 19 22 47 37 16 42 44 13 1 91 8 0 100 0 0 100 0 in the second stage of our research we investigate daily registered monthly logarithmic rates of return from the accounting units in the entire period of analysis and five sub-periods. krzysztof kompa, dorota witkowska dynamic econometric models 14 (2014) 105–124 120 table 8. beta estimates pension funds aeg all avi axa gen ing nor poc pek pzu war period portfolio no. 1 s1 0.483 0.483 0.589 0.507 0.578 0.610 0.456 0.595 0.525 0.463 0.610 bull1 0.645 0.585 0.641 0.626 0.633 0.698 0.651 0.634 0.631 0.671 0.628 bear 0.600 0.546 0.660 0.585 0.574 0.674 0.607 0.591 0.585 0.660 0.607 bull 2 0.542 0.539 0.588 0.546 0.584 0.617 0.544 0.555 0.571 0.636 0.578 s2 0.762 0.773 0.765 0.671 0.742 0.783 0.759 0.778 0.797 0.784 0.748 period portfolio no. 2 s1 0.312 0.315 0.380 0.326 0.375 0.402 0.298 0.386 0.339 0.300 0.390 bull1 0.405 0.379 0.404 0.395 0.400 0.436 0.415 0.401 0.408 0.426 0.403 bear 0.359 0.326 0.399 0.350 0.346 0.403 0.365 0.353 0.352 0.396 0.365 bull 2 0.339 0.335 0.365 0.337 0.363 0.382 0.338 0.344 0.354 0.393 0.359 s2 0.441 0.447 0.443 0.389 0.429 0.453 0.440 0.452 0.462 0.454 0.434 period portfolio no. 3 s1 0.368 0.371 0.442 0.387 0.438 0.471 0.348 0.454 0.398 0.361 0.456 bull1 0.465 0.426 0.465 0.450 0.457 0.506 0.472 0.460 0.454 0.484 0.458 bear 0.435 0.396 0.476 0.424 0.416 0.489 0.439 0.428 0.427 0.476 0.439 bull 2 0.380 0.372 0.405 0.379 0.405 0.425 0.376 0.387 0.395 0.439 0.400 s2 0.506 0.514 0.507 0.446 0.492 0.523 0.506 0.521 0.532 0.523 0.498 note: description of abbreviations: s – stagnation, bull – bullish market, bear – bearish market, aeg – aegon, all – allianz, avi – aviva, gen – generali, nor – nordea, pek – pekao, poc – pocztylion, war – warta. table 9. sharpe and treynor ratios: hypothetical portfolios portfolio 1 ratios portfolio 2 ratios portfolio 3 ratios period sharpe treynor sharpe treynor sharpe treynor s 1 0.027 0.001 0.026 0.002 0.085 0.005 bull 1 0.400 0.011 0.456 0.020 0.472 0.019 bear –0.565 –0.026 –0.607 –0.045 –0.614 –0.040 bull 2 0.390 0.012 0.419 0.020 0.413 0.018 s 2 –0.067 –0.002 –0.069 –0.003 0.034 0.001 note: description of abbreviations: s – stagnation, bull – bullish market, bear – bearish market. applying tests (3)–(7) for the time series representing logarithmic returns in distinguished sub-periods: s1 – stagnation 1, bull1 – bull market 1, bear – bear market, bull2 – bull market 2 and s2 – stagnation 2, we found out that pension funds are characterized by smaller risk in all considered sub-periods. while expected returns are significantly smaller than the ones generated by benchmarks only during two bull market periods, although if the portfolio 1 is taken into account also in these sub-periods some of pension funds generated returns insignificantly smaller than the benchmark (aegon, aviva, axa, generali, ing, pocztylion, pzu, warta in bull market 1 period, ing and pzu in bull market 2 period), while pension funds in poland: efficiency analysis for years 1999–2013 dynamic econometric models 14 (2014) 105–124 121 pekao’s returns were significantly higher than portfolio 1 in the bull market 1 period. table 10. sharpe ratios: pension funds period aeg all avi axa gen ing nor poc pek pzu war s 1 0.522 0.531 0.494 0.487 0.476 0.463 0.587 0.413 0.394 0.551 0.434 bull 1 0.548 0.562 0.569 0.607 0.607 0.529 0.527 0.579 0.676 0.561 0.608 bear –0.454 –0.460 –0.468 –0.445 –0.457 –0.434 –0.463 –0.461 –0.538 –0.435 –0.497 bull 2 0.577 0.599 0.565 0.580 0.559 0.561 0.596 0.571 0.579 0.521 0.571 s 2 0.230 0.259 0.249 0.282 0.253 0.260 0.298 0.221 0.236 0.236 0.266 note: description of abbreviations: s – stagnation, bull – bullish market, bear – bearish market, aeg – aegon, all – allianz, avi – aviva, gen – generali, nor – nordea, pek – pekao, poc – pocztylion, war – warta. table 11. treynor ratios for pension funds portfolio 1 is the market benchmark in the sharpe model period aeg all avi axa gen ing nor poc pek pzu war s 1 0.025 0.027 0.023 0.024 0.022 0.023 0.029 0.020 0.021 0.028 0.021 bull 1 0.016 0.017 0.017 0.018 0.018 0.016 0.016 0.017 0.020 0.017 0.018 bear –0.022 –0.023 –0.023 –0.022 –0.022 –0.021 –0.023 –0.023 –0.027 –0.022 –0.025 bull 2 0.018 0.019 0.018 0.018 0.018 0.018 0.019 0.018 0.018 0.016 0.018 s 2 0.007 0.008 0.007 0.008 0.007 0.008 0.009 0.006 0.007 0.007 0.008 portfolio 2 is the market benchmark in the sharpe model period aeg all avi axa gen ing nor poc pek pzu war s 1 0.038 0.042 0.035 0.037 0.034 0.035 0.045 0.030 0.033 0.043 0.032 bull 1 0.026 0.026 0.027 0.028 0.028 0.025 0.024 0.027 0.031 0.026 0.029 bear –0.037 –0.038 –0.038 –0.037 –0.037 –0.036 –0.038 –0.038 –0.045 –0.036 –0.041 bull 2 0.030 0.031 0.029 0.030 0.028 0.029 0.031 0.029 0.029 0.027 0.029 s 2 0.012 0.013 0.012 0.014 0.013 0.013 0.015 0.011 0.012 0.012 0.013 portfolio 3 is market benchmark in the sharpe model period aeg all avi axa gen ing nor poc pek pzu war s 1 0.032 0.036 0.030 0.031 0.029 0.030 0.039 0.026 0.028 0.036 0.028 bull 1 0.022 0.023 0.023 0.025 0.025 0.022 0.021 0.024 0.028 0.023 0.025 bear –0.031 –0.032 –0.032 –0.030 –0.031 –0.030 –0.032 –0.031 –0.037 –0.030 –0.034 bull 2 0.026 0.028 0.026 0.027 0.025 0.026 0.028 0.026 0.026 0.024 0.026 s 2 0.010 0.011 0.011 0.012 0.011 0.011 0.013 0.010 0.010 0.010 0.012 note: description of abbreviations: s – stagnation, bull – bullish market, bear – bearish market, aeg – aegon, all – allianz, avi – aviva, gen – generali, nor – nordea, pek – pekao, poc – pocztylion, war – warta. the efficiency of the pension funds is measured using sharpe and treynor ratios (8) and (9). to evaluate treynor ratios three versions of singleindex models (10) for each pension fund and every considered sub-period are estimated. these models differ by the market indexes, which are reprekrzysztof kompa, dorota witkowska dynamic econometric models 14 (2014) 105–124 122 sented by constructed portfolios. in our research we employ ols method17. the parameter estimates obtained for three hypothetical benchmarks are presented in table 8, all betas are significantly higher than zero and smaller than one. tables 9 and 10 present results that all pension funds are more effective than constructed benchmarks in terms of sharpe ratios. treynor measures evaluated for pension funds are higher than the ones calculated for the market benchmarks in all analyzed sub-periods, except bear market for the pension fund pekao and portfolios 1 and 2 (tables 11). in other words, analyses of efficiency measures calculated for the hypothetical portfolios shows that portfolio 3 seems to be the most effective especially in terms of sharpe ratio. conclusions demographic structure of the polish population has been changing that causes the increase of old age dependency ratio from 15.26 in 1989 to 17.75 in 1999, and 19.32 in 2011. therefore, general reform of the pension system was necessary and it took place in poland in the year 1999, replacing the pay-as-you-go system, by the three-pillars partly funded system. under the system introduced in poland in 1999, two pillars were universal and mandatory, and the third one – voluntary. the first pillar remained to be pay-asyou-go financed, whereas the second and third pillars were to be funded. in fact, the “old” payg system was downsized and converted to a “notional defined-contribution” system, forming the new first pillar governed by the social insurance institution. in both mandatory pillars, contributions were registered in individual accounts, and the pension benefits depended on contributions paid, not contributions that were due. after the subprime crisis, because of the increase of the budget deficit, polish government introduced the new regulations in the pension system in 2011 and 2013. the most important move was shifting 51.5% of the assets held by the ofes to the state-run payg pension system – zus (affected from february 2014). other changes consisted in changes in:  the retirement age (it has been increasing by a month each quarter beginning from the first quarter of year 2013), 17 discussion of beta estimation has been provided by many researches all around the world however our previous research shows that ols method is appropriate to this purpose (see: tarczyński, witkowska, kompa, 2013). pension funds in poland: efficiency analysis for years 1999–2013 dynamic econometric models 14 (2014) 105–124 123  the share of the contribution of earnings that is saved in both mandatory pillars (affected from may 2011),  the role of the mandatory funded pillar that became voluntary (affected from august 2014),  the scheme of investments i.e. the pension funds’ portfolio composition, especially prohibition of investing in debt securities issued and guaranteed by the state treasury (beginning from 2014). the research, presented in the paper, shows that the performance of pension funds was better than the constructed benchmarks, regardless the general situation in the capital market. it proves that diversified (as it was stated in 1999) portfolio better protects pensioner’s interest than portfolios with limited types of financial instruments. it can be noticed also by analyzing accounting units (table 1), which have been systematically increasing. it proves that new regulations, especially the one concerning structure of the pension funds’ portfolios, does not improve the performance of pension funds. in addition lack of debt securities issued and guaranteed by the state treasury in ofe’s investments will increase the risk exposure of retirement savings. new regulations significantly limited the role of the funded pillar in the pension system. there is also a danger that declining of the pension funds assets will significantly influence polish capital market and the whole economy. references antolin, p. (2008), pension fund performance, oecd working papers on insurance and private pensions, 20, doi: http://dx.doi.org/10.1787/240401404057. bilefsky, d., zurawik, m. (2013), polish plan on pensions arouses sharp criticism, the new york times. october, 9, 2013, http://www.nytimes.com/2013/10/10/business/in ternational/polish-plan-on-pensions-arouses-sharp-criticism.html?_r=0. feldstein, m. (1997), transition to a fully funded pension system: five economic issues, nber working paper, 6149, http://www.nber.org/papers/w6149.pdf. góra, m., rutkowski, m. (2000), the quest for pension reform: poland's security through diversity, , william davidson institute at the university of michigan, working papers, 286, http://deepblue.lib.umich.edu/bitstream/handle/2027.42/39670/wp286.pdf?seque nce=3. hagemejer, j., makarski, k., tyrowicz, j. (2013), unprivatizing the pension system: the case of poland, university of warsaw, faculty of economic sciences, working papers, 26 (111). hausmer, j. (2002), poland security through diversity, in feldstein m. and siebert h. (ed.), social security pension reform in europe, university of chicago press, 349–364, available at: http://www.nber.org/chapters/c10678.pdf. kompa, k. (2014), polish pension system in transition: impact on the investment portfolio construction, indian journal of fundamental and applied life sciences, http:// http://www.cibtech.org/sp.ed/jls/2014/01/jls.htm, 4(s1), 2102–2110. krzysztof kompa, dorota witkowska dynamic econometric models 14 (2014) 105–124 124 kompa, k., wiśniewski, t. (2014), “security through diversity”: portfolio diversification of private pension funds, quantitative methods in economic research, 15(1), 58–65. lewicka-banaszak, e. (2014), against the grain. pension systems in the european union and selected oecd countries, and the direction of changes in the polish open pension funds (pod prąd. systemy emerytalne w unii europejskiej i wybranych krajach oecd, a kierunek zmian w polskich ofe), in biuletyn igte, 1(1), 12–20, http://www.igte.com.pl/files/biuletyn/1_biuletyn_14.pdf (in polish). marcinkiewicz, e., (2011), changing the rules for distribution of pension contributions and the importance of pension funds in the financial market (zmiana zasad dystrybucji składki emerytalnej a znaczenie ofe na rynku finansowym), polityka społeczna. numer specjalny, warszawa, 16–19, (in polish). pelc, p. (2010), polish pension reform – lost decade?, http://www.pawelpelc.pl/index.php? display=5&artykul=167. pensions at a glance 2013: oecd and g20 indicators © oecd 2013. president signs pension reform (2013), warsaw bussiness journal, december, 30, 2013, http://www.wbj.pl/article-64697-president-signs-pension-reform.html rozłucki, w. (2001), the importance of pension funds for the development of the warsaw stock exchange (znaczenie funduszy emerytalnych dla rozwoju warszawskiej giełdy papierów wartościowych), bre bank – case, 57, 37–41 (in polish). security (1997), security through diversity: reform of the pension system in poland. warsaw: office of the government plenipotentiary for social insurance reform. sharpe, w.f. (1966), mutual fund performance, journal of business, 39(1), 119–138. sinn, h.w. (2000), why a funded pension system is useful and why it is not useful, national bureau of economic research, working paper, 7592, http://www.nber.org/pa pers/w7592 tarczyński, w., witkowska d., kompa k. (2013), the beta ratio theory and practice (współczynnik beta teoria i praktyka), warszawa: pielaszek research, (in polish). treynor, j.l. (1965), how to rate management of investment funds, harvard business review, 43(1), 63–75. fundusze emerytalne w polsce: analiza efektywności za lata 1999–2013 z a r y s t r e ś c i. reforma systemu emerytalnego została wprowadzona w polsce w 1999r., kiedy repartycyjny system zdefiniowanego świadczenia zastąpiono systemem zdefiniowanej składki, opartym na 3 filarach. dwa pierwsze filary – repartycyjny i kapitałowy stały się publiczne i obowiązkowe, a trzeci dobrowolny kapitałowy. jednakże problemy związane ze wzrostem deficytu budżetowego spowodowały wprowadzenie przez rząd istotnych zmian systemu emerytalnego w latach 2011 i 2013. regulacje te nie tylko w istotny sposób wpłynęły na podział składek przekazywanych do obu obowiązkowych filarów, ale spowodowały zmiany w strukturze portfeli inwestycyjnych otwartych funduszy emerytalnych (ofe). celem realizowanych badań jest analiza wyników finansowych ofe funkcjonujących w polsce w latach 1999–2013. na podstawie mierników efektywności sharpe’a i treynora pokazano, że dobrze zdywersyfikowane portfele ofe lepiej chronią interesy emerytów niż portfele hipotetyczne, skonstruowane zgodnie z nowymi zasadami. s ł o w a k l u c z o w e: system emerytalny, fundusze ofe, wskaźniki sharpe i treynora. microsoft word dem_2014_51to70.docx © 2014 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2014.003 vol. 14 (2014) 51−70 submitted october 20, 2014 issn accepted december 23, 2014 1234-3862 mariola piłatowska, aneta włodarczyk, marcin zawada* the environmental kuznets curve in poland – evidence from threshold cointegration analysis a b s t r a c t. the article aims to look at the long-run equilibrium relationship between per capita greenhouse gas emissions and per capita real gdp (ekc hypothesis) in an asymmetric framework using the non-linear threshold cointegration and error correction methodology for polish economy during the period 2000 to 2012 (quarterly data). to test the robustness of the results the additional explanatory variable (per capita energy consumption) is added to the ekc model. the ekc hypothesis is tested using threshold autoregressive (tar) and momentum threshold autoregressive (mtar) cointegration method. moreover, the threshold error correction model (tecm) is implemented in order to examine both the short-run and the long-run granger-causal relationship between per capita greenhouse gas emissions and per capita income. we found strong evidence in favour of the ekc hypothesis for the polish case and additionally we confirmed that adjustment of deviations toward the long-run equilibrium is asymmetric. k e y w o r d s: environmental kuznets curve, greenhouse gas emission, energy consumption, growth, threshold cointegration, granger causality. j e l classification: c24, q43, q50. * correspondence to: mariola piłatowska, nicolaus copernicus university, department of econometrics and statistics, 13a gagarina street, 87-100 toruń, poland, e-mail: mariola.pilatowska@umk.pl; aneta włodarczyk, czestochowa university of technology, faculty of management, 19b armii krajowej street, 42-201 czestochowa, poland, e-mail: aneta.w@interia.pl; marcin zawada, czestochowa university of technology, faculty of management, 19b armii krajowej street, 42-201 czestochowa, poland, e-mail: marcinzawada04@gmail.com. mariola piłatowska, aneta włodarczyk and marcin zawada dynamic econometric models 14 (2014) 51–70 52 introduction the harmful effects of climate change made policy makers become increasingly interested in reducing greenhouse gas (ghg) emissions using different policy tools such as environmental taxation and regulation imposing the increased use of renewable energy. at the international level various steps are taken to motivate countries to reduce the emissions of ghg, e.g. the kyoto protocol or the eu energy and climate obligations for member countries. the problem of how the pollutants relate to the economy has been the subject of intense research in the last decades. one of the main developments in understanding the link between the environment and economy was the environmental kuznets curve (ekc). the term 'environmental kuznets curve' was coined almost simultaneously by shafik and bandyopadhyay (1992), grossman and krueger (1995) and panayotou (1993). it refers by analogy to the inverted u-shaped relationship between the level of economic development and the degree of income inequality formulated by kuznets (1955). the ekc hypothesis says that environmental degradation increases with per capita income during the early stages of economic growth, and then declines with per capita income after arriving at a threshold1. hence, the relationship between income per capita and some types of pollutants is approximately an inverted u-shaped. there is a wide stream of researches that has employed cointegration techniques to examine the relationship between per capita income and some types of pollutants, among others, aspergis and payne (2009), halicioglu (2009), soytas and sari (2009), ang (2009), soytas et al. (2007). however, empirical results are mixed and not conclusive to give policy recommendations that can be applied across countries. the common feature of these studies was the linear approach and symmetric cointegration which may be a possible reason for ambiguous results. it has been suggested more recently (balke and fomby, 1997; enders and granger, 1998; enders and siklos, 2001) that the adjustment of deviations toward the long-run equilibrium need not be symmetric and reverting each period. to our knowledge, there are a very few studies that use non-linear (threshold) cointegration techniques for testing the ekc hypothesis, e.g. fosten et. al. (2012), esteve and tamarit (2012). this paper aims to look at testing the ekc hypothesis for the presence of threshold cointegration between per capita greenhouse gas emissions and per 1 see stern (2004), coondoo and dinda (2002), dinda (2004), luzzati and orisni (2009), halicioglu (2009) for extensive review surveys of studies which tested the ekc hypothesis. the environmental kuznets curve in poland – evidence from threshold... dynamic econometric models 14 (2014) 51–70 53 capita income for the polish economy during the period 2000 to 2012 (quarterly data). this approach will allow for a different speed of adjustment to the long-run equilibrium depending on whether emissions of greenhouse gas are above or below the ekc. additionally, another explanatory variable is added, i.e. energy consumption, to test the robustness of the results. the standard ekc hypothesis and its extended version including energy consumption will be tested using threshold autoregressive (tar) and momentum threshold autoregressive (mtar) cointegration method of enders and granger (1998) and enders and siklos (2001). we will also concentrate on the short-run and long-run causal relationship between per capita greenhouse gas emissions and per capita income using threshold error correction models (t-ecm) and momentum threshold error correction model (m-tecm). to our knowledge, there is no such study that uses this approach for the case of poland as if the ekc hypothesis is concerned. the remainder of this paper is organized as follows. section 2 presents the environmental kuznets curve. section 3 describes the methodology employed in the analysis. section 4 describes the data and reports the empirical results. section 5 concludes. 1. the environmental kuznets curve in classical approach to modelling the relationship between environmental degradation and income level the following quadratic function with the turning point occurring at a maximum pollutant level is used (agras and chapman, 1999): ,2210 tttt gdpgdpep   (1) where tep – emissions of some pollutant (per capita), tgdp – real income (per capita), both variables are in logarithms, ,1 ,2 0 – estimated parameters, t error term that may be serially correlated. based on the parameter values we may conclude about the shape of environmental pollution and income linkage. if 01  (or 01  ) while 02  , then the function (1) is a monotonically increasing (decreasing) according to linear function behavior. otherwise, if 01  and 02  , an inverted u-shape describes the situation when the pollution level increases as a country develops until this development reaches a turning point and after that the rising incomes are accompanied by decreasing environmental degradation. the turning point value is approximated by following relation (stern, 2004): mariola piłatowska, aneta włodarczyk and marcin zawada dynamic econometric models 14 (2014) 51–70 54  .2/exp 21 tpgdp (2) in the ekc literature the more sophisticated functional form is also taken into consideration with the third order polynomial for income factor (dinda, 2004): .33 2 210 ttttt gdpgdpgdpep   (3) similarly to the ekc model (2), the parameter estimates in equation (3) indicate the direction and character of the relationship between environmental pollutant and income. in the case of ,03  ,02  01  and 00  the n-shaped function is monotonically increasing with two possible turning points. in the case of opposite signs of cubic polynomial parameters, namely ,03  ,02  01  and ,00  an inverse-n shape is more accurate for describing analyzed relationship. in order to capture the effect of technological progress on pollution emission level the deterministic time trend (squared time trend) and some additional variables tx that may affect tep can be included in equation (1) or (3). in our empirical research of the long-run relationship between greenhouse gas emissions and economic growth first the standard ekc model (1) is assumed and further the ekc model with energy consumption )( te as additional variable is considered to test the robustness of results. in the latter case the model takes the form: ttttt egdpgdpep   2 210 , (4) where  − estimated parameter. 2. methodology the concept of cointegration implicitly assumes linearity and symmetry, what means that the adjustment of the deviations towards the long-run equilibrium is made instantaneously at each period and increases or decreases of the deviations are corrected in the same way. however, the cointegration tests and their extensions are misspecified if adjustment is asymmetric. to take the property of asymmetry into account, enders and siklos (2001) developed the concept of threshold cointegration. this is indeed an extension of residual-based two-stage estimation as developed by engle and granger (1987). the differences between them consist in the formulation of linearity and non-linearity from their second stage of unit root test. the environmental kuznets curve in poland – evidence from threshold... dynamic econometric models 14 (2014) 51–70 55 extracting from the long-run relationship (1) or (4) the disturbance term t (first stage), in the second stage we focus on the coefficient estimates of 1 and 2 (adjustment parameters) in the following regression:    r i titittttt ii 11211 ,)1(  (5) where t is a white noise disturbance. the heaviside indicator function ti is defined to depend on the lagged values of the residuals t :           1 1 0 1 t t t if if i (6) or on the lagged changes in t :           1 1 0 1 t t t if if i (7) where  is a threshold value. equations (5)-(6) are referred to as the tar model (threshold autoregressive model, enders and sikolos, 2001), while equations (5) and (7) are named the mtar model (momentum-threshold autoregressive model; see enders and granger, 1998). petruccelli and woolford (1984) proved that the necessary and sufficient conditions for the stationarity of residuals  t from the ekc model are ,01  02  and 1)1)(1( 21   for any threshold value  (enders and siklos, 2001). if these conditions are satisfied, 0t can be considered as the long run equilibrium value of the sequence. if t is higher than the long-run equilibrium, the adjustment is ,11 t but if t is lower than the long-run equilibrium, the adjustment is .12 t therefore, the equilibrium error behaves like a threshold autoregressive process (tar). the mtar model – according to enders and granger (1998) – is especially valuable when adjustment is asymmetric as the deviations t exhibit more 'momentum' in one direction than in the other. hence, the tar model allows to examine whether the positive deviations )0( t from the long-run equilibrium have different effects on the behavior of emissions than do the negative deviations ),0( t while the mtar model allows to display various amounts of autoregressive decay depending on whether the series is increasmariola piłatowska, aneta włodarczyk and marcin zawada dynamic econometric models 14 (2014) 51–70 56 ing or decreasing (fosten et al., 2012). there is no prescribed rule as to whether to use the tar or mtar model, but it is recommended to select the best adjustment mechanism (tar or mtar) using the aic (akaike information criterion) or sbc (schwarz bayesian criterion) information criteria (enders, chumrusphonlert, 2004). in general, the threshold value  governing the asymmetric behavior is unknown and has to be estimated along with the values of adjustment parameters 1 and .2 in our studies we follow enders and siklos (2001) and yau and nieh (2009) by employing chan’s (1993) methodology2 of searching the consistent estimates of threshold value. however, in many economic applications this value is set to zero, ,0 and then the cointegrating vector coincides with the attractor ( 0t ). once the threshold value  is obtained and the tar or mtar models are estimated, then testing for threshold cointegration can be performed. first, the null hypothesis of no cointegration 0: 210  h is tested, and when it is rejected, then the null hypothesis of symmetric adjustment 210 :  h is verified. to test the null hypothesis of no threshold cointegration, enders and siklos (2001) proposed the  statistics which is the f statistics. as the distribution of  is non-standard, appropriate critical values were tabulated by enders and siklos (2001) and later modified by wane et al. (2004). in the presence of cointegration (rejection of ),0: 210  h the null hypothesis 210 :  h of symmetric adjustment can be tested using the standard f statistics. when the adjustments coefficients are equal (symmetric adjustment), equation (5) converges the standard adf test. rejecting both the null hypotheses of 021   and 21   implies the existence of threshold cointegration with asymmetric adjustment. for instance, if the null of symmetric adjustment is rejected and 21   , then it implies that positive deviations (above threshold) of emissions from the long-run ekc tend to revert quickly towards equilibrium, whereas negative deviations of emissions (below threshold) tend to persist. hence, it suggests faster convergence for positive deviations of emissions (above thresh 2 the chan's method to find the consistent estimate of the threshold value arranges the values 1t or 1 t in ascending order, excludes the smallest and largest 15 percent, and the parameter that yields the smallest sum of squared residuals over the remaining 70 percent is the consistent estimate of the threshold. the environmental kuznets curve in poland – evidence from threshold... dynamic econometric models 14 (2014) 51–70 57 old) than negative deviations (below threshold) from the long-run ekc relationship. given the threshold cointegration is found, the next step proceeds with the granger-causality test using the advanced threshold error correction model (tecm) or momentum-threshold error correction model (m-tecm) by enders and granger (1998) and enders and siklos (2001). the threshold ecm is expressed as the following: ,2 111 1211 321 tit q i iit q i iit q i ittjt vgdpgdpepzzy              (8) or when additional variable te is included , 4 321 1 ` 2 111 1211 tit q i i it q i iit q i iit q i ittjt ve gdpgdpepzzy                   (9) where in equation (8) and (9):  2,, tttjt gdpgdpepy  , and 11 ˆ     ttt iz  and ,ˆ)1( 11     ttt iz  ti – heaviside indicator function determined by (6) or (7), 1ˆ t is obtained from the estimated long-run relationship (1) or (4), and tv is a white noise disturbance. based on the equation (8) and (9) the granger-causality tests are employed. the long-run causality is determined by the significance of adjustment parameters 1 and .2 to test short-run causality (or weak causality) the joint significance of all the coefficients i of ,itep i of ,itgdp i of 2 itgdp or i of ite  is examined using the wald f test. it is also desirable to check whether this two sources of causation (short and long-run) are jointly significant. this can be done by providing the wald f statistics for the interactive terms, i.e. the  terms and the explanatory variables (e.g. ,0: 10  ih ).0: 20  ih the joint test indicates which variables bear the burden of short-run adjustment to re-establish long-run equilibrium, given a shock to the system (asafu-adjaye, 2000; mehrara et al., 2012). this is referred to as the strong granger causality test. mariola piłatowska, aneta włodarczyk and marcin zawada dynamic econometric models 14 (2014) 51–70 58 3. empirical results 3.1. data source the data used in this study consist of greenhouse gas emissions ( tep ) (in tons of co2 equivalent per capita), real gross domestic product per capita ( tgdp ) and energy consumption 3 ( te ) in kilo of oil equivalent per capita in poland. the quarterly data for gdp are obtained from the central statistical office (www.stat.gov.pl) of poland, while annual data describing greenhouse gas emissions and energy consumption in poland were obtained from the eurostat database. to obtain the real gdp some transformations were made, i.e. quarterly nominal gdp data were transformed by the authors into real gdp in 2005 prices using gdp deflator. since the gdp data were characterized by significant quarterly seasonality, the tramo procedure of gretl software was applied to adjust it seasonally. the frequency of real gdp series is quarterly, however, the greenhouse gas emissions and energy consumption data are only available at annual frequency. therefore, we interpolated annual data to quarterly frequency by employing the denton-cholette method (sax, steiner, 2013) in the r software. the sample period is from 2000:q1 to 2012:q4. all variables are employed with their natural logarithms form to reduce heteroscedasticity and to obtain the growth rate of the relevant variables by their differenced logarithms. a) annual data b) disaggregated data to quarterly frequency figure 1. energy use (in kg of oil equivalent per capita) in poland in the period 2000–2012 – for annual data and disaggregated data to quarterly frequency 3 energy consumption covers: consumption by the energy sector itself; distribution and transformation losses; final energy consumption by end users (see epp.eurostat.ec.europa.eu, eurostat glossary). 2200 2300 2400 2500 2600 2700 2000 2002 2004 2006 2008 2010 2012 k g o f oi l eq u iv a le n t 520 540 560 580 600 620 640 660 680 2 0 0 0 q 1 2 0 0 1 q 1 2 0 0 2 q 1 2 0 0 3 q 1 2 0 0 4 q 1 2 0 0 5 q 1 2 0 0 6 q 1 2 0 0 7 q 1 2 0 0 8 q 1 2 0 0 9 q 1 2 0 1 0 q 1 2 0 11 q 1 2 0 1 2 q 1 k g o f oi l eq u iv a le n t the environmental kuznets curve in poland – evidence from threshold... dynamic econometric models 14 (2014) 51–70 59 a) annual data b) disaggregated data to quarterly frequency figure 2. greenhouse gas emissions (in tons of co2 equivalent per capita) in poland in the period 2000–2012 – for annual data and disaggregated data to quarterly frequency a) seasonally unadjusted b) seasonally adjusted figure 3. gdp (in pln per capita, in 2005 prices) in poland, 2000:q1–2012:q4 the evolution of time series data (original and disaggregated) employed in our analysis is shown in figure 1–3. 3.2. cointegration analysis with asymmetric adjustment before performing cointegration analysis, we use the augmented dickey-fuller generalized least squares (adf-gls; elliot et al., 1996) and kwiatkowski-phillips-schmidt-shinn (kpss; kwiatkowski et al., 1992) tests to identify the order of integration for each variable. in table 1, the adf-gls tests show that the unit root hypothesis cannot be rejected at any significant level for each variable in levels. further investigations of the unit root hypothesis indicate that the first differenced variables are stationary at least at the 10% level of significance. we also apply the kpss unit root test based on the null hypothesis of stationarity (or no unit root). the results show that the null hypothesis of stationarity is rejected at least at the 10% significance level. hence, all series are found to be integrated of order i(1). 9,8 10 10,2 10,4 10,6 10,8 11 2000 2002 2004 2006 2008 2010 2012 to n n es o f c o 2 eq u iv a le n t 2,3 2,4 2,5 2,6 2,7 2,8 2 0 0 0 q 1 2 0 0 1 q 1 2 0 0 2 q 1 2 0 0 3 q 1 2 0 0 4 q 1 2 0 0 5 q 1 2 0 0 6 q 1 2 0 0 7 q 1 2 0 0 8 q 1 2 0 0 9 q 1 2 0 1 0 q 1 2 0 11 q 1 2 0 1 2 q 1 to n n es o f c o 2 eq u iv a le n t 4000 5000 6000 7000 8000 9000 10000 2 0 0 0 q 1 2 0 0 1 q 1 2 0 0 2 q 1 2 0 0 3 q 1 2 0 0 4 q 1 2 0 0 5 q 1 2 0 0 6 q 1 2 0 0 7 q 1 2 0 0 8 q 1 2 0 0 9 q 1 2 0 1 0 q 1 2 0 11 q 1 2 0 1 2 q 1 p l n 4000 5000 6000 7000 8000 9000 2 0 0 0 q 1 2 0 0 1 q 1 2 0 0 2 q 1 2 0 0 3 q 1 2 0 0 4 q 1 2 0 0 5 q 1 2 0 0 6 q 1 2 0 0 7 q 1 2 0 0 8 q 1 2 0 0 9 q 1 2 0 1 0 q 1 2 0 11 q 1 2 0 1 2 q 1 p l n mariola piłatowska, aneta włodarczyk and marcin zawada dynamic econometric models 14 (2014) 51–70 60 table 1. the results of unit root tests variables levels differences adf-gls kpss adf-gls kpss ept –1.522 [3] 0.463 [2]* –5.246 [3]*** 0.073 [2] gdpt 0.041 [4] 1.218 [4]*** –2.307 [4]** 0.139 [4] gdp2t –0.144 [5] 1.217 [4]*** –1.676 [5]* 0.132 [4] et –0.957 [2] 1.548 [2]*** –5.368 [2]*** 0.110 [2] note: (***), (**), (*) in adf-gls tests respectively indicate the rejection of the null hypothesis that series has a unit root at 1%, 5% and 10% levels of significance, while in kpss tests indicate the rejection of the null hypothesis that series is stationary. the numbers inside the brackets are the optimum lag lengths determined using aic in adf-gls tests and the bandwidth is used using the newey-west method in kpss tests. table 2 contains cointegration test results of the standard long-run ekc in the form (1) and table 3 – long-run ekc including energy consumption in the form (4), when considering threshold (tar) and momentum adjustment (mtar). the tables report values of the adjustment coefficients 1 and 2 , the  statistics for the null hypothesis of no cointegration (a unit root in t ) against the alternative of cointegration with asymmetric adjustment. the f-test is used to test whether the adjustment back to long-run equilibrium is symmetric 21   . it can be seen that all coefficients 21,  have negative signs and are significant at least at 10% significance level. the necessary and sufficient conditions for cointegration hold in the case of all tar and m-tar models because the null hypothesis 0: 210  h is rejected (at 5% significance level)4 – see table 2. using the standard f-statistics for the restriction 210 :  h it is shown that asymmetric cointegration is strongly significant only in the tar model with ,0169.0 whereas in the tar model with 0 the support for asymmetric cointegration is only at 10% significance level. this evidence favors that the adjustment back to equilibrium between greenhouse gas emissions )( tep and gross domestic product )( tgdp is nonlinear. for choosing the more appropriate adjustment process (tar or m-tar) we will follow enders and chumrusphonlert's (2004) advice in using aic or sbc to select the best adjustment mechanism. the reported sbc for each model shows that the tar model with threshold value 0169.0 is more appropriate adjustment mechanism (the minimum sbc is in bold in table 2). 4 it is worth noting that this null hypothesis is also rejected when critical values from wane et al. (2004) are taken. the environmental kuznets curve in poland – evidence from threshold... dynamic econometric models 14 (2014) 51–70 61 table 2. results of tar and m-tar enders-siklos (e-s) test for cointegration on the standard ekc model 1 2  (1=2=0) f (1=2) lag sbc lb(4) tar =0 –0.124 (–2.372)** –0.199 (–5.036)*** 12.76*** 3.075 [0.087]* 5 –387.6 5.15 [0.273] m-tar =0 –0.125 (–2.489)** –0.191 (–4.804)*** 11.722*** 1.722 [0.197] 5 –386.1 6,5 [0.165] tar =–0.0169 –0.119 (–2.939)** –0.222 (–5.500)*** 15.184*** 6.236 [0.017]** 5 –390.9 4.74 [0.315] m-tar =0.0023 –0.148 (–2.925)** –0.179 (–4.536)*** 10.722*** 0.416 [0.522] 5 –384.6 5.35 [0.253] note: *** (**) (*) indicate significance at 1% (5%) (10%) level. critical values for  statistics from enders, siklos (2001). t-statistics for in  parentheses. p-values in brackets. lb(4) for ljung-box statistics. the lag length is selected such that the aic is minimized. table 3. results of tar and m-tar enders-siklos (e-s) test for cointegration on the ekc model including energy consumption 1 2  (1=2=0) f (1=2) lag sbc lb(4) tar =0 –0.104 (–3.249)** –0.056 (–2.20)* 5.83* 2.155 [0.149] 2 –488.06 6.91 [0.141] m-tar =0 –0.071 (–2.683)* –0.072 (–2.325)* 4.537 0.002 [0.961] 2 –485.77 7.26 [0.123] tar =0.0097 –0.115 (–3.516)** –0.055 (–2.237)* 6.616** 3.462* [0.069] 2 –489.40 6.67 [0.155] m-tar =8.8e-05 –0.064 (–2.36)* –0.082 (–2.74)** 4.74 0.34 [0.563] 2 –486.13 0.03 [0.999] note: ** (*) indicate significance at 5% (10%) level. critical values for  statistics from enders, siklos (2001). t-statistics for  in parentheses. p-values in brackets. lb(4) for ljung-box statistics. the lag length is selected such that the aic is minimized. we can see in table 2 that the point estimates 1 and 2 suggest faster convergence for the deviations from long-run ekc equilibrium when they are below the threshold ( 0169.0t ) than when they are above the threshold because 21   . we see that about 22% of the deviation from equilibrium is corrected in the next period when emissions are falling, compared to about 12% when they are rising. this means that short-run adjustment towards the ekc equilibrium reverts more quickly when the greenhouse gas emissions are decreasing (below the threshold) and tends to persist more when the greenhouse gas emissions are increasing (above the threshold). while the opposite result might be expected, this evidence should not be surprising when we look at the poland's energy profile. heavy reliance on mariola piłatowska, aneta włodarczyk and marcin zawada dynamic econometric models 14 (2014) 51–70 62 coal makes poland a relatively carbon-intensive economy, compared to the iea europe average. the large-scale transition of the polish energy sector, which is characterised by ageing infrastructure5, to a low-carbon economy requires huge long-term investments and an adequate policy and regulatory framework. therefore, from a short run perspective the potential reduction of greenhouse gas emissions should be rather combined with the energy efficiency improvements. figure 4. the fitted values of the estimated ekc results for greenhouse gas emissions table 4. estimated parameters of long-run ekc equation variable estimates of parameters (t-statistics) constant –53.298 (–4.228)*** –23.041 (–2.444)** gdpt 12.225 (4.280)*** 4.345 (1.978)* gdp2t –0.688 (–4.255)*** –0.255 (–2.065)** et 0.863 (7.639)*** note: *** (**) (*) indicate significance at 1% (5%) (10%) level. t-statistics for  in parentheses. estimated parameters for equation (1) and (4). 5 nearly half of today's operating capacity of the polish energy sector is older than 30 years. see: energy policies of iea countries. poland (2011). the environmental kuznets curve in poland – evidence from threshold... dynamic econometric models 14 (2014) 51–70 63 having established the tar cointegration in the long-run relationship for greenhouse gas emissions, it is now justified to analyse the estimation results (table 4) and the implications for the ekc hypothesis. we can see that in terms of the coefficients described in the ekc relation (1), we have ,00  01  and ,02  which implies the inverted u-shaped function. the fitted values of greenhouse gas emissions for the observed values of real gdp are displayed in figure 4. this results show the strong evidence in favour of the ekc hypothesis. the turning point in the observed range of real gdp for greenhouse gas occurs at pln7218.8 per capita which corresponds to around 2007q2. the addition of energy consumption )( te to the standard ekc model has not affected the results in terms of the presence of asymmetric cointegration (table 3), i.e. in the tar framework (with )0097.0 the asymmetric cointegration is observed but only at 10% significance level when critical values from enders and siklos (2001) are used. moreover, the non-linear relationship remains significant and correctly signed ( 1 and 2 are negative), suggesting the relationship is reasonably robust (see table 3). however, when critical values from wane et al. (2004) are taken, then the null hypothesis of non-cointegration )0:( 210  h cannot be rejected. hence, the results behind the threshold cointegration for the ekc model with energy consumption are rather weak and should be treated with caution. also the point estimates 1 and 2 differ with regard to the speed of adjustment process and direction of convergence for deviations above and below the threshold value when compared to the standard ekc model (table 2). now, faster convergence for deviations (from the long-run ekc) above the threshold than for those below the threshold is observed. we can see (table 3) that about 12% of the deviation from equilibrium is corrected when emissions are above the threshold ),0097.0( t compared to only 5.5% when they are below the threshold ).0097.0( t therefore, the short-run adjustment towards the long-run equilibrium reverts more quickly when emissions are increasing (above the threshold) and tends to persist when emissions are decreasing (below the threshold). this result is quite the opposite to that obtained for the standard ekc model what may indicate that energy consumption )( te is an important determinant of greenhouse gas emissions. the faster correction for deviations in emissions, if mariola piłatowska, aneta włodarczyk and marcin zawada dynamic econometric models 14 (2014) 51–70 64 they are too high, indicates that in the presence of environmental regulation6 the pressure to reduce them into their long-run levels was occurred. however, these emissions reductions were to a large extent achieved through the restructuring of polish industry and energy efficiency improvements. 3.3. threshold error correction models the positive finding of cointegration with tar adjustment justifies estimation of threshold error correction model (8) and (9) and testing the granger causality7. the t-ecm models are estimated for changes in greenhouse gas emissions )( tep and gross domestic product ),( tgdp assuming the estimates of threshold values obtained in previous step, i.e. 0169.0 − in model (8) and 0097.0 − in model (9) (see also table 2 and 3). the t-ecm for the te is not estimated since the energy consumption is added into the standard ekc model to test the robustness of the results. based on equations (8) and (9), the granger causality tests are employed to verify whether all the coefficients of ,itep itgdp  or 2 itgdp (or ite  in eq. (9)) are jointly statistically different from zero based on a standard f-test (wald test) and/or whether the coefficients ( 21,  ) of the error correction term are significant. to determine the appropriate lag lengths we apply the sbc criterion, and empirically find that the lag lengths are equal: ,221  qq .332  qq table 5 presents estimates of the error correction parameters along with wald f test statistics regarding granger causality. we will first interpret the results for the t-ecms and then for t-ecms including te (table 5). while the adjustment speed on the exceeding or underlying threshold level in the t-ecm model for tep has the 'right' direction ( 1 and 2 have a negative sign and are significant) by acting to eliminate deviations from the long-run equilibrium, the t-ecm model for tgdp adjusts to the 6 regulation on air pollution has become increasingly stringent, including international protocols such as the oslo protocol, the kyoto protocol and the eu energy and climate obligation for member countries. 7 the t-ecm for 2tgdp is run, but not reported because it has little useful economic interpretation (results are available from authors on request). the environmental kuznets curve in poland – evidence from threshold... dynamic econometric models 14 (2014) 51–70 65 'wrong' direction ( 1 has a positive sign) for one regime, and additionally parameters 1 and 2 are insignificant. in the tep model the adjustment speed responds faster in the lower regime (emissions below the threshold value) than in the higher regime with increasing deviations from the long-run equilibrium (emissions above the threshold value). this is consistent with the results in table 2. now, the greenhouse gas emissions converge to their long-run equilibrium at the rate of 13.2% with a deviation below the threshold and at a lower rate of 9.3% with a deviation above the threshold. table 5. estimates of threshold error correction model (t-ecm) and the wald f statistics for granger causality t-ecm t-ecm including et ept gdpt ept gdpt 1 –0.093* (–2.021) 0.046 (0.693) –0.081 (–1.012) –0.015 (–0.141) 2 –0.132** (–2.610) –0.073 (–1.008) –0.022 (–0.357) –0.163* (–1.961) h0: i=0 1.328 [0.280] 1.519 [0.227] h0: i==0 2.028 [0.111] 1.308 [0.286] h0: i==0 2.952** [0.032] 1.209 [0.324] h0: i=0 1.386 [0.262] 0.562 [0.643] 1.622 [0.202] 1.000 [0.404] h0: i==0 2.059 [0.106] 0.489 [0.744] 1.387 [0.259] 0.750 [0.565] h0: i==0 3.039** [0.029] 0.786 [0.542] 1.293 [0.292] 1.894 [0.133] h0: i=0 5.626** [0.007] 1.162 [0.325] h0: i==0 3.796** [0.018] 1.056 [0.380] h0: i==0 4.487** [0.009] 3.367** [0.029] h0: i=0 3.857** [0.031] 0.129 [0.879] h0: i ==0 2.674* [0.062] 0.187 [0.904] h0: i ==0 2.572* [0.069] 2.012 [0.130] lb(3) 2.60 [0.457] 1.60 [0.659] 2.66 [0.448] 1.95 [0.583] bic –395.72 –362.00 –387.32 –360.22 note: *** (**) (*) indicate significance at 1% (5%) (10%) level. t-statistics for 1, 2 in parentheses. p-values for wald statistics in brackets. lb(3) for ljung-box statistics along with p-values. the results of the granger causality tests show that there is no short-run causality running from real tgdp (and square of tgdp ) to ,tep but there is an unidirectional short-run causality from greenhouse gas emissions to per capita real .tgdp this can be interpreted as the non-rejection of 0:0 ih  and 0:0 ih  in the tep equation and rejection of 0:0 ih  in the tgdp equation (see table 5, 2nd and 3rd column). besides, the strong mariola piłatowska, aneta włodarczyk and marcin zawada dynamic econometric models 14 (2014) 51–70 66 long-run causality running from per capita real gdp (and square of real gdp) to greenhouse gas emissions is found in the regime below the threshold value of −0.00169 (rejection of ,0: 20  ih 0: 20   ih ). the results for the tgdp equation suggest reverse long-run causality running from greenhouse gas emissions to per capita real gdp (rejection of 0: 10  ih and 0: 20  ih ). this implies that deviations from the long-run ekc are corrected not only by movements in greenhouse gas emissions but also by movements in per capita real gdp. in the t-ecm model for tep , including energy consumption (table 5, 4th column), the parameters 21,  are correctly signed, but statistically insignificant8. nonetheless, it is worth noting that the correction back to equilibrium is faster in regime above the threshold value of 0.0097 (since )21   unlike the results obtained in the t-ecm not including energy consumption where faster convergence to equilibrium occurred in the regime below the threshold. this means that the addition of energy consumption into the t-ecm changed the response of greenhouse gas emissions to error correction. the manifested influence of energy consumption to reduce greenhouse gas emissions should be rather attached to the promotion of technological progress and the energy efficiency improvements, but not the large-scale transition to low-carbon economy, as has already been mentioned. further, there is neither short-run granger causality (or weak causality) nor long-run causality from tgdp to tep in the t-ecm including energy consumption (the non-rejection of null hypotheses ,0i ,01  i ,02  i ,0i ,01   i 02   i in table 5, 4th column). the non-significance of the f-statistics for gdp indicates that it is exogenous in the system. however, there exists the short-run and long-run causality from energy consumption to greenhouse gas emissions (the rejection of null hypotheses: ,0i 01  i and 02  i in the tep model including ).te this evidence suggests that energy consumption has an effect on greenhouse gas emissions and bears the burden of short-run adjustment to restore long-run equilibrium after a shock to the system. in the t-ecm for 8 this finding should be rather attached to the effect of small sample and reduction of degrees of freedom in the estimated threshold error correction models due to bigger number of variables and lagged terms in comparison to previous tar estimations (see table 2 and 3). therefore the results in table 5 should be interpreted with caution and validated on the larger sample. the environmental kuznets curve in poland – evidence from threshold... dynamic econometric models 14 (2014) 51–70 67 tgdp (including energy consumption) the rejection of 0: 20  ih suggests the unidirectional long-run causality from greenhouse gas emissions to real gdp in regime below the threshold value of 0.0097. the results from causality analysis based on the t-ecm (including energy consumption) indicate that re-establishing of the long-run equilibrium is carried out by the interaction of greenhouse gas emissions and energy consumption, but the long run effect of gdp is rather weak. conclusions the paper has aimed to investigate the ekc hypothesis for the case of the polish economy during the period 2000-2012. we tested for the presence of threshold cointegration between per capita greenhouse gas emissions and per capita income (real gdp). moreover, to test the robustness of the results, we considered the standard ekc hypothesis with the addition of per capita energy consumption to the model. to address such an issue, we applied the threshold cointegration model which allows the nonlinear adjustment to the long-run equilibrium. in the case of the standard ekc relationship, the results of threshold cointegration indicate that per capita greenhouse gas emissions and per capita real gdp are cointegrated with an asymmetric adjustment process. adjustments towards the long-run equilibrium revert more quickly when emissions are below the threshold value and tend to persist more when emissions are above the threshold value. the evidence of long persistence of adjustments to the equilibrium in higher regime (emissions above the threshold) may be explained by the specific of polish energy sector, namely heavy reliance on coal. as a consequence, the transition to a low-carbon economy will require huge long-term investments and an adequate policy and regulatory framework. therefore, from a short run perspective the reduction of emissions should be rather combined with the energy efficiency improvements. with regard to the granger causality tests the results are following. in the lower regime the bidirectional long-run causality between per capita real gdp and per capita greenhouse gas emissions is found. the short-run dynamics suggests that there is no causal relationship from real gdp to greenhouse gas emissions but there is an unidirectional granger causality from greenhouse gas emissions to real gdp. our results find strong evidence in favour of the ekc hypothesis with per capita greenhouse gas emissions having an inverse u-relation with real gdp per capita. the evidence suggests that the turning point in the observed range of real gdp for greenhouse gas emissions occurred at pln7218.8. as mariola piłatowska, aneta włodarczyk and marcin zawada dynamic econometric models 14 (2014) 51–70 68 a consequence, a decoupling between the two variable appears, i.e. the growth of emissions of some pollutant is slower than the economic growth. the addition of energy consumption to the standard ekc model has not affected the results in terms of the presence of asymmetric cointegration but has changed the direction of convergence for deviations above and below the threshold value. namely, adjustment towards long-run equilibrium is faster in the higher regime (emissions above the threshold) than in the lower regime (emissions below the threshold). this mean that in the presence of environmental regulation the pressure to reduce emissions into their long-run levels is observed. however, it should be emphasized that the abatement of emissions is rather achieved through the restructuring of polish industry, promoting technological progress and energy efficiency improvements, but not the large-scale transition to low-carbon economy. the results of granger causality tests indicate that there is no short-run causality from real gdp to greenhouse gas emissions. however, there is short and long-run causality from energy consumption to greenhouse gas emissions. besides, the unidirectional long-run causality from greenhouse gas emissions to real gdp in regime below the threshold value is observed. the results from causality analysis indicate that re-establishing of the long-run ekc is carried out by the interaction of greenhouse gas emissions and energy consumption, but the long run effect of gdp is rather weak. references acaravci, a., ozturk, i. (2010), on the relationship between energy consumption, co2 emissions and economic growth in europe, energy, 35, 5412–5420, doi: http://dx.doi.org/10.1016/j.energy.2010.07.009. agras, j., chapman, d. (1999), a dynamic approach to the environmental kuznets curve hypothesis, ecological economics, 28, 267–277, doi: http://dx.doi.org/10.1016/s0921-8009(98)00040-8. ang, j. (2007), co2 emissions, energy consumption, and output in france, energy policy, 35, 4772–78, doi: http://dx.doi.org/10.1016/j.enpol.2007.03.032. asafu-adjaye, j. (2000), the relationship between energy consumption, energy prices and economic growth: time series evidence from asian developing countries, energy economics, 22, 615–625, doi: http://dx.doi.org/10.1016/s0140-9883(00)00050-5. aspergis, n., payne, j. e. (2009), co2 emissions, energy usage and output in central america, energy policy, 37, 3282–86, doi: http://dx.doi.org/10.1016/j.enpol.2009.03.048. balke, n. s., fomby, t. b. (1997), threshold cointegration, international economic review, 38, 627–645, doi: http://dx.doi.org/10.2307/2527284. chan, s.-l. (1993), consistency and limiting distribution of the least squares estimator of a threshold autoregressive model, the annals of statistics, 21, 520–533. the environmental kuznets curve in poland – evidence from threshold... dynamic econometric models 14 (2014) 51–70 69 coondoo, d, dinda, s. (2002), causality between income and emissions: a country groupspecific econometric analysis, ecological economics, 40, 351–367, doi: http://dx.doi.org/10.1016/s0921-8009(01)00280-4. dinda, d. (2004), environmental kuznets curve hypothesis: a survey, ecological economics, 49, 431–455, doi: http://dx.doi.org/10.1016/j.ecolecon.2004.02.011. elliott, g., rothenberg, t. j., and stock, j. h. (1996), efficient tests for an autoregressive unit root, econometrica, 64(4) 813–836, doi: http://dx.doi.org/10.2307/2171846. enders, w., chumrunsphonlert, k. (2004), threshold cointegration and purchasing power parity in the pacific nations, applied economics, 36, 889–896, doi: http://dx.doi.org/10.1080/0003684042000233104. engle, f. r., granger, c. w. j. (1987), co-integration and error correction: representation, estimation, and testing, econometrica, 55(2), 251–276, doi: http://dx.doi.org/10.2307/1913236. enders, w., granger, c. w. j. (1998), unit-root tests and asymmetric adjustment with an example using the term structure of interest rates, journal of business economics & statistics, 16, 304–311, doi: http://dx.doi.org/10.2307/1392506. enders, w., siklos, p.l. (2001), cointegration and threshold adjustment, journal of business and economic statistics, 19, 166–176, doi: http://dx.doi.org/10.1198/073500101316970395. energy policies of iea countries. poland (2011), iea review, http://www.iea.org/publications/freepublications/publication/poland2011_web.pdf. esteve, v., tamarit, c. (2012), threshold cointegration and nonlinear adjustment between co2 and income: the environmental kuznets curve in spain, 1857-2007, energy economics, 34, 2148–2156, doi: http://10.1016/j.eneco.2012.03.001. fosten, j., morley, b., taylor, t. (2012), dynamic misspecification in the environmental kuznets curve: evidence from co2 and so2 emissions in the united kingdom, ecological economics, 76, 25–33, doi: http://dx.doi.org/10.1016/j.ecolecon.2012.01.023. grossman, g. m., krueger, a. b. (1995), economic growth and the environment, quarterly journal of economics, 110, 353–377, doi: http://dx.doi.org/10.2307/2118443. halicioglu, f. (2009), an econometric study of co2 emissions, energy consumption, income and foreign trade in turkey, energy policy, 37, 1156–1164, doi: http://dx.doi.org/10.1016/j.enpol.2008.11.012. kuznets, s. (1955), economic growth and income inequality, american economic review, 45(1), 1–28. kwiatkowski, d., phillips, p.c.b., schmidt, p., and shin, y. (1992), testing the null hypothesis of stationarity against the alternative of a unit root, journal of econometrics, 54, 159–178, doi: http://dx.doi.org/10.1016/0304-4076(92)90104-y. luzzati, t., orsini, m. (2009), investigating the energy-environmental kuznets curve, energy, 34, 291–300, doi: http://dx.doi.org/10.1016/j.energy.2008.07.006. mehrara, m., musai, m., nasibparast, s. (2012), the causality between savings and gdp in iran, international journal of advanced research in engineering and applied sciences, 1(6), 43–55. panayotou, t. (1993), empirical tests and policy analysis of environmental degradation at different stages of economic development, world employment programme working paper number wp238, international labor office, geneva. petruccelli, j. woolford, s. w. (1984), a threshold ar(1) model, journal of applied probability, 21(2), 270–286, doi: http://dx.doi.org/10.2307/3213639. sax, c., steiner, p. (2013), temporal disaggregation of time series, the r journal, 5(2), 80– –88, http://journal.r-project.org/archive/2013-2/sax-steiner.pdf. mariola piłatowska, aneta włodarczyk and marcin zawada dynamic econometric models 14 (2014) 51–70 70 shafik, n., bandyopadhyay, s. (1992), economic growth and environmental quality: time series and cross-country evidence, background paper for the world development report 1992, world bank, washington d.c. soytas, u., sari, r. (2009), energy consumption, economic growth and carbon emission: challenges faced by an eu candidate member, ecological economics, 68, 1667–75, doi: http://dx.doi.org/10.1016/j.ecolecon.2007.06.014. soytas, u., sari, r., ewing, b.t. (2007), energy consumption, income and carbon emissions in the united stated, ecological economics, 62, 482–489, doi: http://dx.doi.org/10.1016/j.ecolecon.2006.07.009. stern, d. (2004), the rise and fall of the environmental kuznets curve, world development, 32, 1419–1439, doi: http://dx.doi.org/10.1016/j.worlddev.2004.03.004. wane, k. m., gilbert, s., dibooglu, s. (2004), critical values of the empirical f-distribution for threshold autoregressive and momentum threshold autoregressive models, 2004 discussion papers for the department of economics, southern illinois university at carbondale. yau, h.-y., nieh, c.-c. (2009), testing for cointegration with threshold effect between stock prices and exchange rates in japan and taiwan, japan and world economy, 21, 292– –300, doi: http://dx.doi.org/10.1016/j.japwor.2008.09.001. ekologiczna krzywa kuznetsa dla polski − analiza progowej kointegracji z a r y s t r e ś c i. artykuł przedstawia analizę relacji długookresowej między emisją gazów cieplarnianych w przeliczeniu na mieszkańca a realnym pkb w przeliczeniu na mieszkańca (hipoteza ekc, ekologiczna krzywa kuznetsa) z wykorzystaniem podejścia progowej (asymetrycznej) kointegracji i modelu korekty błędem dla polskiej gospodarki w okresie 20002012 (dane kwartalne). standardowy model ekc został rozszerzony o zużycie energii w celu zbadania wpływu dodatkowych zmiennych na wyniki. hipoteza ekc była sprawdzana z wykorzystaniem progowych modeli autoregresyjnych (tar i mtar). ponadto, dla zbadania krótkookresowej i długookresowej przyczynowości grangera między emisją gazów cieplarnianych w przeliczeniu na mieszkańca a realnym pkb w przeliczeniu na mieszkańca zastosowano progowy model korekty błędem. otrzymane wyniki dostarczają istotnych dowodów na rzecz hipotezy ekc dla przypadku polski oraz wskazują, że tymczasowe odchylenia od ścieżki długookresowej równowagi ekc są korygowane w asymetryczny sposób. s ł o w a k l u c z o w e: ekologiczna krzywa kuznetsa, emisja gazów cieplarnianych, zużycie energii, wzrost gospodarczy, kointegracja progowa, przyczynowość w sensie grangera. microsoft word dem_2014_29to49.docx © 2014 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2014.002 vol. 14 (2014) 29−49 submitted june 25, 2014 issn accepted december 23, 2014 1234-3862 natalia drzewoszewska* searching for the appropriate measure of multilateral trade-resistance terms in the gravity model of bilateral trade flows a b s t r a c t. the aim of the paper is to compare different approximations of multilateral trade-resistance in the gravity model and the influence of their use on estimation results for models of eu-trade. three synthetic variables: for bilateral trade costs, exporter’s and importer’s remoteness are used as an alternative for including time-varying country effects. results indicate significant impact of those variables but not wholly compatible with the theory. estimated coefficients of trade determinants, including euro’s effects, have expected values in both approaches only if the fe estimator is applied. k e y w o r d s: international trade, panel data, gravity model, multilateral trade-resistance terms, bilateral trade costs, globalization in the xxi century, euro‘s effect j e l classification: f10, f14, f15, c23, c24, c26. introduction the gravity model is a common tool for analyzing the flows of international trade. the characteristics of panel data allow for taking into consideration unit specific effects with regard to territorial units covered by the study, as well as time effects referring to the years under analysis. therefore, * correspondence to: natalia drzewoszewska, nicolaus copernicus university, department of statistics and econometrics, ul. gagarina 13a, 87-100 toruń, e-mail: nataliadrzewoszewska@gmail.com. this paper was written during the author's research stay at the chair of statistics and econometrics at justus liebig university giessen. the author would like to thank prof. dr. peter winker for the work conditions, his encouragement and helpful suggestions. natalia drzewoszewska dynamic econometric models 14 (2014) 29–49 30 it assists in controlling for unobserved heterogeneity, which could not be accounted for by explanatory variables in the model, which is useful for such a macroeconomic study. an additional incentive for using the gravity model is that the necessary data is relatively easily available. estimation results of a majority of studies described hereinafter are quite similar for the main variables in the model – differences come from the different test samples and different time periods, as well as from different estimation methods used in the research. focusing on theoretical assumptions of the model can easily explain the inaccuracy of some empirical trade analyses based on the gravity model. according to the theory of anderson and van wincoop (2003), decisions about international trade essentially depend on the relative trade costs, which are however not easy to measure. one of the aims of this study is to identify the best measure of these costs, which will cover the multilateral traderesistance, both for exporter and importer. estimated panel gravity models include typical explanatory variables: national income, measure of bilateral distance and the set of dummy variables for common border, common language and access to the sea. additionally, considering the utility of the gravity model by the test of trade-agreement effect, in former analysis there were the dummy variables used to describe the participation in the economic and monetary union (micco et al., 2002, 2003; maliszewska, 2004). another purpose is the analysis of the international trade between eu countries, which create an integrated, relatively homogenous area, where such variables like tariffs or rates of exchange that do not have to be included in the model. globalization is often defined as the growing integration of economies and societies around the world1 “mainly by free trade and free capital mobility, but also by easy or uncontrolled migration” (daly, 1999), “leading to the notion of a borderless global or planetary economy” (avinash, 2000), which makes the european union a great example of the globalized economies. globalization in the xxi century is a specific time – there are deeper and broader changes in the global economy – spread of the “new economy” as well as the new information and communication technology (ict), what is pointed out in recent studies (ramos and ballell, 2009; farhadi et al., 2012; garcía-muñiz and vicente, 2014). friedman describes 1999 as the year of the internet, when the globalization started a new era, opened for outsourcing, offshoring and other new activities changing the global trade structure 1 definition used by the world bank group, 2001. searching for the appropriate measure of multilateral trade-resistance terms… dynamic econometric models 14 (2014) 29–49 31 (friedman, 1999). that is the reason choosing 1999 the starting year of the analyzing time period. three research hypotheses were put forward within the framework of the carried out objective. the first assumes that the travel time between centroids of countries is a good base for approximation of bilateral trade costs. following the second hypothesis, bilateral trade flows increase if exchange partners are members of the eurozone. the third hypothesis assumes that synthetic variables of bilateral costs and remoteness are accurate approximation of multilateral trade-resistance terms for eu countries. two first parts of the paper discuss the theoretical assumptions of the gravity model for trade flows and the problems with its estimation. the third focuses on the description of the new measures for multilateral traderesistance terms. the final part presents the results of conducted research. 1. theory of the gravity model of bilateral trade flows the first gravity equation was based only on empirical research of tinbergen (1962). inspired by newton's law of universal gravitation, author presented following “traditional” gravity equation for trade2: ,3210  oddood dxxy  (1) where: ody – volume of trade flow from country o (origin) to country d (destination), do x,x – national income of countries o and d (gnp volumes), odd – physical distance between the two countries. more generally, we can describe the gravity model by four forces: g – external (global) factors expressing “gravitational constant”, although it is only held constant in the cross-section, os , dm – specific factors of origin and destination factors expressing their “masses”, and od – negative factors expressing the trade costs, with the following form: ,0 oddod msgy  (2) the gravity model with panel data structure can be written in following logarithmic form: ,,,3,2,10, todtodtdtotod εdxxy   (3) 2 this equation implies that exports have a constant elasticity with respect to each of three explanatory variables – what means that a 1 per cent increase in the gnp of country d always results in an increase of 2 per cent in the exports of the supplying country o. natalia drzewoszewska dynamic econometric models 14 (2014) 29–49 32 where:  nod ,...,2,1, , tod ,y – flow values (log) between regions o,d – object’s number3, t=1,2,...,t – number of time period, t,ox , t,dx – explanatory variables values (log) respectively for origin and destination regions, t,odd – bilateral trade costs, including distance between regions (log), 3210 ,,,  – structural parameters of the model 4, tod ,ε – random component. tinbergen (1962) also extended his model for 18 developed countries by dummy variables of common border, commonwealth preference and benelux preference and, in the second case, by the gini coefficient of export commodity concentration. further research of econometricians was expanded by additional variables and effects, like time effects or country pair effects. nevertheless, the gravity equation still needed the theoretical assumptions, which became a key issue in the following years. the theory of gravity model was proposed by anderson (1979), bergstrand (1989), deardoff (1998), eaton-kortum (2002) and anderson and van wincoop (2003). the last one was named as “the final structural gravity equation” and it passes now for the most accurate description of reality. the most important part relates to the relative trade costs, which are included in the model as multilateral trade-resistance (mtr) terms. namely, these two terms measure the exporter's and importer's joint average trade resistance (in terms of trade barriers), which each of them faces to all their other potential trading partners. for instance, if there is a rise in trade barriers between importing country d and all its other possible trading partners (inward mtr rises), the relative price of the exporting country o’s products will decrease and trade flows between o and d will increase. likewise, if outward mtr rises, overall demand on o’s exported products will slow down, thus reducing the price op , which, under conditions of the constant trade barriers, will consequently increase trade flows between both countries. the new structural gravity equation takes the form of: , 1           do od w do od p t x xx export (4) 3 in case of panel gravity models, which analyze trade flows, a pair of regions represents an object (unit), namely  d,oi  . for n analyzed regions  nn 2 objects are included in the study, i.e. pairs of trading partners. 4 prediction that 11  , 12  leads to the unit-income-elasticity model, what was often assumed by researchers in the studies. searching for the appropriate measure of multilateral trade-resistance terms… dynamic econometric models 14 (2014) 29–49 33 with: outward (exporter’s) multilateral trade-resistance:   , 1 1 w d o d od o x x p t            (5) and inward (importer’s) multilateral trade-resistance:   . 1 1 w o o o od d x xt p              (6) where: ox , dx – national income values of both trading countries, wx – world’s income, odt – trade cost factor reflecting bilateral trade resistance between country o and d,  – elasticity of substitution. the conception of multilateral trade-resistance of trading countries is intuitively convincing since all the countries have a lot of potential alternative trading partners and relationships with them that influence the bilateral trade-resistance. hence the trade impediments between countries should not be approximated only by the bilateral trade costs. moreover, the import and export of more developed and wealthy countries should be easier, which is also expressed in the above form of gravity equation by implementing the income shares in the total world income. omitting the theoretically motivated mtr terms in the gravity models leads to the systematic bias in coefficient estimates of bilateral trade-cost variables. this form of gravity model, acclaimed to be the most accurate one because of using relative differences between countries, was easily expanded to describe another foreign flows, namely migration flows (anderson, 2011). 2. difficulties with empirical research based on the gravity model of trade flows using panel data the multiplicative nature of the gravity equation, quality of available database, characteristics of panel data or the big amount of missing data yield many potential problems with a solid empirical analysis. among the biggest problems occurring by estimating the panel data gravity models are5:  multitude of zero-observations (log-linearization is not feasible in these cases), 5 more problems with trade data are presented in feenstra et al. (2001). for essential reference on panel-data models see hsiao (2003). natalia drzewoszewska dynamic econometric models 14 (2014) 29–49 34  error terms in the usual log-linear form of the gravity equation are heteroscedastic (which violates the assumption that error term should be statistically independent from the regressors, using ols-method after the log-linearization leads to inconsistent estimates of the elasticity of interest, the nls estimator is in turn very inefficient, as it ignores the heteroscedasticity),  variance of the error term is not constant (nls estimator is not optimal)  trade data are suffering from rounding errors (that leads to the bias of estimates),  mtr terms should be included in the gravity model of bilateral flows, but they are not directly observable. there are many potential methods that can more or less overcome the foregoing problems. one way with the first problem is dropping the pairs with zero from the trade-data set, what allows for using ols estimation method. another way is to keep these observations by adding a constant to zero-observations, for instance ( 1ijy ) and use again ols method, what can be found in martinez-zarzoso (2007), westerlund and wilhelmsson (2009), or use tobit model for panel data (soloaga and winters, 2001; baldwin and dinino, 2006; tripathi and leitão, 2013). however, all three of these methods lead to inconsistent estimates (especially by tobit models, where estimation results depend on the chosen constant). to avoid this problem, santos silva and tenreyro (2006) proposed the use of ppml (poisson pseudo maximum likelihood) estimator6 in levels, which not only deals with zero-value observations, but also can be easily adapted in models with endogenous regressors, providing unbiased estimates in the presence of heteroske-dasticity, where all observations are weighted equally. the choice of an accurate estimation method in face of all the problems connected with the gravity model is never infallible; hence the common way is to use several estimation methods, appropriate to considering case of study. every estimator has pros and cons7 and the inference based on the only one method is not advisable. even using the hausman test by pointing out the right version of model between re and fe is not practiced since the form of both models is not the same (the lack of constant variables in femodel) and the assumption about individual fixed effects between trading 6 previously authors used the gamma pml (gpml), which gave good results, but is very sensitive to measurement errors – as it gives an extra weight to the noiser observations. the ppml method was originally proposed by mccullagh and nelder (1989). 7 details about majority of estimation methods of gravity model are presented by gómezherrera (2013). searching for the appropriate measure of multilateral trade-resistance terms… dynamic econometric models 14 (2014) 29–49 35 pairs in this case seems to always be the right one. however, the readiness of researchers to know the coefficients by constant variables leads to implementing more estimation methods. the comparison of the coefficients gives an answer to the questions asked in the hypotheses of the research. interesting research of gómez-herrera (2013) includes a comparison of many estimation methods (truncated ols, ols ( 1ijy ), tobit, probit – with heckman’s approach8, re, fe and ppml), where gravity model, despite of physical distance and dummies (common border, common language, same country and participation in trade agreements)9 among regressors, included also exporter and importer time varying effects. the results of comparison of several techniques with a dataset covering 80% of world trade induce to choose the heckman sample selection model as the preferred estimation method within nonlinear techniques when data are heteroskedastic, but this approach is preferred when the data also contain a significant proportion of zero observations – what is natural by analyzing 80% of the world trade. the need of using mtr terms is the result of new structural gravity equation proposed by anderson and van wincoop (2003), which logarithmic form is following: ,,,4 ,4,3,2,10, todtd totodtdtotod ε π p txxexport     (7) where: 31   . there are two ways to take mtr on board in the gravity model: 1) creating synthetic variables for both countries – remoteness10 – or: 2) including time-varying individual effects for both countries in the gravity model (the dummy variables identifying the exporter and importer) 11. 8 see bikker and de vos (1992), linders and de groot (2006), martin and pham (2008). 9 the formula to compute the effect of dummy-variables is following:   %ibe 1001  , where ib is the estimated coefficient. 10 wei (1996) defined as the log of gdp-weighted average distance to all other countries. 10 the use of simulation method allows to obtain mtr as well. however, because of the complex calculation problem, this method is rarely taken into consideration by researchers. anderson and van wincoop (2003) used non-linear programming to include mtr terms, assuming that elasticity of substitution equal to 8 . however, feenstra (2002) showed that it is possible to apply importer and exporter fixed effects to obtain approximately similar results. alternatively, baier and bergstrand (2009) introduced variables of mr approximations which produce consistent estimates, using taylor approximation. this approach were used also by behar and nelson (2012). natalia drzewoszewska dynamic econometric models 14 (2014) 29–49 36 the first method faces a problem with the choice of the right form of the variable. the implementing of physical distance is not enough to approximate bilateral costs, used then in the remoteness variable, since it doesn’t cover the whole trade costs, is not time-varying and forces to take the assumption about symmetric bilateral trade costs. there also appeared to be another calculation problems, for instance the measure of inter-distance by the formula proposed by head and mayer (2002)12. the literature provides a lot of ways to calculate bilateral trade costs. the most common way, despite using only physical distance, is to create bilateral costs-equation by implementing dummy variables, such as common border, common language, landlocked and others, namely13:   ._...exp 321 dummiesotherlanguageborderdt odododod    (8) however, the equation above is still difficult to calculate and provides still constant and symmetric variable for both countries of the trading pair. the calculation of time-varying bilateral trade costs is possible through using the time-varying specific variables in the equation with some specific weights, like:  ,...ln ,,33,22,11, s tnodns tods tods todtod xwxwxwxwt  (9) where: nw – weights, s t,nodx – standardized values of regressors 14. substantial weakness of this approach is the problem of appropriate weights. taking the arbitrary weights does not seem to be correct in face of the differences between countries and non-theoretical or empirically-based assumptions. the use of the second method – time-varying individual effects – seems to be easier, however, it increased the dimension of the estimated matrix causing calculating problems and does not allow for incorporating specific variables for countries into the model due to collinearity, what leads to a bias15. 12 in this study, the author proposed the approximation of the inter-distance based on literature, namely ≈ square root of land surface*0,4. 13 see baier and bergstrand (2009); baldwin and taglioni (2006). 14 for more details about the method, see drzewoszewska et al. (2013). 15 likewise, the inclusion of the exporter and importer dummies in the model means that inclusion of time invariant exporter and importer characteristics is not possible in this case. see ruiz and vilarrubia (2007). searching for the appropriate measure of multilateral trade-resistance terms… dynamic econometric models 14 (2014) 29–49 37 facing the problems above, there is no standard way to incorporate mtr in the gravity model so far. in the literature, there is a lot of research with exporter and importer effects in gravity model of bilateral trade flows, e.g. rose and wincoop (2001), baltagi (2003), ruiz and vilarrubia (2007). the popular practise is to include country-pair effects as well, eg. glick and rose (2002), baltagi (2003), micco et al. (2003), fratianni and hoon-oh (2007), fidrmuc (2008), bussière and schnatz (2009). furthermore, using time effects in the gravity model is a common issue now, as it replaces global circumstances, shocks, ect. another way could be spatial modeling – in the research of fdi fernández-avilésa et al. (2012) proposed a simple fdibased measure of financial distance with the use of spatial techniques. the remoteness variables for exporting and importing countries used in foregoing studies have different formulas, are both time-varying (baldwin and taglioni, 2006) and fixed (fidrmuc, 2001; ruiz and vilarrubia, 2007). for instance, head (2003) calculates remoteness as a country’s average weighted distance from its trading partners, where weights are the partner countries’ shares of world gdp. the physical distance between trading countries approximates bilateral trade costs since the first application of gravity model. the coefficient of this variable in estimated models is always negative in all the empirical analysis, what makes it a common measure used by researchers. however, the trade costs are created primarily by transport costs, which are depended on the quality of transport infrastructure, tariffs, prices, as well as on the distance. an alternative measure of bilateral trade costs for ue countries is prosed in the empirical part of this study. 3. a new measure of remoteness the new formula of remoteness variables, proposed in this study, allows for using time-varying bilateral costs, which according to the strong assumption in anderson and van wincoop (2003) theory are symmetric. besides, using the distance between countries to describe their bilateral costs leads to the constant remoteness, which is another unreal assumption. the formulas of three synthetic variables – bilateral costs t,odt , exporter’s remoteness t,odrem and importer’s remoteness t,dorem – are following: , _' , , tod od tod opennesssimporter distance t  (10) where: natalia drzewoszewska dynamic econometric models 14 (2014) 29–49 38 . _ _' , ,, , td tdotod tod importtotal exportexport opennesssimporter   (11) bilateral trade costs (10) became time-varying in this approach, which suits better to reality – trading costs are not constant over time and the psychical distance, especially in the era of globalization xxi century, does not lower the trade flows as much as 50 years ago. here the distance between countries, measured by travel time between the centroids of trading countries, is divided by share of bilateral trade exchange in the total import of importing country. moreover, this method reflects the theoretical significance of importer’s demand in the final amount of bilateral trade flows. importer’s demand is also underlined by the following form of exporter’s remoteness variable: , _, , , ,    dok ttk tok tod incomeworldincome t remoteness (12) which is the sum of bilateral costs divided by importer’s income share in the world’s total income. it is expected that a relatively richer importing country will have a larger overall demand, hence the export to this country will be relatively easy (exporter’s remoteness is smaller then). in this approach, importer’s remoteness variable includes analogously the exporter’s income share in the world’s total income as a weight in the weighted average: . _, , , ,    dok ttk tkd tdo incomeworldincome t remoteness (13) however, the denominator of importer’s remoteness variable above underlines exporter’s condition, what (being still potential good weight) does not play substantial role in the demand of importing country16. potentially better weight would be a share of bilateral export from the importer in his total export, since it better expresses importer’s condition and also reflects the interrelation with his trading partner. hence, an alternative measure for importer’s remoteness is the following: 16 in macroeconomic theory, import is defined as a function of the domestic absorption a (total demand for all final marketed goods and services) and the real exchange rate  , taking the form of: ),a(fi  . see burda and wyplosz (2005). searching for the appropriate measure of multilateral trade-resistance terms… dynamic econometric models 14 (2014) 29–49 39 . _, ,, , ,    dok tdtdk tkd tdo exporttotalexport t remoteness (14) comparison of the estimated coefficient’s sign of both above importer’s remoteness synthetic variables would give an answer if the second form, more economically justifiable, contains a better approximation of inward multilateral resistance. according to the theoretical assumptions of anderson and van wincoop (2003), the mtr terms should have a positive impact on bilateral trade flows. according to the theory, estimation results of models with remoteness synthetic terms and models with countries time-varying specific effects should have similar estimates of the rest of the variables. this could confirm that the created synthetic variables are a good approximation of mtr, which allows for estimation of their exact influence, also giving an opportunity to use more estimators, like ppml or ht. the model to compare has the following form: ,,,5 ,43,2,10, todtod todttdtotod εx tiiiexport     (15) where: t,dt,o , ii – time-varying individual effects, ti – time effects, t,odt – bilateral trade costs, t,odx – set of dummies for the trading pair. an easier way to estimate mtr can be the assumption that mtr is constant over time, what allows for using only fixed individual effects for both countries, with lower dimension of the estimating matrix. however, this assumption is advisable in the case of relatively short time period, so it is not considered in this study. 4. gravity model of bilateral trade flows for eu countries in the period of 1999–2011 – empirical results the data used in this study consists of a sample of 25 eu countries, with the following database-sources: comtrade/oecd, wdi and google maps application. in order to analyze the trade in the era of globalization xxi century, the chosen time period of research is opened by “the year of the internet” and includes the last year of available data. variables included in the analysis are presented in table 1. the first step of research was to look for an alternative variable that could replace the physical distance in the traditional gravity model. as a matter of fact, the physical distance is considered as a good approximation natalia drzewoszewska dynamic econometric models 14 (2014) 29–49 40 of bilateral trade costs, however it does not take into account the quality of transport infrastructure, which varies over the countries and influence on the time and costs of transportation. the use of the time travel between centroids of countries became possible owing to free google map application, which time-data was downloaded on 14.03.201417. table 1. variables included in the analysis of international trade flows variable definition measure unit source export export flows in current prices from origin country to destination country usd comtrade/oecd gni gross national income in current prices18 usd wdi dist great circle distance between the national centroids km author’s calculation travel travel time by road between the national centroids19 hour google maps border 1 if two trading countries share a common border and 0 otherwise dummy variable language 1 if two trading countries share a common language and 0 otherwise dummy variable sea 1 if at least one from both trading countries is not landlocked and 0 otherwise dummy variable oneemu 1 if the importer belongs to the economic and monetary union but the exporter does not and 0 otherwise dummy variable bothemu 1 if both of the trading countries in the pair are members of the economic and monetary union and 0 otherwise dummy variable 17 generally, google maps application offers a route planner for traveling by foot, car, bicycle (beta test), or with public transportation. it does not include the information about current traffic in its calculation (this is a property of another application the google traffic). reproducing the calculation in a short time period gives equal results of the travel time by car between two chosen locations. google created the application in 2005, hence it is impossible to find a data with the measurement of travel time across last 13 years. however, the regular collecting of the data generated by google maps could be successfully used in the future research. 18 the use in the study gni instead of gdp variable is intentional, as it measures income received by a country both domestically and from overseas. in fact, there is considered the output from the citizens and companies of a particular nation, regardless of whether they are located within its boundaries or overseas. the first empirical research provided by the author of the gravity equation – tinbergen (1962) included similar measure, namely gnp. 19 great circle distance algorithm was used in the calculation. searching for the appropriate measure of multilateral trade-resistance terms… dynamic econometric models 14 (2014) 29–49 41 table 2. traditional gravity model of trade flows for eu-25 countries20 in 1999– –2011 with physical distance between centroids as approximation of bilateral trade costs – results for alternative estimation methods model a ols ( 1ijy ) re fe ht tobit ( 1ijy ) ppml lngni_o –0.13 1.30*** 1.30*** 1.30*** 1.97*** 0.76*** lngni_d 0.52*** 1.05*** 1.05*** 1.05*** 0.58*** 0.74*** lndist –1.23*** –1.50*** –1.50*** –1.51*** –0.98*** oneemu –0.24 0.08 0.08 0.08*** –0.16* –0.15*** bothemu 0.25** 0.14*** 0.14*** 0.14*** 0.15* –0.02 border 0.21 0.13 0.13 0.05 0.21*** language –0.38 –0.07 –0.06 –0.34 0.47*** sea 0.21 0.33*** 0.33*** 0.33* 0.08** te yes yes yes yes yes yes ce yes yes no yes yes no constant 20.70*** –30.40*** –40.40*** –32.50*** –35.40*** –11.60*** number of state 600 504 504 504 600 504 observations 7800 6533 6533 6533 7800 7800 r2 0.777 note: te – time effects, ce – country effects (separately for exporter and importer); *** p<0.01, ** p<0.05, * p<0.1. table 3. traditional gravity model of trade flows for eu-25 countries in 1999–2011 with travel time between centroids as approximation of bilateral trade costs – results for alternative estimation methods model b ols ( 1ijy ) re fe ht tobit ( 1ijy ) ppml lngni_o –0.13 1.30*** 1.30*** 1.30*** 1.96*** 0.75*** lngni_d 0.52*** 1.05*** 1.05*** 1.05*** 0.59*** 0.73*** lntravel –1.39*** –1.76*** –1.76*** –1.77*** –1.03*** oneemu –0.26* 0.07 0.08 0.07*** –0.19** –0.19*** bothemu 0.27** 0.15*** 0.14*** 0.15*** 0.17* –0.09*** border 0.21 0.09 0.09 0.01 0.19*** language –0.40 –0.13 –0.13 –0.42* 0.38*** sea 0.25 0.35*** 0.35*** 0.35** 0.09*** te yes yes yes yes yes yes ce yes yes no yes yes no constant 16.30*** –35.40*** –40.40*** –38.30*** –40.40*** –15.00*** number of state 600 504 504 504 600 504 observations 7800 6533 6533 6533 7800 7800 r2 0.777 note: te – time effects, ce – country effects (separately for exporter and importer); *** p<0.01, ** p<0.05, * p<0.1. 20 the sample includes all eu countries without malta and cyprus. natalia drzewoszewska dynamic econometric models 14 (2014) 29–49 42 validity of replacing physical distance by travel time in the gravity model was checked by comparison of estimation results of two models: with distance as approximation of bilateral trade costs (model a) and with travel time respectively (model b). tables 2 and 3 show that the gravity model with travel time estimated with several estimation methods – ols ( 1ijy ), re, fe, ht, tobit ( 1ijy ) and ppml – gives similar estimates of other variables as the model including physical distance. the influence of travel time is significant and still negative in all cases, as expected. hence, the travel time between centroids of trading countries is replacing the physical distance in the gravity model in this study. different results for the dummy variable describing the participation of only the importing country in emu have different estimates, however negative signs occur only by the most naïve methods – namely ols and tobit model, where zero-export flows are replaced by the value of 1. unexpected signs occur by ppml method, however, the estimated models do not include country effects, which can lead to the bias in estimates. despite improving the gravity equation by introducing the variable which covers the influence of physical distance and the quality of road infrastructure, the variable of travel time remains still constant, what does not represent the whole reality. then the second step of the research is to create a time-varying synthetic variable describing bilateral trade cost according to the formula (10) and afterwards use it in the next synthetic variables: exporter’s and importer’s remoteness, according to (12), (13) – model 1 – and according to (12), (14) – which reflects model 221. all synthetic variables were used in the gravity model (7), with and without fixed country effects for exporter and importer. the most similar estimates, with higher r2 coefficients as well, were obtained in the models including time and country effects, whose estimation results are shown in table 4. 21 the share in world income in remoteness variable was counted in two ways: through dividing by the total income of ue-25 countries as well as by the total world income. as expected, the estimation results in both cases were almost identical estimates, including the r2 coefficient of estimated fe-model (90%), where the only differences were exposed by the constant. searching for the appropriate measure of multilateral trade-resistance terms… dynamic econometric models 14 (2014) 29–49 43 table 4. the structural gravity models of trade flows for eu-25 countries in 1999– –2011 with remoteness as an approximation of mtr – results of approaches 1 and 2 for alternative estimation methods model 1 re fe ht tobit ppml lngni_o 0.424*** 0.468*** 0.428*** 0.425*** 0.286*** lngni_d 2.259*** 2.215*** 2.256*** 2.258*** 1.449*** lnbtc_od –0.690*** –0.659*** –0.687*** –0.689*** –0.587*** lnrem_od 1.103*** 1.068*** 1.100*** 1.102*** 0.642*** lnrem_1_do –0.523*** –0.515*** –0.523*** –0.523*** –0.326*** oneemu 0.244*** 0.238*** 0.243*** 0.243*** 0.040* bothemu 0.257*** 0.245*** 0.256*** 0.257*** –0.015 border –0.085** –0.078*** –0.083*** 0.038* language –0.078 –0.075 –0.078 0.181*** sea 0.072* 0.076* 0.073** –0.054** constant –53.030*** –51.680*** –53.550*** –53.040*** –25.180*** observations 6533 6533 6533 6533 6533 number of state 504 504 504 504 r2 0.918 model 2 lngni_o 0.484*** 0.546*** 0.494*** 0.486*** 0.256*** lngni_d 1.845*** 1.797*** 1.838*** 1.844*** 1.735*** lnbtc_od –0.638*** –0.596*** –0.632*** –0.637*** –0.622*** lnrem_od 0.757*** 0.710*** 0.750*** 0.756*** 0.960*** lnrem_2_do –0.260*** –0.248*** –0.259*** –0.260*** –0.457*** oneemu 0.256*** 0.248*** 0.255*** 0.256*** 0.018 bothemu 0.292*** 0.275*** 0.290*** 0.292*** –0.031** border –0.005 0.008 –0.002 –0.066*** language –0.036 –0.029 –0.034 0.145*** sea 0.159*** 0.169*** 0.161*** 0.087*** constant –42.910*** –42.350*** –41.310*** –42.910*** –33.890*** observations 5441 5441 5441 5441 5441 number of state 420 420 420 420 420 r2 0.900 te yes yes yes yes yes ce yes no yes yes no note: te – time effects, ce – country effects (separately for exporter and importer); *** p<0.01, ** p<0.05, * p<0.1. table 4 does not show the fully expected results. mainly, the coefficient of importer’s remoteness variable remains negative in all cases, although, due to the anderson and van wincoop’s theory, it covers trade barriers between importing country and all its other potential trading partners, so it is expected to have a positive influence on bilateral import flows from the one considering importer’s partner. the construction of synthetic remoteness variable as weighted average of bilateral costs of trade with other partners is, however specific – not such strongly connected with relative prices as in the natalia drzewoszewska dynamic econometric models 14 (2014) 29–49 44 theoretical approach. in the case of the importer this remoteness could be interpreted more as the importer’s ability to import from other countries, which is not so opposite to the ability to bilateral import, seeing that trading goods are differentiated not only by their place of origin22 and the bilateral trade costs are not symmetric. according to table 4, none of border coefficients are positive, despite the ppml approach, which results in negative influence of sea access instead. model 2 with importer’s remoteness variable calculated with the formula (14), gives more similar estimates for the most of coefficients by using different estimation methods, including ppml. however the weakness of model 2 is a smaller number of state, caused by the importer’s remoteness synthetic formula, which dropped the observations with zero export values. due to calculation problems in stata software, the estimation of ppml model was possible only without the country effects, so the results remain biased, which can be the reason of the negative influence of bothemu and border dummy variables. the different estimates of national incomes (comparing with empirical models of the traditional gravity equation) are the result of synthetic variables formulas, they remain however significantly positive. table 5. the structural gravity model of trade flows for eu-25 countries in 1999– –2011 with time-varying country effects as an approximation of mtr terms (model 3) model 3 re fe ht lnbtc_od –1.000*** –1.000*** –1.000*** oneemu –0.030*** 0.177*** –0.260*** bothemu –0.030*** 0.177*** –0.260*** border –0.845*** –0.839*** language –0.843** –0.489 sea 1.008*** 1.070*** te yes yes yes ce (time-varying) yes yes yes constant 27.10*** 26.25*** 25.78*** observations 6533 6533 6533 number of state 504 504 504 r2 0.999 note: te – time effects, ce – country effects (separately for exporter and importer); *** p<0.01, ** p<0.05, * p<0.1. 22 anderson and van wincoop (2003) assume that each country specializes in the production of one good in the derivation to follow. as a matter of fact, in reality good specific trade resistance varies depending on the product class under consideration, what by estimation of model with aggregated data, like this used in the study, causes a large bias. see anderson and van wincoop (2004), anderson and yotow (2011). searching for the appropriate measure of multilateral trade-resistance terms… dynamic econometric models 14 (2014) 29–49 45 in order to check if the created remoteness synthetic variables can be a good approximation of multilateral trade-resistance, the estimates of the models should be in phase with the estimates of models including timevarying countries effects, which is the next step of study. the estimation results are presented below in table 5. the complexity of calculation (using stata software) of the model with time-varying countries effects (model 3) does allow only for the use of re, fe and ht estimators. the fe-model gives the estimates only for timevarying and non-specific country variables, however it seems to be the most accurate method since its extremely high coefficient of determination and the additional use of time-invariant pair effects, which absorb all timeinvariant determinants of bilateral trade costs, leading to relative small bias in the estimates. furthermore, as the only one estimator, fe results with the same coefficients’ signs in all considering cases. according to these results, bilateral trade costs synthetic variable has a negative influence on the bilateral and the emu-effects are positive. table 6. results of hausman test, sargan-hansen test of overidentifying restrictions and the test for time effects hausman test chi-square p-value te (time effects) ce (const) ce (timevarying)* result model 1 746.92 0.00 + – – fe model 2 475.79 0.00 + – – fe model 3 –16165.56 – + – + fe model 3 760.99 0.00 + – – no answer** test of overidentifying restrictions s-h statistic model 1 2308.25 0.00 + – – fe model 1 588.20 0.00 + + – model 2 5195.58 0.00 + – – fe model 2 4791.49 0.00 + + – model 3 2597.88 0.00 + + – fe model 3 2134.06 0.00 + – – fe test for time effects f statistic model 1 286.85 0.00 + – – te model 2 158.96 0.00 + – – te model 3 2.6e+13 0.00 + – + te note: * test of overidentifying restrictions (fixed vs random effects) for model with time effects (te) and time-varying country effects (ce) is not feasible due to permanent presence of collinearity; ** “no answer” occurs when the matrix was not positive definite. the results of hausman test (table 6), conducted for all three models, show that fe estimators is more preferred than re. however, including time-varying country effects results in negative chi-square statistic. due to natalia drzewoszewska dynamic econometric models 14 (2014) 29–49 46 the investigation of schreiber (2008), this result can happen only if h1 of the test is true – fe is consistent and preferred. moreover, the results of sarganhansen test of overidentifying restrictions confirm the choice of fe estimator. conclusions the purpose of this paper was to analyze the structural gravity model of trade flows with alternative approximations of multilateral trade-resistance terms. the empirical results of two synthetic variables – bilateral trade costs and exporter’s remoteness give significant and expected signs of coefficients. the sign of third created synthetic variable – importer’s remoteness – remains a problematic issue, since the estimates of importer’s remoteness do not respond to the theory of gravity model in any case. the theory of structural gravity equation assumes however symmetric trade barriers and lower differentiation of trade than is observed in the researching sample of eu countries, especially under conditions of globalization in the xxi century. based on the estimation results for statistically preferred fe-model only, it can be concluded that the proposed synthetic remoteness variables are good measures of mtr since including them in the model gives similar results as the model with time-varying country effects. however, it did not allow for unequivocal verification of the third hypothesis. all the results with alternative estimation methods provided grounds for the first research hypothesis verification, confirming the accuracy of using the bilateral trade costs synthetic variable, based on the travel time between country centroids and importer’s openness. the conducted analysis did not allow for verification of the second research hypothesis, according to which bilateral trade flows increase if exchange partners are members of eurozone. different signs of estimated dummies describing the membership in emu, especially in models including time-varying country effects, do not establish the accurate euro effect on the export flows between ue countries in the last 15 years. the specificity of researched sample and time period has definitely influence the deviation from the theoretical suspicions. among the problems still left open for consideration, the following should be mentioned: the extension of the research sample by other global-leading countries, the use of spatial effects and the use of synthetic trade costs and remoteness variables in the model with disaggregated data. searching for the appropriate measure of multilateral trade-resistance terms… dynamic econometric models 14 (2014) 29–49 47 references anderson, j. (1979), a theoretical foundation for the gravity equation, the american economic review, 69(1), 106–116. anderson, j. e., van wincoop, e. (2003), gravity with gravitas: a solution to the border puzzle, the american economic review, 93(1), 170–192, doi: http://dx.doi.org/10.1257/000282803321455214. anderson, j. e., van wincoop, e. (2004), trade costs, journal of economic literature, 42(3), 691–751, doi: http://dx.doi.org/10.1257/0022051042177649. anderson, j. (2011), the gravity model, the annual review of economics, 3(1), 133–160, doi: http://dx.doi.org/10.1146/annurev-economics-111809-125114. anderson, j. e., yotov, y. v. (2012), gold standard gravity, working paper 17835, nber. avinash, j. (2000), background to globalisation, bombay: center for education and documentation. baier, s. l., bergstrand, j. h. (2009), bonus vetus ols: a simple method for approximating international trade cost effects using the gravity equation, journal of international economics, 77(1), 77–85, doi: http://dx.doi.org/10.1016/j.jinteco.2008.10.004. baldwin, r., dinino, v. (2006), euros and zeros: the common currency effect on trade in new goods, hei working paper, 21/2006. baldwin, r. and taglioni, d. (2006), gravity for dummies and dummies for gravity equations, national bureau of economic research working paper, 12516, nber, cambridge . baltagi, b.h., egger, p., pfaffermayr, m. (2003), a generalized design for bilateral trade flow models, economics letters, 80, 391–397, doi: hhttp://dx.doi.org/10.1016/s0165-1765(03)00115-0. behar, a., nelson, b. d (2012), trade flows, multilateral resistance, and firm heterogeneity, imf working paper, wp/12/297. bergstrand, j. h. (1989), the generalized gravity equation, monopolistic competition and the factor-proportions theory in international trade, the review of economics and statistics, 71(1), 143–53, doi: http://dx.doi.org/10.2307/1928061. bikker, j.a., de vos, a. f. (1992), an international trade flow model with zero observations: an extension of the tobit model, brussels economic review, 135, 379-404. burda, m., wyplosz, ch. (2005), macroeconomics: a european text, fourth edition, oxford university press. bussière, m., schnatz, b. (2009), evaluating china’s integration in world trade with a gravity model based benchmark, open economies review, springer, 20(1), 85–111 daly, h.e. (1999), globalization versus internationalization: some implications, global policy forum. deardoff, a. v. (1998), determinants of bilateral trade: does gravity work in a neoclassical world?, in jeffrey a. frankel (ed.), the regionalization of the world economy, chicago, university press, 7–22. drzewoszewska, n., pietrzak, m. b., wilk, j. (2013), gravity model of trade flows between european union countries in the era of globalization, roczniki kolegium analiz ekonomicznych, 30, 187–202. eaton, j., kortum, s. (2002), technology, geography, and trade, econometrica, 70(5): 1741–79, doi: http://dx.doi.org/10.1111/1468-0262.00352. farhadi m, ismail r, fooladi m. (2012), information and communication technology use and economic growth, plos one 7(11), doi: http://dx.doi.org/10.1371/journal.pone.0048903. natalia drzewoszewska dynamic econometric models 14 (2014) 29–49 48 feenstra, r.c. (2002), border effects and the gravity equation: consistent methods for estimation, scottish journal of political economy, 49(5), 491–506, doi: http://dx.doi.org/10.1111/1467-9485.00244. fernández-avilés g., montero j.m., orlov a. (2012), spatial modeling of stock market comovements, finance research letters, 9(4), 202–212, doi: http://dx.doi.org/10.1016/j.frl.2012.05.002. fidrmuc, j., fidrmuc, j. (2001), disintegration and trade, zei working papers b, 24, zei – center for european integration studies, university of bonn. fidrmuc, j. (2009), gravity models in integrated panels, empirical economics, 37, 435–446, doi: http://dx.doi.org/10.1007/s00181-008-0239-5. fiedman, t. (1999), the lexus and the olive tree, new york: farrar, straus and giroux. fratianni, m., hoon oh, ch. (2007), on the relationship between rta expansion and openness, kelley school of business, doi: http://dx.doi.org/10.2139/ssrn.995298. garcía-muñiz, a. s., vicente, m.r. (2014), ict technologies in europe: a study of technological diffusion and economic growth under network theory, telecommunications policy, 38(4), 360–370, doi: http://dx.doi.org/10.1016/j.telpol.2013.12.003. glick, r., rose, a. k. (2002), does a currency union affect trade? the time-series evidence, european economic review, 46 (6), 1125–1151, doi: http://dx.doi.org/10.1016/s0014-2921(01)00202-1. gómez-herrera, e. (2013), comparing alternative methods to estimate gravity models of bilateral trade, empirical economics, 44 (3), 1087–1111. head, k., mayer, t. (2002), illusory border effects: distance mismeasurement inflates estimates of home bias in trade, cepii, working paper, 01. head, k. (2003), gravity for beginners, mimeo, university of british columbia. hsiao, c. (2003), analysis of panel data, second edition, cambridge university press. linders, g. m., de groot, h. l. (2006), estimation of the gravity equation in the presence of zero flows, tinbergen institute discussion paper, 072/3, doi: http://dx.doi.org/10.2139/ssrn.924160. maliszewska, m. a. (2004), new member states trading potential following emu accession: a gravity approach, studies and analyses, case – center for social and economic research, 286, doi: http://dx.doi.org/10.2139/ssrn.1441179. martin, w., pham, c.s. (2008), estimating the gravity model when zero trade flows are frequent, economics series deakin university, faculty of business and law, school of accounting, economics and finance. martínez-zarzoso, i., felicitas nowak-lehmann, d., vollmer, s. (2007), the log of gravity revisited, center for european, governance and economic development research, discussion papers 64. márquez-ramos, l., martínez-zarzoso, i., suárez-burguet, c.( 2007), the role of distance in gravity regressions: is there really a missing globalisation puzzle?, the b.e. journal of economic analysis & policy, 7(1), doi: http://dx.doi.org/10.2202/1935-1682.1557. micco, a., stein, e., ordonez, g. (2002), should the uk join emu?, washington: interamerican development bank. micco, a., stein, e., ordoñez, g. (2003), the currency union effect on trade: early evidence from emu, economic policy, 18(37), 315–356, doi: http://dx.doi.org/10.1111/1468-0327.00109_1. ramos, j., ballell,p. (2009), globalisation, new technologies (ict‘s) and dual labour markets: the case of europe, journal of information, communication and ethics in society, 7 (4), 258–279. searching for the appropriate measure of multilateral trade-resistance terms… dynamic econometric models 14 (2014) 29–49 49 rose, a., van wincoop, e. (2001), national money as a barrier to international trade: the real case for currency union, american economic review, 91(2), 386–90, doi: http://dx.doi.org/10.1257/aer.91.2.386. ruiz, j., vilarrubia, j. m. (2007), the wise use of dummies in gravity models: export potentials in the euromed region, banco de espana working papers 0720, doi: http://dx.doi.org/10.2139/ssrn.997992. santos silva, j., tenreyro, s. (2006), the log of gravity, the review of economics and statistics, 88(4), 641–658, doi: http://dx.doi.org/10.2139/ssrn.380442. schreiber, s. (2008), the hausman test statistic can be negative even asymptotically, journal of economics and statistics, 228(4), 394–405. soloaga, i., winters, a. (2001), regionalism in the nineties: what effect on trade?", north american journal of economics and finance, 12, 1–29, doi: http://dx.doi.org/10.1016/s1062-9408(01)00042-0. tinbergen, j. (1962), shaping the world economy: suggestions for an international economic policy, twentieth century fund, new-york. tripathi, s., leitão, n. c. (2013), india’s trade and gravity model: a static and dynamic panel data, mpra paper, 45502. wei, s. j. (1996), intra-national versus international trade: how stubborn are nations inglobal integration?, national bureau of economic research, working paper, 5531 westerlund, j., wilhelmsson, f. (2006), estimating the gravity model without gravity using panel data, nationalekonomiska institutionen department of economics, working paper, doi: http://dx.doi.org/10.1080/00036840802599784. problem właściwego pomiaru multilateralnego oporu wobec handlu w panelowym modelu grawitacji z a r y s t r e ś c i. celem artykułu jest porównanie różnych metod aproksymacji multilateralnego oporu wobec wymiany międzynarodowej w modelu grawitacyjnym. analizie poddany jest także ich wpływ na wyniki różnych metod estymacji modelu bilateralnych przepływów handlowych w unii europejskiej w latach 1999-2011. jako alternatywę dla zastosowania w modelu zmiennych w czasie indywidualnych efektów dla kraju importera i eksportera, proponowane są trzy zmienne syntetyczne opisujące bilateralne koszty handlu, opór eksportera oraz opór importera. tradycyjna miara odległości w modelu grawitacji, jaką jest dystans fizyczny, zastąpiony został czasem trwania podróży pomiędzy centroidami państw. wyniki estymacji wskazują na istotny statystycznie wpływ proponowanych zmiennych, jednakże znak oceny parametru oporu importera nie odpowiada założeniom teoretycznym modelu grawitacji. wpływ pozostałych zmiennych, w tym efekt strefy euro, jest w pełni zgodny z oczekiwaniami jedynie w przypadku zastosowania estymatora fe. s ł o w a k l u c z o w e: wymiana międzynarodowa, dane panelowe, model grawitacji, multilateralny opór wobec handlu, koszty handlu bilateralnego, globalizacja xxi wieku, strefa euro. microsoft word dem_2014_71to91.docx © 2014 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2014.004 vol. 14 (2014) 71−91 submitted october 20, 2014 issn accepted december 23, 2014 1234-3862 andrzej geise, mariola piłatowska* oil prices, production and inflation in the selected eu countries: threshold cointegration approach a b s t r a c t. this paper applies the threshold cointegration technique developed by enders and siklos (2001) to investigate the impact of an oil price changes on changes in production and inflation in the presence of structural break in seven european union countries. this technique will allow for a different speed of adjustment to the long-run equilibrium depending on whether production in selected economies is above or below the long-run relationship. given the presence of asymmetric cointegration between oil prices, production and inflation, we estimate threshold error correction models to examine longand short-run granger causality. we found evidence for cointegration with asymmetric adjustment in the case of france, denmark and the total eu. k e y w o r d s: asymmetric adjustment, oil price shocks, threshold cointegration, nonlinearity, threshold error correction model. j e l classification: c32, e23, e32, q43. introduction the relationship between oil prices and macroeconomy has drawn attention in many recent studies. oil shocks have important effects on economic activity and macroeconomic policy of many countries. barsky and kilian * correspondence to: andrzej geise, nicolaus copernicus university, faculty of economic sciences and management, 13a gagarina street, 87-100 toruń, poland, e-mail: a.geise@doktorant.umk.pl; mariola piłatowska, nicolaus copernicus university, faculty of economic sciences and management, 13a gagarina street, 87-100 toruń, poland, e-mail: mariola.pilatowska@umk.pl.  this work was financed from faculty of economic sciences and management research grant no. 1844-e. andrzej geise, mariola piłatowska dynamic econometric models 14 (2014) 71–91 72 (2004) shows that the economic slowdowns and increases of inflation rates can be leaded by huge and sudden increases in oil prices. also, brown and yucel (2002) and lardic and mignon (2006) presented explanations of the oil price shocks on economic activity. the rises in the crude oil prices are transferred on the energy prices through the price of petroleum products. the rise in the energy price causes an decrease in productivity, which is passed on to real wages, unemployment, inflation, investment or stock prices. therefore, it is important to investigate how the oil price shock are transmitted into economic activity, both in the short-run and long-run perspective. the main goal of this study is to examine the dynamics between brent oil prices and economic activity (described by the production and inflation) in the presence of structural break due to financial crisis. the error correction model with threshold cointegration is employed for the analysis of short-run relationship with non-linear long-term adjustment between production, inflation and oil prices. we assume two long-run relationships: first equation without taking into account a structural break and second equation including a structural break due to financial crisis in 2008. the analysis uses the nonlinear threshold cointegration elaborated by enders and granger (1998) and enders and siklos (2001). this technique will allow for a different speed of adjustment to the long-run equilibrium depending on whether production in selected economies is above or below the long-run relationship. at the end, an error correction model with threshold cointegration is utilized to analyze the short-run and long-run granger causality. in the analysis the monthly data of brent crude oil prices from 1995:01 to 2014:04, industry production index and consumer price index for the european union economy and six european countries: germany, france, denmark, nederland, poland and czech republic are used. the data were taken from the u.s. energy information administration database, the oecd database and eurostat database. these economies have been selected on the basis of three criteria: level of economic growth, structure of net imports of crude oil and gas, and structure of final energy consumption in industry. characteristics of countries according to selected criteria is presented in table 1. besides these six economies, the total eu was taken into account when investigating the relationship between oil prices, production and inflation. this will allow to compare the results for chosen six economies and total average for the eu. oil prices, production and inflation in the selected eu countries... dynamic econometric models 14 (2014) 71–91 73 table 1. characteristics of countries according to selected criteria country description germany (de) france (fr) two highly developed countries in the eu with the highest gdp in 2012, the highest net imports of crude oil and gas in 2012 and the highest final energy consumption in industry (germany 61.15 mtoe; france 29.58 mtoe). nederland (nl) denmark (dn) two highly developed countries in the eu economies with different structure of imports and exports of crude oil and/or gas. nederland is a gas net exporter and crude oil net importer. denmark is a net exporter of both, crude oil and gas. poland (pl) czech republic (cz) two developing countries in the eu with the highest economic growth in 2012 among central and eastern europe countries (cee). poland and czech republic are countries with the highest net imports of crude oil and gas in 2012 and the highest final energy consumption in industry among the cee countries. the plan of this paper is as follows. section 1 presents the empirical literature review. section 2 reviews the methodology applied in this paper. empirical results of threshold cointegration and granger-causality test based on threshold error correction model are presented in section 3 and section 4. the last section draws some conclusions. 1. a brief overview of the empirical studies many studies are available which offer different theoretical explanations for the relationship between oil price changes and the level of economic activity. hamilton (1983) using post-war data found a statistically significant relationship between oil price changes and gdp in us economy. gisser and godwin (1986) and burbridge and harrison (1984), among others, confirm the hamilton’s results. they identified the causal relationship between oil prices and economic variables, i.e. real gdp, general price level, rate of unemployment and real investment. gisser and goodwin (1986) indicated for the analyzed period from 1961 to 1982 that the oil prices had not lost its potential to predict gdp growth. they showed also, that oil shocks have an impact on production by other means than inflationary cost-push effects. they confirmed hamilton’s (1983) results about negative correlation between oil prices and real output in united states. burbidge and harrison (1984) found similar correlations for other industrialized countries. a number of authors have suggested that the relation between oil price shocks and economic variables is nonlinear. linear approximation to the andrzej geise, mariola piłatowska dynamic econometric models 14 (2014) 71–91 74 relation between oil price shock and macroeconomic activity may appear unstable over time. it is caused by shifts in the process generating oil prices (mork, 1989; hamilton, 1996; davis and haltiwanger, 2001). this means that the economic activity may respond asymmetrically to positive and negative oil price shocks, i.e. the increases of oil prices slow the economy more than the decreases of oil prices stimulate it (brown and yucel, 2002). the impact of oil price decreases is not always positive, indeed these decreases may slow output growth down. mork (1989) shows that the relationship between oil price changes and real gnp growth for the u.s. economy breaks down when the analysis is extended to include the oil price collapse of 1986. hence, mork (1989) decided to test the influence of declines and increases in oil price on the economic growth, he showed that the coefficient on oil price increases are highly significant and negative. in another work, mork et al. (1994) showed that for the most of european countries a negative relationship between oil price increases and gdp growth occurs. however, evidence for the causal relationship between oil prices and economic growth was not always confirmed in empirical studies. for example hooker (1996) found some evidence that oil prices are no longer granger cause of gdp. in direct response to this empirical work, hamilton (1996) showed new measure (net oil price increases-nopidifference between oil price level and the maximum price of the previous four quarters) which was able to detect a significant relationship between oil prices and real gdp. the nopimeasure from assumption is the measure of rapid and huge changes in oil prices. according to most empirical studies the rising oil prices lead to real gdp loses and it is consistent with the economic theory, though, it is possible to find the positive response of production to changes in oil prices. this kind of findings, however, should not be treated as inconsistent with economic theory since, as bernanke et. al (1997) showed on the example of the u.s. economy, the response of production to an oil price shock may be different depending on the response of monetary policy to oil prices shocks (i.e. whether the federal funds rate is constrained to be constant than in the case in which monetary policy is unconstrained). if the federal funds rate is held constant, a positive oil price shock leads to an increase in real gdp. in the unconstrained case, a positive oil prices shock leads to a decline in real gdp. similar findings concern the response of gdp to changes in inflation, i.e. although the economic theory indicates that movements in inflation cause a decrease in production, there are some empirical findings indicating the opposite direction of relationship. the reason why increases in inflation have positive influence on economic growth can be explained in the framework oil prices, production and inflation in the selected eu countries... dynamic econometric models 14 (2014) 71–91 75 of the optimal rate of inflation (friedman, 1976; belka, 1986; barro, 1995). if the inflation is below the optimal level, then the reaction of production is positive (slight increase to the optimal level of inflation stimulates the economic growth). while, the impact of inflation on production is negative (an increase in inflation causes a decrease in production) when consumer price level is higher than the optimal one. most of empirical literature on relationships between oil prices and economic activity focuses on the us economy. europe, despite of economic size and huge oil import, has received relatively less attention. only few studies focus on this issue, e.g. scholtens and yurtsever (2012) and arouri (2011) investigated how european industries responded to oil price shock. they found that the impact of oil price shocks substantially differed along the different industries. they also found that the significance of the result differs along the various oil price specifications. to studies considering this relationship for some european countries belong the papers by jimenezrodriguez and sanchez (2005) and papapetrou (2001). the former confirmed positive effects of oil price decreases on output for japan, germany, france, canada, norway and the united kingdom. and the latter found causal relationship from oil prices to industrial production, employment and share prices in greek economy. to our knowledge, there is no such study which considers the relationship between oil prices changes and economic activity using threshold cointegration methodology with non-linear adjustment for the countries studied in this paper. 2. methodology to test the relationship between brent oil prices and economic activity (described by the production and inflation) in the asymmetric framework the threshold cointegration technique developed by enders and granger (1998) and enders and siklos (2001) is used. similarly to engle-granger (1987) procedure, this is indeed a residual-based two-stage estimation. in the first stage the long-run relationship between production, inflation and oil prices is considered, e.g. in the following form: ,210 tttt inbp   (1) where tp − stands for production, tb − for oil prices and tin − for inflation, t − disturbance term. the second stage focuses on the coefficient estimates of 1 and 2 in the following regression: andrzej geise, mariola piłatowska dynamic econometric models 14 (2014) 71–91 76 ,)1( 11211 t r i itittttt ii      (2) where t is a white noise disturbance. term it is the heaviside indicator function ti such that 1ti if  1t and 0ti if ,1  t where  is the threshold value. then, equation (2) is a threshold autoregressive (tar) model of the disequilibrium error. if the heaviside indicator depends not on levels but on changes of t−1, then it is specified as 1ti if   1t and 0ti if .1   t this model is termed the momentum threshold autoregressive (m-tar) model. the tar framework is designed to capture potential asymmetric movements in residuals t when they are above or below the long-run equilibrium, while the m-tar framework − the direction of these movements, in other words their momentum (fosten et. al, 2012). the necessary and sufficient condition for t to be stationary is: ,01  ,02  1)1)(1( 21   for any threshold value  (petruccelli, woolford, 1984). if these conditions are satisfied and threshold value  is set to zero (as occurs in many economic applications), 0t can be considered as the long-run equilibrium value of the sequence. if t is higher than the longrun equilibrium, the adjustment is ,11 t but if t is lower than the longrun equilibrium, the adjustment is .12 t in general, the threshold value  has to be estimated along with the values of adjustment parameters 1 and .2 in our studies we follow enders and siklos (2001) and yau and nieh (2009) by employing chan's (1993) methodology of searching the consistent estimates of threshold value. testing for threshold cointegration is performed in two steps. first, the null hypothesis of no cointegration 0: 210  h is tested, and when it is rejected, then the null hypothesis of symmetric adjustment, ,: 210  h is verified. to test the first hypothesis enders and siklos (2001) proposed the  statistics which under the null of no cointegration has a non standard distribution. the critical values for this non standard  statistics are tabulated in their paper. the second null hypothesis of asymmetric adjustment is tested using the standard f statistics. rejecting both the null hypotheses implies the existence of threshold cointegration with asymmetric adjustment. given the threshold cointegration is found, the next step proceeds with the granger-causality test using threshold (or momentum-threshold) error oil prices, production and inflation in the selected eu countries... dynamic econometric models 14 (2014) 71–91 77 correction model (tar-ecm or m-tar-ecm) (enders, granger, 1998; enders, siklos, 2001). this model is expressed as the following: t q i itji q i itji q i itjitjtjjt vbinpzzy           321 1111211  (3) where ),,,( tttjt binpy  ,3,2,1j 11     ttt iz  and ,)1( 11     ttt iz  ti − heaviside indicator function, tv is a white noise disturbance. the longrun causality is determined by the parameters 1 and .2 the short-run causality is governed by the parameters ,i i and i and may come either from its own history of lagged dynamics or from the lagged effects of changes in real gdp (and square real gdp) and/or some additional explanatory variables (e.g. energy consumption). it is also desirable to check whether this two sources of causation are jointly significant. 3. threshold cointegration – empirical analysis in this section, an integration and threshold cointegration analysis between brent oil prices, production and inflation in german, france, nederland, denmark, poland, czech republic and european union economies are discussed. the data consist of seasonally adjusted monthly industrial production index at constant prices of 2010, consumer price index for the eu countries (corresponding month of previous year = 100) and brent crude oil prices from january 1995 to april 2014. for seasonal adjustment the tramo/seats procedure was used. each time series has been transformed using the natural logarithm. then empirical threshold error correction models for oil prices, inflation and production are constructed to study short and long-run causality. to assess the time series properties of the data, we examine the order of integration for all variables involved. we run the conventional unit root test, the augmented dickey-fuller test. findings from the adf test, as shown in table 2, reveal that each series is integrated of first order, i (1), at least at the 5% significance level. since the studied time span includes the period 2008-2009 when the financial crisis occurred1 and the presence of a structural break may change the nature of the long-run relationship between production, inflation and oil prices, from the very start we assume both the log-run relationship without 1 visual inspections of production suggests that the shift in mean occurred about september 2008. andrzej geise, mariola piłatowska dynamic econometric models 14 (2014) 71–91 78 taking into account a structural break (eq. (4)) and long-run relationship in the presence of structural break (eq. (5)). the first model takes the following form: tttt utimeinbp  3210  , (4) where tp , ti denote respectively the industrial production index and consumer price index in selected economies, tb stands for brent crude oil prices; all variables are in natural logarithms; and the second model has the form: ttttt dttimeinbp   )( * 43210 , (5) where )(* tdt is the dummy variable for the break in constant term of production from the chosen exogenously breakpoint  so 1)(* tdt if 2008:09t  and 0 otherwise. table 2. results of the adf test country variable adf levels first differences c only c and t c only – brent oil price –1.04 [1] –3.23 [1]* –9.51 [1]*** germany (de) production –0.96 [1] –2.18 [1] –9.89 [1]*** inflation –3.03 [1]** –3.07 [1] –10.51 [1]*** france (fr) production –1.14 [1] –1.58 [1] –11.12 [1]*** inflation –3.02 [1]** –3.00 [1] –10.01 [1]*** nederland (nl) production –2.18 [1] –2.23 [1] –15.69 [1]*** inflation –2.57 [1]* –2.63 [1] –10.54 [1]*** denmark (dn) production –2.51 [1] –2.43 [1] –13.34 [1]*** inflation –2.82 [1]* –2.95 [1] –9.95 [1]*** poland (pl) production –0.80 [1] –2.19 [1] –11.52 [1]*** inflation –3.13 [1]** –3.15 [1]* –7.71 [1]*** czech republic (cz) production –0.76 [1] –2.02 [1] –13.04 [1]*** inflation –2.10 [1] –2.50 [1] –7.71 [1]*** european union (eu) production –1.12 [1] –1.69 [1] –6.09 [1]*** inflation –2.97 [1]** –3.05 [1] –8.61 [1]*** note: asymptotical critical values for the adfmax test statistics: –3.46 (1%); –2.88 (5%); –2.57 (10%) (adf regression with drift – c only); asymptotical critical values for the adfmax test statistics: –3.99 (1%); –3.43 (5%); –3.13 (10%) (adf regression with an intercept and trend – c and t); ***, ** and * denote significant at 1%, 5% and 10% levels, respectively. oil prices, production and inflation in the selected eu countries... dynamic econometric models 14 (2014) 71–91 79 table 3. estimated parameters of long-run equilibrium for production country estimated parameters of long-run equation adf const. bt int time dtt(λ) levels (c+t) long-run relationship without structural break (4) germany (de) –17.02 (–7.63)*** 0.020 (1.65)* 4.616 (9.48)*** 0.001 (10.27)*** – –3.29* france (fr) –10.839 (–4.21)*** –0.022 (–1.38) 3.373 (6.00)*** 0.0001 (19.8)*** – –2.05 nederland (nl) 0.936 (0.74) 0.053 (6.49)*** 0.718 (2.62)*** 0.001 (5.99)*** – –4.64*** denmark (dn) –1.965 (–0.54) 0.003 (0.14) 1.428 (1.81)* 0.0001 (0.62) – –2.48 poland (pl) 5.083 (15.31)*** 0.110 (9.20)*** –0.351 (–4.85)*** 0.003 (21.92)*** – –2.78 czech republic (cz) 1.605 (2.12)** 0.132 (8.87)*** 0.466 (2.84)*** 0.002 (8.47)*** – –2.81 european union (eu) 9.645 (6.7)*** 0.132 (10.98)*** –1.172 (–3.74)*** –0.001 (–7.12)*** – –3.73** long-run relationship with structural break (5) germany (de) –11.67 (–5.97)*** 0.017 (1.67)* 3.455 (8.11)*** 0.002 (15.11)*** –0.077 (–9.78)*** –3.65 *** france (fr) 0.509 (0.40) –0.031 (–4.13)*** 0.909 (3.27)*** 0.001 (11.65)*** –0.017 (–28.3)*** –3.78*** nederland (nl) 1.387 (1.29) 0.036 (4.97)*** 0.626 (2.67)*** 0.001 (11.33)*** –0.063 (–9.61)*** –5.94*** denmark (dn) –6.748 (–3.97)*** –0.069 (–6.92)*** 2.484 (6.73)*** 0.002 (17.12)*** –0.245 (–28.67) –6.82*** poland (pl) 3.287 (6.05)*** 0.064 (3.93)*** 0.047 (0.4) 0.005 (14.49)*** –0.07 (–4.1)*** –2.73 czech republic (cz) –1.012 (–1.84)* 0.074 (6.75)*** 1.046 (8.74)*** 0.003 (19.32)*** –0.154 (–15.67)*** –4.48*** european union (eu) 1.771 (1.95)* 0.025 (2.91)*** 0.576 (2.90)*** 0.001 (8.14)*** –0.142 (–20.47)*** –4.37*** note: ***, ** and * denote significance at 1%, 5% and 10% level respectively; in parentheses the tstatistics are given. table 3 presents the estimation results of eq. (4) and eq. (5). it is worth noting that for the long-run equation without a structural break only in the case of france the response of production to the change in oil prices is negative ( 01  ). having included the dummy variable for a structural break in eq. (5), except france, only in the case of denmark the negative response of production is additionally observed and for the rest of countries (germany, nederland, poland, czech republic and european union) the positive response of production to changes in oil prices is found. the response of gdp to changes in inflation is positive too. positive impact of oil prices on proandrzej geise, mariola piłatowska dynamic econometric models 14 (2014) 71–91 80 duction can be explained by the monetary policy carried out by the european central bank (ecb). while the ecb reacting to the financial crisis in 2008 increased the interest rates and then in recession phase reduced them, the interest rates before the financial crisis were held constant. this may explain why the response of production to oil price shocks (table 3) is positive as suggested by bernanke et. al (1997) − see comments in section 1. but the positive response of gdp to changes in inflation can be explained by low inflation rate, near to optimal one (the general price level was comparatively stable, except 2008 when inflation increased near to 4 per cent (increased by 4 percentage points in euro zone) – see comments in section 1. while in the case of long-run equilibrium relationship without a structural break the cointegration is found for germany, nederland and the eu, in the case of long-run relationship allowing for a structural break the cointegration is found for germany, france, netherlands, denmark, czech republic and the european union. however, the standard cointegration framework (engle-granger approach) assuming symmetric adjustment toward equilibrium is misspecified if the adjustment is asymmetric. therefore, to test for cointegration with non-linear (asymmetric) adjustment, we use the threshold cointegration approach proposed by enders and siklos (2001). in further analysis we use only the results based on residuals from long-run relationship with a structural break in 2008. table 4 contains cointegration test results based on the long-run equilibrium relationships for production in selected countries in the form of equation (5), when considering threshold and momentum adjustment. we consider threshold autoregressive (tar) model and momentum threshold autoregressive (m-tar) model with zero-value of threshold,  = 0, and also with non-zero value of threshold   0 which should be estimated. the table reports values of the adjustment coefficients 1 and 2 , the  statistics for the null hypothesis of no cointegration against the alternative of cointegration with asymmetric adjustment, the f statistics for testing the symmetric adjustment, aic and ljung-box statistics. the results for the threshold cointegration tests present some interesting relations among oil prices, production and inflation in selected economies. oil prices, production and inflation in the selected eu countries... dynamic econometric models 14 (2014) 71–91 81 table 4. results of tar and m-tar test for threshold cointegration on the longrun equation with one structural break in production threshold cointegration test  = 0   0 tar m-tar tar m-tar germany (de)  0 0 –0.016 –0.013 1 –0.123 (2.23)** –0.037 (0.69) –0.089 (1.65)* –0.100 (2.30)** 2 –0.161 (3.06)*** –0.243 (4.64)*** –0.196 (3.68)*** –0.286 (3.66)***  6.882** 10.865** 7.859** 9.057** f(1–2=0) 0.266 [0.61] 7.875 [0.01]** 2.109 [0.15] 4.447 [0.04]** lag 1 1 1 1 aic –1207.7 –1215.2 –1209.5 –1211.9 lb(4) 3.87 [0.42] 4.88 [0.30] 4.13 [0.39] 4.40 [0.36] france (fr)  0 0 –0.022 –0.008 1 –0.114 (2.01)** –0.067 (1.07) –0.118 (2.19)** –0.096 (1.91)* 2 –0.197 (3.05)*** –0.219 (3.76)*** –0.207 (2.94)*** –0.290 (3.72)***  6.208** 7.423** 6.247* 8.17** f(1–2=0) 1.048 [0.31] 3.41 [0.07]* 1.121 [0.29] 4.834 [0.03]** lag 2 2 2 2 aic –1281.3 –1283.7 –1281.4 –1285.1 lb(4) 1.47 [0.83] 1.75 [0.78] 1.44 [0.84] 1.26 [0.87] netherlands (nl)  0 0 0.011 0.022 1 –0.337 (3.93)*** –0.278 (3.04)*** –0.334 (3.87)*** –0.373 (2.69)** 2 –0.318 (3.54)*** –0.371 (4.36)*** –0.322 (3.61)*** –0.321 (4.52)***  11.404** 11.715** 11.395** 12.191** f(1–2=0) 0.031 [0.86] 0.688 [0.41] 0.013 [0.91] 1.551 [0.21] lag 2 2 2 2 aic –1090.3 –1091 –1090.3 –1090.5 lb(4) 0.80 [0.94] 0.77 [0.94] 0.81 [0.94] 0.84 [0.93] denmark (dn)  0 0 0.031 –0.020 1 –0.38 (4.69)*** –0.47 (4.73)*** –0.386 (4.48)*** –0.357 (5.14)*** 2 –0.458 (5.56)*** –0.394 (5.53)*** –0.443 (5.52)*** –0.572 (5.52)***  23.671** 23.517** 23.524** 25.26** f(1-2=0) 0.517 [0.47] 0.431 [0.51] 0.272 [0.60] 3.323 [0.07]* lag 1 1 1 1 aic –975 –974.9 –974.8 –977.8 lb(4) 1.96 [0.74] 2.54 [0.64] 2.05 [0.73] 4.50 [0.83] andrzej geise, mariola piłatowska dynamic econometric models 14 (2014) 71–91 82 table 4 (continued) poland (pl)  0 0 -0.012 0.015 1 –0.126 (2.67)*** –0.136 (2.84)*** –0.134 (2.85)*** –0.124 (1.67)* 2 –0.061 (1.26) –0.053 (1.13) –0.052 (1.08) –0.086 (2.27)**  4.256 4.546 4.546 3.846 f(1–2=0) 0.972 [0.33] 1.572 [0.21] 1.533 [0.22] 0.212 [0.65] lag 1 1 1 1 aic –1136.2 –1136.8 –1136.8 –1135.4 lb(4) 0.05 [0.99] 0.08 [0.99] 0.05 [0.99] 0.05 [0.99] czech republic (cz)  0 0 0.03 –0.017 1 –0.184 (2.79)*** –0.24 (3.48)*** –0.203 (2.87)*** –0.192 (3.65)*** 2 –0.192 (2.95)*** –0.148 (2.37)** –0.179 (2.88)*** –0.173 (1.68)*  7.437** 8.092** 7.408** 7.353** f(1–2=0) 0.005 [0.94] 1.296 [0.26] 0.074 [0.79] 0.032 [0.86] lag 2 2 2 2 aic –1026.2 –1010.1 –1026.3 –1026.2 lb(4) 0.11 [0.99] 0.07 [0.99] 0.10 [0.99] 0.11 [0.99] european union (eu)  0 0 0.003 –0.006 1 –0.155 (2.84)*** –0.061 (1.07) –0.153 (2.79)*** –0.082 (1.75)* 2 –0.200 (3.44)*** –0.274 (5.21)*** –0.203 (3.48)*** –0.368 (5.61)***  9.901** 14.117** 9.942** 17.037** f(1–2=0) 0.333 [0.56] 8.126 [0.004]*** 0.407 [0.52] 13.465 [<0.01]*** lag 1 1 1 1 aic –1238 –1245.6 –1238.1 –1250.7 lb(4) 2.76 [0.60] 2.59 [0.63] 2.47 [0.65] 3.30 [0.51] note: ***, ** and * denote significance at 1%, 5% and 10% level respectively; in parentheses the tstatistics are given; in brackets the p-values are given. it can be seen that both the null hypotheses of 021   and 021   (the null of no cointegration and the null of symmetric adjustment respectively) are rejected for germany, france, denmark and european union. this finding implies the existence of threshold cointegration with asymmetric adjustment for these countries and the eu what means that the adjustment back to equilibrium between production, inflation and oil prices is non-linear. to select the best adjustment mechanism (tar or m-tar) the akaike information criterion (aic) is used (the minimum aic is reported in bold in table 4). in case of germany the m-tar model with zero-value of threshold is selected, and in case of france, denmark and the eu − the mtar model with non-zero threshold value. selection of m-tar model indicates that the direction in which production is moving (its momentum) matoil prices, production and inflation in the selected eu countries... dynamic econometric models 14 (2014) 71–91 83 ters more than whether production is above or below the equilibrium (as in tar model). it is worth noting that all adjustment coefficients 1( and )2 have negative signs acting to eliminate deviations from the long-run relationship and are significant for either or both regimes. the point estimates 1 and 2 in the case of above mentioned countries indicate that deviations below the threshold adjust faster toward the long-run equilibrium than the deviations above the threshold (since ).21   in other words, deviations below the threshold from the long-run equilibrium resulting from decreases in production or increases in inflation and oil prices are corrected (eliminated) more quickly than deviations above the threshold). these models differ with regard to the magnitude of  terms in regimes (below and above the threshold value. the largest discrepancy between the elimination of below and above threshold deviations occurs for germany, france and the eu, e.g. for the eu the deviations below the threshold are eliminated at 36.8% rate per month, while deviations above the threshold are eliminated only at a rate of 8.2%. although for nederland and czech republic the null hypothesis of no cointegration is rejected, no support is found for cointegration with asymmetric adjustment (since the null of symmetric adjustment is not rejected, see f statistics in table 4). the results for polish economy show that both the null hypothesis of no cointegration and symmetric adjustment are not rejected, suggesting oil prices, production and inflation in poland are not cointegrated. these results showed that the threshold cointegration occurs only for the developed countries (i.e. germany, france, denmark) those contribution to the total production of the eu is the highest. this is the reason why for the total eu the threshold cointegration has been also confirmed. since the speed of adjustment is faster in the lower regime (deviations in production below the threshold), these countries and the eu as a whole have the mechanism that enable them to quickly revert to equilibrium after moving away from it or in other words, economies remain shorter period of time in the regime of slower economic activity. for poland and czech republic belonging to the developing countries with lower level of gdp than averagely for the total eu or developed countries no evidence in favor of the threshold cointegration with asymmetric adjustment has been found. andrzej geise, mariola piłatowska dynamic econometric models 14 (2014) 71–91 84 4. granger causality analysis form threshold error correction model given the threshold cointegration results found in the previous section, the next step proceeds with granger causality test using the momentum threshold error correction model (m-tar-ecm). in our case, the m-tarecm models take the following forms: ttitit i ittt binpzzp          11111 3 1 1112111 , (6) ttitit i ittt binpzzin          12121 3 1 2122121 , (7) ttitit i ittt binpzzb          13131 3 1 3132131 , (8) where tp , tin denote log first differences of production and inflation in germany (de), france (fr), denmark (dn) and european union (eu) respectively, tb stands for the log differences of brent crude oil prices. here, 11     ttt iz  and ,)1( 11     ttt iz  given 1ti if ,1   t and 0ti if .1   t furthermore, t is the residual from long-run equation (5) for a given country and ttt v,, are the white noise disturbance. this approach allows us to distinguish between short-run and long-run granger causality. since granger causality tests are very sensitive to the selection of lag length, we apply the aic criterion to determinate the appropriate lag lengths. it is found that the lag length of production equals 3 and for both inflation and brent oil prices is equal to one for all the economies. the wald f-statistics for granger causality is employed to examine whether all the coefficients of a given first differenced variable are jointly statistically different from zero (short-run causality) and t-statistics for the 21 , coefficients of error correction term to test their significance (long-run causality). moreover, the jointly significance of the  coefficients and all the coefficients of a given explanatory variable is tested in order to indicate which variables bear the burden of short-run adjustment to restore the longrun equilibrium given a shock to the system. tables 5 and 6 present the results of the granger-causality test based on m-tar-ecm models for oil prices, inflation and production. oil prices, production and inflation in the selected eu countries... dynamic econometric models 14 (2014) 71–91 85 table 5. estimates of threshold ecm models and granger-causality analysis (case of germany and france) germany (de) m-tar-ecm ( = 0) france (fr) m-tar-ecm ( = –0.008) bt int pt bt int pt 1 z+t-1 0.027 (0.099) 0.0257 (2.979)*** –0.158 (–3.549)*** –0.242 (–0.793) 0.0042 (0.462) –0.052 (–1.269) 2 z-t-1 –0.2363 (–0.863) 0.0051 (0.596) –0.101 (–2.325)*** –0.651 (–0.443) 0.005 (0.38) –0.244 (–4.104)*** h0: 1=2=0 0.3821 [0.683] 4.544 [0.012]** 8.6267 [<0.01]*** 1.398 [0.249] 0.677 [0.509] 9.246 [<0.01]*** h0: 1=2 0.479 [0.49] 3.039 [0.08]* 0.853 [0.357] 0.573 [0.45] 0.0023 [0.96] 7.028 [<0.01]*** h0: 1=2=3=0 0.931 [0.336] 0.0275 [0.869] 2.973 [0.086]* 0.0303 [0.862] h0: 1=2=3=1=0 0.5061 [0.604] 5.383 [<0.01]*** 1.615 [0.201] 0.673 [0.511] h0: 1=2=3=2=0 0.6792 [0.508] 0.276 [0.759] 2.588 [0.077]* 0.123 [0.884] h0: 1=0 0.0026 [0.96] 3.286 [0.071]* 0.025 [0.874] 0.006 [0.938] h0: 1=1=0 0.0056 [0.99] 9.066 [<0.01]*** 0.3187 [0.728] 0.806 [0.448] h0: 1=2=0 0.378 [0.69] 4.583 [0.011]** 1.115 [0.329] 8.496 [<0.01]*** h0: 1=0 17.378 [<0.01]*** 4.589 [0.033]** 8.35 [<0.01]*** 7.017 [<0.01]*** h0: 1=1=0 12.062 [<0.01]*** 9.393 [<0.01]*** 4.19 [0.016]** 4.697 [0.011]** h0: 1=2=0 8.697 [<0.01]*** 5.609 [<0.01]*** 4.189 [0.018]** 12.808 [<0.01]*** lb(4) 0.91 [0.92] 1.84 [0.77] 3.31 [0.51] 0.85 [0.93] 5.28 [0.26] 1.22 [0.75] note: *** (**) (*) indicate significance at 1% (5%) (10%) level. t-statistics for 1, 2 in parentheses. p-values for wald statistics in brackets. lb(4) for ljung-box statistics along with p-values. it is clear (table 5 and 6) that the point estimates of 1 and 2 in all mtar-ecm models for tp have a negative sign but only for germany and denmark both are significant (for france and the eu only the coefficient 2 is significant). in the case of france, denmark and the eu, similarly to the findings in table 4, the production deviations below the threshold adjust faster toward the long-run relationship than the deviations above the thresh andrzej geise, mariola piłatowska dynamic econometric models 14 (2014) 71–91 86 table 6. estimates of threshold ecm models and granger-causality analysis (case of denmark and the total eu) denmark (dn) m-tar-ecm ( = –0.020) european union (eu) m-tar-ecm ( = –0.006) bt int pt bt int pt 1 z+t-1 –0.79 (–3.58)*** –0.013 (–2.044)** –0.23 (–3.466)*** 0.481 (1.251) 0.016 (1.645) –0.044 (–1.388) 2 z-t-1 –0.167 (–0.514) –0.016 (–1.707)* –0.513 (–5.242)*** –0.992 (–1.943)* –0.0089 (–0.686) –0.2701 (–6.48)*** h0: 1=2=0 6.4086 [<0.01]*** 3.073 [0.048]** 17.318 [<0.01]*** 2.594 [0.077]* 1.553 [0.214] 22.277 [<0.01]*** h0: 1=2 2.9376 [0.088]* 0.0823 [0.774] 6.708 [0.011]** 5.781 [0.017]** 2.574 [0.11] 16.528 [<0.01]*** h0: 1=2=3=0 1.835 [0.177] 5.426 [0.021]** 0.893 [0.346] 0.571 [0.451] h0: 1=2=3=1=0 5.804 [0.003]*** 3.097 [0.047]** 2.006 [0.137] 1.904 [0.152] h0: 1=2=3=2=0 0.9423 [0.391] 3.095 [0.047]** 2.564 [0.079]* 0.704 [0.496] h0: 1=0 0.171 [0.697] 2.228 [0.137] 0.172 [0.679] 0.1897 [0.664] h0: 1=1=0 7.299 [<0.01]*** 6.226 [<0.01]*** 0.872 [0.42] 1.054 [0.351] h0: 1=2=0 0.248 [0.781] 14.104 [<0.01]*** 1.996 [0.139] 21.175 [<0.01]*** h0: 1=0 6.127 [0.014]** 1.512 (0.22) 5.732 [0.018]** 5.435 [0.021]** h0: 1=1=0 5.523 [<0.01]*** 6.504 [<0.01]*** 4.06 [0.019]** 3.825 [0.023]** h0: 1=2=0 4.98 [<0.01]*** 14.001 [<0.01]*** 3.534 [0.031]** 27.508 [<0.01]*** lb(4) 2.38 [0.67] 1.85 [0.76] 0.54 [0.97] 0.66 [0.96] 2.68 [0.61] 4.36 [0.36] note: see table 5. old (since ).21   we can see that respectively 24.4%, 51.3% and 27.01% of the production deviations from equilibrium is corrected in the next period when they are below the threshold, but only 5.2%, 23% and 4.4% when they are above the threshold. in other words, deviations below the threshold from the long-run equilibrium resulting from decreases in production or increases in inflation and oil prices are corrected more quickly than deviations above the threshold. however, for germany, unlike the previous results in table 4, the speed of adjustment is faster in the regime with deviations in production above the threshold but on the other hand both oil prices, production and inflation in the selected eu countries... dynamic econometric models 14 (2014) 71–91 87 coefficients are similar in magnitude (the hypothesis of symmetric adjustment is not rejected, f=0.853 with p-value=0.357). this rather symmetric adjustment of production may indicate that germany in general is not negatively influenced by higher energy prices. although it is one of the important oil importing countries, at the same time it is the leading goods exporting economy with a strong investment goods industry and is currently becoming the leader for some renewable energies and energy efficiency goods. this allows germany to level off the negative impact of increasing oil prices − see lutz, meyer (2009). the m-tar-ecm results for denmark show that the deviation away from the long-run relationship is corrected not only by movements in production, but also by movements in inflation and oil prices when the production in denmark is temporarily above the long-run equilibrium. this can be seen by the negative signs and significance of the error correction parameter 1 in the m-tar-ecms for tp , tin and tb . when the production in denmark is temporarily below the long-run equilibrium, the deviation away from long-run relation is corrected by movements in production and inflation (significance of 2 in equation for tp and tin ). the m-tar-ecm models for tin and tb in germany, france and eu adjust to the ‘wrong’ direction (the  coefficients have positive signs) in either or both regimes, and additionally parameters 1 and 2 are not always significant – see table 5. summing up, the results from threshold error correction model confirmed the previous findings (table 4) with regard to cointegration with asymmetric adjustment for deviations in production in the case of france, denmark and the total eu. this can be seen by statistical significance of both or either adjustment coefficients () which are properly signed (are negative). while in the case of germany the cointegration between production, oil prices and inflation occurs, production adjustments toward equilibrium seem to be symmetric. the results of granger-causality test, based on the m-tar-ecm models for all economies show that there is short-run unidirectional causal relationship running from oil prices )( tb to production )( tp and inflation )( tin (rejection of 0: 10 h in equations for tp and tin with exception of denmark for which unidirectional short-run causal relationship running from )( tb to )( tp was not confirmed). bidirectional short-run causal relationship between oil prices and production exists for france (the rejection of 0: 10 h in equation for tp and 0: 3210  h in equation for andrzej geise, mariola piłatowska dynamic econometric models 14 (2014) 71–91 88 ,tb but the latter only at the 10% significance level). in terms of long-run situation, a unidirectional strong causal relationship running from oil prices tb to production tp is found for both regimes in the case of germany and denmark, and in lower regime (below the threshold) in the case of france and the eu (rejection of 0: 110  h and/or 0: 210  h ). this means that oil prices, among explanatory variables, contributes most to the short-run adjustment to re-establish the long-run equilibrium in the case of all analyzed economies. conclusions this paper investigated the relationship between oil prices, production and inflation in an asymmetric framework in selected european union economies we tested for the threshold cointegration assuming two types of longrun relationship: firstly, the long-run equation without a structural break and then the long-run relationship taking into account a structural break in 2008 due to financial crisis. while the in the case of long-run equilibrium relationship without a structural break the cointegration was not found (except germany, nederland and the total eu), we proceeded with testing threshold cointegration based only on the long-run relationship with a structural break. the results provided evidence that production, oil prices and inflation are cointegrated with non-linear (asymmetric) adjustment process in the case of germany, denmark, france and the eu, i.e. for developed countries with the high level of economic growth. adjustments toward the long-run relationship revert faster when production is below the threshold value and tend to persist more when production is above the threshold. this indicate that all these economies have the mechanism allowing them to quickly revert to equilibrium after moving away from it or in other words, economies remain shorter period of time in the regime of slower economic activity. however, when estimating the threshold error correction models, the findings of threshold cointegration with asymmetric adjustment was fully confirmed in the case of france, denmark and the total eu. the estimated threshold error correction models and granger causality tests provide evidence that there is unidirectional short run causality running form oil prices to production and inflation in the case of germany, denmark, france and the eu. besides, the support for unidirectional long-run causal relationship from oil prices to production is found for germany, denmark (in both regimes) and for france and the eu. this evidence suggests that oil prices play an important role in driving economic growth. oil prices, production and inflation in the selected eu countries... dynamic econometric models 14 (2014) 71–91 89 in this paper we focused on testing causality using threshold error correction model. in further studies we will concentrate on investigating the relationship between production, oil prices and inflation in the framework of threshold vector error correction model. references albiński, p. (2014), kryzys a polityka stabilizacyjna w unii europejskiej (crisis and stabilization policy in european union), oficyna wydawnicza sgh, warszawa. arouri, m. (2011), does crude oil move stock markets in europe? a sector investigation, economic modelling, 28, 1716–1725, doi: http://dx.doi.org/10.1016/j.econmod.2011.02.039. barro, r. (1995), inflation and economic growth, nber working paper, no. 5326, www.nber.org (15.01.2015). barsky, r. b., kilian, l. (2004), oil and macroeconomy since the 1970s., journal of economic perspectives, 18 (4), 115–134, doi: http://dx.doi.org/10.1257/0895330042632708. belka, m. (1986), doktryna społeczno-ekonomiczna miltona freadmana (milton’s friedman socio-economic doctrine), pwn, warszawa. bernanke, b. s., gertler, m., watson, m. (1997), systematic monetary policy and the effects of oil price shocks, brooking papers on economic activity, vol. 1997, no. 1, 91–157, doi: http://dx.doi.org/10.2307/2534702. brown, s. p. a., yucel, m. k. (2002) energy prices and aggregate economic activity and interpretative survey, the quarterly review of economic and finance, 42, 193–208, doi: http://dx.doi.org/10.1016/s1062-9769(02)00138-2. chan, k. s. (1993), consistency and limiting distribution of the least squares estimator of a threshold autoregressive model, the annals of statistics, 21, 520–533, doi: http://dx.doi.org/10.1214/aos/1176349040. davis, s. j., halitwanger, j. (2001), sectoral job creation and destruction responses to oil price changes, journal of monetary economics, 48, 465–512, doi: http://dx.doi.org/10.1016/s0304-3932(01)00086-1. ebc (2008), 10 rocznica ebc, biuletyn miesięczny, europejski bank centralny, frankfurt am main, http://www.ecb.int/pub/pdf/other/10thanniversaryoftheecbmb200806pl.pdf (15.01.2015). enders, w., granger, c. w. j. (1998), unit-root tests and asymmetric adjustment with an example using the term structure of interest rates, journal of business and economic statistics, 16, 304–311, doi: http://dx.doi.org/10.2307/1392506. enders, w., siklos, p. (2001), cointegration and threshold adjustment, journal of business and economic statistics, 19 (2), 166–176, doi: http://dx.doi.org/10.1198/073500101316970395. engle, r. f., granger, c. w. j. (1987) cointegration and error correction representation. estimation and testing, econometrica, 55, 251–276, doi: 10.1111/j.1368-423x.2005.00149.x. fosten, j., morley, b., taylor, t. (2012), dynamic misspecification in the environmental kuznets curve: evidence from co2 and so2 emissions in the united kingdom, ecological economics, 76, 25–33, doi: http://dx.doi.org/10.1016/j.ecolecon.2012.01.023. andrzej geise, mariola piłatowska dynamic econometric models 14 (2014) 71–91 90 friedman, m. (1976), inflation and unemployment, nobel memorial lecture, http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1976/friedmanlecture.pdf (15.01.2015). hamilton, j. d. (1983), oil and the macroeconomy since world war ii, journal of political economy, 91, 228–248, doi: http://dx.doi.org/10.1086/261140. hamilton, j. d. (1996), this is what happened to oil price-macroeconomy relationship, journal of monetary economics, 38, 215–220, doi: http://dx.doi.org/10.1016/s0304-3932(96)01282-2. hooker, m. (1996), what happened to the oil price-macroeconomy relationship?, journal of monetary economics, 38, 195–213, doi: http://dx.doi.org/10.1016/s0304-3932(96)01281-0. jimenez-rodriguez, r., sanchez, m. (2005), oil price shocks and real gdp growth: empirical evidance for some oecd countries, applied economics, 37, 201–228, doi: http://dx.doi.org/10.1080/0003684042000281561. lardic, s., mignon, v. (2006), the impact of oil prices on gdp in european countries: an empirical invetigation based on asymmetric cointegration, energy policy, 34, 3910– 3915, doi: http://dx.doi.org/10.1016/j.enpol.2005.09.019. lutz, c., mayer, b. (2009), economic impacts of higher oil and gas prices. the role of international trade for germany, energy economics, 31, 882–887, doi: http://dx.doi.org/10.1016/j.eneco.2009.05.009. mork, k. a. (1989) oil and the macroeconomy when prices go up and down: an extention of hamilton’s results, journal of political economy, 91, 740–744, doi: http://dx.doi.org/10.1086/261625. mork, k.a., olsen, o., mysen, h.t. (1994), macroeconomic responses to oil price increases and decreases in seven oecd countries, energy journal, 15, 19–35, doi: http://dx.doi.org/10.5547/issn0195-6574-ej-vol15-no4-2. papapetrou, e. (2001), oil price shocks a, stock market, economic activity and employment in greece, energy economics, 23 (5), 511–532, doi: http://dx.doi.org/10.1016/s0140-9883(01)00078-0. petruccelli, j., woolford, s. w. (1984), a threshold ar(1) model, journal of applied probability, 21(2), 270–286, doi: http://dx.doi.org/10.2307/3213639. scholtens, b., yurtsever, c. (2012), oil price shocks and european industries, energy economics, 34(4), 1187–1195, doi: http://dx.doi.org/10.1016/j.eneco.2011.10.012. yau, h-y., nieh, c. (2009), testing for cointegration with threshold effect between stock prices and exchange rates in japan and taiwan, japan and the world economy, 21(3), 292–300, doi: http://dx.doi.org/10.1016/j.japwor.2008.09.001. ceny ropy naftowej, produkcja oraz inflacja w gospodarkach unii europejskiej podejście kointegracji progowej z a r y s t r e ś c i. w artykule wykorzystuje się podejście kointegracji progowej, które zostało opracowane przez endersa i siklosa (2001) w celu zbadania zależności między cenami ropy a aktywnością gospodarczą (wyrażoną przez produkcję i inflację) z uwzględnieniem wystąpienia zmiany strukturalnej dla sześciu wybranych państw ue. podejście to pozwala na różną szybkość dostosowań do długookresowej równowagi w zależności, czy produkcja jest powyżej czy poniżej długookresowej zależności. ponadto, dla zbadania krótkoi długookresowej przyczynowości między produkcją, cenami ropy i inflacją, zostały oszacowane progooil prices, production and inflation in the selected eu countries... dynamic econometric models 14 (2014) 71–91 91 we modele korekty błędem. otrzymane wyniki wskazują, że dla francji, danii i całej unii europejskiej zmienne te są skointegrowane, przy czym proces dostosowawczy ma asymetrycznych charakter. s ł o w a k l u c z o w e: asymetryczne dostosowania, cenowe szoki naftowe, kointegracja progowa, nieliniowość, progowy model korekty błędem. microsoft word dem_2015_111to128.docx © 2015 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2015.007 vol. 15 (2015) 111−128 submitted october 23, 2015 issn (online) 2450-7067 accepted december 15, 2015 issn (print) 1234-3862 sylwester bejger* testing parallel pricing behavior in the polish wholesale fuel market: an ardl – bound testing approach a b s t r a c t. in this study, we investigated whether the observed series of fuel prices can be compatible with a specific theoretical model of strategic player interaction. our primary interest is in determining whether a parallel pricing policy, implied by a theoretical model of strategic interactions, can be an industry-observed pricing mechanism. therefore, we first calculated various descriptive statistics of the price series to discover any common patterns of individual series. next, we determined whether parallel co-movement of the price levels exist using an ardl – bound testing approach. this study finds that if we restricted our research to the described pricing mechanism (ipp pricing based on previous day fundamentals), the players will have chosen the levels of price in a parallel mode; this excludes 2007, when lotos appeared to be the price leader. k e y w o r d s: wholesale fuel market, parallel pricing, cointegration. j e l classification: l1, l7, c22. introduction the general objective of this paper was to discover particular, strategic patterns in the price behavior of the players in the polish wholesale fuel market in a sample period of 2004 to 2013. this study is shaped as an intersection of monitoring and verification (to use the terminology of harrington, 2008) in the context of behavioral screening. however, this study is more broadly understood because we did not constitute tests for verification of the * correspondence to: sylwester bejger, nicolaus copernicus university, department of econometrics and statistics, ul. gagarina 13a, 87-100 toruń, poland, e-mail: sylw@umk.pl. sylwesterbejger dynamic econometric models 15 (2015) 111–128 112 hypothesis of collusion (tacit or overt) existence in the industry on the basis of a specific marker1. instead, we strive to answer the question regarding whether the observed series of price levels can be compatible with the known model of the strategic interaction of players (may be part of the equilibrium of a game). on the basis of industry2 research, important factors that influence the strategic conduct of the market participants can be enumerated: a duopolistic market with high concentration, homogeneous products, high barriers to entry, capacity constraints for domestic production, inelastic demand, possible ipp3 pricing mechanism and major players’ full price transparency. based on these characteristics, it is possible to restrict the set of adequate models of noncooperative games by the specification of the essential elements of a game model. these specifications include the following: the price should be considered as a main strategic variable for the players, there were no threats of significant entry to the market in the sample period, and production capacities should be treated as exogenous parameters in the game period. through these restrictions, one can isolate a single period game as a non-zero-sum, simultaneous moves finite game in pure action spaces. this set of assumptions leads to a choice of a standard bertrand model as a building block of certain one-shot games; this is enriched with additional elements to eliminate equilibrium in marginal costs as the sole solution. models based on the well-known edgeworth model (1925) would be particularly well-suited in this instance. edgeworth’s model modifications have led to various models, for example, those by levitan and shubik (1972), kreps and scheinkman (1983), osborne and pitchik (1986), deneckere and kovenock (1992). because those models are essentially static, one can consider a proper dynamic specification. in this case, a highly popular maskin and tirole (1988) model and a supergame approach to oligopolistic competition could be considered. as a reference work in that topic, one can point to tirole (1998) and vives (2001). for our empirical research, the works of brock and scheinkman (1985), lambson (1987), green and porter (1984), rotemberg and saloner (1986), dudey (1992), lu and wright (2010) could be inspirational because of the choice of price as a strategic variable and because of the imposed capacity constraints in certain games. however, analyzing the abovementioned models, we can conclude that their structures do not explain 1 refer to: harringtopn (2006, 2008) or aberantes-metz (2011, 2013), abrantes-metz and bajari (2012). 2 details of that study will be published in an another paper. 3 import parity pricing. testing parallel pricing behavior in the polish wholesale fuel market… dynamic econometric models 15 (2015) 111–128 113 the very important market’s phenomenon, i.e., ipp pricing mechanism. in our view, such a mechanism leads to a concept of a reference point in strategic interactions that could be described as a focal point or ceiling point. there is substantial empirical evidence of the use of focal point prices by firms. scherer (1980) found price lining is widespread at the retail level in the us. using data from the us-credit card market during the 1980s, knittel and stango (2003) show that nonbinding price ceilings, which serve as external focal points, increase the probability that firms engage in tacit collusion. in faber, janssen (2011), they investigated the focal point effects of oil companies that suggest petrol and diesel prices to their retailers. we wanted to utilize the concept then but formulate the question slightly differently: what were the strategic implications of ipp treated as the “focal price” for player’s daily actions (price levels)? we then decided to construct a game theory model of pricing behavior that is suitable for the industry’s parameters4. the main conclusions from our theoretical model are the following:  it is strategically possible to use the same (or very close) price levels by both players in daily interactions; therefore, regarding the parallel pricing phenomenon, we can state,  the common price level for both players should be very close to the ipp price level (or a proxy of it), if we assume the ability of players to properly calculate it on the basis of commonly known factors. thus, the primary purpose of this article concerns the empirical verification of the first of the above implications. we want to determine whether the observable data reflects parallel pricing strategic behavior in a market and, as an implication, if the rational conduct of the players is coherent with the equilibrium strategies of a game theory model. few studies of broad price movements and dependencies in the polish refining sector in general (bejger and bruzda, 2002; miłobędzki, 2008; leszkiewicz-kędzior, 2012) exist; in addition, no such studies (to our knowledge) are devoted to analyzing the pricing behavior of both players in parallel, focused on the strategic interactions among them. the methodological construction of the research first contains the basic statistical measures of the empirical distributions of prices used; finally, it examines the cointegration between the prices of players and the existence of the long-term and short-term relations (if any) for the two series. to investigate the construction, the ardl – bound testing approach was used because it is fairly universal and insensitive to the misspecification of integration order. the remainder of this paper is structured as follows. section 2 briefly 4 bejger (2015), forthcoming. sylwesterbejger dynamic econometric models 15 (2015) 111–128 114 outlines the data. section 3 reviews the estimation techniques and contains the empirical analysis. section 4 presents the study’s conclusions. 2. data description the main data set covers wholesale (unregular) daily prices on pkn pb95, pkn on, lotos pb95 and lotos on in pln for cubic meters and the 01.01.2004–31.12.2013 sample period. these four time series have been obtained from official websites of the players5. we decided to conduct research on two slightly different data sets from the basic sample; we examined “raw” series (of different lengths) to reveal their individual properties. next, we synchronized these data sets to study cointegration. in the raw data analysis, we did not subtract the excise tax and fuel duty because the reactions of price levels to the changes in these duties could contain information regarding the strategic behavior of players. for the purpose of synchronization of the observations, we used the dates of observations of pkn’s gasoline price as reference. missing daily observations of lotos were replaced by simple extrapolation between the values observed immediately before and after; therefore, both final series have 2281 observations. such transformation of lotos’ data did not harm the statistical properties of original series substantially (integration order unchanged, descriptive statistics that are closer to pkn in values). additionally, we simplified the names of the synchronized series to lotospb and pknpb to use them as variable names. figure 1 shows raw data. there are obvious similarities in the time evolution of the prices of the same type of fuel in both players and in the differences in the price processes of different fuels. as a preliminary examination, the basic descriptive statistics of the (raw) series were calculated. table 1 contains the results of this step for the individual series (in levels)6. from the results, one can observe the similarities of moments for the same type of fuels and the strong rejection of the normality of empirical distributions in all cases. 5 http://www.orlen.pl/pl/dlabiznesu/hurtowecenypaliw; http://www.lotos.pl/144/dla_biznesu/hurtowe_ceny_paliw. 6 as we can observe, the series are of different length. for statistical or econometric research, this difference is an obvious disadvantage; however, we rely strongly on daily observations such as those that could reveal strategic patterns in player’s behavior. testing parallel pricing behavior in the polish wholesale fuel market… dynamic econometric models 15 (2015) 111–128 115 figure 1. time series of prices (wholesale, fuels) table 1. descriptive statistics pkn pb95 pkn on lotos pb95 lotos on mean* 3521.64 3400.70 3530.19 3375.92 median* 3374.00 3184.00 3381.00 3153.00 maximum* 4757.00 4663.00 4768.00 4667.00 minimum* 2425.00 2117.00 2428.00 2121.00 std. dev.* 573.84 657.40 572.05 664.37 coef. of var. 16.29% 19.33% 16.20% 19.68% skewness 0.36 0.45 0.35 0.45 kurtosis 1.98 1.81 1.97 1.87 jarque-bera 149.24 194.44 149.95 204.37 p-value 0.00 0.00 0.00 0.00 number of obs. 2281 2094 2343 2343 note: * – values in pln.   sylwesterbejger dynamic econometric models 15 (2015) 111–128 116 3. estimation techniques and empirical analysis as stated in the introduction, the main objective for this research was an empirical verification of the pricing schema that was implied by the theoretical model of strategic interaction (the focal price game), parallel pricing strategic behavior. at first, the model has to specify what statistical and econometric measures can be used to verify such pricing. if one understands movements of prices as parallel pricing, buccirossi (2006) stated, on the basis of a simple game, that: “concise representation of the degree of price parallelism is provided by the correlation between prices”7. a testable proposition (partly adequate to our case) is: “if that the market is perturbed only by shocks on costs, then: if shocks are perfectly common, the correlation between prices equals 1 in both the competitive and the collusive equilibrium”8. of course, such an ascertainment is consistent with intuition of course; however, in our opinion, simple correlation is not a sufficient tool for time series of very complex structure. we can begin from this point; however, we test other measures of similarity of series, as well. however, first, we want to determine whether the parallel movement could be observed not only in the commoncomovement of the levels. parallelism could mean similar patterns in the empirical distributions of price changes and price grids, for example. we want to research that possibilities at first and move to the examination of co-movements next. to limit the size of the analysis, and for data availability, we focused solely on gasoline prices. we began with raw data (daily observations not paired) to obtain as much information as possible. 3.1 statistical analysis of the gasoline price changes to check the movements of the price changes more closely, we calculated the positive and negative average changes of values (in pln) and their relation to the price level. the results are shown in table 2. at the beginning of the sample period (2004 and 2005), a higher relative average change can be noticed, which could be connected to higher grid price levels in that period for lotos (figure 2). 7 buccirossi (2006), p. 92. 8 proposition 3, buccirossi (2006), p. 94. testing parallel pricing behavior in the polish wholesale fuel market… dynamic econometric models 15 (2015) 111–128 117 table 2. average positive and negative changes of prices by years lotos pb95 pkn pb95 year avg. change avg. change as % of price avg. change avg. change as % of price abs. difference of % change 2004 37.95 1.374% 39.40 1.426% 0.052% –37.55 –1.348% –41.53 –1.488% 0.140% 2005 38.08 1.280% 46.83 1.575% 0.296% –40.60 –1.315% –55.71 –1.801% 0.486% 2006 22.18 0.729% 12.04 0.397% 0.332% –21.79 –0.712% –14.32 –0.472% 0.241% 2007 20.47 0.652% 13.47 0.432% 0.220% –15.05 –0.466% –10.37 –0.322% 0.144% 2008 18.42 0.552% 13.59 0.410% 0.143% –22.58 –0.709% –18.69 –0.594% 0.115% 2009 25.47 0.821% 23.80 0.771% 0.049% –20.60 –0.632% –15.02 –0.461% 0.170% 2010 18.21 0.519% 17.71 0.505% 0.015% –14.90 –0.422% –12.83 –0.363% 0.059% 2011 20.39 0.508% 19.43 0.484% 0.024% –20.40 –0.505% –19.35 –0.479% 0.026% 2012 17.05 0.383% 16.33 0.368% 0.015% –22.98 –0.515% –21.59 –0.485% 0.031% 2013 19.99 0.468% 19.08 0.447% 0.021% –18.88 –0.439% –17.53 –0.408% 0.031% note: bolded numbers mean the rejection of h0 regarding the equality of a mean positive or negative change of prices between lotos and pkn in a particular year. figure 2. empirical distribution of price grids – pkn pb95 (left panel) and lotos pb95 (right panel) pkn’s grid was more evenly distributed; however, grid 5 was dominant in 2004 and 2005, as well. in accordance with table 2, 2006–2008 were very sylwesterbejger dynamic econometric models 15 (2015) 111–128 118 different than the remainder of the sample because the average changes (of both signs) decreased but became significantly different for both players. those changes signal disturbances in the behavior of players in that period. as can be observed, the beginning of this change corresponds to lotos’ introduction of values other than 5 pln of the price grid and the synchronization of the announcement of prices. from 2009 to the end of the sample average price, the changes became statistically equal; however, we can subjectively (not in the statistical sense) observe that considering relative changes, upward movements were closer. the last interesting analysis of raw data is contained in table 3. table 3. daily distributions of price changes lotos pb95 pkn pb95 lotos pb95 pkn pb95 lotos pb95 change pkn pb95 change up down no change up down no change mon 8 3 0.3% 0.1% 0.3% 0.2% 0.1% 0.2% 0.0% 0.5% tue 432 441 18.6% 19.6% 19.1% 18.0% 17.8% 20.1% 19.1% 16.7% wen 439 446 18.2% 19.5% 17.8% 18.7% 21.7% 17.2% 22.0% 19.8% thur 457 469 19.9% 20.6% 18.0% 21.9% 17.2% 21.0% 20.2% 19.3% frid 461 456 19.8% 19.8% 20.7% 18.9% 18.9% 20.4% 19.2% 22.4% sat 476 462 20.7% 20.3% 21.6% 19.8% 18.1% 21.0% 19.5% 19.8% sun 70 4 2.5% 0.0% 2.5% 2.6% 5.6% 0.1% 0.0% 1.6% note: columns up, down, no change contain contribution of daily up, down, no change price movements in whole number of movements of a given type. highest values bolded. table 3 refers to the daily distributions of price movements of both players; there are certain differences. however, after checking pairs of empirical distributions of the same type (i.e., lotos up – pkn up) using the kolmogorov-smirnov test, we could not reject the hypothesis of the equality of cumulative distributions. therefore, the days of movements of prices are largely the same. 3.3. cointegration analysis 3.3.1. ardl – bounds testing procedure with that test, we want to move to the next step of our research, testing the common movements of both price series. preliminarily, we checked the correlation coefficient for both series. as we expected, correlations were very strong, with a magnitude of 0.99 in the entire sample and not below 0.97 in the sub-samples that encompass each year. however, for the possible cointegrated series, we need a more precise testing parallel pricing behavior in the polish wholesale fuel market… dynamic econometric models 15 (2015) 111–128 119 evaluation of the co-movement. we were interested in determining whether a cointegration exists between the prices of players, and what type of longrun and short-run relations (if any) exist for the two series. to study these relations, we decided to use the ardl-bound testing approach. the ardl modeling approach was originally introduced by pesaran and shin (1999) and extended by pesaran et al. (2001). the ardl cointegration approach has numerous advantages over conventional cointegration testing. the approach yields consistent estimates of the long-run coefficients that are asymptotically normal irrespective of whether the underlying regressors are i(1) or i(0).the approach involves a single-equation set-up, which is simple to implement and interpret; different lag-lengths can be assigned to model variables. in accordance with pesaran, shin (1999), we consider a general ardl(p; q) model: ∑ ∑ ∗∆ , (1) ∆ ∆ ∆ ⋯ ∆ . (2) the scalar disturbance, ut in the model (1) is iid(0; ). we do not want to impose any integration assumption9. the ardl model used in this study is a version of eq. (1), which could be called an unrestricted ecm or, as in pesaran et al. (2001), a conditional ecm. the model is as follows: δlotospb α ∑ δlotospb ∑ ∆ lotospb , (3) where we assume the lotos price as “dependent”, in accordance with miłobędzki (2008), when he concluded that pkn is a “price leader” in the industry. 3.3.2. empirical analysis first, a test for order of integration was conducted. although the bounds test for cointegration allows variables to be a mixture of i(1) and i(0), it is important to conduct stationarity tests to ensure that the variables are not all i(0) and not i(2). the results reported in table 4 confirm that all series have one unit root, i.e. all are i(1). to exclude the possibility of distinct leader-follower strategic behavior in the entire sample, we next tested for causality between the variables. us 9 the order of integration was preliminary tested (table 4); however, those were usual tests (adf and kpss). we did not test the order of integration more thoroughly (in the presence of breaks in intercept and/or trend, for example). sylwesterbejger dynamic econometric models 15 (2015) 111–128 120 ing the toda-yamamoto procedure (toda and yamamoto, 1995), the granger causality in both directions between the price series was found. table 5 contains the final wald test of the granger non-causality for var with 13 lags (optimal lag length selected for var was 12 + 1 lag to explain i(1) in each variable, treated here as exogenous). table 4. adf test for unit root level 1st difference series adf test statistics p-value* lag adf test statistics p-value* lag pkn pb95 0.484 0.819 2 –21.875 0.000 2 pkn on 0.845 0.893 2 –24.608 0.000 1 lotos pb95 0.509 0.825 2 –20.218 0.000 3 lotos on 0.887 0.899 2 –22.859 0.000 2 note: adf test for h0: series has a unit root. test's critical value at the 1% level: –2.56607. we conducted the kpss test for all series with the same results (all series i(1)).*mackinnon (1996) one-sided p-values. table 5. wald tests for granger causality dependent variable: lotospb excluded chi-sq df p-value pknpb 284.433 12 0.000 all 284.433 12 0.000 dependent variable: pknpb excluded chi-sq df p-value lotospb 105.109 12 0.000 all 105.109 12 0.000 the null hypothesis of no causality in both directions must be rejected. regarding the specifications of the model, the lag structure of (3) has been defined. the ardlbound: an eviews add-in by tarverdi m. yashar (2014) was utilized here. the parameters of the ardl (1,2) model were subsequently estimated. the residuals were tested for a serial correlation next because the key assumption in the ardl / bounds testing methodology is that the errors of equation (3) must be serially independent. table 6 summarizes the results of those steps. the estimation results for the ardl(1,2) specification of choice are satisfactory; all coefficients (except one) are statistically significant (at the 0.05 level), and the residuals are serially independent. having the parameters’ estimates, we next apply a bounds f-test to eq (3). the long run relation between lotospb and pknpb series was sought. as usually occurs in testing parallel pricing behavior in the polish wholesale fuel market… dynamic econometric models 15 (2015) 111–128 121 cointegration testing, the absence of a long-run equilibrium relation between the variables was tested for. the hypothesis structure was as follows: h0: 0, h1: . table 6. ardl (p, q) model selection and estimation ardl model aic sc ardl(1,1) 9.2751 9.2852 ardl(1,2) 9.2721 9.2847 ardl(1,3) 9.2732 9.2883 variable coefficient std. error t-statistic p -value c 6.7113 3.2528 2.0632 0.0392 d(lotospb(–1)) –0.0374 0.0408 –0.9160 0.3598 d(pknpb(–1)) 0.3875 0.0408 9.4960 0 d(pknpb(–2)) 0.0675 0.0216 3.1240 0.0018 lotospb(–1) –0.2645 0.0438 –6.0310 0 pknpb(–1) 0.2629 0.0438 5.9938 0 breusch-godfrey serial correlation lm test: durbin-watson statistic f-statistic 0.7081 prob. f(3,2269) 0.5471 obs*r-squared 2.1309 prob. chi-square(3) 0.5457 durbin-watson stat 1.9988 note: aic and sc information criteria for the ardl preferred structure are in bold. variables d(*) are the first differences of the series of levels. a rejection of h0 implies that there is a long-run relation between series. the f-statistics in the form presented in pesaran et al. (2001), p. 297, equation (21) was used. because this statistic has non-standard distributions, in accordance with the so-called bounds procedure proposed by pesaran et al. (2001), we tested h0 within the conditional ecm (3). the researchers provided bounds on the critical values for the asymptotic distribution of the fstatistic for various specifications of ecm for sizes 0.100, 0.050, 0.025 and 0.010 of the test. the lower bound values assume that the forcing variables are purely i(0), and the upper bound values assume that variables are purely i(1). if the computed f-statistics fall outside the critical value bounds, a conclusive decision results without needing to know the cointegration rank r of the variables. however, if the wald or f-statistic falls within these bounds, the inference would be inconclusive. in such circumstances, knowledge of the cointegration rank r of the forcing variables is required to proceed further10. additionally, a bounds t-test of the θ0 estimated value was performed: 10 pesaran et al (2001), p. 299. sylwesterbejger dynamic econometric models 15 (2015) 111–128 122 h0 : 0, h1 0 . asymptotic critical t-statistic’s values for this test are provided by pesaran et al. (2001, p. 303–304) in a form of similar to the f-statistic’s i(0)–i(1) bounds, calculated for various specifications. inference is similar, either the calculated t-value is smaller than the lower bound, which means stationarity of the data; a t-value higher than i(1) bound supports the hypothesis of a long run relation. table 7 concludes this examination: table 7. bounds testing test statistic calculated value   f-statistic 19.6814 t-statistic –6.0310 significance level 0.1 0.05 0.01 f-stat. bounds 4.04; 4.78 4.94; 5.73 6.84; 7.84 t-stat.bounds –2.57; –2.91 –2.86; –3.22 –3.43; –3.82 note: bound values from pesaran et al. (2001), p.300, p.303 for unrestricted intercept, no trend specification and k = 1. it can be stated that because the value of our f-statistic and t-statistic exceeds the upper bound at the 1% significance level, there was evidence of a long-run relation between the lotospb and pknpb. the estimated longrun multiplier between pknpb and lotospb is –(0.262933/–0.264599)) = 0.9937; therefore, in the long run, an increase of 1 unit in pknpb will lead to an increase of 0.9937 units in lotospb. after confirming cointegration, there was a possibility to estimate the longrun relation meaningfully: lotospb α α , (4) and use the ols residuals series from the (4) as an error correction term to estimate the restricted ecm of the form: δlotospb α ∑ δlotospb ∑ ∆ , (5) where: (6), and a0, a1 are ols estimates of parameters in (6). table 8 contains estimated values of parameters in (5). testing parallel pricing behavior in the polish wholesale fuel market… dynamic econometric models 15 (2015) 111–128 123 table 8. estimation of parameters of restricted ecm variable coefficient std. error t-statistic p-value c 0.3822 0.5224 0.7316 0.4644 d(lotospb(–1)) –0.0367 0.0408 –0.8988 0.3688 d(pknpb(–1)) 0.3866 0.0408 9.4719 0.0000 d(pknpb(–2)) 0.0666 0.0216 3.0800 0.0021 ec(–1) –0.2649 0.0438 –6.0368 0.0000 note: variables d(*) are the first differences of the series in levels. it can be observed that the estimated coefficient of the error-correction term, ect-1, is negative, as expected, and significant. the absolute value of this coefficient implies that approximately 27% of any disequilibrium between the lotospb price level and the pknpb price level is corrected within one day. 3.4. preliminary analysis of leader-follower behavior in subsamples the last problem, which was preliminarily examined, was the leaderfollower behavior of a more complicated nature (as replications of certain patterns in price changes and price announcements), which was exhibited in the sub-samples. to begin, the sub-samples were examined in the most intuitive manner, consistent with the assumed pricing mechanism, i.e., by testing the differences between the same day prices of both players, the current price of one of the players and the lagged price of the other. figure 3 provides the possibility to visually inspect various series of differences of prices of the players: figure 3. absolute values of differences between levels of prices of players and the entire sample the differences of the same day prices appear to have the smallest mean and variance; however, an interesting period (it covers nearly precisely all observations of 2007) in the series of differences of the pknpb and lagged 1-day 0 50 100 150 200 04 05 06 07 08 09 10 11 12 13 lotospb minus pknpb (abs v alues) 0 50 100 150 200 04 05 06 07 08 09 10 11 12 13 lotospb minus pknpbt-1 (abs v alues) 0 50 100 150 200 04 05 06 07 08 09 10 11 12 13 lotospb minus pknpbt+1 (abs v alues) sylwesterbejger dynamic econometric models 15 (2015) 111–128 124 table 9. price’s differences analysis descriptive statistics – whole sample descriptive statistics – year 2007 subsample s1 = yt – xt s2 = yt – xt–1 s3 = xt – yt–1 s1 = yt – xt s3 = xt – yt–1 mean 9.749 20.873 19.386 11.530 8.724 median 6 16 14 10 7 std. dev. 10.755 19.391 19.853 7.980 6.814 skewness 1.992 2.529 2.802 0.341 1.077 kurtosis 10.060 17.332 18.119 2.348 4.251 test – equality of means e(s1) = e(s2) test test stat value p-value test stat value p-value t-test –23.9695 0.0000 anova f-test 574.5346 0.0000 test – equality of means e(s1) = e(s3) t-test –20.4223 0.0000 4.2020 0.0000 anova f-test 417.0719 0.0000 17.6569 0.0000 test – equality of medians me(s1) = me(s2) wilcoxon/mann-whitney (tieadj.) 24.3783 0.0000 adj. med. chi-square 441.7884 0.0000 van der waerden 613.5586 0.0000 test – equality of medians me(s1) = me(s3) wilcoxon/mann-whitney (tie-adj.) 21.4293 0.0000 3.8747 0.0001 adj. med. chi-square 346.6815 0.0000 18.6559 0.0000 van der waerden 502.5227 0.0000 11.8360 0.0006 test – equality of variances var(s1) = var (s2) f-test 3.4100 0.0000 bartlett 808.2732 0.0000 brown-forsythe 201.9853 0.0000 test – equality of variances var(s1) = var (s3) f-test 3.2498 0.0000 1.371368 0.0135 bartlett 749.3125 0.0000 6.096009 0.0135 brown-forsythe 282.6512 0.0000 9.518529 0.0021 note: series x – pknpb in levels, series y – lotospb in levels. all the series in pln/m3. price of lotos (rightmost panel)11 can be observed. descriptive statistics and various test statistics are reported in table 9. these statistics confirm that the measures of the average level and dispersion were significantly different in a case of three series, s1, s2 and s3, and that the smallest values of 11 to ensure coherent interpretation, we use differences lotos – pkn in every case; however, for absolute values, x – yt+1 = y – xt-1 testing parallel pricing behavior in the polish wholesale fuel market… dynamic econometric models 15 (2015) 111–128 125 the average and dispersion measures can be observed in a case of differences between the same day prices in an entire sample. this observance excludes the subsample from 2007, when the smallest values of those measures for the difference between lotos’ day t price and pkn’s day t+1 price were noted. these results lead to the conclusion that, if the research is restricted to ipp pricing based on previous day fundamentals, the players will, in fact, choose levels of prices in a parallel mode; this excludes 2007, when lotos appeared to be the price leader12. conclusions to inspect the parallel pricing policy, certain descriptive statistics for the raw price series (not paired) were calculated to evaluate how consistent the pricing policies were. this study found that for the entire sample, an average positive and negative value of prices change were statistically equal (on standard 0.05 significance level); however, the mean changes between players were significantly different. analysis of the yearly subsamples revealed that the years 2006–2008 were highly different than the remainder of the sample because the average changes (of both signs) decreased but became significantly different for the both players. that finding signals certain disturbances in the behavior of players in that period. the beginning of this change corresponded to lotos’ introduction of other than 5 pln values of the price grid and the synchronization of the price announcement. from 2009 to the end of the sample, the average price changes became statistically equal. analyzing the empirical distributions of price grids and the absolute values of price changes, other differences in pricing policies were detected. however, an examination of the empirical distributions of daily price movements has confirmed that the daily price changes policy of both players was not significantly different over the entire sample period. next, the casual dependencies and the cointegration of gasoline prices was examined. on the basis of the estimated ardl (1, 2) model, the bound testing procedure of cointegration testing was used. f-test and t-test results have confirmed the existence of a long-run relation between the wholesale prices of gasoline at the 1% significance level. the estimated long-run multiplier between pknpb and lotospb was 0.9937; therefore, in the long run, an increase 12 we have checked the medians and means equality hypothesis in all pairs of the three series of differences for all the year’s subsamples and have not confirmed the phenomenon of 2007 in the remaining years. sylwesterbejger dynamic econometric models 15 (2015) 111–128 126 of 1 pln of pknpb will lead to an increase of 0.9937 pln in lotospb. the estimation of the regular ecm model provided information regarding the short-run adjustments of prices. the absolute value of the estimated coefficient of the error-correction term implies that approximately 27% of any disequilibrium between the lotospb price level and the pknpb price level was corrected within one day. finally, the simple test of the leaderfollower pricing behavior over the subsamples was performed. to check that test in a manner consistent with the assumed pricing mechanism, the differences between the same day prices of both players, the current price of one of the players and the lagged price of the other were examined. overall, this study finds that, if we restricted our research to described pricing mechanism (ipp pricing based on previous day fundamentals) players really chose the levels of price in a parallel mode, with the exclusion of year 2007 when lotos appeared to be the price leader. in terms of the assessment of the players’ strategic behavior, this study confirmed that the publically observable pricing conduct of the players in the polish wholesale fuel market could be consistent with an equilibrium of an assumed game theory model. this finding, in turn, implies that the players’ observed behavior was coherent with the assumed competition model and did not exhibit disturbances (unreasonable conduct), which could reflect competition’s distortions. however, it should be emphasized that for a decisive “screening” of conclusions based on the model, the test of the second behavioral implication of the theoretical model, i.e., the focal role of the ipp price, must be examined. this test and examination will be a subject of the subsequent research. references abrantes-metz, r. (2011), design and implementation of screens and their use by defendants, antitrust chronicle, competition policy international, 9, 2–11. abrantes-metz, r., bajari, p. (2012), screens for conspiracies and their multiple applications, antitrust magazine, 24(1), 23–35. bejger, s. (2015), theoretical model of pricing behavior on the polish wholesale fuel market, univeristy of szczecin, forthcoming. bejger, s., bruzda, j. (2002), identification of market power using test for asymmetric pricing – an example of polish petrochemical industry, dynamic econometric models, 5, 135–146. brock, w. a., scheinkman, j. a. (1985), price setting supergames with capacity constraints, the review of economic studies, 52, 371–382, doi: http://dx.doi.org/10.2307/2297659. buccirossi, p. (2006), does parallel behavior provide some evidence of collusion? review of law and economics, 2:1, 86–102, doi: http://dx.doi.org/10.2202/1555-5879.1027. testing parallel pricing behavior in the polish wholesale fuel market… dynamic econometric models 15 (2015) 111–128 127 deneckere, r. j., kovenock, d. (1992), price leadership, the review of economic studies, 59, 143–162, doi: http://dx.doi.org/10.2307/2297930. dudey, m. (1992), dynamic edgeworth-bertrand competition, the quarterly journal of economics, 107(4), 1461–1477, doi: http://dx.doi.org/10.2307/2118397. edgeworth, f. y. (1925), the pure theory of monopoly, papers relating to political economy, 1, 111–142. faber, r. p., janssen, m. c. w. (2011), on the effects of suggested prices in gasoline markets, working paper, cpb netherlands bureau for economic policy analysis, department of competition and regulation, doi: http://dx.doi.org/10.2139/ssrn.1309608. green, e., porter, r. (1984), non – cooperative collusion under imperfect price information, econometrica, 52, 87–100. harrington, j. e. (2006), behavioral screening and the detection of cartels, european competition law annual, 234–251. harrington, j. e. (2008), detecting cartels, in: buccirossi (ed.) handbook of antitrust economics, the mit press. knittel, c. r., stango, v. (2003), price ceilings as focal points for tacit collusion: evidence from credit cards, american economic review, 93(5), 1703–1729, doi: http://dx.doi.org/10.1257/000282803322655509. kreps, d., scheinkman, j. a. (1983), quantity precommitment and bertrand competition yield cournot outcomes, bell journal of economics, 14, 326–337, doi: http://dx.doi.org/10.2307/3003636. lambson, v. e. (1987), optimal penal codes in price-setting supergames with capacity constraints, the review of economic studies, 54(3), 385–397, doi: http://dx.doi.org/10.2307/2297565. leszkiewicz-kędzior, k. (2012), modelling fuel prices. an i(1) analysis; central european journal of economic modelling and econometrics, 3, 75–95. levitan, r., shubik, m. (1972), price duopoly and capacity constraints, international economic review, 13, 111–123, doi: http://dx.doi.org/10.2307/2525908. maskin, e., tirole, j. (1988), a theory of dynamic oligopoly ii, econometrica, 56, 571–599, doi: http://dx.doi.org/10.2307/1913726. miłobędzki, p. (2008), orlen or lotos? which is setting prices at the wholesale market for unleaded petrol in poland?, dynamic econometric models, 8, 37–43. osborne, m., pitchik, c. (1986), price competition in a capacity-constrained duopoly, journal of economic theory, 38, 238–260, doi: http://dx.doi.org/10.1016/0022-0531(86)90117-1. pesaran, h. m., shin, y. c., smith, j. r. (2001), bounds testing approaches to the analysis of level relationships, journal of applied econometrics, 16, 289–326, doi: http://dx.doi.org/10.1002/jae.616. pesaran, m. h., shin, y. (1999), an autoregressive distributed lag modelling approach to cointegration analysis, in s. strom (ed.), econometrics and economic theory in the 20th century: the ragnar frisch centennial symposium, cambridge university press, cambridge, doi: http://dx.doi.org/10.1017/ccol521633230. rotemberg, j., saloner, g. (1986), a supergame theoretic model of business cycles and price wars during booms, american economic review, 76, 390–407. scherer, f. m. (1980), industrial market structure and performance, 2 ed., chicago: rand mcnally. tarverdi, m. y. (2014), ardlbound: an eviews add-in to perform ardl bound test based on pesaran et al. (2001), electronic document provided with program. tirole, j. (1998), the theory of industrial organization, the mit press, england. sylwesterbejger dynamic econometric models 15 (2015) 111–128 128 toda, h. y., yamamoto, t. (1995), statistical inferences in vector autoregressions with possibly integrated processes, journal of econometrics, 66, 225–250, doi: http://dx.doi.org/10.1016/0304-4076(94)01616-8. vives, x. (2001), oligopoly pricing. old ideas and new tools, the mit press, cambridge. lu, y., wright, j. (2010), tacit collusion with price-matching punishments, international journal of industrial organization, 28, 298–306, doi: http://dx.doi.org/10.1016/j.ijindorg.2009.10.001. test paralelnych zachowań cenowych na hurtowym rynku paliw płynnych w polsce oparty na podejściu ardl /bound testing z a r y s t r e ś c i. artykuł dotyczy badania, czy obserwowane cen paliw płynnych na rynku hurtowym mogą być zgodne z określonym modelem teoretycznym interakcji strategicznych graczy. głównym celem badawczym było określanie, czy paralelne zachowania cenowe, implikowane równowagą teoretycznego modelu interakcji strategicznych, mogą być mechanizmem ustalania cen faktycznie obserwowanym na rynku. aby to stwierdzić wyznaczono statystyki opisowe indywidualnych szeregów czasowych cen dla ustalenia podobieństw. następnie zbadano współzależność ruchów cenowych graczy wykorzystując podejście ardl /bound testing. stwierdzono, że jeśli gracze wykorzystują mechanizm cenowy ceny parytetu importowego (przyjęty w modelu teoretycznym), ich zachowania replikują paralelne zachowanie cenowe, za wyjątkiem roku 2007, w którym lotos wydaje się być liderem cenowym. s ł o w a k l u c z o w e:hurtowy rynek paliw, zachowania paralelne, kointegracja. dem_2015_157to165 © 2015 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2015.008 vol. 15 (2015) 157−165 submitted october 24, 2015 issn (online) 2450-7067 accepted december 15, 2015 issn (print) 1234-3862 tomasz groszkowski, tomasz stryjewski* the econometric analysis of entrepreneurship determinants in polish voivodeships in the years 2004–2013 a b s t r a c t. this article presents research results and describes and clarifies differences in a level of the entrepreneurship index observed between polish voivodeships in the period from 2004 to 2013. the expected results were confirmed with the fixed effect (fe)/random effect (re) panel data model. the analysis showed that the level of entrepreneurship varies depending on a voivodeship, and that this differentiation is structural and permanent. the applied model also confirmed the expected determinants of entrepreneurship. k e y w o r d s: entrepreneurship, development factors, panel data model, fe/re model. j e l classification: c23, c26, l26, r12. introduction entrepreneurship is undoubtedly one of the fundamental economic development factors, through its influence on development of new products, new markets, creating jobs, and raising the social welfare in general. it is noticeable that some areas are growing fast through establishment of new companies, when others remain far behind. the concept of entrepreneurship is a complex issue, and it is permanently inscribed into different branches of science. a wide range of studies shown that a large and continuously increasing number of classifications and descriptions of entrepreneurship re * corresponding author: tomasz stryjewski, the university of computer science and economics in olsztyn, e-mail: tomasz.stryjewski@erbud.pl; tomasz groszkowski, the university of computer science and economics in olsztyn. tomasz groszkowski,tomasz stryjewski dynamic econometric models 15 (2015) 157–165 158 sults in a lack of a uniform theory of entrepreneurship and associated measures, implying possible difficulties in their assessment (szarecki, 2008, p. 181). entrepreneurship is multidimensional and consists of many elements that should be considered when determining the characteristics of entrepreneurship (kalkan, kaygusuz, 2012). a management sphere pays particular attention to this aspect, mainly for functional reasons, describing it as the process of organizing and running a business in conditions of risks associated with those activities (gryffin, 1997, p. 730–731). entrepreneurship can therefore be considered as a process creating or identifying opportunities, which later are frequently used regardless of resources possessed at a given time. entrepreneurship understood this way is a stimulating factor for a creative entrepreneur who finds energy to establish and build a company or organization. therefore, the entrepreneur is more than just a passive observer of the situation in which he is positioned (timmons, 1990, p. 5). entrepreneurship can therefore be described as an organized process consisting of successive phases, occurring in specific conditions and aiming at using innovative ideas to achieve certain benefits, while considering risks involved in this process (adamczyk, 1995, p. 9–10). entrepreneurship is one of the basic factors of economic, social and cultural development. it is associated with activities of different types of businesses, from microenterprises, throughout the whole sme sector, up to large domestic and international enterprises. certainly, companies play an important role in the polish and in the european economies. they create a space where human skills and entrepreneurial attitudes are revealed and concentrated. furthermore, they are a source of innovations and employment. the experience of well-developed countries shows that entrepreneurship plays a significant role in the economy, stimulating economic growth, influencing employment, and providing goods to the market. therefore, companies are a very important factor for economic growth of a relevant area. entrepreneurship and businesses are also stimulating factors for growth, and the number of enterprises is frequently considered to be an indicator of economic development. 1. entrepreneurship factors entrepreneurial individuals, stimulated into action in appropriate conditions, establishing businesses, creating new jobs and additional sources of income for themselves and the local population, can be found in every community, also at a local level. the econometric analysis of entrepreneurship determinants… dynamic econometric models 15 (2015) 157–165 159 however, an answer to a question about factors influencing entrepreneurial behavior is not so clear and obvious. in numerous studies on entrepreneurship, selection of factors that affect the level of entrepreneurship is a frequently encountered problem. this article focuses on economic factors, describing the structural and economic determinants of entrepreneurship and quantitative aspects of that issue. entrepreneurship is determined by the certain socio-economic and political factors of intensity varying in time and space. those factors may act in two ways: by bringing positive effects to the economy in form of incentives for business and economic growth stimulators or, on contrary, as means restraining or even inhibiting creation of new and development of already existing business entities. these factors are classified according to various criteria (chrapek, 2009, p. 321). their effects on a recovery of a given area can either be positive or adverse, through their potential to create conditions for formation and development of entrepreneurship. providing suitable conditions for development of entrepreneurship should therefore be one of the most important priorities for the regional development for authorities at any level (jezierska-thole, 2010, p. 129). processes associated with entrepreneurial behavior occur in specific socio-economic circumstances. gdp, an economic situation, revenues or investments are just examples of certain macroeconomic indicators affecting development of enterprises. considering the entrepreneurship development conditions, it should be noted that every company operates in a certain environment with which it interacts. in entrepreneurial research, one of the common problems is identifying factors determining an entrepreneurial activity. a type of analyzed data, constrained a limitations in very narrow definition of entrepreneurship as the number of formal registered companies. in this study, it was decided to apply a division into regions defined as established territorial units of a relatively large surface area and population, and with a specific economic policy prevailing. in polish economic reality, this region can correspond to a voivodeship. this approach is based on factors including their independently established policies, independent local authorities, and own budget. with specifics of each voivodeship, an endogenous development potential can be developed, influencing opportunities and barriers to the development and growth of enterprises in that region. according to the subject of this article, individual features of the region, having a direct influence on the investment level and the profitability of the tomasz groszkowski,tomasz stryjewski dynamic econometric models 15 (2015) 157–165 160 business should be emphasized (godlewska, 2001, p. 14). if the region is understood as an environment consisting of humans, other companies and institutions, having certain features that may or may not be attractive to new businesses, then it is important to communicate conditions prevailing in that region, which will affect the company's position, and to ensure future entrepreneurs are familiar with and have access to that information. 2. research methodology and analysis results in this paper, the main part of the space-time analysis of entrepreneurship in poland aimed at finding information about factors determining specific indicators, and reasons why these indicators vary in individual voivodeships. a panel approach of the fixed effect/random effect (fe/re hereafter) type is suitable for such kind of modeling (greene, 2005). the analysis covered annual data for a period from 2004 to 2013 for 16 polish voivodeships. 15 factors possibly affecting the entrepreneurship development in poland1 were selected, following a detailed substantive analysis: gross domestic product per capita (gdp per capita), enterprises investment outlays per capita (inv), retail sales per capita (sprzed), new apartments put in use per thousand inhabitants (mieszk), population density (gest), percentage of population living in cities (urbanizacja), number of cities in a region (miasta), roads outside built-up areas per 100 km2 (drogi), employment rate (zatr), average monthly disposable income per capita (doch), number of tourist facilities per capita (turyst), number of people with higher education per thousand inhabitants (wykszt), net migration per thousand inhabitants (migr1000), and municipal revenues per capita (doch per capita). starting with a framework of the congruent modeling concept, an internal structure of specific processes was analyzed during the first stage of the study (talaga, zieliński, 1989). a congruent panel data model exhibits the same harmonic structure on both sides of the equation in dynamic terms. the analysis showed the significance of a linear trend in all tested processes and autoregressive relationships over time. however, due to a small sample size in the researched period, only the first-order autoregression was considered. the main problem associated with the use of models of this class is that they belong to a group of static methods. modeling the dynamic relationship with these methods results in both ols and gls coefficients bias (wooldridge, 2010). 1 understood as the number of companies per 10000 people of employable age (entrepreneurship index – wskpz). the econometric figure 1. entrepreneurship index 2004–2013 unfortunately, dynamics of terms of acf and pacf functions) ble is not allowed in the fe/re models priate instrumental variable that the search for an give any meaningful cial" variable correspond an instrumental variable for the entrep ed with a dynamic panel model based on (model hypothesis): itwskpz α= 0 where: ������� – spatial entrepren trend, � ,��,��,� econometric analysis of entrepreneurship determinants… dynamic econometric models 15 (2015) entrepreneurship index – time series for voivodeships in the years dynamics of all processes exhibits strong autocorrelation terms of acf and pacf functions). since use of a lagged dependent vari in the fe/re models, it was necessary to select an priate instrumental variable, so no information was lost. it should be noted an instrument within a specified group of variables meaningful results. therefore, it was necessary to create an corresponding to a character of the real instrument. nstrumental variable for the entrepreneurship indicator was a dynamic panel model based on a structure of internal variables : ittiti twskpzwskpz ηααα ++++ −− 32,21,10 spatial entrepreneurship indicator in period t, � – evaluation parameters, η �� – random spatial comp ) 157–165 161 time series for voivodeships in the years autocorrelation (in dependent variaselect an approit should be noted specified group of variables did not an "artifireneurship indicator was evaluatof internal variables (1) – linear random spatial compotomasz groszkowski,tomasz stryjewski dynamic econometric models 15 (2015) 157–165 162 nent in period t. the following model was developed with a two-step estimation method2 (2-step method). table 1. a dynamic panel model for a spatial entrepreneurship indicator (blundell and bond estimator) dependent variable: wskpz coefficient sd z p-value wskpz(-1) 0.9835 0.0385 25.5764 <0.00001 *** const. 1.4756 5.0447 0.2925 0.7699 time 0.4533 0.1128 4.0188 0.00006 *** sargan over-identification test: chi-square(43) = 15.7061 test for ar(1) errors: z = –2.27161 [0.0231] test for ar(2) errors: z = –0.713604 [0.4755] the fitted values from the above model created an instrumental variable for a dynamic relationship in fe/re models. further research focused on identifying determinants for the spatial entrepreneurship indicator and its spatial differentiation described by a neighborhood matrix. for this purpose, a group of potential exogenous variables was determined and a pooled ols model was estimated. results of this estimation are shown in table 2 (only statistically significant variables). the next step of the analysis was verification which approach is correct: pooled ols or fe/re. results of this verification are shown in table 3. table 2. the data panel model estimation using pooled ols dependent variable: wskpz coefficient sd t-ratio p-value const. 4.6527 3.59303 1.2949 0.1978 urbanizacja −16.5172 4.61591 –3.5783 0.0005 *** turyst(-1) 0.0625 0.0300 2.0804 0.0396 ** doch per capita 0.0090 0.0018 4.9687 <0.00001 *** time −0.4339 0.1932 -2.2453 0.0266 ** yhat17(-1) 0.9631 0.0297 32.3751 <0.00001 *** mean dependent var 149.7844 s.d. dependent var 25.9606 r-squared 0.9824 adjusted r-squared 0.9817 log-likelihood −339.4414 akaike criterion 690.8827 schwarz criterion 707.9949 hannan-quinn 697.8355 rho 0.3696 durbin-watson 1.0856 note: that 17 – instrumental variable from table 1 model. 2 see footnote 1. the econometric analysis of entrepreneurship determinants… dynamic econometric models 15 (2015) 157–165 163 table 3. panel models diagnostics3 f test (fixed effect) the null hypothesis that the pooled ols model is adequate, in favor of the fixed effects alternative f(15, 107) = 2.6292 p-value = 0.0021 breusch-pagan test (random effect) the null hypothesis that the pooled ols model is adequate, in favor of the random effects alternative lm = 1.7714 p-value = prob(chi-square(1) > 1.77137) = 0.1832 hausman test the null hypothesis that the random effects model is consistent, in favor of the fixed effects model h = 14.5289 p-value = prob(chi-square(5) > 14.5289) = 0.0126 results presented in table 3 imply that a fixed effects-fe model is an appropriate approach. this means that the spatial differentiation in the entrepreneurship index level is of a structural and permanent character. therefore, the next step of the study was to estimate and validate the fe model. the model presented in table 2 became a model hypothesis and the estimated (after verification) fe model is presented in table 4. table 4. the fe model (dependent variable: wskpz) – lsdv estimator coefficient sd t-ratio p-value const 31.9812 9.3363 3.4255 0.00086 *** doch per capita 0.0091 0.0017 5.4573 <0.00001 *** yhat17(-1) 0.7004 0.0728 9.6190 <0.00001 *** mean dependent var 149.7844 s.d. dependent var 25.9606 sum squared resid 1127.241 s.e. of regression 3.2012 lsdv r-squared 0.9868 within r-squared 0.7427 lsdv f(17, 110) 484.8430 p-value(f) 4.79e-95 log-likelihood −320.8560 akaike criterion 677.7120 schwarz criterion 729.0486 hannan-quinn 698.5703 rho 0.2742 durbin-watson 1.1935 the fe model has better statistical properties than the model estimated by pooled ols. in particular, significant reduction in the first-order random component autocorrelation was possible in the fe model. however, the study also showed that the analysis of the entrepreneurship spatial differentiation using the spatial entrepreneurship indicator is biased and, in conse 3 see footnote 1. tomasz groszkowski,tomasz stryjewski dynamic econometric models 15 (2015) 157–165 164 quence, has some disadvantages. particularly, this indicator is strongly autoregressive, indicating an autonomous nature of changes in that process. it may also imply that variables used in the analysis were unsuitable. however, the results presented in table 4 confirm the economic theory about the entrepreneurship development index. the main determinants of entrepreneurship remain the same, being the entrepreneurship indicator from the previous period and the municipal revenues per capita. these two variables positively influence the entrepreneurship rate, although they strengthen the autoregressive character of that process. conclusions a study on entrepreneurial activities based on their index led to three main conclusions. first, the analysis showed that the problem of entrepreneurship development in poland is a structural problem. this means that differences in the levels in individual regions are stable and result from internal conditions, as well as from the regional polarization processes. therefore, this problem cannot be solved locally, but only as part of a coherent regional policy at the governmental or european level. second, the entrepreneurship index determinants in the studied period were identified. the main determinant is the municipal revenues per capita. however, the entrepreneurship index is a strongly autonomous variable and, therefore, the autoregressive component is important in the analyzed model. the main causes of the dependent variable formation are the regional economic situation (autoregressive process) and public revenues. public revenues affect the dependent variable in two ways: as a determinant (indicator) of regional economic situation and as the basis for public expenditures which, in turn, influence small local enterprises. third, an important part of the study is focusing on a selection of suitable explanatory variables. the spatial entrepreneurship indicator is highly autocorrelated, and this causes problems with identification of its determinants. in the further research, other measures should be considered for exploration and selection, better describing the investigated process. references adamczyk, w. (1995), ewolucja form i typów przedsiębiorczości (the evolution of forms and types of entrepreneurship), zeszyty naukowe akademii ekonomicznej w poznaniu, 236. the econometric analysis of entrepreneurship determinants… dynamic econometric models 15 (2015) 157–165 165 chrapek, g. (2009), przedsiębiorczość osób fizycznych na obszarach wiejskich podkarpacia, [w:] rola przedsiębiorczości w kształtowaniu społeczeństwa informacyjnego (entrepreneurship individuals in rural areas of podkarpacie, [in:] the role of business in shaping the information socjety), seria przedsiębiorczość – edukacja, warszawakraków. godlewska, h. (2001), lokalizacja działalności gospodarczej (the location of economic activity), dom wydawniczy elipsa, warszawa. greene, w. h. (2005), econometric analysis, prentice hall, new york. gryffin, r. w. (1997), podstawy zarządzania organizacjami (fundamentals of organizational management), pwn, warszawa. jezierska-thole, a. (2010), zmiany poziomu infrastruktury i jej wpływ na rozwój przedsiębiorczości na obszarach wiejskich na przykładzie województwa kujawskopomorskiego i pomorskiego (changes in the level of infrastructure and its impact on development of entrepreneurship in rural areas on the example of kujawskopomorskie and pomorskie), acta acientum polonorium, oeconomia, 9(3). kalkan, m., kaygusuz, c. (2012), the psyhology of enterpreneurship [in:] entrepreneurship – born, made and educated, ed. t. burger-helmchen, rijeka. szarecki, a. (2008), przedsiębiorczość jako forma kultury (entrepreneurship as a form of culture), problemy zarządzania, 2. talaga, l., zieliński, z. (1986), analiza spektralna w modelowaniu ekonometrycznym (spectral analysis in econometric modeling), pwn, warszawa. timmons, j. (1990), new venture creation, irvin, boston. wooldridge, j. m. (2010), econometric analysis of cross section and panel data, massachusetts institute of technology. ekonometryczna analiza determinant przedsiębiorczości w polskich województwach w latach 2004–2013 z a r y s t r e ś c i. w artykule przedstawiono wyniki badań, opisujące i wyjaśniające różnice w poziomie wskaźnika przedsiębiorczości pomiędzy województwami polski w latach 2004– –2013. w celu potwierdzenia zakładanych rezultatów, użyto modelu panelowego z efektami stałymi i losowymi. analiza wykazała, że wskaźnik przedsiębiorczości jest zróżnicowany pomiędzy województwami, a zróżnicowanie to ma charakter strukturalny i stały. model potwierdził również oczekiwane determinanty przedsiębiorczości. s ł o w a k l u c z o w e: przedsiębiorczość, czynniki rozwoju, model panelowy, model fe/re pusta strona dem_2015_71to87 © 2015 nicolaus copernicus university press. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2015.004 vol. 15 (2015) 71−87 submitted october 30, 2015 issn (online) 2450-7067 accepted december 15, 2015 issn (print) 1234-3862 błażej mazur* density forecasts based on disaggregate data: nowcasting polish inflation∗∗ a b s t r a c t. the paper investigates gains in performance of density forecasts from models using disaggregate data when forecasting aggregate series. the problem is considered within a restricted var framework with alternative sets of exclusion restrictions. empirical analysis of polish cpi m-o-m inflation rate (using its 14 sub-categories for disaggregate modelling) is presented. exclusion restrictions are shown to improve density forecasting performance (as evaluated using log-score and crps criteria) relatively to aggregate and also disaggregate unrestricted models. k e y w o r d s: prediction, model comparison, density forecasting, inflation, var models, shrinkage. j e l classification: e31, e37, c53, c32. introduction the paper focuses on the question whether point and in particular density forecasts of univariate series can be improved using disaggregate information (assuming that it is available) – or more generally, whether economic fluctuations are more accurately modeled at the aggregate or at the disaggregate level. the crucial assumption being relevant here is that a multivariate model for disaggregate data is used only as a tool for obtaining the implied univariate forecast of the aggregate series. the aggregating weights are as * correspondence to: błażej mazur, cracow university of economics, chair of econometrics and operations research, e-mail: eomazur@cyfronet.pl. ∗∗ this research was supported by the polish national science center (ncn) based on decision number dec-2013/09/b/hs4/01945. błażej mazur dynamic econometric models 15 (2015) 71–87 72 sumed to be known and fixed. there is no direct and necessary relationship between forecasting power in the disaggregate (multivariate) context and the validity of the implied univariate (aggregate) forecast. therefore the usual statistical procedures used for forecast evaluation or model comparison applied at the disaggregate level have to be adjusted in order to take the context into account. the probabilistic approach to inference have been gaining popularity in the systematic manner over time. in particular events like the recent economic crisis have made it evident that uncertainty quantification is an inherent part of the forecasting process. some problems that seem to be uncomplicated when point forecast perspective is taken become more complex when one has the density forecasting perspective in mind. the paper investigates consequences of the aggregation for uncertainty of the aggregate forecast. however, an alternative perspective on the problems is the general question whether economic fluctuations are better modeled at the aggregate or the disaggregate level, given some specific model classes. a framework for formal investigation of the problems mentioned above builds on a gaussian var specification with restrictions. usage of such relatively simple models is dictated by complexity that increases rapidly with the dimension (i.e. disaggregation level). moreover, as the problem considered here can be interpreted in the context of variable selection (resulting in a very large number of possible exclusion restrictions), numerical complexity becomes quite considerable even within the relatively simple model class consisting of gaussian vars. the objective of the paper is to investigate empirical evidence supporting the use of disaggregate data for aggregate forecasting. the application under consideration deals with forecasting polish monthly cpi inflation rate (relative to the previous month) using 164 observations on growth rates of 14 price sub-indices. a pseudo real-time forecasting experiment is performed and out-of-sample predictive performance of competing specifications is evaluated over a verification window consisting of the last 44 observations. all the criteria (those relevant for point forecast and density forecasts) show the evidence of substantial gains from considering a large menu of competing specifications at the disaggregate level. however, as the total number of alternative restricted specifications in enormous, a stochastic search algorithm was used to explore the model space. the results suggest that the problem of aggregate vs. disaggregate modeling for the sake of aggregate forecasting is not trivial and illustrates a potential for gains in predictive performance, but this requires some form of promoting parametric parsimony – as attributing non-zero probability to density forecasts based on disaggregate data: nowcasting polish inflation dynamic econometric models 15 (2015) 71–87 73 models with zero restrictions might be interpreted as introducing some form of shrinkage within the overall (unrestricted) model. the rest of the paper is organized in the following way: firstly, the general problem of forecasting of the aggregate series is presented in a more detailed way. secondly, a formal model framework used here is described, with the relevant forecasting methodology. thirdly, methods for evaluation of density forecasts are outlined. fourthly, the empirical illustration is provided. a summary and some remarks about possible directions for further developments conclude. 1. aggregate vs. disaggregate approach to forecasting a natural approach to forecasting is to use a model formulated directly in terms of the quantity of interest itself. however, if one considers the context of e.g. inflation forecasting, an approach that is often taken by practitioners is to obtain the aggregate forecast using individual disaggregate inflation forecasts for a number of sub-categories. the aggregate approach seems to be conceptually simpler and more elegant. moreover, due to the lack of the dimensionality problem, at the aggregate level it is possible to make use of more sophisticated models. aggregation could have a regularizing impact as well. for example, one could expect that in certain cases the assumption of say homoscedastic gaussian errors is more likely to hold at the aggregate level. on the contrary, at the disaggregate level one has to deal with increased heterogeneity. another issue is that of modeling dependencies across the variables in a multivariate process which might be challenging. however, there is more information available, and sometimes an expert knowledge can be utilized at the disaggregate level only. in the specific context considered here, the only objective of the analysis is the aggregate forecast. it is not obvious that the efforts made to deal with heterogeneity and dependence modeling that improve goodness of fit at the disaggregate level would lead to an improved forecasting performance of the aggregate series. this is even more true when one has in mind the density forecasting context. in general it is possible to obtain a point forecast of the aggregate series using disaggregate point forecasts obtained from individual, unrelated univariate models for the sub-aggregates. however, if one makes an attempt to generate a density forecast in the same way, the omitted dependence between sub-aggregates might result in uncertainty misspecification for the aggregate forecast, resulting in poor density predictive performance. the błażej mazur dynamic econometric models 15 (2015) 71–87 74 tradeoffs between various aspects of heterogeneity and dependence modelling and possible gains in the aggregate density forecasting performance are among the points of interest in the paper. the essential empirical issue under consideration is whether there exist potential gains from working at the disaggregate level in terms of the aggregate density predictive performance. moreover, one might be interested in verifying to what extent the aggregate predictive performance deteriorates as one neglects e.g. the stochastic dependence among the series in the model for sub-aggregates. the problems of disaggregate vs. aggregate modeling and forecasting (also with focus on inflation forecasting) has been considered by many authors. in particular hubrich (2005) considers similar inflation forecasting problem, but does not deal with density forecasts. at a more general level, the question was addressed by hendry and hubrich (2011). some theoretical developments are related to the work of giacomini and granger (2004); lütkepohl (2009) considers aggregation within an interesting class of disaggregate dgps. contributions involving also empirical applications, especially on inflation forecasting, include aron and muellbauer (2013), castle and hendry (2010) and faust and wright (2013) among others. the distinctive features of the approach taken here include the focus on density forecasting and the role of exclusion restrictions at the disaggregate level. 2. model framework in order to establish the notation used below, some fairly standard results concerning estimation of multivariate linear models are recalled. consider a gaussian var(p) model for m variables: ,...11 tpptt s t εayayαy ++++= −− ,,...,1 tt = (1) where: ty is m-dimensional row vector (corresponding to period t), s α is m-vector of seasonal intercept terms (consisting of ms unknown parameters in total, where s is the number of different seasons) 1a ,…, pa are mm × matrices of parameters, with pa including at least one non-zero element, { }tε is a gaussian vector white noise process with covariance matrix .σ assuming m = 1 leads to a gaussian autoregressive process ar(p). however, if one assumes that 1a ,…, pa and σ are diagonal, (1) describes a set of m unrelated gaussian ar(p)-type processes. density forecasts based on disaggregate data: nowcasting polish inflation dynamic econometric models 15 (2015) 71–87 75 assuming σ to be positive definite and otherwise unrestricted and 1a ,…, pa to be unrestricted (i.e. containing no zero restrictions) results in a standard var formulation, where ols estimates of the parameters are also mles and thus are asymptotically effective. however, for unrestricted σ if 1a ,…, pa include zero restrictions (i.e. the set of explanatory variables is not the same in all the equations), ols estimates are no longer equivalent to ml estimates (and are not asymptotically efficient anymore). issues of non-stationarity (existence of unit or explosive roots in the characteristic polynomial of (1)) are not considered here, the autoregressive parameters are assumed to be unrestricted for the sake of simplicity. the initial conditions 0y are assumed to be fixed and the values are taken from the data, as the further analysis is conditional on 0y . in order to consider asymptotically efficient estimation of the model parameters in special cases (with exclusion restrictions), the following sur-type form of (1) is considered: ,~ ~~ εβxy += (2) where: ( )' ... ~ 21 tyyyy = is a tm-dimensional column vector, β is a k-dimensional column vector consisting of unrestricted elements of 1a ,…, pa and 1 α ,…, ,sα x ~ is ktm × matrix consisting of rows and columns arranged in a way that matches the convention assumed for ,~y ( )' ... ~ 21 tεεεε = is tm-dimensional column vector of gaussian error terms with zero mean and covariance matrix of the form σi ⊗ . asymptotically efficient estimation of β can be achieved by means of a zellner-type estimator of the form: ( )[ ] ( ) ,~'~~'~ˆ 111 yσixxσixβ σ −−− ⊗⊗= (3) with σ replaced by its consistent estimate, β s , based on residuals (e.g. obtained in the previous iteration of the procedure, as the procedure could be iterated): ,'1 ees β −= t (4) błażej mazur dynamic econometric models 15 (2015) 71–87 76 where e is mt × -dimensional matrix of residuals. in order to initialize the procedure, the first estimate of β can be obtained as ( ) .~'~~'~ˆ 1 yxxxβ −=ols the procedure is assumed to stop once some estimate of σ is considered the final one, here denoted by σ̂ , with σ β ˆ ˆ being the final estimate of β (being also an approximation of mles for β ). 3. forecasting methodology for the purpose of one-period-ahead prediction, it is assumed that the point forecast of ' 1+ty (its estimated conditional expectation) is given by: ,ˆ~ˆ ˆ ' 1 σ βxy ft =+ (5) where: fx~ is an km × matrix consisting of known constants and lagged values of the dependent variables (all these are readily available for onestep-ahead forecast), preserving the structure of x ~ . for the sake of density forecasting, the predictive distribution of ' 1+ty considered here is m-dimensional gaussian, with mean given by (5) and variance-covariance matrix given as: ( )[ ] .'~~ˆ'~~ˆˆ 111 fftv xxσixxσy −−+ ⊗+= (6) consequently, a gaussian distribution with mean and covariance matrix given by (5) and (6) respectively is perceived as an approximation to the bayesian predictive distribution obtained with diffuse priors. the approximation can be quite a satisfactory one, especially if the number of observations is not very small and the prior information on parameters is not very strong. a motivation for the use of the approximation instead of the exact bayesian results is twofold. firstly, the recursive of out-of-sample prediction experiment with extensive specification search imposes a considerable numerical burden, which requires some simplifications for the feasibility reasons. secondly, one of the reasons often stated in favor of the bayesian approach in var forecasting (e.g. with continuous minnesota-type priors) is that the approach introduces some form of shrinkage, which is beneficial for the forecasting performance. in the paper the counterpart of shrinkage is the specification search described below. it could be perceived as analogous to density forecasts based on disaggregate data: nowcasting polish inflation dynamic econometric models 15 (2015) 71–87 77 so-called stochastic search variable selection approach, which in turn imposes very strong (“hard”) shrinkage by means of discrete-continuous prior distributions. the paper focuses on the results obtained using the exclusion restrictions (similar to “hard” shrinkage) without imposing any “soft” restrictions (which usually imply reduction of the prior variance for certain parameters). the question whether imposing additional “soft” shrinkage could improve forecasting performance even more is left for further research. it is assumed here that the variable of interest (labeled tz ) is a linear combination of variables in ty with known, fixed weights tc : ,'tttz yc= (7) where tc is a m-dimensional row vector of weights. predictive distribution of 1+tz is therefore univariate gaussian with mean given by: ,ˆ~ˆ ˆ11 σβxc f ttz ++ = (8) and variance of the form: ( )[ ] '.'~~ˆ'~~'ˆrâv 1111111 +−−++++ ⊗+= tfftttzt cxxσixxccσc (9) the above formulas simplify in certain cases, in particular when σ is assumed to be diagonal. the restriction leads to a considerable reduction of the numerical burden. empirical verification of its predictive consequences is therefore of great practical importance. it is obvious that ex post (point) forecast error of 1+tz is given by: ( ),ˆ~'ˆ~'ˆ ˆ11ˆ11111 σς βxycβxcyc fttfttttt zz −=−=− +++++++ (10) so it is a linear combination of forecasts errors of 1+ty . minimization of ex ante forecast error does not necessarily require the forecast error of 1+ty to be minimized (in ‘mean squared’ sense for instance). in particular, it is possible that the disaggregate forecast errors just cancel out in the aggregate. moreover, within a given model class, the forecast error resulting from application of the model directly to tz could be higher compared to the one obtained for implicit forecast based on the analogous model for the disaggregate data ty . błażej mazur dynamic econometric models 15 (2015) 71–87 78 as density forecasts are considered, it is of course crucial to provide adequate description of uncertainty: formulas (6) and (9) take into account potential stochastic dependence between the model equations (represented by estimated σ ) and correlations within the joint distribution of the estimator for whole β in all the model equations (or a joint posterior for all the structural parameters). therefore in general no ‘limited information’ approximation is used, unless it follows directly from the model structure (e.g. as a special case in which a full information method is equivalent to a limited information one). another advantage of taking the general perspective of multivariate modeling is that it provides a framework in which the “naïve” disaggregate forecast strategy, based on separate sub-models, can be evaluated against more complicated alternatives. the crucial issue is that of model choice (or comparison) at the disaggregate level. the approach used here is based on two basic premises. firstly, importance of the exclusion restrictions is emphasized. this amounts to recognizing the fact that applying some reasonable variable selection or model selection procedures can lead to huge gains in predictive performance compared to a basic, unrestricted gaussian var model. the related idea of stochastic search variable selection (see e.g. frühwirth-schnatter and wagner, 2010) can be interpreted in terms of approximating the results of a bayesian inference pooling experiment, dealing with the problem of finding the relevant model reducing restrictions. introducing some form of parsimony is particularly important in the case of var models that are potentially heavily overparametrized. secondly, having in mind the model pooling or model comparison context, it seems to be reasonable to consider the criteria that measure predictive performance with respect to tz . consequently, though individual models are estimated on the disaggregate data, meaning that the estimation procedure itself aims at optimizing goodness of fit at the disaggregate level, the model comparison (or model selection) is undertaken based on predictive criteria at the aggregate level. similar issues to the ones presented above are formally considered by lütkepohl (2009) within a broader model class (though lütkepohl does not attempt to introduce parsimony and does not consider density forecasts). in order to describe the above procedure in detail, an overview of some criteria for density forecast evaluation is provided below. density forecasts based on disaggregate data: nowcasting polish inflation dynamic econometric models 15 (2015) 71–87 79 4. density forecast evaluation the awareness of the necessity of more careful investigation of the forecast uncertainty has become widespread in the econometric literature (see e.g. clark and ravazzolo, 2015). the discussion on ex ante predictability of the recent economic crisis has also contributed to the recognition of importance of the probabilistic approach to forecasting. tasks such as evaluation of probability of extreme events are no longer considered non-standard. however, the strand of research is deeply rooted in history of the statistical inference methods and some of the ideas can be traced back to classical texts. however, for the purpose of the paper just a brief review of the selected techniques is presented. standard ways of reporting ex post forecasting performance for point forecasts include considering mean forecast error, mean absolute forecast error or root mean squared forecast error (rmsfe). however, the concept of absolute forecast error can be generalized into a density context, leading to so-called continuous ranked probability score (crps). the generalization is based on the fact that the crps formula simplifies to that of absolute error if the predictive distribution is represented by point mass. the definition of crps, together with a closed-form analytical formula for the cases where predictive distribution is gaussian is given by gneiting and raftery (2007, p. 367). here the negatively-oriented version of crps is used (labeled crps* by the authors), which takes positive values and directly generalizes absolute error. for two predictive distributions with location parameter corresponding exactly to the actual outturn, crps (intuitively, ‘a density forecast error’) will be higher for the density that is more dispersed. for the sake of ex post predictive performance analysis, the averaged crps is reported. another popular measure is so-called log score, being just a logarithm of the value of probability density function corresponding to the predictive distribution at the point being the actual outturn. log-score has a bayesian interpretation, because a sum of the one-period-ahead log-scores in a recursive predictive experiment on expanding sample can be perceived as related to so-called predictive bayes factor, being a modification of the bayes factor, which in turn is a basic method of formal bayesian model comparison. relative to the crps measure, log-score (and the bayes factor) is considered to be sensitive to tail outcomes. differences between (log of) values of gaussian pdf say two, three or four standard deviations away from the mean do not increase proportionally, therefore in case of mis-predicted outturn log-score will discriminate models stronger compared to crps. details and błażej mazur dynamic econometric models 15 (2015) 71–87 80 more elaborate theoretical justifications of the above measures are summarized by gneiting and raftery (2007). as the analysis conducted here is intended to provide an approximation to the results of bayesian inference, the log-score will be used as the main criterion used for the model choice. 5. model comparison framework the model framework used here has the advantage of nesting a simple “practitioner’s approach” of the following form. consider the problem of inflation forecasting. a simple practitioner’s solution would be to use disaggregate data on sub-indices and apply say ar(p) forecasting models to each of the individual series. the total cpi forecast would be then obtained making use of the fact that weights of the sub-aggregates in the total index are usually known. such an approach accounts for heterogeneity to some extent, but neglects possible dependence. individual predictive models are chosen based on say goodness of fit for the individual series. however, an advantage of the framework provided here is that it highlights the fact that the collection of such individual processes can be perceived as a single model for total cpi forecasting. from such a perspective the models for subaggregates should be evaluated jointly rather than separately. consequently, neither of the sub-models should be evaluated without taking actual combination of the remaining ones into account. this of course results in a big increase in the number of specifications being considered. the approach makes some generalizations of the “simple practitioner’s approach” readily available. for instance it is easy to impose stochastic linkages between equations by allowing the contemporaneous variancecovariance matrix to be non-diagonal. another option would be to include lags of the other sub aggregates into the individual equations. the framework allows for considering of a broad menu of models that differ substantially. one end of the model spectrum would be the simplistic approach using individual processes (which is likely to be too restrictive), whereas the other end would correspond to an unrestricted var model (which is very far from being parsimonious). a solution proposed here is to explore options that are somewhere in between, meaning that these are less restrictive than the simple approach, but introduce “hard” shrinkage (parsimony by exclusion restrictions) compared to full, unrestricted vars. this would correspond to including only some lags of some variables in the equations at the disaggregate level. density forecasts based on disaggregate data: nowcasting polish inflation dynamic econometric models 15 (2015) 71–87 81 alternative models (or sets of restrictions) should be compared by the predictive accuracy of the aggregate, here measured with log-score. the number of possible combinations of the restrictions is way too large to allow for any systematic specification search. instead, a random-search algorithm is applied. even if it is not capable of achieving the global optimum, it is interesting to check whether it can find a combination of restrictions that results in a model with predictive performance clearly dominating the simple special cases already mentioned. if such a specification can be found within reasonable computational time, the strategy would be empirically attractive and the issue should be investigated in more detail. the objective of the paper is to present a strategy that is feasible for more than ten disaggregate variables. for each model (a set of exclusion restriction that actually corresponds to some particular form of x ~ ) a predictive experiment is conducted, with recursive model reestimation and one step ahead prediction (with the number of repetitions denoted by n , being also a number of one-step-ahead forecasts obtained from one specification). results of the repeated out-of-sample predictive exercise are summarized (here by a sum of log-score values) and the best specification is retained. a modification of the algorithm that aims at inference pooling instead of model selection could be also considered, though the possibility is not explored here. as the stochastic specification search requires thousands of model specifications to be checked, the computational burden is substantial. the computational limitations are the reason for which the attention here is restricted to simple gaussian var models only (instead of say varma class members), for analogous applications see george, sun and ni (2008). one more thing should be kept in mind: if n is not large, the specification search might result in overfitting issues. this might show a spurious predictive gains that are not necessarily observed out-of-sample, resulting from using the statistical noise to obtain perfect prediction in the verification period. in order to consider possible empirical consequences in depth, it would be necessary to add another data window over which performance is not maximized but just analyzed. however, for polish macroeconomic time series the number of available observations is not large enough to do so. in order to avoid overfitting problems one might want to promote specification parsimony (imposing e.g. a limit on the number of non-zero autoregressive parameters, or using some penalty function). błażej mazur dynamic econometric models 15 (2015) 71–87 82 6. empirical analysis: nowcasting polish cpi inflation rate in order to illustrate practical applicability of the approach and possible gains in terms of forecast performance an empirical analysis of polish inflation is provided1. the analysis makes use of month-over-month cpi inflation rate in polish economy for the period 2002m01–2015m08, with t = 164, (see figure 1.). the last n = 44 observations are treated as a verification window2. it is important to notice that the verification period is quite challenging, as it includes a period of deflation unprecedented in polish economy. figure 1. cpi m-o-m inflation rate [%] in poland, 2000–2015 as for the data at the disaggregate level, polish statistical office (gus) reports disaggregation into 10 main price groups. however, as two of the components include both energy and non-energy prices, an effort has been made to separate the price indices. moreover, housing expenditures and alcoholic beverages with tobacco are also separated from food, resulting in a total of 14 categories. the unrestricted model would therefore correspond to a var for 14 variables, having extra 196 parameters for each additional lag in the unrestricted version. 1 other studies examining similar problems include clark (2006), dees and güntner (2014), huwiler and kaufmann (2013) and ibarra (2012), see also stock and watson (2015). 2 all the calculations were conducted using own routines written in ox (see doornik and ooms, 2003), details of the specification search procedure are available from author by request. -1 -0,5 0 0,5 1 1,5 2 2000 2002 2004 2006 2008 2010 2012 2014 density forecasts based on disaggregate data: nowcasting polish inflation dynamic econometric models 15 (2015) 71–87 83 the predictive exercise is pseudo-real time, which means that no revisions are taken into account (though for inflation the revisions are rather minor, occurring once a year only). it is assumed that there are no aggregation errors (that is that the percentage m-o-m change of the total cpi is equal to weighted sum of changes of the sub-aggregate categories), which is in accordance with (7). moreover, the weights are assumed to be known, which is not always the case in real time, as the weights for a calendar year are published in march. the disaggregate categories are listed in table 1, together with average weights in the sample period. table 1. disaggregate cpi sub-categories used in the empirical analysis no. code category label average weight 1 1a food and non-alcoholic beverages 0.258 2 1b alcoholic beverages, tobacco 0.060 3 2 clothing and footwear 0.052 4 3a dwelling: housing, water (excluding energy) 0.116 5 3b dwelling: electricity, gas and other fuels 0.089 6 3c dwelling: furnishings, household equipment and routine maintenance of the house 0.049 7 4 health 0.050 8 5a transport: fuels for personal transport equipment 0.045 9 5b transport: excluding fuels 0.044 10 6 communication 0.049 11 7 recreation and culture 0.070 12 8 education 0.013 13 9 restaurants and hotels 0.052 14 10 miscellaneous goods and services 0.053 note: the average weight is computed as average over the years included in the sample (as the sample end does not correspond to december). the benchmark models considered in the comparison are stationary and include unrestricted var models with one and two lags, as well as ar(1) process applied at the aggregate level, and a result of the specification search applied to ar(p) process for the aggregate data (resulting in omission of certain own lags of inflation). for the stochastic specification search algorithm, the maximum number of autoregressive parameters is limited to 42 (which reflects the idea that on average there are 3 parameters per equation, though these need not be uniformly allocated) and the maximum lag order is set to be equal to 13. additional parameters (not included in the parameter count mentioned here) are the seasonal dummy variables, though the stochastic algorithm also switches between inclusion and exclusion of the seasonal dummies in each equation. moreover, as for the contemporaneous covariance matrix two options are separately considered. in the first version, błażej mazur dynamic econometric models 15 (2015) 71–87 84 the σ matrix is assumed to be unrestricted (being of course positivedefinite), in the other version it is assumed to be diagonal. the results are reported in table 2., including the number of parameters (with sub-total for intercept/seasonal terms). the criteria of forecasting performance include rmsfe and average crps (the lower the better) and summed up (decimal) log-score (the higher the better). all the criteria show consistent results (neither is contradicting the other ones). table 2. comparison of the results from alternative specifications: predictive performance for aggregate inflation based on aggregate/disaggregate data model σ st. dev. rmsfe crps logscore no. of parameters total intercepts ar(1) for total cpi – 0.248 0.250 0.142 –0.726 14 12 ar(2) for total cpi – 0.249 0.249 0.142 –0.657 15 12 ar with variable selection total cpi – 0.294 0.224 0.129 0.306 7 1 var(1) nd 0.268 0.280 0.156 –2.283 455 168 var(2) nd 0.280 0.348 0.180 –4.949 651 168 var with variable selection d 0.228 0.174 0.100 5.058 147 91 var with variable selection nd 0.247 0.187 0.107 3.668 224 102 note: full estimation results are available from the author by request. verification period: 2012m01– 2015m08; the forecasts are one-period-ahead (which amounts to nowcasting due to the publication lag). crps (continuous ranked probability score) is averaged over the realized forecasts, for log-score the sum of decimal logs is taken. for ar(1), ar(2), var(1),var(2) seasonal dummies are included. the second column indicates assumptions regarding the contemporaneous variance-covariance matrix (relevant for disaggregate models only), with ‘d’ denoting a diagonal matrix, whereas ‘nd’ denotes nondiagonal one; ‘st. dev.’ denotes ex ante forecast error (standard deviation of the predictive distribution) averaged over all forecasts. the column labeled ‘intercepts’ takes into account parameters corresponding to seasonality modelling as well. unrestricted vars for disaggregate data show relatively poor performance, deteriorating with the number of lags. ar(1) and ar(2) models are similar, though ar with variable selection is clearly better. poor performance of the ar(1), ar(2), var(1) and var(2) models results from restrictive assumptions about the highest lag order allowed, though var(2) is much worse than ar(2), so the other aspects seem to matter as well. vars with variable selection are clearly the best models, strongly dominating the other specifications by all the criteria. moreover, the model assuming diagonal contemporaneous covariance matrix is getting slightly better results. in both cases ex ante errors are large compared to the rmsfe, consequently the forecast densities seems to be overdispersed. one caveat should be mentioned here: as the models with non-diagonal covariance matrix of shocks and exclusion restrictions are computationally more demanding, the specification search algorithm might be less efficient density forecasts based on disaggregate data: nowcasting polish inflation dynamic econometric models 15 (2015) 71–87 85 there. it is therefore possible that the algorithm used here gets trapped in a local optimum, which is not the global one, and the result could be somewhat improved upon a more intensive specification search. however, the practical conclusion would be that it is easier to find a useful model assuming uncorrelated shocks. such a simplification is quite important from computational point of view as well. the predictive gains obtained here are non-negligible, though the attention is restricted to one-period-ahead forecasts. however, the disaggregate information is likely to be useful in rather short horizons only. moreover, it seems that the dependence between variables induced by incorporation of lags of other variables into equations is more important than the dependence represented by contemporaneous correlations of errors (at least from the viewpoint of quality of the forecasts of the aggregate series). conclusions the paper aims at demonstrating potential gains in terms of density forecasting that can be obtained by allowing for the use of disaggregate data. the analysis of density forecasts reflects the importance of the issue of uncertainty quantification in current econometric literature. the explicit objective of forecasting of the aggregate series only is introduced here. consequently, from a theoretical point of view it might be plausible to consider all the models for sub-aggregates jointly, as one model (even if the individual models are fully stochastically independent). the comparison of such models (or the corresponding sets of exclusion restrictions) is based on the predictive performance of the aggregate series. the results obtained for polish cpi inflation series confirm that considering multivariate models for the disaggregate data might result in substantial improvements in terms of prediction of the aggregate inflation. the conclusion remains true for various criteria for both point and density ex post forecast evaluation. however, the result stems from introducing parsimonyoriented exclusion restrictions. some cross-variable dependence matters, as the best performing models include lags of other variables as well. on the other hand, there is no evidence that allowing for contemporaneous correlations of the shocks in disaggregate series is really important for the aggregate predictive performance in the case under consideration. there are many possibilities to extend the scope of the analysis presented here, in particular by considering more general model classes. besides that, two aspects are worth pointing out. błażej mazur dynamic econometric models 15 (2015) 71–87 86 firstly, a fully consistent theoretical framework for statistical inference in such situations should be developed. it was pointed out to the author that such a development could be based on a reparametrization of the observation space (at the disaggregate level) in such a way that the aggregate variable is explicitly considered3. this opens possibility for considering specifications taking advantage of concepts like the bayesian cut. secondly, the stochastic search algorithm used here to explore the model space (or, equivalently, the space containing various sets of exclusion restrictions) was rather simple and heuristic, with no theoretical premises for its efficiency. perhaps some solutions used in the stochastic search variable selection setup could be adopted to match the specification search problem considered here. the most important conclusion from the empirical analysis provided is that even within quite a simple class of multivariate models for disaggregate data a substantial improvement in predictive performance (over the standard unrestricted specifications) is possible. however, it requires a thorough specification search in terms of variable selection, which might be non-trivial when the dimension is large, which corresponds to detailed disaggregation. references aron, j., muellbauer, j. (2013), new methods for forecasting inflation, applied to the us*, oxford bulletin of economics and statistics 75(5), 637–661, doi: http://dx.doi.org/10.1111/j.1468-0084.2012.00728.x. castle, j. l., hendry, d. f. (2010), nowcasting from disaggregates in the face of location shifts, journal of forecasting, 29(1–2), 200–214, doi: http://dx.doi.org/10.1002/for.1140. clark, t. (2006), disaggregate evidence on the persistence of consumer price inflation, journal of applied econometrics, 21(5), 563–587, doi: http://dx.doi.org/10.1002/jae.859. clark, t., ravazzolo, f. (2015), macroeconomic forecasting performance under alternative specifications of time-varying volatility, journal of applied econometrics, 30(4), 551–575, doi: http://dx.doi.org/10.1002/jae.2379. dees, s., guntner, j. (2014), analysing and forecasting price dynamics across euro area countries and sectors: a panel var approach, economics working papers 2014-10, department of economics, johannes kepler university linz, austria. doornik, j. a., ooms, m. (2003), computational aspects of maximum likelihood estimation of autoregressive fractionally integrated moving average models, computational statistics & data analysis, 42(3), 333–348, doi: http://dx.doi.org/10.1016/s0167-9473(02)00212-8. 3 the author would like to thank jacek osiewalski for the valuable suggestion. density forecasts based on disaggregate data: nowcasting polish inflation dynamic econometric models 15 (2015) 71–87 87 faust, j., wright, j. h. (2013), forecasting inflation, in elliott g., timmermann. a. (eds.), handbook of economic forecasting, vol. 2a, amsterdam, north holland, doi: http://dx.doi.org/10.1016/b978-0-444-53683-9.00001-3. frühwirth-schnatter s., wagner. h. (2010), stochastic model specification search for gaussian and partial non-gaussian state space models, journal of econometrics, 154, 85– –100, doi: http://dx.doi.org/10.1016/j.jeconom.2009.07.003. george, e. i., sun, d., ni. s. (2008), bayesian stochastic search for var model restrictions, journal of econometrics, 142(1), 553–580, doi: http://dx.doi.org/10.1016/j.jeconom.2007.08.017. giacomini, r., granger, c. (2004), aggregation of space-time processes, journal of econometrics, 118(1-2), 7–26, doi: http://dx.doi.org/10.1016/s0304-4076(03)00132-5. gneiting, t., raftery, a. (2007), strictly proper scoring rules, prediction, and estimation, journal of the american statistical association, 102(477), 359–378, doi: http://dx.doi.org/10.1198/016214506000001437. hendry, d. f., hubrich, k. (2011), combining disaggregate forecasts or combining disaggregate information to forecast an aggregate, journal of business & economic statistics, 29(2), 216–227, doi: http://dx.doi.org/10.1198/jbes.2009.07112. hubrich, k.. (2005), forecasting euro area inflation: does aggregating forecasts by hicp component improve forecast accuracy?, international journal of forecasting, 21(1), 119–136, doi: http://dx.doi.org/10.1016/j.ijforecast.2004.04.005. huwiler, m., kaufmann, d. (2013), combining disaggregate forecasts for inflation: the snb's arima model, economic studies 2013-07, swiss national bank. ibarra, r. (2012), do disaggregated cpi data improve the accuracy of inflation forecasts?, economic modelling, 29(4), 1305–1313, doi: http://dx.doi.org/10.1016/j.econmod.2012.04.017. lütkepohl, h. (2009), forecasting aggregated time series variables: a survey, economics working papers eco2009/17, european university institute. stock, j. h., watson, m. (2015), core inflation and trend inflation, nber working paper no. 21282. własności rozkładów predyktywnych z modeli dla danych zdezagregowanych: prognoza inflacji w polsce z a r y s t r e ś c i. w artykule podjęto kwestię weryfikacji występowania korzyści w zakresie poprawy jakości prognostycznej (oceniając także trafność ex post rozkładów prognoz) w przypadku prognozowania agregatu na podstawie modeli dla danych zdezagregowanych. problem ten rozpatrywano w ramach modelu wektorowej autoregresji z restrykcjami, przy czym alternatywne specyfikacje odpowiadały różnym układom restrykcji zerowych. zaprezentowano empiryczną analizę stopy inflacji cpi m/m w polsce (rozpatrując 14 podkategorii dla danych zdezagregowanych). modele z restrykcjami dla danych zdezagregowanych prowadziły do lepszych prognoz agregatu w porównaniu do modeli dla danych zagregowanych i modeli dla danych zdezagregowanych bez restrykcji (do porównania prognoz wykorzystano kryteria dla rozkładów prognoz takie jak crps oraz logarytm gęstości predyktywnej). s ł o w a k l u c z o w e: predykcja, porównanie modeli, rozkład predyktywny, inflacja, modele var pusta strona dem_2015_5to26 © 2015 nicolaus copernicus university press. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2015.001 vol. 15 (2015) 5−26 submitted september 11, 2015 issn (online) 2450-7067 accepted december 15, 2015 issn (print) 1234-3862 elżbieta szulc*, dagna wleklińska spatio-temporal analysis of convergence of development level of selected stock exchanges in the period of 2004–2012 a b s t r a c t. the paper concerns the convergence of selected stock exchanges from the point of view of their development. it presents the methodological approach which points up taking into account spatial and economic connections among stock markets in convergence analyses. in this analysis the need for division of the stock exchanges according to a spatial regimes is pointed up as well. the research includes 42 largest trading floors analyzed in the period of 2004-2012. the empirical data refer to six diagnostic variables acknowledged as the important determinants of the development of stock markets. k e y w o r d s: stock exchanges, convergence, spatial regimes, physical and economic distance, connectivity matrix, spatial panel models. j e l classification: c10, c12, c58, g15. introduction the paper concerns the convergence of selected stock exchanges, with european stock exchanges on the one hand and the asian and american on the other, from the point of view of their development. the study is a continuation of our previous investigation, the results of which were published in dynamic econometric models, 2014 (14) (szulc et al., 2014, pp. 125– –144). the main findings of the quoted work were as follows: (1) including the linkages that result from physical and/or economic distance between * correspondence to: elżbieta szulc, nicolaus copernicus university, department of econometrics and statistics, ul. gagarina 13a, 87-100 toruń, poland, e-mail: eszulc@umk.pl elżbieta szulc, dagna wleklińska dynamic econometric models 15 (2015) 5–26 6 stock exchanges in their convergence models is justified and crucial for the analysis of this phenomenon. (2) as a result, it is possible to define the influence of the distance between exchanges on their economic development, the estimates of convergence parameter are more precise, and some statistical properties of the models are better. (3) due to the heteroskedasticity, the empirical panel models for the exchanges investigated as a whole were not entirely satisfactory. it means that there are differentials in relationship between objects considered and their speed of convergence. (4) in some empirical models which we obtained there appeared the problem of autocorrelation of residuals. motivated by the desire to improve the properties of the empirical models, firstly we decided to establish some spatial regimes and then repeat the research with the division of stock exchanges. the applied categorization involved placing european stock exchanges on one side, and the american and asian stock markets on the other. the validity of this choice was confirmed by the results of the chow test on spatial variability of the model parameters (arbia 2006, p. 133) presented in section 4 of this paper. the discussion on the convergence of stock exchanges is associated with one of the directions of the analysis of the relationship between capital markets, which searches out the ever-increasing convergence of these markets from the point of view of their specific characteristics. this process, which can be referred to as convergence of stock exchanges is associated with an integration of the financial markets, and their growing interdependence, which in turn is associated with the liberalization of capital flows and technological progress. these processes are favorable for further development of stock markets, and thus the distinctions between them are becoming increasingly blurred over time. the problem of convergence of equity markets has already been considered in the literature on the subject, e.g. aspergis, christou, miller (2014), koralun-bereźnicka (2008), fraser, helliar, power (1994), caporale, erdogan, kuzin (2009). in particular, the papers presenting the analysis of convergence of the stock exchanges with the prospect of space are the most interesting, e.g. asgharian, hess, liu (2013), suchecka, łaszkiewicz (2011), wójcik (2009). the literature indicates the validity of the analysis of the relationship between capital markets having regard on their location in the geographic space, and also takes into account the so-called economic distance between these markets. this paper refers to such a methodological approach. the aim of the paper is to investigate whether, in the light of the current empirical analyses, one may observe the process of convergence of the main spatio-temporal analysis of convergence of development level… dynamic econometric models 15 (2015) 5–26 7 european as well as american and asian stock markets. in addition, the importance of distance between the markets for the process is evaluated. particularly, the role of economic distance is considered. the research is aimed to verify the hypothesis that the relative location of a stock market in the geographic as well as a particular economic space affects its growth rate. the achievement of the objective and verification of the research hypothesis was enabled through: defining a taxonomic measure of development of exchanges, then building of empirical dynamic models of stock exchanges’ convergence for pooled time series and cross-sectional data and for panel data in the traditional version, which ignore spatial and economic linkages between stock exchanges and dynamic spatial models (including spatial panel models), i.e. models with regard to the described relationship, and finally the comparison of statistical properties and the interpretation values of model’ parameters in various versions. 1. subject and range of the investigation the subject of the investigation contains the selected european, asian and american stock exchanges, characterized in terms of their level of development. the study included 42 largest trading floors in the period of 2004–2012. the specification of the exchanges with the assignment to the relevant country is presented in table 1. the level of stock exchange development was defined by a synthetic measure based on six diagnostic variables, i.e. x1 − the capitalization of domestic shares, x2 − the capitalization of newly listed domestic shares, x3 − the total value of share trading, x4 − gdp per capita, x5 − the top 10 most heavily capitalized domestic companies, x6 − the ratio of market capitalization to gdp. it was recognized that, in the light of theory and empirical analyses, the specified variables are important determinants of the development of stock exchanges (see e.g. demirgur-kunt and levine, 1996; levine and zelvos, 1996; łuniewska and tarczyński, 2006; szulc et al., 2014; wiśniewski, 2003). taking into account the connections of the capital market with the economy of the country of its location was also an important issue for the specification of the diagnostic variables. the range of information provided by the world federation of exchanges (www.worldexchanges.org), whence we get the data, played a significant role as well. elżbieta szulc, dagna wleklińska dynamic econometric models 15 (2015) 5–26 8 table 1. specification of the stock exchanges considered north.south america and asia brazil bm&bovespa (bov) chile santiago se (sse) canada tmx group (tmx) colombia colombia se (cse) mexico mexican exchange (bmv) bermuda bermuda se (bsx) argentina buenos aires se (bcba) peru lima se (bvl) united states nasdaq omx (nasdaq) nyse euronext (us) (nyse) singapore singapore se (se) thailand thailand se (thse) philippines phillippine se (pse) china honk kong se (sehk) shanghai se (shse) shenzen se (szse) taiwan se corp. (tsec) japan osaca se (ose) tokyo se group (tse) india national se india (nse) indonesia indonesia se (ise) south korea korea exchange (krx) malaysia bursa malaysia (bm) sri lanka colombo se (clse) europe austrian wiener borse (ag) cyprus cyprus se (cpse) egypt cairo&alexandria se (egx)* greece athens exchange (athex) spain bme spanish exchange (bme) netherlands nyse euronext (europe) (nee) turkey istanbul se (isse) ireland irish se (irse) israel tel aviv se (tase)* luxemburg luxemburg se (lxse) malta malta se (mse) hungary budapest se (bdse) germany deutsche borse (db) norway oslo bors (obe) poland warsaw se (wse) great britain london se (lse) switzerland six swiss exchange (six) sweden nasdaq omx nordic exchange (nomx) note: stock exchanges signed by * have been included in the european capital markets for the reason of their geographical proximity to the continent and their economic similarity as well. 2. methodology the research was conducted in relation to the aggregate characteristic of the stock exchanges in the form of a taxonomic measure of development. this indicator is understood as a synthetic normalized formula expressed by (see hellwig, 1968): spatio-temporal analysis of convergence of development level… dynamic econometric models 15 (2015) 5–26 9 , 2 1' q i i sq q q + −= (1) where: iq − the synthetic variable determining the level of development of the i-th exchange in relations to a development standard, q − the average value of the synthetic variable, qs − the standard deviation of the variable. in this approach the values of the synthetic variable iq are calculated according to the formula: ( ) , 1 2 0∑ = −= m j jiji zzq (2) where: ijz − the value of j-th diagnostic variable for i-th exchange standardized to 0–1, jz0 − the value of j-th diagnostic variable for the standard of development standardized to 0–1. thus, iq means a distance between i-th exchange and the development standard. through the use of the taxonomic measure of stock exchanges’ development it is possible to present the rankings of exchanges and their changes in time, the evaluation of the correlation between stock exchanges in terms of development, the identification of linkages between markets in an economic space, and finally, the analysis of the stock exchanges’ convergence, which is meant as equalizing their development levels. in this paper we focus on the examination of the concept of exchanges’ convergence, in the light of which the stock market with an initial lower level of development showed a faster growth rate in the considered measure of development. the analysis of the stock exchanges' convergence was based on econometric models of β-convergence, in particular, the spatial models for pooled time series and cross-sectional data (tscs) as well as spatial panel models. the same classes of models were used previously (szulc et al., 2014).the premises for the application of spatial models are as follows: − the result of including the spatial linkages between stock exchanges in the models of their convergence is better evaluation of the convergence phenomenon on the grounds of β parameter estimated. the estimate of elżbieta szulc, dagna wleklińska dynamic econometric models 15 (2015) 5–26 10 the parameter reflects more accurately the impact of the base level of development of a given stock exchange on the growth rate of the stock exchange characteristics in question because such estimate is not influenced by omitting spatial relationship. − the use of spatial models provides the opportunity to measure and interpret the impact of connections of a given stock exchange with others on its development. the spatial models for pooled time series and cross-sectional data (tscs) are presented by formulas 3 and 4, whereas the formulas 5 and 6 refer to the spatial panel models. the model tscs with spatial component takes the form of the spatial autoregressive model (sar_pooled), i.e.: [ ] ,lnlnln ' 1 ' ' 1' 1 ' it jt jt ij ijit it it q q wq q q ερβα +         ++=      −≠ − − ∑ (3) or of the model with spatial autoregressive residuals (se_pooled), i.e.: [ ] ,lnln ' 1' 1 ' itit it it q q q ηβα ++=      − − .itjt ij ijit w εηλη += ∑ ≠ (4) the spatial panel models used in the investigation were as follows: [ ] ,lnlnln ' 1 ' ' 1' 1 ' it jt jt ij ijiti it it q q wq q q ερβα +         ++=      −≠ − − ∑ (5) i.e. the spatial autoregressive panel model with individual fixed effects (the spatial autoregressive fixed-effect model) (sar_fe_ind) and [ ] ,lnln ' 1' 1 ' ititi it it q q q ηβα ++=      − − ,itjt ij ijit w εηλη += ∑ ≠ (6) i.e. the spatial error panel model with individual fixed effects (se_fe_ind). elements wij in the formulas (3)–(6) come from connectivity matrix w which refers to the linkages between exchanges considered. various types of weights wij may be pointed out according to the established criteria (see e.g. haining, 2005, pp. 83–84). in this paper the linkages between stock exchanges will be defined with the use of two approaches. the first uses a matrix of connections with weights established on the basis of the physical distance between the centers of the countries where the stock exchanges are located. the second consists spatio-temporal analysis of convergence of development level… dynamic econometric models 15 (2015) 5–26 11 in the consideration of the economic distance in the matrix of connections. the essence of the second approach is to establish similarity of the exchanges on the basis of the value of the taxonomic measure of exchanges’ development. the quantification of the spatial linkages between stock exchanges on the basis of the geographical distance was carried out according to the following scheme: 1. determining the spatial relationships using the linkages matrix s, with elements:     = ≠ = ,if0, if, 1 ki ki ds ikik (7) where: ikd − the physical distance between capitals of the countries where the i-th and the k-th stock exchanges are located. 2. row standardization of the connectivity matrix to one, i.e.: . 1 ∑ = = n k ik ik ik s s w (8) 3. construction of the block matrix of connections, i.e.: , 9 2 1             = w00 0w0 00w w l momm l l (9) where: 921 ... www === – matrixes of the spatial connections based on the physical distance, the same for all the considered years. in the second approach, i.e. with the use of an economic distance between stock exchanges, the following scheme was used: 1. determining the linkages between stock exchanges with the use of an economic distance, expressed by the formula: ( ) , 1 2 ∑ = −= m j kjijik zzd (10) where: ikd − the economic distance between i-th and k-th stock exchange, elżbieta szulc, dagna wleklińska dynamic econometric models 15 (2015) 5–26 12 kjij zz , − the values of standardized diagnostic variables for each i-th and k-th stock exchange, j = 1, 2, …, 6 − the number of the diagnostic variable. 2. construction of the matrix of linkages between stock exchanges, with elements:     = ≠ = .if0, if, 1 ki ki dw ikik (11) 3. row standardization of the connectivity matrix to one, i.e.: . 1 ∑ = ∗ = n k ik ik ik w w w (12) 4. construction of the block matrix of cross-sectional and time connections which may be described in the following form: , 9 2 1               = ∗ ∗ ∗ ∗ w00 0 0w0 00w w l mom l l (13) where: 921 ... ∗∗∗ ≠≠≠ www – matrixes of connections, taking into account the economic distance between exchanges, different for successive years. in order to evaluate the quality of the empirical models in the investigation the following tools were used: the moran test for verifying spatial independence of the residuals, the lagrange multiplier tests (lmlag, lmerr) and their robust versions (rlmlag, rlmerr) as spatial dependence diagnostics, the likelihood ratio test (lr) for testing the significance of the spatial dependence, the breusch-pagan heteroskedasticity test, the chow test for verifying the spatial changeability of β parameters and the need for including fixed effects in the spatial panel models (on the tools see e.g. arbia, 2006; millo and piras, 2012; mutl and pfaffermayr, 2011; baltagi et al., 2003; suchecki (ed.), 2012). all calculations were performed with r (version 3.0.1) and the graphical illustrations – with the use of mapviever and corel. spatio-temporal analysis of convergence of development level… dynamic econometric models 15 (2015) 5–26 13 3. preliminary data analysis figure 1 shows locations of the investigated exchanges on the world map and bar charts of taxonomic measure of development (tmd) in the years 2004–2012. this presentation allows us to observe changes in the level of development of the individual stock exchanges and a comparison of the dynamics of changes by their spatial location as well. it is worth noting that most of the developing economies’ stock exchanges, both on the european continent, as well as american and asian, are characterized by a relatively stable level of the taxonomic measure of development throughout the whole adopted time horizon, even during a sharp slowdown in the economic conditions caused by the global financial crisis. this finding is particularly evident in relation to such exchanges as e.g. bcba, bvl, sse, bdse, eez, clse, ise, pse. figure 1. bar charts of tmd for the investigated stock exchanges in the years 2004–2012 figures 2 and 3 show the value of the taxonomic measure of development (surface of the wheel) for each stock exchange included in the study for the year 2004 and 2012, respectively. this graphical presentation is useful for a preliminary assessment of changes in the global capital market over the considered period. in 2004, two dominant financial centers are clearly visible. in the west, it is nyse and nasdaq, while in central europe, the london stock exchange and nyse euronext europe stand out in particular. in turn, in 2012 a slight strengthening of the position of the two largest us stock exchanges: nyse and nasdaq may be observed. however, the most spectacular changes can be seen in the case of the nomx central european stock exchange. nomx has strengthened at the expense of two neighboring elżbieta szulc, dagna wleklińska dynamic econometric models 15 (2015) 5–26 14 stock exchanges lse and nee, gaining a leading position in 2012 and clearly outperforming their level of development. with regard to the second group of the analyzed exchanges, there were no significant changes in the values of the synthetic measure of development. therefore nyse and nasdaq are again placed in the dominant position of the ranking. figure 2. the taxonomic measure of stock exchanges’ development in 2004 figure 3. the taxonomic measure of stock exchanges’ development in 2012 4. results of the econometric analysis in order to justify the division of the considered stock exchanges into two groups the chow test of spatial changeability of β parameters was applied. the results are presented in table 2. the hypothesis that parameters in β-convergence models estimated in the investigation are constant should be rejected. this leads to the identification of spatial regimes and means that the spatio-temporal analysis of convergence of development level… dynamic econometric models 15 (2015) 5–26 15 convergence of the european and asian/american stock markets should be investigated separately. table 2. results of the tests for spatial invariance of the β-convergence parameters models linear regression spatial autoregressive model spatial error model variant i variant ii variant i variant ii values of chow test 210.709 113.517 117.036 113.557 121.783 p-value 0.0000 0.0000 0.0000 0.0000 0.0000 the successive tables presented below contain information on the usefulness of various methodological concepts expressed by the spatial models presented in section 2, in comparison with the linear regression model, i.e. the traditional model not including the spatial effects. tables 3–6 refer to the empirical models obtained for the european stock exchanges, and tables 7– –10 for the asian and american stock exchanges. in tables 3 and 4 there are presented the results of the estimation and verification of the three models for pooled time series and cross-sectional data: the linear regression model (tscs), the spatial autoregressive model (sar_pooled) and the spatial error model (se_pooled). table 3 contains the results obtained in the case when, for the purpose of quantification of the connections among the investigated exchanges, the matrix w of the physical distance between them was used (variant i). table 4 presents the analogical results, but in the spatial models there was used the connectivity matrix w* of the economic distance between the exchanges (variant ii). the classical model estimated with the use of the pooled time series and cross-sectional data does not satisfy the fundamental criteria of statistical verification. the main drawback of this model is autocorrelation of residuals, which is confirmed by the result of the moran test (see tables 3 and 4). in order to propose an alternative opposed to the classical model the lagrange multiplier tests (lm) were used (see tables 3 and 4). the lm tests for the linear model for the pooled time series and cross-sectional data used consider the spatial lag model (spatial autoregressive) and the spatial error model as alternatives (lmlag and lmerr, respectively). tables 3 and 4 report the results of using the robust tests (rlmlag, in which h0: ρ = 0 under the assumption that λ ≠ 0 and rlmerr, where h0: λ = 0 under the assumption that ρ ≠ 0) as well. elżbieta szulc, dagna wleklińska dynamic econometric models 15 (2015) 5–26 16 table 3. results of the estimation and verification of β-convergence models for pooled time series and cross-sectional data, obtained for european stock exchanges – variant i linear regression spatial autoregressive model spatial error model parameters α β ρ λ –0.2087 (0.0055) –0.1356 (0.0007) – – –0.1316 (0.0275) –0.0789 (0.0128) 0.6779 (0.0000) – –0.0533 (0.4338) –0.0527 (0.0704) – 0.6964 (0.0000) goodness of fit adjusted r2 aic 0.0711 –60.5170 – –109.7800 – –107.0800 heteroskedasticity breuch-pagan test 3.2874 (0.0698) 2.2977 (0.1296) 2.7537 (0.0970) autocorrelation of residuals moran test 11.9203 (0.0000) –1.1350 (0.1282) –0.6776 (0.2490) spatial dependence lr lmlag lmerr rlmlag rlmerr – 145.0560 (0.0000) 116.1595 (0.0000) – – 51.2580 (0.0000) – – 35.9081 (0.0000) – 48.5600 (0.0000) – – – 7.0116 (0.0081) speed of convergence half-life 0.0182 38,05 0.0103 67.47 0.0068 102.42 note: numbers in brackets refer to the p-values. since the lmlag tests are more significant than the lmerr, and the rlmlag tests are more significant than the rlmerr, the spatial lag models should be preferred. subsequently, the significance of the spatial effects in the sar and se models using the likelihood ratio test (lr) was confirmed (see tables 3 and 4). the results show that the statistical properties of the obtained empirical models are the same (the spatial autocorrelation of residuals of linear regression model, significant lm statistics and significant spatial spatio-temporal analysis of convergence of development level… dynamic econometric models 15 (2015) 5–26 17 effects confirmed by the lr test), irrespective of which connectivity matrix (of physical or of economic distance) was used in the spatial models. table 4. results of the estimation and verification of β-convergence models for pooled time series and cross-sectional data, obtained for european stock exchanges – variant ii linear regression spatial autoregressive model spatial error model parameters α β ρ λ –0.2087 (0.0055) –0.1356 (0.0007) – – –0.1219 (0.0336) –0.0747 (0.0144) 0.7442 (0.0000) – –0.0638 (0.4084) –0.0800 (0.02534) – 0.7920 (0.0000) goodness of fit adjusted r2 aic 0.0711 –60.5170 – –118.4300 – –117.5600 heteroskedasticity breuch-pagan test 3.2874 (0.0698) 2.2312 (0.1353) 2.1093 (0.1464) autocorrelation of residuals moran test 12.9587 (0.0000) 1.4475 (0.0739) 1.7664 (0.0387) spatial dependence lr lmlag lmerr rlmlag rlmerr – 156.2542 (0.0000) 137.3223 (0.0000) − − 59.9100 (0.0000) – – 27.8846 (0.0000) – 59.0380 (0.0000) – – – 8.9527 (0.0028) speed of convergence half-life 0.0182 38.05 0.0097 71.42 0.0104 66.50 note: numbers in brackets refer to the p-values. similarly, irrespective of which the connectivity matrix was applied in the spatial models, parameters ρ and λ are statistically significant. it is worth noting that the fact of including the connectivity matrixes in the considered models has a crucial impact on convergence parameters (β). absolute values elżbieta szulc, dagna wleklińska dynamic econometric models 15 (2015) 5–26 18 of the parameters for the sar and se models are lower than for the traditional model which does not take into account the connections across the investigated stock exchanges. table 5. results of the estimation and verification of panel models with fixed effects obtained for the european stock exchanges – variant i fe_ind sar_fe_ind se_fe_ind parameters α β ρ λ –1.6250 (0.0000) –0.8927 (0.0000) – – –1.2397 (0.0000) –0.6785 (0.0000) 0.3543 (0.0006) – –1.5821 (0.0000) –0.8696 (0.0000) – 0.5218 (0.0000) goodness of fit adjusted r2 aic 0.4375 –117.1200 – –127.0400 – –132.3600 heteroskedasticity breuch-pagan test 30.4129 (0.0336) 27.5709 (0.0689) 28.4229 (0.0559) autocorrelation of residuals moran test 7.0928 (0.0000) 2.0868 (0.0185) –0.4411 (0.3296) spatial dependence lr lmlag lmerr rlmlag rlmerr – 16.7366 (0.0000) 38.3239 (0.0000) − − 11.9190 (0.0006) – – 0.1360 (0.7123) – 17.2380 (0.0000) – – – 21.7233 (0.0000) chow test f – 65.9038 (0.0000) 83.4606 (0.0000) speed of convergence half-life 0.2790 2.48 0.1418 4.89 0.2546 2.72 note: numbers in brackets refer to the p-values. tables 5 and 6 contain the results of the estimation and verification of exemplary panel models used in the investigation, i.e. the panel model with fixed effects without the spatial component (fe_ind), the spatial autospatio-temporal analysis of convergence of development level… dynamic econometric models 15 (2015) 5–26 19 regressive panel model with fixed effects (sar_fe_ind), and the spatial error panel model with fixed effects (se_fe_ind). just as in the pooled time and cross-sectional data models also in the panel data models the connections among the stock exchanges in two variants (connections according to physical/economic distance) were taken into account. fixed effects are significant in the considered models. it means that individual characteristics of every exchange are valid for their convergence. table 6. results of the estimation and verification of panel models with fixed effects obtained for the european stock exchanges – variant ii fe_ind sar_fe_ind se_fe_ind parameters α β ρ λ –1.6250 (0.0000) –0.8927 (0.0000) – – –1.1442 (0.0000) –0.6248 (0.0000) 0.4542 (0.0000) – –1.6095 (0.0000) –0.8894 (0.0000) – 0.6309 (0.0000) goodness of fit adjusted r2 aic 0.4375 –117.1200 – –132.0000 – –138.2700 heteroskedasticity breuch-pagan test 30.4129 (0.0336) 27.3471 (0.0727) 27.4311 (0.0713) autocorrelation of residuals moran test 7.4093 (0.0000) 3.0568 (0.0011) 1.2405 (0.1074) spatial dependence lr lmlag lmerr rlmlag rlmerr – 22.4971 (0.0000) 41.8167 (0.0000) – – 16.8810 (0.0000) – – 0.4012 (0.5265) – 23.1530 (0.0000) – – – 19.7207 (0.0000) chow test f – 22.0179 (0.0000) 28.7812 (0.0000) speed of convergence half-life 0.2790 2.48 0.1225 5.66 0.2752 2.52 note: numbers in brackets refer to the p-values. elżbieta szulc, dagna wleklińska dynamic econometric models 15 (2015) 5–26 20 the diagnostics of the considered models suggests that the classical panel model is the worst among them. in this case, the breusch-pagan statistic is significant (at the level of significance γ = 0.05), leading to the rejection of the model assumption of homoskedasticity. in addition, on the basis of the moran test the hypothesis of the independence of the model residuals should be rejected (comp. the tscs model). table 7. results of the estimation and verification of β-convergence models for pooled time series and cross-sectional data, obtained for asian and american stock exchanges – variant i linear regression spatial autoregressive model spatial error model parameters α β ρ λ –0.4678 (0.0000) –0.2084 (0.0000) – – –0.4181 (0.0000) –0.1845 (0.0000) 0.3065 (0.0000) – –0.3749 (0.0000) –0.1751 (0.0000) – 0.3188 (0.0000) goodness of fit adjusted r2 aic 0.0987 –31.8650 – –51.2710 – –49.7970 heteroskedasticity breuch-pagan test 18.7771 (0.0000) 19.1601 (0.0000) 18.4701 (0.0000) autocorrelation of residuals moran test 6.1474 (0.0000) –0.5729 (0.2834) –0.3759 (0.3535) spatial dependence lr lmlag lmerr rlmlag rlmerr – 39.7261 (0.0000) 32.3990 (0.0000) – – 21.4060 (0.0000) – – 12.2581 (0.0005) − 19.9320 (0.0000) – – – 4.9310 (0.0264) speed of convergence half-life 0.0292 23.73 0.0255 27.19 0.0241 28.81 note: numbers in brackets refer to the p-values. spatio-temporal analysis of convergence of development level… dynamic econometric models 15 (2015) 5–26 21 the necessity of model re-specifications towards the spatial panel models was also confirmed with the lagrange multiplier tests. all the tests are statistically significant, except for the robust version (rlmlag), which suggests that the spatial error panel model should be preferred. moreover, the significance of the spatial effects with the aid of the lr test has been confirmed (see tables 5 and 6). table 8. results of the estimation and verification of β-convergence models for pooled time series and cross-sectional data, obtained for asian and american stock exchanges – variant ii linear regression spatial autoregressive model spatial error model parameters α β ρ λ –0.4678 (0.0000) –0.2084 (0.0000) – – –0.3915 (0.0000) –0.1751 (0.0000) 0.6866 (0.0000) – –0.3812 (0.0002) –0.2014 (0.0000) – 0.7268 (0.0000) goodness of fit adjusted r2 aic 0.0987 –31.8650 – –73.7960 – –76.4300 heteroskedasticity breuch-pagan test 18.7771 (0.0000) 18.4305 (0.0000) 17.8364 (0.0000) autocorrelation of residuals moran test 11.8902 (0.0000) 1.7776 (0.0377) 1.8976 (0.0289) spatial dependence lr lmlag lmerr rlmlag rlmerr – 120.0009 (0.0000) 117.0748 (0.0000) − − 43.9310 (0.0000) – – 3.2764 (0.0703) – 46.5650 (0.0000) – – – 0.3503 (0.5540) speed of convergence half-life 0.0292 23.73 0.0241 28.81 0.0281 24.66 note: numbers in brackets refer to the p-values. elżbieta szulc, dagna wleklińska dynamic econometric models 15 (2015) 5–26 22 for the purpose of investigating the reasonableness of including the fixed effects in the spatial models there was applied the chow test which considers the spatial model for pooled tscs data vs. the spatial panel model with fixed effects. the results of the chow test have pointed out the statistical significance of the fixed effects in the spatial autoregressive panel model, as well as in the panel spatial error model (see tables 5 and 6). table 9. results of the estimation and verification of panel models with fixed effects obtained for asian and american stock exchanges – variant i fe_ind sar_fe_ind se_fe_ind parameters α β ρ λ –1.5180 (0.0000) –0.9213 (0.0000) – – –1.4074 (0.0000) –0.8537 (0.0000) 0.2192 (0.0000) – –1.4489 (0.0000) –0.8805 (0.0000) – 0.2941 (0.0000) goodness of fit adjusted r2 aic 0.4167 –94.2000 – –107.3700 – –107.9300 heteroskedasticity breuch-pagan test 43.7765 (0.0081) 47.0493 (0.0033) 46.0142 (0.0044) autocorrelation of residuals moran test 5.4701 (0.0000) 0.7071 (0.2397) –0.0850 (0.4661) spatial dependence lr lmlag lmerr rlmlag rlmerr – 23.3580 (0.0000) 24.4627 (0.0000) − − 15.1720 (0.0000) – – 3.2420 (0.0718) – 15.7270 (0.0000) – – – 4.3467 (0.0371) chow test f – 123.4199 (0.0000) 129.7126 (0.0000) speed of convergence half-life 0.3178 2.18 0.2403 2.88 0.2656 2.61 note: numbers in brackets refer to the p-values. spatio-temporal analysis of convergence of development level… dynamic econometric models 15 (2015) 5–26 23 taking into account the geographical connections (variant i) among the european stock exchanges investigated, in the panel convergence models there has been removed the problem of autocorrelation of the residuals (in the spatial autoregressive panel model at the level of significance γ = 0.01). however, in the case of using the matrix of economic distance (variant ii) the residual autocorrelation has been eliminated only from the spatial error panel model. table 10. results of the estimation and verification of panel models with fixed effects obtained for asian and american stock exchanges – variant ii fe_ind sar_fe_ind se_fe_ind parameters α β ρ λ –1.5180 (0.0000) –0.9213 (0.0000) – – –1.3046 (0.0000) –0.7933 (0.0000) 0.5110 (0.0000) – –1.3980 (0.0000) –0.8588 (0.0000) – 0.6899 (0.0000) goodness of fit adjusted r2 aic 0.4167 –94.2000 – –121.4400 – –127.2300 heteroskedasticity breuch-pagan test 43.7765 (0.0081) 47.7040 (0.0027) 44.2658 (0.0071) autocorrelation of residuals moran test 9.8500 (0.0000) 3.3626 (0.0004) 1.4042 (0.0801) spatial dependence lr lmlag lmerr rlmlag rlmerr – 50.3592 (0.0000) 76.4993 (0.0000) – – 29.2450 (0.0000) – – 3.6813 (0.0550) – 35.0310 (0.0000) – – – 29.8214 (0.0000) chow test f – 41.4627 (0.0000) 45.2704 (0.0000) speed of convergence half-life 0.3178 2.18 0.1971 3.52 0.2447 2.83 note: numbers in brackets refer to the p-values. elżbieta szulc, dagna wleklińska dynamic econometric models 15 (2015) 5–26 24 generally, we may say that the statistical properties of the convergence models of the european stock exchanges are better than of the models obtained in our first study mentioned earlier, in which exchanges from different parts of the world were taken into account. the other tables show the results of the estimation and verification of the convergence models obtained for asian and american stock exchanges. table 7 shows the characteristics of the models obtained for the pooled time series and cross-sectional data in the classical and spatial version (the spatial autoregressive model and the spatial error model), respectively. the spatial components in the spatial models are included through the matrix quantifying the physical distance between the stock exchanges. the characteristics of the spatial models using the matrix of the economic distance in comparison with the characteristics of the model without the spatial connections are presented in table 8. in both variants of quantification of relationships between the stock exchanges spatial models are better than the models which do not take into account the connections, in terms of the autocorrelation of residuals. unfortunately, all the models obtained are characterized by heteroskedasticity of variance. the panel models obtained for asian and american stock exchanges have the analogical fault (see tables 9 and 10). conclusions the analysis confirms the earlier findings that the inclusion of the linkages which result from physical and/or economic distance between the stock exchanges in the models of their convergence is justified. in other words, the results of the investigation provide another evidence for the existence of spatial effects in the empirical models of stock exchanges' convergence. the earlier study (szulc et al., 2014) found that the geographical distance has less impact on the process of equalizing differentiation of stock markets then the economic distance between them. in this study the finding was not revealed as clearly. the empirical models of convergence obtained for the european stock exchanges satisfy the basic criteria of statistical verification. unfortunately, this is not the case of the models obtained for the asian and american stock exchanges. the analyses of the process of convergence of stock exchanges should be further continued in terms of methodology as well as for the purpose of searching of properly established spatial regimes. spatio-temporal analysis of convergence of development level… dynamic econometric models 15 (2015) 5–26 25 references arbia, g. (2006), spatial econometrics. statistical foundations and applications to regional convergence, springer-verlag, berlin heidelberg, doi: http://dx.doi.org/10.1007/3-540-32305-8. asgharian, h., hess, w., liu, l. (2013), a spatial analysis of international stock market linkages, journal of banking & finance, 37(12), 4738–4754, doi: http://dx.doi.org/10.1016/j.jbankfin.2013.08.015. aspergis, n., christou, c., miller, s. m. (2014), country and industry convergence of equity markets: international evidence from club convergence and clustering, the north american journal of economics and finance, 29(c), 36–58. baltagi, b. h., song, s. h., koch, w. (2003), testing panel data regression models with spatial error correlation, journal of econometrics, 117(1), 123–150, doi: http://dx.doi.org/10.1016/s0304-4076(03)00120-9. caporale, g. m., erdogan, b., kuzin, v. (2009), testing for convergence in stock markets: a non-linear factor approach, cesifo working paper series no. 2845, doi: http://dx.doi.org/10.2139/ssrn.1496819. demirgus-kunt, a., levine, r. (1996), stock market, corporate finance and economic growth: an overview, the world bank economic review, 10(2), 223–239, doi: http://dx.doi.org/10.1093/wber/10.2.223. fraser, p., helliar, c. v., power, d. m. (1994), an empirical investigation of convergence among european equity markets, applied financial economics, 4(2), 149–57, doi: http://dx.doi.org/10.1080/758523959. haining r. (2005), spatial data analysis. theory and practice, cambridge university press, 3th ed., cambridge. hellwig, z. (1968), zastosowanie metody taksonomicznej do typologicznego podziału krajów ze względu na poziom ich rozwoju oraz zasoby i strukturę wykwalifikowanych kadr (the application of the taxonomic method to the typological division of a countries due to their level of development, resources and structure of qualified personnel), przegląd statystyczny (statistical survey), z. 4, 307–327. koralun-bereźnicka, j. (2008), zjawisko konwergencji rynków kapitałowych na rynkach europejskich (the convergence of the capital markets in the european markets), studia i prace kolegium zarządzania i finansów (working papers of collegium of finance and management), szkoła główna handlowa w warszawie, dom wydawniczy elipsa, z. 87, 87–98. levine, r., zelvos, s. (1996), stock market development and long-run growth, the world bank economic review, 10(2), 323–339, doi: http://dx.doi.org/10.1093/wber/10.2.323. login, b., solnik, f. (2001), extreme correlation of international equity markets, journal of finance, 56(2), 649–676, doi: http://dx.doi.org/10.1111/0022-1082.00340. łuniewska, m., tarczyński, w. (2006), metody wielowymiarowej analizy porównawczej na rynku kapitałowym (methods of multidimensional comparative analysis in the capital market), pwn, warszawa. millo, g., piras, g. (2012), splm: spatial panel data models in r, journal of statistical software, 47(1), 2−38. mutl, j., pfaffermayr, m. (2011), the hausman test in a cliff and ord panel model, econometrics journal, 14(1), 48–76, doi: http://dx.doi.org/10.1111/j.1368-423x.2010.00325.x. elżbieta szulc, dagna wleklińska dynamic econometric models 15 (2015) 5–26 26 suchecka, j., łaszkiewicz, e. (2011), the influence of spatial and economic distance on changes in the relationships between european stock markets during the crisis of 2007–2009, acta universitatis lodziensis, folia oeconomica, 252, wydawnictwo uniwersytetu łódzkiego, 69–84. suchecki, b. (ed.) (2012), ekonometria przestrzenna ii. modele zaawansowane (spatial econometrics ii. advanced models), wydawnictwo c.h.beck, warszawa. szulc, e., wleklińska, d., górna, k., górna, j. (2014), the significance of distance between stock exchanges undergoing the process of convergence: an analysis of selected world stock exchanges during the period of 2004–2012, dynamic econometric models, 14, 125–144, doi: http://dx.doi.org/10.12775/dem.2014.007 wiśniewski, t. (2003), do unii europejskiej – giełdy krajów kandydackich, (for the european union – stock exchanges of candidate countries), nasz rynek kapitałowy (our capital market), 6, 22–25. wójcik, d. (2009), the role of proximity in secondary equity markets, in clark, g. l., dixon, a. d., and monk, a. h. b. (eds.), managing financial risk. from global to local, oxford university press. oxford, 140–162, doi: http://dx.doi.org/10.1093/acprof:oso/9780199557431.003. przestrzenno-czasowa analiza konwergencji poziomu rozwoju wybranych giełd papierów wartościowych w okresie 2004–2012 z a r y s t r e ś c i. artykuł dotyczy analizy konwergencji wybranych giełd papierów wartościowych z punktu widzenia poziomu ich rozwoju. przedstawia podejście, które wskazuje na potrzebę uwzględniania przestrzennych i ekonomicznych powiązań między rynkami giełdowymi w analizach ich konwergencji. przeprowadzone badanie pokazuje także, że analiza konwergencji giełd w ustalonym zakresie przestrzennym wymaga podziału rozważanych giełd zgodnie z ustalonymi reżimami przestrzennymi badanie obejmuje 42 wybrane parkiety, analizowane w okresie 2004–2012. dane empiryczne odnoszą się do 6 zmiennych diagnostycznych, uznanych jako ważne determinanty rozwoju rynków giełdowych. s ł o w a k l u c z o w e: giełda papierów wartościowych, konwergencja, reżimy przestrzenne, odległość fizyczna, odległość ekonomiczna, macierz sąsiedztwa, przestrzenne modele panelowe. dem_2015_49to69 d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2015.003 vol. 15 (2015) 49−69 submitted september 4, 2015 issn (online) 2450-7067 accepted december 15, 2015 issn (print) 1234-3862 sabina nowak, joanna olbryś* day-of-the-week effects in liquidity on the warsaw stock exchange a b s t r a c t. the purpose of this study is to explore the day-of-the-week patterns in liquidity on the warsaw stock exchange (wse) using daily turnover as a liquidity measure. the existence of an inverted u-shape in the stock turnover across the trading days is examined. the research sample covers 2502 daily observations in the period january 2005 – december 2014. 53 wse-listed companies divided into three size groups are investigated. in the study the ols method with the hac covariance matrix estimation and the garch-type models are employed. the results indicate that liquidity on the wse tends to be significantly lower on mondays and higher on wednesdays in comparison with the other days of the week. however, the inverted u-shape in daily turnover occurs only among the companies with the largest market capitalization. k e y w o r d s: market microstructure, day-of-the-week effect, liquidity, turnover, hac, garch, warsaw stock exchange. j e l classification: c10, c58, g10, g12. introduction the day-of-the-week patterns in returns and volatility on stocks and stock market indices rank among the most common seasonality anomalies. there is a growing body of empirical literature on that issue, also for the polish capital market (see e.g. fiszeder, 2009 and the references therein). on * correspondence to: sabina nowak, university of gdansk, faculty of management, armii krajowej 101, 81-824 sopot, poland, e-mail: sabina.nowak@ug.edu.pl; joanna olbryś, bialystok university of technology, faculty of computer science, e-mail: j.olbrys@pb.edu.pl. sabina nowak, joanna olbryś dynamic econometric models 15 (2015) 49–69 50 the contrary, relatively little empirical research has been conducted on the day-of-the-week effects in liquidity on equity markets (e.g. jain, joh, 1988; foster, viswanathan, 1993; chordia et al., 2001; chordia et al., 2005; hameed et al., 2010; alrabadi, 2012; karolyi et al., 2012). the goal of this study is to examine day-of-the-week patterns in liquidity on the warsaw stock exchange (wse) using daily turnover as a liquidity measure. to address this issue, the ols method with hac covariance matrix estimator (newey, west, 1987) and the garch-type models are employed. the research covers the sample period january 2005 – december 2014, during which 53 wse-listed companies divided into three size groups are investigated. an inverted u-shape in the stock turnover across trading days is examined. this effect means that the trading volume tends to be at its lowest on monday and friday, while the most active periods are in the middle of the week (jain, joh, 1988). to the best of the authors’ knowledge, the empirical results concerning day-of-the-week effects in liquidity on the wse are novel and have not been presented in the literature thus far. the remainder of the study is organized as follows. section 1 specifies a methodological background and a brief literature review. in section 2, we present and discuss the empirical results of the day-of-the-week patterns in liquidity on the wse. section 3 recalls the main findings and concludes. 1. methodological background 1.1. brief literature review the presence of calendar anomalies has been investigated extensively since the nineteen seventies. the existence of seasonal behavior in returns and volatility has been widely documented in the finance literature. some of the fundamental and most broadly citied papers are e.g. (fama, 1965; french, 1980; gibbons, hess, 1981; rogalski, 1984; french, roll, 1986, etc). as the aim of this research is to investigate day-of-the-week patterns in liquidity and trading activity, we focus our analysis of previous literature on the studies related mostly to that issue. among others, jain and joh (1988) studied the trading volume and returns on the new york stock exchange (nyse). they showed significant differences in the trading volume within and across days. the authors provided evidence of an inverted u-shape in volume across days, i.e. monday and friday had the lowest volume, and the most active periods were in the middle of the week. day-of-the-week effects in liquidity on the warsaw stock exchange dynamic econometric models 15 (2015) 49–69 51 in their theoretical research, foster and viswanathan (1990) proposed a model to explain time-dependent patterns in securities trading. they analysed inter-day trading where an informed trader and a subset of the liquidity traders act strategically. in their model, the informed trader receives information each day, but this information becomes less valuable through time, because there is a public announcement of some portion of the private information. the authors predicted a weekend effect in the trading volume and return volatility. they showed that the trading volume should be lower on monday than tuesday, and the trading costs are highest on monday. in another study, foster and viswanathan (1993) introduced empirical tests to document changes in the trading volume within and between days. they tested the null hypothesis that the trading volume is uniform through time. for the interday case, the authors used daily turnover as a measure of trading activity. they found that the monday trading volume is significantly lower than the tuesday and wednesday trading volume for the most actively traded firms. these findings were consistent with the predictions of the fosterviswanathan (1990) theoretical model. foerster and keim (1993) explored the frequency of non-trading for nyse and amex stocks, in the sample period 1973–1990. among other results, they found an interesting day-of-the-week pattern: non-trading increased monotonically through the week. chordia et al. (2001) documented strong day-of-the-week effects in trading activity for the u.s. stock market. they found that fridays accompanied a significant decrease in trading activity and liquidity, while tuesdays displayed the opposite pattern. in another paper, chordia et al. (2005) investigated the u.s. stock and bond markets and they found distinct seasonal patterns in stock and bond liquidities. both stock and bond market liquidities were higher at the beginning of the week compared with friday. hameed et al. (2010) used bid/ask spread as a measure of liquidity. they regressed the quoted spread of the stocks on a set of variables known to capture seasonal variation in liquidity. the estimated parameters showed calendar effects in the liquidity measure. alrabadi (2012) investigated day-of-the-week regularities on the amman stock exchange (ase). the author confirmed the significant seasonal patterns in aggregate market liquidity on the ase but, in contrast to the u.s. evidence of chordia et al. (2001), trading activity reached its minimum in the middle of the week and was significantly higher on thursdays. probably, the contradictory results arose from the nature of the ase as an emerging market. sabina nowak, joanna olbryś dynamic econometric models 15 (2015) 49–69 52 examining commonality in liquidity around the world, karolyi et al. (2012) followed the approach taken by hameed et al. (2010) and they measured whether fluctuations in liquidity of individual stocks are correlated within a country. the authors employed the model with day-of-the-week dummies and ran regressions for each stock using two different liquidity measures. the residuals of these regressions were treated as the daily innovations in liquidity and they were subsequently used as the endogenous variables in the monthly regressions. their coefficients of determination were regarded as the measures of commonality in liquidity of individual stocks. 1.2. measuring of liquidity as lesmond (2005) emphasized, liquidity, by its very nature, is difficult to define and even more difficult to estimate. given the uncertainty surrounding liquidity estimation, some measures are especially often advocated in the literature to provide empirical research in liquidity/illiquidity effects (e.g. olbryś, 2014a; 2014b). the popular measures of trading activity, i.e. volume, dollar trading volume, and share or market turnover are among them. the raw trading volume is the number of shares traded. the stock turnover is defined as the ratio of the number of shares traded in a day to the number of shares outstanding at the end of the day. it is worthwhile to note that using turnover disentangles the effect of firm size from the trading volume. in this research, we compute daily turnover as a measure of liquidity for stock i on day d: , , , , di di di nso v t = (1) where dit , is the turnover of stock i on day d, div , is the trading volume of stock i on day d, dinso , is the number of shares outstanding at the beginning of the quarter for stock i on day d. 1.3. econometric analysis of day-of-the-week effects many studies investigating day-of-the-week effects employ the standard ols methodology by regressing an endogenous variable on daily dummy variables. however, using that methodology has two disadvantages. firstly, errors in the model may be autocorrelated resulting in misleading inferences. day-of-the-week effects in liquidity on the warsaw stock exchange dynamic econometric models 15 (2015) 49–69 53 secondly, error variances may not be constant over time (kiymaz, berument, 2003). hamilton (2008) stresses that even if the researcher’s primary interest is in estimating the conditional mean, having a correct description of the conditional variance can still be quite important. by incorporating the observed features of the heteroskedasticity into the estimation of the conditional mean, substantially more efficient estimates of the conditional mean can be obtained. to account for daily seasonality, dummy variables are incorporated into the model. in order to avoid the dummy variable trap (foster, viswanathan, 1993), one selected dummy is always excluded from the regression. as the main aim of the research is to examine an inverted u-shape in the stock turnover on the wse, three different versions of the model are employed. the first version excludes the dummy variable for monday, the second for wednesday, and the third for friday. finally, for each stock, daily turnover (1) on day t is composed of a fixed effect for monday (wednesday, friday) ( 0a ), an inter-day adjustment for days other than monday (wednesday, friday), and an idiosyncratic error term with zero expected value ( tε ). the model describing the effect for monday is given as follows: ∑ = ++= 5 2 ,0 j ttjjt dbat ε (2) where 5,4,3,2,1,, =jd tj are the dummy variables for monday (j=1), tuesday (j=2), wednesday (j=3), thursday (j=4) and friday (j=5) at time t, respectively, 5,4,3,2,1, =jb j are the corresponding coefficients. the respective models for wednesday and friday effects are also considered1. the values of the coefficients of the dummy variables are central to test for inter-day variations in daily turnover. we initially estimate day-of-the 1 in the first version of the research in order to address the autocorrelation problem the lagged values of the variable t t were included as the explanatory variables in the model (2). in order to capture a one week delay, the lag equal to 5 was considered. however, employing ar part in formula (2) did not substantially improve the values of the lagrange multiplier statistics of the breusch-godfrey test for autocorrelation of order 5 – the hypothesis of no autocorrelation was continually rejected. the ols estimator was therefore inconsistent. sabina nowak, joanna olbryś dynamic econometric models 15 (2015) 49–69 54 week effects in eq. (2) by using the ols method and the robust hac estimates2. however, the newey-west corrections may not fully correct for the influence problems introduced by the arch effect. for this reason, the estimation of the day-of-the-week effect model as a garch-type model is appropriate for this study. to test for the arch effect, the test of engle (1982) with the lagrange multiplier (lm) statistic is employed. in order to test the presence of seasonality anomalies in stock returns, volatility or liquidity, various versions of garch-type models (bollerslev, 1986) have been applied in the literature (e.g. choudhry, 2000; franses, paap, 2000; berument, kiymaz, 2001; kiymaz, berument, 2003; apolinario et al., 2006; žikeš, bubák, 2006; alrabadi, 2012). in this research, the garch(p, q) model is utilised. according to the literature, the lower order garch(p, q), p, q = 1, 2, models are used in most applications (tsay, 2010). the garch(p, q) models are usually compared and selected by the akaike (aic) and schwarz (sc) information criteria3. the garch(p, q) model, with the excluded monday dummy variable, is given by eq. (3): ,0,,,1,0,0,,,1,0,0 , ),1,0(~, 0 11 2 0 5 2 ,0 ≥=≥>=≥> ⋅+⋅+= = ++= ∑∑ ∑ = − = − = pplqqk hh nzhz dbat lk p l ltl q k ktkt tttt j ttjjt kk βαα βεαα ε ε (3) where tε is the innovation in a linear regression with ,)( 2σε =v th is the variance function, and remaining notation like in eq. (2). similarly, the garch(p, q) model with excluded dummy variable for either wednesday or friday could be written respectively. 2 hac – heteroscedasticity and autocorrelation consistent covariance matrix estimation (newey, west, 1987). 3 when the values of the information criteria aic or sc for different variants of the garch(p, q) models are almost equal, the statistical significance of the parameters in the conditional mean and conditional variance equation of the garch(p, q) model could be analysed to choose the appropriate model. day-of-the-week effects in liquidity on the warsaw stock exchange dynamic econometric models 15 (2015) 49–69 55 the parameters of garch(p, q) models are almost invariably estimated via maximum likelihood (ml) or quasi-maximum likelihood (qml) (bollerslev, wooldridge, 1992) methods4. 2. empirical results on the warsaw stock exchange in this research, a database containing data for the wse-listed stocks for the period from january 2, 2005 to december 30, 2014 was utilised. when forming the database, we included only those securities which existed on the wse for the whole sample period since december 31, 2004, and were not suspended. the stock daily trading volumes (in items) were obtained from the website http://www.gpwinfostrefa.pl. the data of the number of shares outstanding are coming from the notoria serwis. all companies entered into the database (147) were sorted according to their market capitalization at the end of each year. next, the stocks were divided into three size groups based on the breakpoints for the bottom 30% (small – 44 companies), middle 40% (medium – 59 companies) and top 30% (big – 44 companies) (e.g. fama, french, 1993). the companies that remained in the same group during the period investigated were selected. finally, the 53 wse companies were entered into separate, representative groups, specifically: 8 firms into the small group, 18 firms into the medium group and 27 firms into the big group (nowak, olbryś, 2015). we computed daily turnover dit , given by eq. (1), providing 2,502 observations for each company. to avoid numerical problems, the data were rescaled by multiplying by 104 (lucchetti, balietti, 2014). all calculations were done using gretl 1.10.1 software (adkins, 2014). first we detected stationarity of the analysed daily turnover series for 53 stocks included in the size groups. we employed the adf-gls test (elliott et al., 1996) and we proved that the unit-root hypothesis can be rejected for all series at 5 per cent significance level. second, in order to carry out an initial assessment of the existence of an inverted u-shape in the stock turnover on the wse, the graphs showing the average level of the stock turnover on each day of the week were created. on such basis we noticed the occurrence of an inverted u-shape in the turnover across the majority of the analysed big companies. their level of turnover turned out to be the lowest on mondays, increasing on tuesdays and the 4 to choose the conditional distribution of innovations, various variants of the model (3) were estimated. unfortunately, they did not yield satisfactory results and even estimation failed in many cases. therefore, the distribution for the innovations is supposed to be normal. dynamic econometric 56 highest on wednesdays. however, it subsequently decreased on thursdays and finally achieved on fridays the level close to the level of mondays. it is pertinent to mention that the inverted u rarely for the medium and small companies. average daily turnover bos, bph, kgh, kty, lpp, net, opl, pkn. figure 1. an inverted u in the next step, ters of three versions of monday, wednesday and friday, estimated, comprising small companies. due to the existence of five or sabina nowak, joanna olbryś conometric models 15 (2015) 49–69 highest on wednesdays. however, it subsequently decreased on thursdays and finally achieved on fridays the level close to the level of mondays. it is pertinent to mention that the inverted u-shape in the turnover occurred very rarely for the medium and small companies. in figure 1, one can observe the average daily turnover of the nine selected big companies, namely bdx, bos, bph, kgh, kty, lpp, net, opl, pkn. an inverted u-shape in the average daily turnover of selected wse step, we employed the ols method to estimate the ersions of the model (2), excluding the dummy variable f monday, wednesday and friday, respectively. in total, 159 models were comprising 81 models for big, 54 for medium and companies. due to the existence of five order serial autocorrelation highest on wednesdays. however, it subsequently decreased on thursdays and finally achieved on fridays the level close to the level of mondays. it is shape in the turnover occurred very one can observe the of the nine selected big companies, namely bdx, turnover of selected wse-stocks employed the ols method to estimate the paramethe dummy variable for respectively. in total, 159 models were and 24 for der serial autocorrelation day-of-the-week effects in liquidity on the warsaw stock exchange dynamic econometric models 15 (2015) 49–69 57 and – in some cases – also the heteroscedasticity of residuals, the neweywest covariance matrix estimator (1987) was employed. the results of the model (2) estimation with the monday dummy variable excluded are presented in tables 2 –3 in appendix5. in the case of 35 ‘without monday’ models (comprising 16 models for big, 11 for medium and 8 for small companies), the arch effect in residuals was detected. therefore, for those 35 companies the garch (p, q), p, q = 1, 2, models were estimated. the number of lags p, q was selected on the basis of the akaike (aic) and schwarz (sc) information criteria. the results of the estimation of the ‘without monday’ model (3) are reported in tables 4 –6 in appendix6. table 1 presents a brief summary of the major day-of-the-week effects in daily turnover on the wse. table 1. summarized day-of-the-week effects in daily turnover on the wse effect in daily turnover big group medium group small group monday effect bph, bos, bdx, bhw, bzw, ech, gtc, gtn, ing, kgh, kty, lpp, mbk, mil, mol, net, opl, orb, peo, pkn, pko, sns, stp, tvn cng, mci, mni, stx vst mza wednesday effect bph, bos, bdx, bhw, bzw, gtc, gtn, ing, kgh, kty, lpp, mbk, mil, net, opl, peo, pkn, pko, tvn mni, stf, stx mza partial inverted u-shape bph, bos, bdx, bhw, bzw, gtc, gtn, ing, kgh, kty, lpp, mbk, mil, net, opl, peo, pkn, pko, tvn mni, stx mza full inverted u-shape bph, mil – – note: the monday effect means that the level of daily turnover is statistically significantly lower on mondays than on the other days of the week; the wednesday effect means that the level of daily turnover is statistically significantly higher on wednesdays in comparison with mondays and/or the other days of the week; the partial inverted u-shape in turnover means the presence of the monday and wednesday effects, but the absence of the friday effect; the full inverted u-shape in turnover means the presence of the monday, wednesday and friday effects at the same time. the obtained estimation results of the models (2) and (3) led to the conclusion that the ‘monday effect’ was the most frequently present in the stock turnover time series. in the case of 24 big and 3 medium firms, the level of daily turnover was statistically significantly lower on mondays than on the other days of the week7. the ‘wednesday effect’ (recognized in those cases when the level of daily turnover was statistically significantly higher on 5 the estimation results of the models with wednesday and friday dummies excluded are available upon request. 6 see footnote 3. 7 at the 5 per cent level. sabina nowak, joanna olbryś dynamic econometric models 15 (2015) 49–69 58 wednesdays in comparison with mondays and/or the other days of the week) occurred in the case of 19 big companies. the findings of the ‘friday effect’ existence were ambiguous. the results mentioned above justify the preliminary conclusions from analysing the graphs, where a pronounced ‘monday effect’ accompanied by a weaker ‘wednesday effect’ were detected. besides, both effects occurred more often in the case of big companies. continuing the analysis of the model (2) estimation results, we observed the presence of the so-called ‘partial inverted u-shape’ in daily turnover related to the presence of the monday and wednesday effect, but not the friday effect, in the case of 19 out of 27 big firms (namely bph, bos, bdx, bhw, bzw, gtc, gtn, ing, kgh, kty, lpp, mbk, mil, net, opl, peo, pkn, pko, tvn). for these companies, the level of daily turnover turned out to be either statistically significantly lower on mondays than on wednesdays and/or the other days of the week, or statistically significantly higher on wednesdays than on mondays. moreover, on the basis of the model (3) estimation, we found the ‘partial inverted u-shape’ in daily turnover for 14 companies, including 11 big (bph, gtc, ing, kgh, mbk, mil, net, opl, peo, pkn, pko), 2 medium (mni, stx) and 1 small firm (mza). furthermore, we did not confirm the occurrence of the ‘full inverted u-shape’ in daily turnover of the companies analysed, since in the majority of the cases the level on friday turnover was not statistically significantly lower than on the other days of the week. for 17 big companies (bhw, bzw, ech, gtc, gtn, ing, kgh, kty, lpp, mbk, mil, net, opl, peo, pkn, pko, tvn) the level of turnover on fridays was statistically significantly higher compared with the level on mondays. only for 2 big companies (bph and opl) the daily turnover level turned out to be statistically significantly lower on fridays in comparison with wednesdays. barely for 2 companies (bph and mil) we can venture the conclusion of the existence of the ‘full inverted u-shape’ in daily turnover (involving monday, wednesday and friday effect at the same time). in the case of the bph company, on the basis of the model (2) estimation, we noted that the turnover was simultaneously: (i) statistically significantly lower on mondays compared with tuesdays, wednesdays and thursdays; (ii) statistically significantly higher on wednesdays compared with mondays, thursdays and fridays; (iii) statistically significantly lower on fridays compared with tuesdays and wednesdays. those findings were only partially confirmed by the results of the estimation of the model (3), which showed rather the existence of the ‘partial inverted u-shape’ in daily turnover of the bph. conday-of-the-week effects in liquidity on the warsaw stock exchange dynamic econometric models 15 (2015) 49–69 59 versely, in the case of the mil company, the results of the model (2) estimation proved the occurrence of the ‘partial inverted u-shape’, whereas the results of model (3) estimation – the existence of the ‘full inverted u-shape’ in the stock turnover. conclusions the main goal of this paper was to explore and to document day-of-theweek effects in liquidity on the wse, using daily turnover as a liquidity measure. to address this issue, we employed the ols method with the hac covariance matrix estimation and the garch-type models. to account for daily seasonality in the turnover, dummy variables were incorporated into the models. we investigated 53 wse-listed stocks from three size groups. our research provided evidence for pronounced monday and wednesday effects in daily turnover on the wse, especially in the big group. furthermore, we observed the so-called ‘partial inverted u-shape’ in daily turnover in the case of 22 out of 53 firms. moreover, the graphs showing the average daily stock turnover on each day of the week were created and they revealed an inverted u-shape in some cases. although relatively little empirical research has been conducted on the day-of-the-week effects in liquidity on stock markets in the world, our findings are rather consistent with the existing literature. in light of our empirical results, it seems that the trading volume on the wse is usually the lowest on mondays, but the most active periods are in the middle of the week. from an investor’s point of view it is important that these findings are also in accordance with the investor’s intuition. it is worth stressing that our study situates itself in the broad strand of literature concerning commonality in liquidity, which is nowadays the centre of attention of many empirical research papers (e.g. olbryś, 2014a; 2014b; karolyi et al., 2012). given the importance of the topic, one of the possible directions for further investigation could be to examine day-of-the-week patterns in liquidity on the wse following the methodology proposed by franses and paap (2000) or žikeš and bubák (2006). the former authors employ the par-pgarch model to investigate the seasonality in the s&p 500 index, while the latter use the same model to explore daily returns on the central european stock markets. sabina nowak, joanna olbryś dynamic econometric models 15 (2015) 49–69 60 references adkins, l. c. (2014), using gretl for principles of econometrics, 4th edition, version 1.041. alrabadi, d. w. h. (2012), an analysis of aggregate market liquidity: the case of amman stock exchange, international business research, 5(5), 184–194, doi: http://dx.doi.org/10.5539/ibr.v5n5p184. apolinario, r. m. c, santana, o. m., sales, l. j. (2006), day of the week effect on european stock markets, international research journal of finance and economics, 2, 53–70. berument, h., kiymaz, h. (2001), the day of the week effect on stock market volatility, journal of economics and finance, 25(2), 181–193, doi: http://10.1016/s1058-3300(03)00038-7. bollerslev, t. (1986), generalized autoregressive conditional heteroskedasticity, journal of econometrics, 31, 307–327, doi: http://10.1016/0304-4076(86)90063-1. bollerslev, t., wooldridge, j. m. (1992), quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances, econometric reviews, 11, 143–179, doi: http://10.1080/07474939208800229. chordia, t., roll, r., subrahmanyam, a. (2001), market liquidity and trading activity, journal of finance, 56(2), 501–530, doi: http://10.1111/0022-1082.00335. chordia, t., sarkar, a., subrahmanyam, a. (2005), an empirical analysis of stock and bond market liquidity, review of financial studies, 18(1), 85–129, doi: http://10.1093/rfs/hhi010. choudhry, t. (2000), day of the week effect in emerging asian stock markets: evidence from the garch model, applied financial economics, 10, 235–242, doi: http://10.1080/096031000331653. elliott, g., rothenberg, t. j., stock, j. h. (1996), efficient tests for an autoregressive unit root, econometrica, 64(4), 813–836, doi: http://dx.doi.org/10.2307/2171846. engle, r. f. (1982), autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflations, econometrica, 50, 987–1007, doi: http://dx.doi.org/10.2307/1912773. fama, e. f. (1965), the behaviour of stock market prices, journal of business, 38, 34–105. fama, e. f., french, k.r. (1993), common risk factors in the returns on stocks and bonds, journal of financial economics, 33(1), 3–56. fiszeder, p. (2009), modele klasy garch w empirycznych badaniach finansowych (the class of garch models in empirical finance research), nicolaus copernicus university press, torun. foerster, s., keim, d. (1993), direct evidence of non–trading of nyse and amex stocks, working paper, university of pennsylvania. foster, f. d., viswanathan, s. (1990), a theory of the interday variations in volume, variances, and trading cost in securities market, review of financial studies, 3(4), 593–624, doi: http://dx.doi.org/10.1093/rfs/3.4.593. foster, f. d., viswanathan, s. (1993), variations in trading volume, return volatility, and trading costs: evidence on recent price formation models, journal of finance, 48(1), 187–211, doi: http://dx.doi.org/10.2307/2328886. franses, p. h., paap, r. (2000), modelling day-of-the-week seasonality in the s&p 500 index, applied financial economics, 10, 483–488, doi: http://dx.doi.org/10.1080/096031000416352. french, k. r. (1980), stock returns and the weekend effect, journal of financial economics, 8, 55–69, doi: http://dx.doi.org/10.1016/0304-405x(80)90021-5. day-of-the-week effects in liquidity on the warsaw stock exchange dynamic econometric models 15 (2015) 49–69 61 french, k. r., roll, r. (1986), stock returns variances: the arrival of information of the reaction of traders, journal of financial economics, 17, 5–26. gibbons, m., hess, p. (1981), day of the week effects and asset returns, journal of business, 54, 579–596, doi: http://dx.doi.org/10.1086/296147. hameed, a., kang, w., viswanathan, s. (2010), stock market declines and liquidity, journal of finance, 65(1), 257–293, doi: http://dx.doi.org/10.1111/j.1540-6261.2009.01529.x. hamilton, j. d. (2008), macroeconomics and arch, working paper 14151, nber working paper series, cambridge. jain, p. c., joh, g.-h. (1988), the dependence between hourly prices and trading volume, journal of financial and quantitative analysis, 23(3), 269–284. karolyi, g. a., lee, k.-h., van dijk, m. a. (2012), understanding commonality in liquidity around the world, journal of financial economics, 105(1), 82–112, doi http://dx.doi.org/10.1016/j.jfineco.2011.12.008. kiymaz, h., berument, h. (2003), the day of the week effect on stock market volatility and volume: international evidence, review of financial economics, 12(4), 363–380, doi: http://dx.doi.org/10.1016/s1058-3300(03)00038-7. lesmond, d. a. (2005), liquidity of emerging markets, journal of financial economics, 77(2), 411 –452, doi: http://dx.doi.org/10.1016/j.jfineco.2004.01.005. lucchetti, j., balietti, s. (2014), the gig package, version 2.14. newey, w. k., west, k. d. (1987), a simple, positive semi-define, heteroskesticity and autocorrelation consistent covariance matrix, econometrica, 55(3), 703–708, doi: http://dx.doi.org/10.2307/1913610. nowak, s., olbryś, j. (2015), autokorelacja stóp zwrotu spółek giełdowych w kontekście zakłóceń w procesach transakcyjnych (serial correlation of individual stock returns in the context of friction in trading processes), zeszyty naukowe uniwersytetu szczecińskiego no. 854. finanse, rynki finansowe, ubezpieczenia, 73, 721 –734. olbryś, j. (2014a), is illiquidity risk priced? the case of the polish medium-size emerging stock market, bank i kredyt, 45(6), 513–536. olbryś, j. (2014b), wycena aktywów kapitałowych na rynku z zakłóceniami w procesach transakcyjnych (capital asset pricing on market with frictions in trading processes), difin press, warszawa. rogalski, r. j. (1984), new findings regatding day-of-the-week returns over trading and nontrading periods: a note, journal of finance, 35, 1603–1614. tsay, r. s. (2010), analysis of financial time series, john wiley, new york. žikeš, f., bubák, v. (2006), seasonality and the non-trading effect on central european stock markets, czech journal of economics and finance, 56, 69–79. analiza efektu dnia tygodnia w płynności spółek notowanych na giełdzie papierów wartościowych w warszawie s.a. z a r y s t r e ś c i. celem artykułu jest analiza występowania efektu dnia tygodnia w płynności spółek notowanych na giełdzie papierów wartościowych w warszawie s.a., z wykorzystaniem dziennych wartości względnego wolumenu jako miary płynności. badaniu poddano w szczególności występowanie tzw. efektu odwróconego u w dziennym względnym wolumenie 53 spółek, z podziałem na grupy według wartości rynkowej, w okresie od stycznia 2005 r. do grudnia 2014 r. w badaniu wykorzystano modele ols-hac oraz garch. na sabina nowak, joanna olbryś dynamic econometric models 15 (2015) 49–69 62 podstawie uzyskanych wyników stwierdzono, że dzienny względny wolumen na giełdzie warszawskiej jest generalnie istotnie niższy w poniedziałki (tzw. efekt poniedziałku) oraz istotnie wyższy w środy (tzw. efekt środy) w porównaniu do pozostałych dni tygodnia. ponadto zaobserwowano, że częściowy efekt odwróconego u występuje głównie w dziennym względnym wolumenie spółek o największej kapitalizacji. efekt ten oznacza jednoczesną obecność efektów poniedziałku i środy, przy braku tzw. efektu piątku, czyli spadku dziennego względnego wolumenu w piątek do poziomu z początku tygodnia. s ł o w a k l u c z o w e: mikrostruktura rynku, efekt dnia tygodnia, płynność, względny wolumen, hac, garch, giełda papierów wartościowych w warszawie s.a. appendix table 2. estimation results of model (2), ols-hac, monday dummy variable excluded, the big group big group bph bnp bos bdx bzw dbc ech gtn gtc bhw ing kty kgh lpp 2b 1.196*** 0.093 0.346** 1.590 1.850*** 0.977 1.657*** 2.052*** 2.432*** 1.406* 0.152* 4.653*** 7.685*** 0.470 3b 1.802*** –0.010 0.606** 2.466** 2.176*** 1.270* 0.755* 3.415*** 4.006*** 1.306** 0.372** 4.991*** 11.346*** 2.403*** 4b 0.804** –0.042 0.179 1.147 1.512** 1.589* 1.255*** 2.767*** 5.197*** 0.578 0.283** 5.359*** 12.181*** 1.809** 5b 0.265 –0.012 0.139 1.376 2.476*** 1.190 1.448*** 4.077*** 4.390*** 1.387** 0.184** 4.926*** 12.150*** 2.483** lm 47.640 [0.000] 874.406 [0.000] 3.037 [0.694] 11.683 [0.039] 5.826 [0.324] 0.489 [0.993] 4.562 [0.472] 240.258 [0.000] 13.985 [0.016] 11.028 [0.051] 23.055 [0.000] 0.051 [0.999] 279.072 [0.000] 0.748 [0.980] 2tr 605.480 [0.000] 606.167 [0.000] 248.992 [0.000] 198.401 [0.000] 580.037 [0.000] 128.216 [0.000] 231.766 [0.000] 764.311 [0.000] 589.677 [0.000] 206.991 [0.000] 391.061 [0.000] 108.312 [0.000] 661.087 [0.000] 85.203 [0.000] w 13.352 [0.010] 2.752 [0.600] 5.947 [0.203] 3.715 [0.446] 5.378 [0.251] 3.969 [0.410] 6.425 [0.170] 6.298 [0.178] 4.444 [0.349] 2.386 [0.665] 3.189 [0.527] 4.435 [0.350] 5.903 [0.207] 5.023 [0.285] table 2 cont. estimation results of model (2), ols-hac, monday dummy variable excluded, the big group big group mbk mil mol net orb peo pkn pko stp sns opl tvn zwc 2b 1.664*** 1.979*** 0.034 5.087*** 0.409 3.884*** 4.587*** 3.230*** 0.553 1.529** 3.973*** 1.993*** 0.033 3b 2.834*** 2.565*** 0.101* 5.478*** 1.248 4.791*** 6.311*** 5.246*** 0.420 0.824* 6.573*** 2.052*** 0.013 4b 2.181*** 2.762*** 0.197*** 4.224*** 2.971** 4.651*** 7.472*** 5.610*** 1.000** 0.214 4.668*** 2.001*** 0.000 5b 2.308*** 2.018*** 0.168* 3.644*** 1.683 4.363*** 7.437*** 5.262*** 0.271 3.078 3.967*** 4.161*** 0.009 lm 184.953 [0.000] 33.166 [0.000] 3.164 [0.675] 16.050 [0.007] 2.372 [0.796] 542.272 [0.000] 74.756 [0.000] 471.023 [0.000] 16.104 [0.007] 0.008 [1.000] 36.749 [0.000] 0.851 [0.974] 195.769 [0.000] 727.798 [0.000] 277.598 [0.000] 298.113 [0.000] 537.436 [0.000] 46.544 [0.000] 616.530 [0.000] 580.538 [0.000] 922.911 [0.000] 147.017 [0.000] 79.600 [0.000] 390.676 [0.000] 484.002 [0.000] 461.129 [0.000] w 9.385 [0.052] 5.842 [0.211] 6.865 [0.143] 4.434 [0.350] 3.801 [0.434] 5.465 [0.243] 14.533 [0.006] 9.897 [0.042] 2.515 [0.642] 4.168 [0.384] 3.358 [0.500] 5.732 [0.220] 2.572 [0.632] note: 5432 , , , bbbb – the estimates of the model (2) coefficients using the ols method with the hac covariance matrix estimator (newey, west, 1987); ** (***, *) – indicates statistical significance at 5 per cent (1 per cent, 10 per cent) significance level; lm – the lagrange multiplier statistic of the engle’s test (1982), order lag equal to 5; 2r – the adjusted 2r , 2tr – the lagrange multiplier statistic of the breusch-godfrey test for autocorrelation of order 5; w – the white statistic; relevant p-values in brackets under the estimates. 2tr table 3. estimation results of model (2), ols-hac, monday dummy variable excluded, the medium group medium group alm amc atm atg col ipl ind ltx mci mni cng pek ska stx 2b 0.683 –0.161 –0.038 0.787 0.310 –0.030 1.632 0.237 4.375** 0.949 3.768** 0.191 2.644* 0.003 3b 1.331 0.199 0.124 0.144 0.024 1.688 0.684 –0.348 4.873* 4.299 3.331* 1.162 0.689 0.301 4b 0.128 1.252 1.229 –0.098 0.272 1.647 2.898* –1.140 7.143** 3.708 4.117** 1.011 1.906* 1.630 5b 0.672 0.379 –0.456 1.016 0.758 1.191 0.720 –1.106 2.718 2.797 2.487* 0.353 1.138 0.978 lm 21.847 [0.001] 67.675 [0.000] 41.616 [0.000] 1.936 [0.858] 84.352 [0.000] 4.026 [0.546] 4.568 [0.471] 569.325 [0.000] 501.096 [0.000] 438.636 [0.000] 7.008 [0.220] 38.001 [0.000] 4.068 [0.540] 692.530 [0.000] 2tr 298.859 [0.000] 551.466 [0.000] 205.035 [0.000] 293.065 [0.000] 664.712 [0.000] 305.668 [0.000] 107.632 [0.000] 1245.440 [0.000] 1175.298 [0.000] 964.227 [0.000] 269.080 [0.000] 314.258 [0.000] 52.412 [0.000] 1189.473 [0.000] w 2.351 [0.671] 0.975 [0.914] 2.566 [0.633] 2.921 [0.571] 3.700 [0.448] 3.811 [0.432] 6.533 [0.163] 0.903 [0.924] 2.354 [0.671] 1.316 [0.859] 3.668 [0.453] 2.186 [0.702] 5.264 [0.261] 1.610 [0.807] table 3 cont. estimation results of model (2), ols-hac, monday dummy variable excluded, the medium and small groups medium group small group stf tim vst pue apl bdl efk enp kmp mza pla sme 2b 2.262* 1.997 4.450** 1.426 17.500 –0.062 3.036* –3.250 1.984 8.368* –8.366 –0.459 3b –0.382 1.979* 2.782 3.657* 12.106 0.176 1.021 6.769 1.071 12.335 –3.698 0.978 4b 0.718 2.126 5.249 3.567 9.275 0.033 2.840 –3.308 –2.865 11.356 –5.348 0.985 5b –1.033 0.104 0.689 1.011 –2.722 0.269 2.103 –2.614 –7.695* 5.771 –1.797 –0.155 lm 426.069 [0.000] 0.153 [0.999] 613.836 [0.000] 0.173 [0.999] 279.249 [0.000] 1245.050 [0.000] 1029.52 [0.000] 380.345 [0.000] 665.594 [0.000] 619.648 [0.000] 565.978 [0.000] 929.439 [0.000] 2tr 777.058 [0.000] 58.392 [0.000] 1106.698 [0.000] 34.729 [0.000] 987.168 [0.000] 1159.090 [0.000] 1095.110 [0.000] 1135.479 [0.000] 1153.429 [0.000] 1224.741 [0.000] 1196.819 [0.000] 1370.149 [0.000] w 3.111 [0.539] 3.244 [0.518] 3.694 [0.449] 2.987 [0.560] 4.153 [0.386] 1.408 [0.843] 1.450 [0.835] 1.977 [0.740] 3.193 [0.526] 3.555 [0.470] 3.043 [0.551] 0.498 [0.974] note: see table 2 for explanation. table 4. estimation results of model (3), garch(p, q), monday dummy variable excluded, the big group big group bph bdx gtn gtc ing kgh mbk mil net peo pkn pko stp opl zwc conditional mean equation 2b 0.81 5.61** 1.19* 1.84 0.11 6.05*** 0.88 1.73** 1.88 3.60*** 4.10*** 2.53*** 0.81 5.86*** 0.07*** 3b 1.30** 1.83 2.06*** 3.70*** 0.26*** 10.30*** 1.85*** 6.67*** 8.38*** 4.20*** 5.77*** 3.97*** –0.16 5.80*** 0.02** 4b 1.46** 3.65 1.11 2.99*** 0.15 11.73*** 2.50** –0.85 6.53*** 3.61*** 6.77*** 4.17*** 0.26 5.04*** 0.01 5b 2.59 –0.55 2.83*** 3.57 0.16** 10.31*** 2.21*** 0.93 4.68** 4.05*** 6.80*** 3.73*** –0.16 5.17*** 0.10*** conditional variance equation 7.09** 12.99 9.19* 2.49 0.002 18.04*** 0.05 4.43* 6.04 1.03* 117.67*** 1.11 7.88 64.49 0.002* 1.07 1.25*** 0.12*** 0.61*** 1.02*** 0.29*** 0.63*** 1.12*** 0.93 0.25*** 0.41*** 0.57*** 0.36 0.44** 0.15** – –1.06*** – –0.56*** –0.98*** –0.24*** –0.63*** –1.01*** –0.88* –0.21*** – –0.55*** – – 5.10*** 0.44*** 0.87*** 0.86*** 1.22*** 0.98*** 0.93*** 1.11*** 0.92*** 1.08*** 1.02*** 0.21 0.98*** 0.02 0.55*** 0.04* – – – –0.27* – – –0.12* – –0.11 –0.07 – – 0.60 – 0.10*** ll –8207 –10416 –10081 –10413 –3702 –11568 –8918 –9601 –11323 –9286 –10357 –9822 –8045 –10613 206 note: the variance-covariance matrix of the estimated parameters of the model (3) is based on the qml algorithm (bollerslev, wooldridge, 1992); the distribution for the innovations is supposed to be normal; ** (***, *) – indicates statistical significance at 5 per cent (1 per cent, 10 per cent) significance level. 0α 1α 2α 1β 2β table 5. estimation results of model (3), garch(p, q), monday dummy variable excluded, the medium group medium group alm amc atm col ltx mci mni pek stx stf vst conditional mean equation 2b 0.356 0.915 1.307 0.146 2.545 1.344 4.570 1.322 –1.637 2.092** 0.436 3b –1.222 2.788 1.678* –0.723 0.246 2.157 5.974 0.843 9.686*** –0.127 –0.013 4b 0.202 3.672 1.873** –0.022 –1.214 0.025 13.307 1.051 –0.607 2.837*** –0.067 5b 0.293 0.984 0.182 –0.166 –0.887 0.825 1.569 0.772 1.870 1.392 0.755 conditional variance equation 22.65*** 300.9*** 1.773 46.40** 3.142* 1.647 17.07*** 8.132*** 20.584 8.344 0.102 0.663 0.760** 0.020* 0.488*** 1.033*** 0.575*** 1.002*** 0.058*** 2.405** 0.118* 0.120 –0.529 – 0.197** – –0.919*** –0.548*** –0.930*** – – – 0.771 0.854*** –0.006 0.019 –0.009*** 0.916*** 1.183*** 0.944*** –0.009** –0.001 0.349*** 0.020 – 0.278** 0.862*** 0.203 – –0.208 – 0.929*** 0.389** 0.529*** 0.711 ll –10551 –11844 –9794 –8995 –9651 –12393 –12368 –10248 –11052 –10560 –11631 note: see table 4 for explanation. 0α 1α 2α 1β 2β table 6. estimation results of model (3), garch(p, q), monday dummy variable excluded, the small group small group apl bdl efk enp kmp mza pla sme conditional mean equation 2b 0.867 –0.104* –0.471 0.370 –0.679 2.822** –2.956 –1.948 3b 1.454 –0.170** –0.876 0.449 0.394 4.426*** –1.930 0.203 4b 1.094 0.040 –0.365 3.465* 0.906 1.736 –8.793*** 8.155*** 5b –0.097 –0.018 –0.267 2.289* 0.678 1.725 –4.354** –1.315 conditional variance equation 4.418 0.009 0.536* 2.309** 1.141* 1.806 9.578** 60.21 2.724*** 1.906** 0.712** 0.879*** 0.993*** 1.021*** 1.544*** 0.877** –0.810** – –0.639** –0.855*** –0.847** – –1.063*** 1.814 0.302** 0.017 0.893*** 0.979*** 0.916*** 0.042 0.815*** 0.029 0.322*** 0.478*** 0.057 – – 0.639*** – 0.193 ll –14474 –3394 –10205 –12904 –12458 –12211 –12636 –12340 note: see table 4 for explanation. 0α 1α 2α 1β 2β pusta strona dem_2015_27to48 © 2015 nicolaus copernicus university press. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2015.002 vol. 15 (2015) 27−47 submitted october 25, 2015 issn (online) 2450-7067 accepted december 15, 2015 issn (print) 1234-3862 łukasz lenart* discrete spectral analysis. the case of industrial production in selected european countries∗∗ a b s t r a c t. the aim of this paper is to show the usefulness the discrete spectral analysis in identification cyclical fluctuations. the subsampling procedure was applied to construct the asymptotically consistent test for fourier coefficient and frequency significance. the case of monthly production in industry in european countries (thirty countries) was considered. using proposed approach the frequencies concerning business fluctuations, seasonal fluctuations and trading-day effects fluctuations were recognized in considered data sets. the comparison with existing procedures was shown. k e y w o r d s: discrete spectral analysis, almost periodic function, frequency identification, graphical test. j e l classification: c14, c46, e32. introduction the main part of monthly or quarterly macroeconomic time series concerning industry, trade, service, national accounts, prices, etc. exhibit both: seasonal fluctuations and business cycle fluctuations. the problem of analysing these cyclical fluctuations is widely considered in the literature using different statistical tools. one popular nonparametric approach is based on a spectral analysis. * correspondence to łukasz lenart, cracow university of economics, department of mathematics, e-mail: lenartl@uek.krakow.pl ∗∗ this research was supported by research grant dec-2013/09/b/hs4/01945 from the national science centre łukasz lenart dynamic econometric models 15 (2015) 27–47 28 the spectral analysis of macroeconomic time series is considered mainly in continuous counterpart (see ftiti, 2010; metz, 2009; orlov, 2006; orlov, 2009; pakko, 2004; mcadam and mestre, 2008; uebele and ritschl, 2009). under stationarity assumption the continuous function called spectral density function is defined. based on spectral density function (and the definition of harmonizable time series) the popular spectral characteristics are defined: modulus of coherency function, dynamic correlation, dynamic correlation on a frequency band, cohesion, cohesion within the frequency band, phase shift (see priestley, 1981; hamilton, 1994; croux, 2001 for more details). these measures are broadly used in analysing the business cycle fluctuations. they are estimated under fundamental assumption that time series is zero mean. but this assumption is not supported by any formal statistical test in most empirical macroeconomic real data analysis. in this paper the more general assumption is formulated concerning nontrivial mean function. this general assumption was considered in lenart, 2011; lenart and pipień, 2013a; lenart and pipień, 2013b (in univariate case) and recently in lenart and pipień, 2015 (in multivariate case) with application to macroeconomic time series. in this paper we show that the parameters of the discrete spectrum can be identified not only with the seasonal and business cycle fluctuations but additionally with the trading-day effect. in the section 2, based on lenart and pipień, 2013a and lenart and pipień, 2015 the model was formulated and illustrative example was presented. in the next part the empirical analysis was presented. the production in industry – monthly data (mining and quarrying; manufacturing; electricity, gas, steam and air conditioning supply) from feb. 2000 to dec. 2014 was considered. in the first subsection the graphical methods to identify/recognize the frequency (concerning to business fluctuations, seasonal fluctuations and trading-day effect fluctuations) was presented. such graphical methods to identify ’periodic phenomena’ in the autocovariance function in class of almost periodically correlated time series were presented in hurd and gerr, 1991 and recently in lenart, 2011. finally in the second subsection formal statistical test for frequency significance was applied to data sets. 1. model specification let ty be macroeconomic time series (index, gross data) with possible: seasonal fluctuations with period t , business fluctuations and trading-day discrete spectral analysis. the case of industrial production… dynamic econometric models 15 (2015) 27–47 29 fluctuations. let us denote the natural logarithm: )(ln= ~ tt yy . based on lenart and pipień, 2013a we assume that ),(),(=) ~ ( ttfye t µβ + (1) where )(tµ is almost periodic function (ap in short) of the form tiemt ψ ψ ψµ )(=)( ∑ ψ∈ , where .)( 1 lim=)( 1= ti n tn et n m ψµψ − →∞ ∑ we assume that the set of frequencies 0}=|)(:|)[0,2{= /∈ψ ψπψ m is finite and unknown. this set ψ can be decompose in natural way via: 321= ψ∪ψ∪ψψ , where 1ψ corresponds to business fluctuations, 1}0,1,=,/{22 −⊂ψ tktkπ corresponds to seasonal fluctuations and 3ψ is a set of remaining frequencies (corresponding to interaction between seasonal and business fluctuations and frequencies corresponding to trading-day effects). equivalently, the model (1) can be written via: ),()(),(=) ~ ( 21 tttfye t µµβ ++ (2) where )(1 tµ is a periodic function with period t which represents the seasonal fluctuations and tiemt ψ ψ ψµ )(=)( 2\ 2 ∑ ψψ∈ . the function )(1 tµ can be equivalently represented by the vector (sequence of seasonal values) ' 121 ][= −tµµµ kµ , and )(= 121 −+++− tt µµµµ k . the sequence 1 ~~ = −− ttt yyx represents the monthly log growth rate. if ),( βtf is a polynomial of order one then ),(~)(~=)( 21 ttxe t µµ + (3) where 1)()(=)(~ 111 −− ttt µµµ is periodic function that corresponds to seasonal pattern and tiemttt ψ ψ ψµµµ )(~=1)()(=)(~ 2\ 222 ∑ ψψ∈ −− . the sequence 1 ~~ =' −− ttt yyx represents the annual log growth rate. if ),( βtf is a polynomial of order one then ),('~=)'( 2 txe t µ (3) łukasz lenart dynamic econometric models 15 (2015) 27–47 30 where tiemttt ψ ψ ψµµµ )('~=12)()(=)('~ 2\ 222 ∑ ψψ∈ −− . in future work we assume that the autocovariance function of the time series }:{ z∈tx t (or }:'{ z∈tx t ) is a periodic function with the same period t . this is a natural generalization of the assumption concerning second order stationarity. this assumption follows from the natural hypothesis that for monthly data at some months the variability can be higher than at another month. with no loss of generality we assume that exists natural m such that msn = . then the time series }:{ z∈tx t can be represented as a second order stationary t -valued time series with almost periodic mean function. more precisely the time series ttsststt xxxy ][= 1)(21)(1 k−+−+ is t values second order stationary time series with almost periodic mean function. this mean function can be decomposed (in natural way) to two main parts: periodic function (that corresponds to seasonal frequencies 2ψ ) and almost periodic function (that corresponds to frequencies 31 ψ∪ψ ) the natural estimator of kµ (k=1,2,...,t–1) based on sample },,,{ 21 nxxx k where mtn = has the following form . 1 =ˆ 1)( 1= , tjk m j nk x m −+∑µ (4) the estimator '1,2,1, ]ˆˆˆ[=ˆ ntnnn −µµµ kµ is asymptotically normally distributed with known variance covariance matrix. theorem 2.1 assume that there exist constants 0>δ , ∞∆ < and ∞<k such that a) ,<||sup 2 ∆+∈ δ||z tt x b) kk k ≤+ ∞ ∑ δ δ α 2 0= )( . then )(0,)ˆ( 1 σ→− −t d nm nµµ (5) proof. the proof can be found in the appendix. discrete spectral analysis. the case of industrial production… dynamic econometric models 15 (2015) 27–47 31 the problem ψ∈ψ is equivalent to 0=|)(~| /ψm (or 0=|)('~| /ψm ). the natural estimator of fourier coefficient based on sample nxxx ,,, 21 k has the following form: . 1 =)(ˆ 1= ti t n t n ex n m ψψ −∑ (6) this estimator is asymptotically normally distributed (see lenart and pipień, 2013a). the frequencies from the set 2\ ψψ are estimated in nonparametric way using procedure introduced in lenart and pipień, 2013a. this procedure is based on subsampling approach applied for (nonstationary) almost periodically correlated time series (for more details see to lenart, 2013; lenart and pipień, 2013a). note that using similar technique the problem of testing seasonality effect (in nonparametric framework) in macroeconomic time series was considered in lenart and pipień, 2013b. the multivariate case of the model (1) was considered in lenart and pipień, 2015, where the testing procedure for common length of the business cycle was considered in selected european countries. to show the impact of frequencies from the sets: 1ψ , 2ψ , 3ψ to the dynamic of time series we consider the following illustrative example. example 2.1 let us consider time series }:{ z∈tx t of the form: ,)(= )( t t ti t emx ηψ µ ψ ψ +∑ ψ∈ 444 3444 21 (7) where }:{ z∈ttη is zero mean time series with finite variance on integer line. it is assumed (in this example) that the considered time series is not necessary second order stationary. to explain the impact of the frequencies from the sets: 1ψ , 2ψ , 3ψ (under decomposition 321= ψ+ψ+ψψ ) we consider four cases (t1–t4): t1 ∅ψ =1 , ∅ψ =3 , ,5}1,2,=:/12{2=2 kkkπψ , with ttt εηη +−10.95= , where }:{ z∈ttε is gaussian white noise with zero mean and variance equal one. in this case only seasonal pattern (with period 12=t ) is present in mean function. t2 {0.15}=1ψ , ∅ψ =3 , ,5}1,2,=:/12{2=2 kkkπψ , with ttt t εηπη +− −1|)) 6 (sin|0.5(0.95= , where }:{ z∈ttε is gaussian white noise with zero mean and variance equal one. in this case considered łukasz lenart dynamic econometric models 15 (2015) 27–47 32 time series has periodic autocovariance function with period equal to 12 and seasonal pattern (in mean function) with length of the season equal 12 . in the mean function we put additional frequency (equal to 0.15) that corresponds to cyclical fluctuations with period approx. to 3.5 years. t3 ∅ψ =1 , 0.1}/3{2=3 +ψ π , ,5}1,2,=:/12{2=2 kkkπψ , with ttt εηη +− −10.6= , where }:{ z∈ttε is gaussian white noise with zero mean and variance equal one. in this case the additional frequency 0.1/32 +π was put in mean function. t4 {0.15}=1ψ , 0.1}/3{2=3 +ψ π , ,5}1,2,=:/12{2=2 kkkπψ , with ttt εηη +−10.5= , where }:{ z∈ttε is gaussian white noise with zero mean and variance equal one. in this case we consider the case where both set: 1ψ and 3ψ are not empty. figure 1. the realizations of }:{ z∈ttη for considered cases, 180=n -3 -2 -1 1 2 3 t3 -2 -1 1 2 3 t4 -8 -6 -4 -2 2 4 t1 -2 2 4 t2 discrete spectral analysis. the case of industrial production… dynamic econometric models 15 (2015) 27–47 33 figure 2. the realizations of }:{ z∈tx t for considered cases, 180=n figure 3. the estimated magnitude of fourier transforms |)(ˆ| ψnm for considered cases, ][0,πψ ∈ -5 5 10 15 t3 -10 -5 5 10 t4 -15 -10 -5 5 10 15 t1 -10 -5 5 10 15 t2 0.5 1 1.5 2 2.5 3 0.25 0.5 0.75 1 1.25 1.5 1.75 t3 0.5 1 1.5 2 2.5 3 0.5 1 1.5 t4 0.5 1 1.5 2 2.5 3 0.5 1 1.5 t1 0.5 1 1.5 2 2.5 3 0.5 1 1.5 t2 łukasz lenart dynamic econometric models 15 (2015) 27–47 34 in all cases we clearly observe peaks at seasonal frequencies ,5}1,2,=:/12{2 kkkπ (see figure 3). note that the fourier coefficient at frequency π equals zero. in the case t1 considered time series is second order stationary and the peak close to zero corresponds to second order properties of considered time series. in the case t2 and t4 the peak at frequency 0.15 is observed while the frequency from the set 3ψ ( 0.1/32 +π ) can be recognized easily only in the case t4. 2. empirical analysis we consider production in industry – monthly data (mining and quarrying; manufacturing; electricity, gas, steam and air conditioning supply) from feb 2000 to dec 2014. we consider monthly and annual rate of change (gross data and data adjusted by working days). the production in industry for thirty countries were considered (with ordinal numbers): belgium (1); bulgaria (2); czech republic (3); denmark (4); germany (5); estonia (6); greece (7); spain (8); france (9); croatia(10); italy (11); cyprus (12); latvia (13); lithuania (14); luxembourg (15); hungary (16); malta (17); netherlands (18); austria (19); poland (20); portugal (21); romania (22); slovenia (23); slovakia (24); finland (25); sweden (26); united kingdom (27); norway (28); former yugoslav republic of macedonia, (29); serbia (30); 2.1. graphical methods for frequency identification in this section we introduce two graphical methods (visual methods) for recognise cyclical fluctuations. the first method is based directly on observing the estimates of magnitude of fourier coefficient |)(| ψm (as a function of frequency). in this graphical method we try to recognise maximum local values (peaks). the second method is based on theorem 2.1. in this method we consider only the m-o-m change cantered at each month by appropriate estimates nk ,µ̂ of seasonal coefficient kµ . the aim of this method is to recognize any cyclical fluctuations around estimates nk ,µ̂ . in the first step we consider monthly rate of change (gross data – data not adjusted by working days). for each considered economies the test presented in lenart and pipień, 2013b reject the null hypothesis concerning empty set of seasonal frequencies 2ψ . in this case the estimated magnitude of fourier coefficients (for ][0,πψ ∈ ) clearly support the existence of the peaks at seasonal frequencies (see figure 8). the peaks at frequencies that correspond to business cycle fluctuations are not observed jointly for consi discrete spectral analysis. the case of industrial production… dynamic econometric models 15 (2015) 27–47 35 figure 4. cantered seasonal values at their estimates (january–june). data not adjusted by working days january february march april may june -0,2 -0,1 0 0,1 0,2 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,2 -0,1 0 0,1 0,2 0,3 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,2 -0,1 0 0,1 0,2 0,3 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,2 -0,1 0 0,1 0,2 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,2 -0,1 0 0,1 0,2 0,3 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 łukasz lenart dynamic econometric models 15 (2015) 27–47 36 figure 5. cantered seasonal values at their estimates (july–december). data not adjusted by working days july august september october november december -0,15 -0,1 -0,05 0 0,05 0,1 0,15 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,2 -0,1 0 0,1 0,2 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,4 -0,2 0 0,2 0,4 0,6 0,8 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,2 -0,1 0 0,1 0,2 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,2 -0,1 0 0,1 0,2 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 discrete spectral analysis. the case of industrial production… dynamic econometric models 15 (2015) 27–47 37 figure 6. cantered seasonal values at their estimates (january–june). data adjusted by working days january february march april may june -0,2 -0,1 0 0,1 0,2 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,2 -0,1 0 0,1 0,2 0,3 0,4 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,2 -0,1 0 0,1 0,2 0,3 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,2 -0,1 0 0,1 0,2 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,3 -0,2 -0,1 0 0,1 0,2 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 łukasz lenart dynamic econometric models 15 (2015) 27–47 38 figure 7. cantered seasonal values at their estimates (july–december). data adjusted by working days july august september october november december -0,2 -0,1 0 0,1 0,2 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,2 -0,1 0 0,1 0,2 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,4 -0,2 0 0,2 0,4 0,6 0,8 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,2 -0,1 0 0,1 0,2 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -0,2 -0,1 0 0,1 0,2 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 discrete spectral analysis. the case of industrial production… dynamic econometric models 15 (2015) 27–47 39 dered countries. but the additional frequency that belongs to the set 3ψ is clearly observed. this frequency is approx. to 2.19 and is connected with trading-day effect in considered data. this frequency is a main theoretical frequency associated with a monthly trading-day effect (see ladiray, 2012, page 262). the peak at this frequency is observed for almost all considered data sets also in the case of annual rate of change (see figure 10). when the estimates of fourier coefficient |)(ˆ| ψnm are considered for data adjusted by working days (monthly and annual rate of change) the peaks at frequency approx. 2.19 are not observed. figure 8. the estimated magnitude of fourier transforms |)(ˆ| ψnm for m-o-m data – not adjusted by working days, ][0,πψ ∈ using second graphical method, the trading-day effect is clearly observed. the figure 4 and 5 presents the results of mean subtraction separately at each different month. obtained results clearly show the common fluctuations (around their estimates) for almost each countries. note that the phase of this fluctuations (above or below mean) differs in different consecutive month at the same year (compare for example april with may in the same years). therefore, such fluctuation seems not to be the effected by 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 belgium 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 bulgaria 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 czech republic 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 denmark 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 germany 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 estonia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 greece 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 spain 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 france 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 croatia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 italy 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 cyprus 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 latvia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 lithuania 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 luxembourg 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 hungary 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 malta 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 netherlands 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 austria 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 poland 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 portugal 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 romania 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 slovenia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 slovakia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 finland 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 sweden 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 united kingdom 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 norway 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 former yugoslav republic of macedonia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 serbia łukasz lenart dynamic econometric models 15 (2015) 27–47 40 business cycles. finally, these fluctuations are not observed for the data after working day adjustment (see figure 6 and 7). figure 9. the estimated magnitude of fourier transforms |)(ˆ| ψnm for m-o-m data – adjusted by working days, ][0,πψ ∈ 2.2. testing for frequency significance in this part we consider formal statistical tools to detect significant frequencies in mean function for considered data sets. we use nonparametric test presented in lenart, 2013; lenart and pipień, 2013a and we compare our results with parametric estimates obtained by so-called contraction method (cm in short) introduced by li and song, 2002. based on lenart, 2013; lenart and pipień, 2013a we consider the following testing problem: ψ∈ ψ∈/ ψ ψ : : 1 0 h h (8) 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 belgium 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 bulgaria 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 czech republic 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 denmark 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 germany (until 1990 former territory of the frg) 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 estonia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 ireland 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 greece 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 spain 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 france 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 croatia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 italy 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 cyprus 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 latvia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 lithuania 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 luxembourg 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 hungary 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 malta 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 netherlands 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 austria 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 poland 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 portugal 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 romania 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 slovenia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 slovakia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 finland 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 sweden 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 united kingdom 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 norway 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 former yugoslav republic of macedonia, the discrete spectral analysis. the case of industrial production… dynamic econometric models 15 (2015) 27–47 41 with the test statistics |)(ˆ| ψnmn and critical values of the test calculated by subsampling approach (we take nb 2.5= ). based on the test (8) and estimation procedure proposed in lenart, 2013 the set of frequency was estimated. in our analysis we consider both: gross data and data adjusted by working days for both m-o-m and y-o-y case. to compare our results we use parametric cm. in this case the number of iterations is 100 on the interval ),0( π (with the resolution equal to 300). the bandwidth parameters are: 96.0=1η for 20<m , 98.0=1η for 40<m , 99.0=1η for 60<m , 995.0=1η for 80<m . figure 10. the estimated magnitude of fourier transforms |)(ˆ| ψnm for y-o-y data – not adjusted by working days, ][0,πψ ∈ in the case of m-o-m data sets method based on subsampling procedure clearly confirm the existence of seasonal frequencies for data not adjusted and adjusted by working days (see figure 12 (a)–(b)). in the case of data not adjusted by working days the significant frequency approx. 2.19 is observed for most considered countries, while in the case of data adjusted by working days this frequency is clearly not significant on considered significance level. in the case of y-o-y data sets not adjusted by working days the method 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 belgium 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 bulgaria 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 czech republic 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 denmark 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 germany 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 estonia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 greece 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 spain 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 france 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 croatia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 italy 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 cyprus 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 latvia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 lithuania 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 luxembourg 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 hungary 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 malta 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 netherlands 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 austria 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 poland 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 portugal 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 romania 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 slovenia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 slovakia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 finland 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 sweden 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 united kingdom 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 norway 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 former yugoslav republic of macedonia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 serbia łukasz lenart dynamic econometric models 15 (2015) 27–47 42 based on subsampling procedure also confirm the significance of frequency approx. 2.19 for most considered countries (see figure 12 (c)–(d)). in the case of data adjusted by working days the frequency approx. 2.19 is clearly not significant (under considered significance level). figure 11. the estimated magnitude of fourier transforms |)(ˆ| ψnm for y-o-y data – adjusted by working days, ][0,πψ ∈ the procedure proposed by li and song, 2002 gives similar results for frequency approx. 2.19. but for the frequency that corresponds to business cycle fluctuations the cm gives different results (in comparison with method based on subsampling). in the case of cm for almost all countries the frequencies that correspond to business cycle fluctuations are included in estimated set. it should be emphasized that in the case of subsampling the nonparametric test is applied – based on resampling procedure, while in the case of cm the only estimation procedure (based on parametric model) is used without any test procedure (see estimation procedure in li and song, 2002). therefore figure 12 a)–d) presents results of testing procedure while e)–f) only results of the estimation procedure. 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 belgium 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 bulgaria 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 czech republic 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 denmark 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 germany 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 estonia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 greece 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 spain 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 france 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 croatia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 italy 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 cyprus 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 latvia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 lithuania 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 luxembourg 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 hungary 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 malta 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 netherlands 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 austria 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 poland 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 portugal 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 romania 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 slovenia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 slovakia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 finland 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 sweden 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 united kingdom 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 norway 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 former yugoslav republic of macedonia 0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1 serbia discrete spectral analysis. the case of industrial production… dynamic econometric models 15 (2015) 27–47 43 (a) subsampling for m-o-m data, not adjusted by working days (b) subsampling for m-o-m data; adjusted by working days (c) subsampling for y-o-y data, not adjusted by working days (d) subsampling for y-o-y data, adjusted by working days (e) cm for y-o-y, data not adjusted by working days (f) cm for y-o-y, data adjusted by working days figure 12. (a)–(d) estimated frequency sets (dots). horizontal axis – estimated frequency; vertical axis – the ordinal number of considered country. (a)–(d) – nonparametric approach based on subsampling methodology introduced in lenart, 2013; (e)–(f) cm introduced by li and song, 2002 0 5 10 15 20 25 30 0,00 0,52 1,05 1,57 2,10 2,62 3,14 0 5 10 15 20 25 30 0,00 0,52 1,05 1,57 2,10 2,62 3,14 0 5 10 15 20 25 30 0,00 0,52 1,05 1,57 2,10 2,62 3,14 0 5 10 15 20 25 30 0,00 0,52 1,05 1,57 2,10 2,62 3,14 0 5 10 15 20 25 30 0,00 0,52 1,05 1,57 2,10 2,62 3,14 0 5 10 15 20 25 30 0,00 0,52 1,05 1,57 2,10 2,62 3,14 łukasz lenart dynamic econometric models 15 (2015) 27–47 44 conclusions the discrete spectral analysis can be useful supplementary analysis to usual spectral analysis based on continuous spectrum. presented methods (graphical and formal statistical test) are useful in recognizing frequencies that corresponds to business fluctuations, seasonal fluctuations and tradingday effects. one of the main open problems is a formal statistical test for trading-day effect. corresponding test for business fluctuations was introduced in lenart and pipień, 2013a and in lenart and pipień, 2013b for seasonal pattern. note that in the literature the popular so-called visual test (see ladiary, 2012) for trading-day frequencies plays a central role. appendix proof of the theorem 2.1. we start from auxiliary lemma. lemma a.1 if assumptions of the theorem 2.1 hold then for any 1},{1,2,, 21 −∈ tkk k the limit         −− −+−+ ∞→ ∑∑ ))(( 1 lim 21)2(211)1(1 1=21=1 ktjkktjk m j m jm xx m e µµ (9) exists and it’s finite. proof of the lemma. notice that ),(cov || 1= ),(cov 1 = ))(( 1 21 1 1= 1)2(21)1(1 1=21=1 21)2(211)1(1 1=21=1 tkk m m tjktjk m j m j ktjkktjk m j m j xx m xx m xx m e τ τ τ µµ + − +− −+−+ −+−+       −         −− ∑ ∑∑ ∑∑ (10) using now theorem 3, page 9 form doukhan, 1994 we get inequality |)(|8|),(cov| 12 22 21 tkkxx tkk τα δ δ τ +−∆≤ ++ . since (10) is a feyèr’s sum at a point 0=0x the limit for ∞→m exists. this finish the proof of the lemma. discrete spectral analysis. the case of industrial production… dynamic econometric models 15 (2015) 27–47 45 we use the same technique as presented in lenart, 2013 in proof of theorem 2.1. let us consider the decomposition: .))ˆ(())ˆ(ˆ(=)ˆ( )(2)(1 4444 34444 214444 34444 21 nr n nr nnn ememm µµµµµµ −+−− using the same arguments it is sufficient to show the convergence: )(0,)( 11 σ→ −t d nr n (11) and 0.)(2 →nr (12) to show (11) it is enough to use the cramér-wold device, theorem 3.3.1 from guyon, 1995 and lemma 5.1. to prove (12) notice that for any 1,1,2,= −tk k we have 0. 1 = 1 2 )(~ 1 )(~ 1 = )1)((~ 1 = )( 1 =|)ˆ(| 2\ )1)(( 2\1= 2 1= 1)( 1= , →         − ≤ −+ −− ∑ ∑∑ ∑ ∑ ψψ∈ −+ ψψ∈ −+ m o e m m em m tjk m xe m mem ti tjki m j m j ktjk m j knk ψ ψ ψ ψ ψ ψ µ µµµ this completes the proof. references croux, ch., forni, m., reichlin, l. (2001), a measure of covomement for economic variables: theory and empirics, the review of ecomonics and statistics, 83(2), 232–241, doi: http://dx.doi.org/10.1162/00346530151143770. doukhan, p. (1994), mixing: properties and examples, springer-verlag, new york, ftiti, z. (2010), the macroeconomic performance of the inflation targeting policy: an approach based on the evolutionary co–spectral analysis (extension for the case of a multivariate process), economic modelling, 27, 468–476. guyon, x. (1995), random fields on a network. springer-verlag, new york. hamilton, j. d., (1994), time series analysis, princeton university press, new jersey. łukasz lenart dynamic econometric models 15 (2015) 27–47 46 hurd, h., gerr, l. (1991), graphical methods for determining the presence of periodic correlation, journal of time series analysis, 12(4), 337–350, doi: http://dx.doi.org/10.1111/j.1467-9892.1991.tb00088.x. ladiray, d. (2012), theoretical and real trading-day frequencies, in bell w.r., holan s.h. and mcerloy t.s. (ed.), economic time series: modeling and seasonality, 255–279. doi: http://dx.doi.org/10.1201/b11823-16. lenart, ł, pipień, m. (2013a), almost periodically correlated time series in business fluctuations analysis, acta physica polonica a, 123(3), 567–583, doi: http://dx.doi.org/10.12693/aphyspola.123.567. lenart, ł, pipień, m. (2013b), seasonality revisited – statistical testing for almost periodically correlated processes, central european journal of economic modelling and econometrics, 5, 85–102. lenart, ł, pipień, m. (2015), testing the common length of the business cycles with discrete spectral analysis and subsampling approach, submited to journal of time series analysis. lenart, ł. (2011), asymptotic distributions and subsampling in spectral analysis for almost periodically correlated time series, bernoulli, 17(1), 290–319, doi: http://dx.doi.org/10.3150/10-bej269. lenart, ł. (2013), non-parametric frequency identification and estimation in mean function for almost periodically correlated time series, journal of multivariate analysis, 115, 252–269. doi: http://dx.doi.org/10.1016/j.jmva.2012.10.006. li, t. h., song, k. s. (2002), asymptotic analysis of a fast algorithm for efficient multiple frequency estimation, ieee transactions on information theory, 48(10), 2709 – 2720. mcadam, p., mestre, r. (2008), evaluating macro–economic models in the frequency domain: a note. economic modelling, 25, 1137–1143. metz, r. (2009), comment on ”stock markets and business cycle comovement in germany before world war i: evidence from spectral analysis”, journal of macroeconomics, 31, 58–67, doi: http://dx.doi.org/10.1016/j.jmacro.2008.01.004. orlov, a. g. (2006), capital controls and stock market volatility in frequency domain, economics letters, 91, 222–228, doi: http://dx.doi.org/10.1016/j.econlet.2005.09.014. orlov, a. g. (2009), a cospectral analysis of exchange rate comovements during asian financial crisis, journal of international financial markets, institutions and money, 19, 742–758, doi: http://dx.doi.org/10.1016/j.intfin.2008.12.004. pakko, m. r. (2004), a spectral analysis of the cross–country consumption correlation puzzle, economics letters, 84, 341–347. priestley, m. b. (1981), spectral analysis and time series, academic press, london, doi: http://dx.doi.org/10.2307/2983035. uebele, m., ritschl, a. (2009), stock markets and business cycle comovement in germany before world war i: evidence from spectral analysis, journal of macroeconomics, 31, 35–57. analiza spektrum dyskretnego na przykładzie produkcji przemysłowej w wybranych krajach europejskich z a r y s t r e ś c i. artykuł przedstawia analizę spectrum dyskretnego dla produkcji przemysłowej wybranych gospodarek europejskich. analizie poddano dane o częstotliwości miesięcznej. przyjęto założenie o prawie okresowej postaci funkcji wartości oczekiwanej wokół discrete spectral analysis. the case of industrial production… dynamic econometric models 15 (2015) 27–47 47 długookresowej tendencji rozwojowej. w artykule przedstawiono dwie metody graficzne identyfikacji wahań cyklicznych (koniunkturalnych, sezonowych oraz wynikających z różnej liczby dni roboczych w miesiącu). pierwsza z nich bazuje na identyfikacji najwyższych wartości estymatora współczynnika fouriera, jako funkcji częstotliwości. metoda ta może być stosowana dla miesięcznej oraz rocznej stopy wzrostu. druga metoda graficzna może być zastosowana jedynie dla miesięcznej stopy wzrostu. bazuje ona na graficznej identyfikacji wahań cyklicznych w danych po odjęciu od nich wartości estymatora współczynników sezonowych. w kolejnej części artykułu przedstawiono wyniki formalnego testu istotności częstotliwości w oparciu o prace: lenart, 2013 oraz lenart oraz pipień, 2013a. warto zaznaczyć, iż test ten pozwolił na formalne zidentyfikowanie częstotliwości korespondującej do wahań związanych z elektem różnej liczby dni roboczych w kolejnych miesiącach roku. s ł o w a k l u c z o w e: analiza spectrum dyskretnego, funkcja prawie okresowa, identyfikacja częstotliwości, test graficzny. pusta strona dem_2015_89to109 © 2015 nicolaus copernicus university. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2015.005 vol. 15 (2015) 89−109 submitted september 29, 2015 issn (online) 2450-7067 accepted december 15, 2015 issn (print) 1234-3862 katarzyna andrzejczak, agata kliber* the model of french development assistance – who gets the help?∗∗ a b s t r a c t. development cooperation is an important element of international relations because it influences the power balance between major players on the world markets and in the political debate. the aim of the article was to analyze the french development assistance model based upon the amount of help sent to africa over the period 2001–2012. the motivation of donor country is a crucial factor of development assistance, which influence not only the relations between donors and recipients, but also the effectiveness of aid. we estimated a series of dynamic panel models to assess whether the poverty-related factors play a dominant role in the distribution of help. on the contrary, we found that the most important variables appeared to be the political and economic dependencies, among others: colonial history and oil/gas reserves. k e y w o r d s: development assistance; bilateral aid; development cooperation; developing country; france, dynamic panel model; j e l classification: f53, f54, i32, o15. introduction the role and effectiveness of development aid have been repeatedly questioned ever since the system was established. the growing income dis * correspondence to: katarzyna andrzejczak, poznan university of economics e-mail: katarzyna.andrzejczak@ue.poznan.pl, agata kliber, poznan university of economics, e-mail: agata.kliber@ue.poznan.pl. ∗∗ this work was supported by the polish ministry of science and higher education through the project sonata: 2013//09/d/hs4/01849 (mechanizmy wykorzystania technologii w afryce subsaharyjskiej). katarzyna andrzejczak, agata kliber dynamic econometric models 15 (2015) 89–109 90 parities between the north and the south, energy crisis in 1970s, debt crisis and structural adjustment in the late 1980s, numerous political and military conflicts in the 1990s, the number of people living below the poverty line, as well as interference in political situation in beneficiary countries are only a few examples of system dysfunction, repeatedly analyzed in the development literature (e.g. chenery, carter, 1973; bonne, 1996; kosack, 2003; amprou et al., 2007; easterly, pfutze 2007; doucouliagos, paldam, 2008; roodman, 2008). at the same time, the economic and social context, as well as the main character of development cooperation changed over the years. the critics of aid and donors as well as the appearance of new donors on the scene lead to official redefinition of development cooperation in year 2000 (united nations millennium development goals declaration). the malfunctioning of the development cooperation system has been officially recognized and the aid has been declared to serve development issues, not the donors’ interests. in the meantime, the emergence and spreading of global communication means and channels allowed more open debate on migrations, detriment of environment, health and natural disasters as well as terrorism threats issues. this increased public opinion’s role in evaluation of the northern countries approach towards less developed regions. as a consequence, the role of grass root level of cooperation has been noticed and its importance raised in the development agenda. a philosophy to direct the support towards local communities in need with the omission of state administration often affected by corruption and ineffectiveness (tanburn, 2008) has been widely recognized the landscape of development cooperation has undergone major changes in the last decade especially due to the changes in the “donors’ club”. this means that the donors represent very different (not only political) contradictory interests and approaches, and that they start to “compete” as donors. the appearance of the “generous” nordic states and multilat-eral organizations, such as european union, then the so called south countries, compelled the “old donors” to revise their international policy in development cooperation (mcewan, mawdsley, 2012). the appearance of new actors in the system was a main driving force of its evolution. the architecture of aid relations known in 20th century and based on organization for economic cooperation and development assistance committee (hereafter dac), world bank and international monetary fund is therefore being replaced by a more complex system of actors and approaches (gore, 2013). the appearance of emerging economies and private actors and organizations has changed the international scene and requires rethinking of the traditional development assistance schemes (eyben, 2012). main aid donors are no the model of french development assistance – who gets the help? dynamic econometric models 15 (2015) 89–109 91 longer only the triada countries. old donors, which placed the themes of democracy and the market economy as pivotal for aid policy agenda are now only a group in a more complex donor group (kim, lightfoot, 2011). the dac-ability concept, which embraced the philosophy of aid giving and the very definition of what aid is (official development aid term), is now challenged by the non-dac donors (ndd). there is an increasing volume of development aid provided by the emerging economies. this aid influences international relations between states and redesigns the cooperation scene, influences and interests. among the symbolic signs of change, the fourth high level forum on aid effectiveness was organized at busan in south korea. the changes in aid system are undeniable, however, apart from the optimistic declarations, it is debatable to say that the system has evolved for the benefit of aid recipients. it is therefore doubtful that the donors stopped focusing on realization of their external policy goals in the scope of development cooperation. because of this, the dilemmas that undermine the very concept of development cooperation need to be addressed. the critics of aid donors persist. the ineffectiveness problem remains unresolved, as the paris declaration agenda of 2005 does not seem to be fulfilled. the relations between development agents tend to rely on one-sided domination, not partnership. rather than development aid creating growth, doucouliagos and paldam (2013), suggest it is possible that economic growth itself may influence donor aid allocation decisions. since donor dysfunctions burden aid efficiency (simplice, 2014), in the analysis of aid effects in recipient country, the issue of donors motivation in decision making process is crucial. development assistance system is given, so the research should concentrate on finding the means to make it more effective and enhance countries’ capacities to develop. for these effects to appear, we assumed that they are more probable, once the aid is provided with such an aim. the goal of this paper is, therefore, to analyze the bilateral development cooperation in the context of the donor motivation to provide assistance. we chose france as a model country representing traditional donors, since it has the characteristics of traditional donors. however, we are aware that the other traditional countries need to be tested for compliance with the model in further research. we concentrate on the aid for african countries. first, we discuss current literature on development assistance cooperation and next we introduce the traditional model of aid, based on qualitative analysis of cooperation dimensions. methods of comparative and system analysis were applied in order to elaborate theoretical foundations for the models of cooperation. we used the system-gmm approach for dynamic panel models to verikatarzyna andrzejczak, agata kliber dynamic econometric models 15 (2015) 89–109 92 fy the motivation of the donor in the aid distribution over the period 2001– –2012. it appeared that the poverty-related factors do not play any significant role in the aid distribution, but rather the political and economic ones. the robustness of the results was confirmed by the non-linear correlation test (kendall’s tau). in the last section we discuss the implication of the results. 1. traditional development aid donors development cooperation system is a post-second world war phenomenon, which was initiated by the usa as a response to the economic situation and cold war in europe (overseas development assistance). it’s further evolution was a consequence of the decolonization process and the independence of african states (williams, 2014). in the oecd terminology, the aid is associated with official development assistance (oda) public flows to beneficiaries placed on the dac (development assistance committee) list of oda recipients and to multilateral development institutions. these transfers are concessional in character – they convey a grant element of at least 25 per cent (calculated at a rate of discount of 10 percent per annum) and are administered with the promotion of the economic development and the welfare of developing countries as the main objective (oecd, 2014). that leaves out a great deal of development initiatives by private entities (enterprises, ngos) but also public actors (non-oda initiatives, local communities cooperation). moreover, currently some major donors, such as china, do not report their flows to oecd at all. the character of these flows is often unknown, so there is no possibility to conclude if they meet the requirements of oda or not. because of that, the term development cooperation should not be limited purely to oda transfers recognized by the dac anymore. development aid gains a wider meaning, which include oda and other development purposes transfers, not reported to dac. in a broad sense, aid is neither limited to the type of transfer nor to the type of agents in the relation. however, it still does not include some “foreign aid” elements, such as military support. in this article we concentrate on development aid, treated as an instrument of a state foreign policy towards less developed countries. the broadening range of donor countries is a consequence of major changes in world economy. while the group of aid recipients is shrinking, the group of donors grows (the good news about the aid system is just about it). the enlarged donor club is currently an amalgam of different countries. verifying whether their aid is instrumentally subordinated to the long and short term strategic goals of foreign policy is an important aim. analysis of the model of french development assistance – who gets the help? dynamic econometric models 15 (2015) 89–109 93 donors motivation requires more than dividing them into dac and ndd groups, because there are differences between the donors in both of the groups. the division logic may be based on the geography, their public willingness to adhere to dac standards, level of economic development, political regime, and others. there are oecd countries that are not members of dac–mexico, turkey and several european countries; new eu members; middle east and organization of the petroleum exporting countries, and non-oecd donors from neither of these groups: brazil, china, india and russia (kim, lightfoot, 2011). in the dac there are traditional donors, scandinavian states (generous), new-dac members, including new eu members. such classifications are not exhaustive, but sufficient to understand the complexity of interests, which may be related with development cooperation. all these groups are specific in character and realize different development aid policy. in this paper we concentrate on the practice of traditional dac donors, based on french example. we are interested to see, weather the emergence of new donors affected their actions or only the declarations. we suspect, that as cumming and chaffer wrote: “european member states are driven more by their desire to enhance their own relative power within the international system than by any overriding need to help african countries” (2011: 212). traditional dac donors based their foreign aid policy mainly on bilateral relations. major contributors such as usa, france and uk were therefore criticized for supporting dictators in favor of their geopolitical goals in the bipolarized reality of the second half of xxth century (williams, 2014). today, their image of cold war players is still in force, but the war against terrorism and the peace maintenance in strategic regions of the world are of increased importance (chou, 2012). the altruistic motivation of traditional donors involvement in any international security issue is questioned, as well as their true willingness to decrease poverty. traditional donors approach has been first contested by the aid agenda of scandinavian countries and multilateral organizations. generally, sweden and other nordic states are considered not only most generous donors (barczak, 2008) but also donors more focused on the idea of effective use of help, on building a civil society in developing countries, and on directing their assistance towards realization of millennium development goals (thiele et al., 2007). however, even the nordic states are claimed to enhance their relative power and punch above their weight in international donor circles (cumming, chaffer, 2011). multiparty organizations, on the other hand, are theoretically devoid of egoistic national interests and oriented tokatarzyna andrzejczak, agata kliber dynamic econometric models 15 (2015) 89–109 94 wards democracy promotion and human rights protection in recipient states (der-chin, 2003). their policy stance was supposed to reflect the coherent interests of developed and developing countries. because of this, the increased share of multilateral disbursements in the total donors’ aid volume was considered as higher aid quality. in reality, multilateral aid is biased for the western world with the omission of national labels (see: charbonneau, 2008). ohler and nunnenkamp (2014) found that the multilateral institutions do not take regional needs into account while directing aid, instead favoritism plays an important role for location choices. traditional donors are influential in the international organizations and may profit from their activities indirectly. therefore, the emergence of south-south development cooperation has influenced the position of traditional donors in a much serious way, than the recommendation to increase the multiparty aid element. some research values south-originated aid as multidimensional and encompassing the intrinsic and non‐economic roles of development and hence very practical for african economies (babaci‐wilhite et al., 2013). realization of sustainable development goals investments in joint investment promotion mechanisms, joint programs for absorptive capacity and joint public-private partnership models are said to be more likely to happen in the scope of regional and south-south cooperation (un wir, 2014). however, the chinese development cooperation practice is in many ways similar to french or american model (chaponniere et al., 2009). the major accusations towards traditional donors are their limited involvement in cooperation and self-oriented approach – despite helping to close savings gap, based on the neo-classical theory, aid serves donors’ interests (page, te velde, 2004; cumming, chaffer, 2011). however, for china and india energy, land and raw materials imports from africa are equally important (bearce et al., 2009; walker, 2008, p. 21). what is especially interesting, is that the appearance of new donors, which negotiate with african countries without the colonial history burden, and their increased presence in the region, influenced the need of a change in the traditional donors’ policies. both individually and as a member of organizations, france has always played leading role in development cooperation system. in the past france was realizing its neo-colonial françafrique strategy, partly created by jaques foccart and supported by the “cellule africaine” in presidential palace, whoever was its resident since charles de gaulle. historically france kept close relations with the countries of the so called champ and even closer with the group of pre carré, especially in africa. the strategy embraced economic, political and diplomatic goals (fuchs, 1993). for africa – or more precisely the model of french development assistance – who gets the help? dynamic econometric models 15 (2015) 89–109 95 – for african leaders these relations have been a source of material, political and diplomatic power, while for france – more literally of resources and comparative advantage in international relations (lancaster, 1999). france was more direct in their policy goals realization than the united kingdom, for which african policy was a source of relative power – soft influence. french were less liberal than the british and took a more interventionist stance in african relations (cumming, chaffer, 2011). the element shared by france and uk was the establishment of language communities, which were enforcing the relations and creating another dimension for a closer link (hugon, 2008). today, according to the french development policy, main components of development agenda: economy, society and environment are subordinated to fighting with poverty and sustainable development promotion. francphone terrirtories are the main area of development aid focus. french cooperation policy aims to address four mutually-supportive issues, promoting peace, stability, human rights and gender equality (1); equity, social justice and human development (2); sustainable, job-rich economic development (3); protecting the environment and global public goods (4). strategic partners of french development cooperation are: benin, burkina faso, burundi, djibouti, comoros, ghana, guinea, madagascar, mali, mauretania, niger, centralafrican republic, democratic republic of congo, chad, togo, senegal (mae, 2015). among them only two not being a former french colony, both rich in natural resources. such a selection of partners indicates that the realization of national interests can actually remain a major issue for the development policy. in the next sections, the analysis of french practice will be studied in order to evaluate the motivation of france as a traditional aid donor. 2. methodology panel data modeling is a statistical methodology allowing to use information included both in time and cross-sectional dimension. the regression is therefore run on the two (or more) dimensions. depending on the assumptions made about the error term, one can talk about fixedor random-effect models (see e.g. hsiao, 2014; greene, 2011). let us consider a panel model of the following form: ,,,, tititi uy += βx (1) katarzyna andrzejczak, agata kliber dynamic econometric models 15 (2015) 89–109 96 where ity is the dependent variable observed at time t for the i-th crosssectional unit, while ti ,x – the matrix of explanatory variables. depending on the error decomposition we receive different model specification. for instance: ,itiitit uy ++= ηβx (2) is a “fixed-effect” panel model, that can be estimated e.g. through leastsquare dummy variable (lsdv) estimator. such specification is justified when we assume that apart from the time-varying explanatory variables there are also time-invariant individual effects that influence the dependent variable. another possible specification is the random-effect one: .itiitit uy +++= νηβx (3) in this formulation iν is a group-specific random element similar to itu , except that for each group there is but a single draw that enters the regression identically in each period (greene, 2011). if a lagged dependent variable is included in the model, we talk about the dynamic panel data models. in order to utilize all available information: both changes of variables in time, as well as across countries, we composed a panel of the data, including 12 time periods (2001–2012) and 53 cross-section units (53 countries). since the number of time periods in our model was relatively small (t=12) compared to the number of cross-sectional units (n=53), we used the systemgmm approach for dynamic panel models. let us consider a dynamic panel model of the form: tiitititi uηyy ,,1,, +++= − βxα , (4) where tiy , denotes amount of financial help sent from donor (france) to the ith african country at time t. in our case t = 1,…,12, while i= 1,…,53. β is a 1×k vector of the coefficients, α is the vector of autoregressive coefficients, ti ,x is a k ×1 vector of explanatory variables observed for country i at time t, iη are so called individual effects (time-invariant), while it u – the disturbance specific for country i at time t. the model can be estimated using the so called “difference estimator”. the main idea of the approach is to first difference the data. then, one obtains: titititititi ,,,,1,, uγwuxβyαy ∆+=∆+∆+∆=∆ − . (5) the model of french development assistance – who gets the help? dynamic econometric models 15 (2015) 89–109 97 the error term of the equation (2) is by definition autocorrelated and also correlated with the lagged dependent variable. all values of ktiy −, with k>1 can be used as instruments for 1, −∆ tiy . one-step estimator of equation (5) amounts to computing (cottrel, lucchetti, 2014):             ∆                        = ∑∑∑∑ == − == n i iin n i ii n i iin n i ii 1 ' 1 ' 1 1 ' 1 ' ˆ yzazwwzazwγ , (6) where: [ ] . , 0...000 .................. 0...0 0...00 , ... ... ,,..., 1 1 ' , 4,2,1, 3,1, ,3, 1,2, ,3, − = −       =               ∆ ∆ ∆ =       ∆∆ ∆∆ = ∆∆=∆ ∑ n i iin ti iii ii i tii tii i tiii x xyy xy xx yy yyy hzza z w once the 1-step estimator is computed, the 2-step estimated are obtained through replacing the matrix h with the sample covariance matrix of the estimated residuals. the 2-step estimator is consistent and asymptotically efficient. in our paper we used the so-called “system” estimator that complements the differenced data with data in levels, so the lagged differences are used as instruments (see: blundell and bond, 1998). the key equation of the system estimator is as follows (cottrel, lucchetti, 2014): ,~ ~~~~~~~~ 1 ' 1 ' 1 1 ' 1 '                 ∆                               = ∑∑∑∑ == − == n i iin n i ii n i iin n i ii yzazwwzazwγ (7) katarzyna andrzejczak, agata kliber dynamic econometric models 15 (2015) 89–109 98 where: [ ] .~~ , ...00...000 ........................... 0...00000 ........................... 0...0...000 ........................... 0000...0 0...00...00 ~ , ...... ......~ ,......~ 1 1 *' ,1, 3,2, ,2, 4,2,1, 3,1, ,3,,3, 1,2,1,2, ,3,,3, − = − − −−       =                           ∆ ∆ ∆ ∆ ∆ =       ∆ ∆ = ∆∆=∆ ∑ n i iin titi ii titi iii ii i tiitii tiitii i tiitiii xy xy xy xyy xy xxxx yyyy yyyy zhza z w y the choice of matrix *h is not trivial. the details are presented for instance in roodman (2009) or hsiao (2014). see also: dańska-borsiak (2009). for more detailed description of panel data concept and modelling we refer the readers to e.g. hsiao (2014), longhi and nandi (2014) or gruszczyński et al. (2012). 3. french oda – the data the amount of french oda sent to africa is very heterogeneous and it seems to depend both on the period and on the receiver country. we observe an enormous growth over the period 2004–2006 of the amount of help received by nigeria, as well as high transfers to the democratic republic of congo in 2003, to congo in 2005 and 2010 and periodical higher transfers to ivory coast. in order to select a relatively homogeneous group, we excluded from the analysis those countries, where the high jumps in data were present (more precisely: we deleted from the sample those countries, where the standard deviation value was higher than the mean value). we filtered out: congo, the democratic republic of congo, cote d’ivore, liberia, mozambique, nigeria, seychelles, sierra leone, the united republic of tanzania and zambia. the model of french development assistance – who gets the help? dynamic econometric models 15 (2015) 89–109 99 in table 1 we present the descriptive statistics of oda sent from france to africa over the period 2001–2012 in the filtered group of countries. the mean value of the help in us 2012 mln dollars amounted to 54.168, but the median only to 20.159. we divided the full sample into two subsamples: colony and non-colony countries as well as oil/gas vs. nonoil/gas ones. the differences in the amount of help received in each group is striking. in the case of the previous colonies the mean value of help amounted to 104.380 mln usd, while in the case of the non-colonies – only to 9.868. in the case of the countries possessing oil/gas reserves the amount of oda received equaled 73.246, while in the case of the remaining ones – 36.247. the cochran-cox test for equality of mean values rejected the null hypothesis in the case of both pairs. therefore, we can say that the amount of financial help sent to african countries is larger in the case of the ones that possess natural resources. the analysis of the “within” and “between” standard deviations reveals that the non-colony as well as the nonoil/gas group are more homogeneous than the remaining ones. in all subsamples the data is right-skewed. table 1. descriptive statistics of financial help from france to the african countries full sample colony non-colony oil/gas non-oil/gas mean 54.168 104.380 9.868 73.246 36.247 median 20.159 74.089 5.779 44.145 13.754 minimum 0.025 1.350 0.025 0.580 0.025 maximum 811.650 811.650 121.390 811.650 310.700 std.dev. 84.443 101.430 13.032 101.650 59.037 vol.factor 1.559 0.972 1.321 1.388 1.629 skewness 3.780 3.167 4.451 3.566 2.624 kurtosis 22.150 15.013 29.524 18.033 7.094 5% percentile 0.670 14.011 0.469 2.195 0.445 95% percentile 210.170 268.570 27.311 232.060 189.310 q3–q1 66.111 88.910 12.570 93.550 38.716 missing obs. 4 0 4 4 0 obs. no. 512 240 272 248 264 within s.d. 49.744 71.795 10.320 68.146 20.964 between s.d. 70.302 76.229 8.596 79.419 56.705 3.1. explanatory variables in order to model the amount of financial help received from france, we chose the following set of explanatory variables: − migration – number of immigrants in a given year; − fdi – foreign direct investment of the donor country in the receiver country; katarzyna andrzejczak, agata kliber dynamic econometric models 15 (2015) 89–109 100 − import_fuel – value of import of fuels, lubricates and related products (import from africa); − export_fuel – value of export of fuels, lubricates and related products (export from france/great britain); − import_crude – value of import of crude materials, inedible except fuel; − export_crude – value of export of crude materials, inedible except fuel; − import – total import from the receiver country; − export – value of the total export to the receiver country; − political_stability – index of political stability in the receiver country; − external_debt – value of the external debt of the receiver country (as % of gdp); − corupt_control – value of the index of corruption control; − girls_out_of_ps – number of girls out of primary school; − mortality_rate – infant mortality rate (deaths per 1000 live births); − life_expectancy – expected length of life in the receiver country; − gdp_per_capita – value of the gdp per capita in the receiver country; − oil – binary variable, taking one for the 12 african countries that have documented oil reserves: algeria, angola, chad, egypt, equatorial guinea, gabon, libya, ,sudan1; − oil_gas – binary variable taking value 1 for the african countries that have documented gas and/or oil reserves, the gas-producers are: angola, benin, cameroon, chad, equatorial guinea, ethiopia, gabon, ghana, kenya, liberia, madagascar, malawi, mauritania, namibia, niger, senegal, sudan, uganda. the source of the data were the following databases: oecd, and afmi (african financial markets initiative). the descriptive statistics of the explanatory variables are given in table 8 in the appendix. the values of import, export and gdp are given in 2012 us dollars. the dependent variable was oda – the amount of help received from the donor. all the data were collected for the time period from 2001 to 2012. the computations were performed using gretl (cottrel, lucchetti, 2014, kufel, 2011). 1 we excluded south sudan from the whole analysis due to the lack of data for most of the indicators. the model of french development assistance – who gets the help? dynamic econometric models 15 (2015) 89–109 101 4. results in table 2 we present the results of the estimation of the model for the full sample. we observe that the amount of help sent from france to africa depends on political factors and trade links. first of all, the higher the migration rate from a given country, the higher the amount of help. the amount of help sent seems to depend positively on the amount of export sent to the given country. this suggests that the trade partners receive higher support. eventually, the historical colonies receive significantly more than the other countries. table 2. results of the dynamic panel model estimation for france – the full sample variable coefficient std.error t statistics p-value oda(–1) 0.318 0.070 4.572 0.000 constant −62.219 24.406 −2.549 0.011 migration (–1) 2.972 0.920 3.231 0.001 colony 38.331 9.309 4.118 0.000 log(eksport)(–1) 3.970 1.448 2.742 0.006 note: sargant test statistics of overidentification amounted to 28.4 (p-value: 1), z-statistics for ar(1) test: –1.77 (p-value=0.07), for ar(2): –0.17 (p-value:0.87). joint wald test statistics: 168 (p-value:0). standard error of residuals: 69.5. table 3. results of the dynamic panel model for france – the oil&gas countries variable name estimate std.error t-statistics p-value oda(–1) 0.294 0.064 4.575 0.000 const −93.3005 47.477 −1.965 0.049 log(import) 5.424 2.583 2.100 0.036 colony 78.742 18.543 4.247 0.000 note: sargant test statistics of overidentification amounted to 16.24 (p-value: 1.00), z-statistics for ar(1) test: –1.73 (p-value=0.08), for ar(2): –0.28 (p-value:0.77). joint wald test statistics: 84.27 (p-value:0). standard error of residuals: 79.19. in table 3 we present the results obtained for the group of countries that have proven oil and gas reserves. the group is not homogeneous, as the descriptive statistics in table 1 show. the heterogeneity of the group can be explained by the political instability and internal conflicts, large debt forgiveness events, and the delayed wealth effect. moreover, the countries that exploit natural resources since many years are more wealthy than the countries in which the oil/gas reserves have been found only recently. thus, the explanatory power of the model is very weak. however, it shows the main drivers for the help distribution. it seems that the factors that influence generosity is again the amount of import from france. supporting outlets for french products is an important goal, which at the same time serves internal katarzyna andrzejczak, agata kliber dynamic econometric models 15 (2015) 89–109 102 policy, by the promotion of french industry and companies. moreover, the countries that used to be french colonies receive on average 70 mln usd more than the other countries. being a former colony increases the benefit of possession of natural resources. in table 4 we present the results obtained for the more homogenous group – the countries that do not possess natural resources. from table 1 we know that the amount of help received by them is significantly lower than those of the countries that have oil and/or gas. we observe again that political factors do play a role in generosity of the donor – the amount of oda received grows together with the migration rate. the residuals from the model have the smallest standard deviation from all estimated ones – 29.51. providing help to the countries of origin of diasporas living in france allows to realize domestic policy goals at the same time indeed. table 4. results of the dynamic panel model for france – the non-oil&gas countries variable name estimate std.error t-statistics p-value oda(–1) 0.228 0.103 2.207 0.027 const 18.660 5.422 3.441 0.001 migration2 0.333 0.047 7.125 0.000 note: sargant test statistics of overidentification amounted to 21.41 (p-value: 1.00), z-statistics for ar(1) test: –1.82 (p-value=0.07), for ar(2): 1.34 (p-value:0.18). joint wald test statistics: 1749.29 (p-value:<0.001). standard error of residuals: 29.51. next, we estimated the models for the groups: historical colonies and others. in table 5 we present the results of the model for colonies. we again observe that the amount of help depends on political dependencies (migration) and trade (the more france imports from the donor, the more help is sent in return there). as the reason for the colonies was to ensure the access to certain resources, the same logic explains the development agenda. table 5. results of the dynamic panel model for france – the historical colony group variable name estimate std.error t-statistics p-value oda(–1) 0.311 0.063 4.969 0.000 const −113.774 60.898 −1.868 0.062 migration(–1) 2.326 1.386 1.679 0.093 log(import)(–1) 9.452 3.772 2.506 0.012 note: sargant test statistics of overidentification amounted to 14.7 (p-value: 1.00), z-statistics for ar(1) test: –1.91(p-value=0.05), for ar(2): –0.12 (p-value:0.90). joint wald test statistics: 67.7 (p-value:0). standard error of residuals: 82.36 eventually, in the case of the group of non-colonies we estimated the static fixed-effect model, since the data did not exhibit any autocorrelation. this the model of french development assistance – who gets the help? dynamic econometric models 15 (2015) 89–109 103 can suggest that french policy is not consequent in this group of countries. the lsdv r2 amounted to 0.47 (the “within” one – to 0.07). in table 6 we present the fixed-effect model with time-effect. the time effect for the period 2009–2012 was insignificant, so we conclude that the average value of oda received by the group was the same as in 2001 (in real value). moreover – the amount of help received by the countries from the group depended on the recipient and the year (this model was the best one from all other estimated models, including those with additional explanatory variables). each recipient received different and specific amount of help which varied in different years but did not depend on any other factors – neither political nor poverty-related ones. we observe that in 2004 and 2006 the average amount of oda was higher, while the lowest – in 2002. all the values presented in the table should be interpreted as the surplus in comparison with 2001. table 6. results of the static fixed-effect panel model with time effect for france – the non-colony group. variable name estimate std.error t-statistics p-value const 6.457 1.179 5.476 0.000 dt_ 2002 1.444 0.462 3.124 0.002 dt_2003 2.286 0.845 2.706 0.007 dt_2004 7.234 2.845 2.543 0.012 dt_2005 6.921 4.040 1.713 0.088 dt_2006 5.308 2.769 1.917 0.057 dt_2007 5.673 1.733 3.273 0.001 dt_2008 4.311 1.423 3.029 0.003 dt_2009 6.400 4.787 1.337 0.183 dt_2010 0.672 0.947 0.710 0.478 dt_2011 0.884 0.919 0.962 0.337 dt_2012 −0.320 0.909 −0.352 0.725 note: lsdv r2 amounted to 0.47, while the within r2 to 0.08. lsdv f(33,238)=6.43(p-value= 0), f-test statistics for named regressors: f(11,238) =1.8 (p-value=0.05).durbin-watson statistics to 1.77. the null hypothesis for common constant in groups was rejected at p-value 4.22e-20. the p-value of the wald test for common significancy of dummy time effects was equal to 1.75e-13. standard error of residuals amounted to 10.14. 4.1 robustness check – kendall tau in table 7 we present the kendall τ computed for the value of financial help from france and the remaining variables in the set. the τ is the rankbased correlation, indicating the non-linear dependencies in the data. the data in table is sorted according to the decreasing value of τ. according to this simple analysis, the political factors influence the amount of help sent the most. france seems to support mainly its own historical colonies and trade partners or – more precisely – the countries of the highest value of katarzyna andrzejczak, agata kliber dynamic econometric models 15 (2015) 89–109 104 import from france. a very important factor seems to be also the migration number, as well as the fact of being a historical colony of france. the fact of possessing oil reserves is also significant, but of lesser importance. france is using the nuclear power to quite a large extend, so this may be one of the explanations. from the poverty-related factors only the illiteracy is significantly related to the amount of oda received. political stability factor is also significantly related to the amount of oda, but the relationship is inverse (the less stable the country, the higher amount of help should it receive). this is neither coherent with the burnside and dolar paradigm of aid effectiveness in sound policy environment nor with the general conditionality of help based on democracy promoted by the dac. the conclusions from the panel models seem to be robust. table 7. kendall-tau for financial help and other considered variables correlation between oda and: tau p-value colony 0.576 <0.001 migration 0.551 0 export_crude 0.531 0 export 0.529 0 export_fuel 0.362 0 import_crude 0.361 0 import 0.323 0 fdi 0.328 0 oil/gas 0.238 0 girls out of primary school 0.272 0 life expectancy 0.073 0.006 gdp_per_capita 0.046 0.086 mortality rate 0.017 0.517 corruption control –0.036 0.172 import_fuel –0.05 0.246 external debt –0.03 0.206 political stability –0.116 0 note: insignificant variables (p-value higher than 0.05) are put in italics. 5. conclusions most of development aid research concentrate on the aid effectiveness in the context of recipient performance. however, the motives of donors are also important. the non-profit and development-driven character of foreign aid flows are questionable. from the early stages of development cooperation former colonial powers enjoyed the access to the commodities from newly created states. the economic ties established during colonialism were not sealed off. despite the reluctance towards former powers, the reconstruction of the economy towards independence was a more complicated process, the model of french development assistance – who gets the help? dynamic econometric models 15 (2015) 89–109 105 than assumed. moreover, the increasing political instability in many countries and numerous military conflicts, made many african states fragile and explained traditional aid donors presence in the politics. establishment of aid structures in african states, allowed donors’ governments to insert national business (especially of the key branches such as telecommunication, energy, minig etc.) on the key local markets. companies without government support or local experience wouldn’t easily decide to invest due to high risks. the countries which were politically involved with african leaders, either for the past reasons (france) or the new ones (usa), were automatically predestinated to gain access to the resources of the region. the traditional development cooperation model, represented here by france, is characterized by longtime relationship at relatively high intensity. the evolution of development assistance is the evolution of traditional model policy declaration – from neocolonial relations to development agenda and sustainable development. this policy means both: keep using aid as an instrument of their foreign policy (votes in un, peacekeeping, access to natural resources), and engage in the initiatives of millennium development goals and the democracy building. they claim to be driven by moral obligation for the past wrongdoing, but at the same time are perceived to use their superior position in negotiations. our study suggest, that the traditional approach to development cooperation is based on donors’ interests in 21st century, just as it was in the precedent one. the analysis of recipient structure shows that the aid volume is correlated positively with oil reserves, both export and import and with migration. to a lesser extent literacy rate and mortality rate were depicted as important. the motivation of traditional donor is therefore self-oriented. development cooperation remains instrumental to realization of foreign and internal policy goals of donors. governments of donor countries, responding to their voters, tend to subordinate development cooperation to increase realization of national and regional interests. securing the access to markets and low cost imports supports both donor country’s entrepreneurs going abroad as the consumers inside the country. the orientation towards “migration countries” may imply the will to keep stable relations, possibly i.a. for the security reasons. also, the importance of cultural relations for the development cooperation, especially because of the use of common language is recognized. we conclude, that despite the official redefinition of the development cooperation goals, the system tends to serve the traditional donor’s agenda in the first place, and next the recipient’s needs are considered. practically, for the african states, it means that a stronger negotiation approach with the donors is needed. at the same time the improvement of internal katarzyna andrzejczak, agata kliber dynamic econometric models 15 (2015) 89–109 106 transparency to ensure local politicians accountability for the development cooperation agreements could positively influence the system functioning in the future. references babaci‐wilhite, z., macleans geo‐jaja, a., shizhou, l. (2013), china's aid to africa: competitor or alternative to the oecd aid architecture?, international journal of social economics, 40(8),729–743, doi: http://dx.doi.org/10.1108/ijse-09-2012-0172. barczak, i. (2008), pomoc szwecji w rozwoju społeczeństwa obywatelskiego w krajach rozwijających się (supporting civil society development with swedish development aid), in przybylska-kapuścińska w. (ed), gospodarka, finanse i społeczeństwo (economy, finances and society), poznań, wyd. ae poznań, 241–245. bearce, d. h., floros, k. m., mckibben, h. (2009), the shadow of the future and international bargaining: the occurrence of bargaining in a three-phase cooperation framework, journal of politics, 71(2), 719–732, doi: http://dx.doi.org/10.1017/s0022381609090562. blundel, r., bond, s. (1998), initial conditions and moment restrictions in dynamic panel data models, journal of econometrics, 87(1), 115–143. bonne, p. (1994), the impact of foreign aid on savings and growth, london school of economics, london. brautigam, d. (2011), aid with chinese characteristics: chinese foreign aid and development finance meet the oecd-dac aid regime, journal of international development, 23(5), 752–764, doi: http://dx.doi.org/10.1002/jid.1798. chaponniere, j., comolet, e., jacquet, p. (2009), les pays emergents et l'aide au developpement (emerging countries and foreign aid policy. with english summary), revue d'economie financiere, 95, 173–188. chou, t. (2012), does development assistance reduce violence? evidence from afghanistan, economics of peace and security journal, 7(2), 5–13. corkin, l. (2011), uneasy allies: china's evolving relations with angola, journal of contemporary african studies, 29(2), 16 –180. cottrel, a., lucchetti, r. j. (2014), gretl user’s guide, gnu regression, econometrics and time-series library web-doc, http://gretl.sourceforge.net/gretl-help/gretl-guide.pdf (25.09.2015). crescenzi, m. c., enterline, a. j., long, s. b. (2008), bringing cooperation back in: a dynamic model of interstate interaction, conflict management and peace science, 25(3), 264–280. chafer, t., cumming, g. (2011), from rivalry to partnership? new approaches to the challenges of africa, ashgate publishing company, burlington. dehart, m. (2012), remodelling the global development landscape: the china model and south–south cooperation in latin america, third world quarterly, 33(7), 1359–1375. dańska-borsiak, b. (2009), zastosowania panelowych modeli dynamicznych w badaniach mikroekonomicznych i makroekonomicznych (dynamic panel data models in microeconomic and macroeconomic research), statistical review, lvi(2), 25–41. charbonneau, b. (2008), dreams of empire: france, europe, and the new interventionism in africa, modern & contemporary france, 16(3), 279–295, doi: http://dx.doi.org/10.1080/09639480802201560. the model of french development assistance – who gets the help? dynamic econometric models 15 (2015) 89–109 107 der-chin horng, a. (2003), the human rights clause in the european union's external trade and development agreements, european law journal, 9(5), 677–701. deszczyński, p. (2001), kraje rozwijające się w koncepcjach ekonomicznych spd. doktryna i praktyka, (spd concept of developing countries. doctrine and practice), wyd. ae poznań, poznań. doucouliagos, h., paldam, m. (2008), aid effectiveness on growth: a meta study, european journal of political economy, 24(1), 1–24, doi: http://dx.doi.org/10.1016/j.ejpoleco.2007.06.002. easterly, w. (2006), les pays pauvres sont-ils condamnés à le rester, mondes en développement, 3(135), 139–140, doi: http://dx.doi.org/10.3917/med.135.0139. easterly, w., pfutze, t. (2008), where does the money go? best and worst practices in foreign aid, journal of economic perspectives, 22(2), 29–52, doi: http://dx.doi.org/10.2139/ssrn.1156890. eyben, r. (2012), struggles in paris: the dac and the purposes of development aid. european journal of development research 25(1), 78–91, doi: http://dx.doi.org/10.1057/ejdr.2012.49. fuchs, j. p. (1993), pour une politique de développement efficace, mâitrisée et transparent: rapport au premier ministre, web-doc. http://lesrapports.ladocumentationfrancaise.fr/brp/954089600/0000.pdf (25.09.2015) gore, c. (2013), the new development cooperation landscape: actors, approaches, architecture, journal of international development, 25, 769–786, doi: http://dx.doi.org/10.1002/jid.2940. greene, w. h. (2011), econometric analysis. 7-th edition, prentice hall. gruszczyński, m., bazyl, m., książek, m., owczarczuk, m., szulc, a., wiśniowski, a., witkowski, b. (2012), mikroekonometria (microeconometrics), wolters kluwer, warszawa. hsiao, ch. (2014), analysis of panel data. third etition, cambridge university press, cambridge, doi: http://dx.doi.org/10.1017/cbo9781139839327. hughes, a., wheeler, j., eyben, r. (2005), rights and power: the challenge for international development agencies, ids bulletin, 36, 63–72, doi: http://dx.doi.org/10.1111/j.1759-5436.2005.tb00179.x. hugon, p. (2008), l’economie du développement et la pensée francophone, editions des archives contemporaines, paris. kim, s., lightfoot, s. (2011), does ‘dac-ability’ really matter? the emergence of nondac donors: introduction to policy arena, journal of international development, 23, 711–721, doi: http://dx.doi.org/10.1002/jid.1795. kmita, j. (1991), essays on the theory of scientific cognition, pwn – polish scientific publishers, kluwer academic publishers, warszawa. kragelund, p. (2008), the return of the non-dac donors to africa: new prospects for african development, development policy review, 26(5), 555–84, doi: http://dx.doi.org/10.1111/j.14677679.2008.00423.x. kufel, t. (2011), ekonometria. rozwiązywanie problemów z wykorzystaniem program gretl (econometrics. problems solving using gretl), pwn, warszawa. lancaster, c. (1999), aid to africa, university of chicago press, chicago. longhi, s., nandi, a. (2014), a practical guide to using panel data, sage, london. mawdsley, e., savage, l., kim, s.-.m. (2013), a 'post-aid world'? paradigm shift in foreign aid and development cooperation at the 2011 busan high level forum. geographical journal, 180, 27–38, doi: http://dx.doi.org/10.1111/j.1475-4959.2012.00490.x. katarzyna andrzejczak, agata kliber dynamic econometric models 15 (2015) 89–109 108 mcewan, c., mawdsley, e. (2012), trilateral development cooperation: power and politics in emerging aid relationships, development and change, 43(6), 1185–1209. oecd (2012b), strategy for development, web-doc, http://www.oecd.org/development/50452316.pdf (23.06.2013). ohler, h., nunnenkamp, p. (2014), needs-based targeting or favoritism? the regional allocation of multilateral aid within recipient countries, kyklos, 67(3), 420–446. osei b., how (2005). aid tying can impose additional cost, african development review, 17(3), 348–365, doi: http://dx.doi.org/10.1111/j.1017-6772.2006.00119.x. ouattara b. (2007). foreign aid, public savings displacement and aid dependency in cote d’ivoire: an aid disagregation approach, oxford development studies, 35(1), 33–46, doi: http://dx.doi.org/10.1080/13600810601167579. page, s., willem te velde, d. (2004), foreign direct investment by african countries, webdoc, http://www.odi.org/sites/odi.org.uk/files/odi-assets/publications-opinionfiles/5739.pdf (25.09.2015). papanek, g. f.(1972), the effect of aid and other resource transfer on savings and growth in less developed economies, the economic journal, 82, 934–950. przeworski, a. (2004), democracy and economic development, in: mansfield e.d. and sisson r.(eds.), the evolution of political knowledge, columbus, ohio state university press. quadir, f. (2013), rising donors and the new narrative of ‘south–south’ cooperation: what prospects for changing the landscape of development assistance programmes?, third world quarterly, 34(2), 321–338, doi: http://dx.doi.org/10.1080/01436597.2013.775788. radelet, s., clemens, m., bhavnani, r. (2004), aid and growth: the current debate and some new evidence, centre for global development working paper, https://www.imf.org/external/np/seminars/eng/2005/famm/pdf/radele.pdf (25.09.2015) roodman, d. (2009), how to do xtabond2: an introduction to difference and system gmm in stata, the stata journal, 9(1), 86–136, doi: http://dx.doi.org/10.2139/ssrn.982943. rosen, h. (1977), technology transfer to developing nations, journal of technology transfer, 1(2), 93–104. sanfilippo, m. (2010), chinese fdi to africa: what is the nexus with foreign economic cooperation?, african development review/revue africaine de developpement, 22599–614, doi: http://dx.doi.org/10.1111/j.1467-8268.2010.00261.x. shirazi, n. s., mannap, t. a., ali, m. (2009), effectiveness of foreign aid and human development, pakistan development review, 48(4), 853–862. simplice, a. (2014), development thresholds of foreign aid effectiveness in africa, international journal of social economics, 41(11), 1113–1155, doi: http://dx.doi.org/10.1108/ijse-01-2013-0014. tajoli, l. (1999), the impact of tied aid on trade flows between donor and recipient countries, journal of international trade and economic development, 8 (4),373–388. tanburn, j. (2008), the 2008 reader on private sector development, web-doc, http://www.ilo.org/wcmsp5/groups/public/---ed_emp/---emp_ent/--ifp_seed/documents/publication/wcms_143158.pdf (25.09.2015). thiele, r., nunnenkamp, p., dreher, a. (2007), do donors target aid in line with the millennium development goals? a sector perspective of aid allocation. review of world economics, weltwirtschaftliches archiv, 143(4), 596–630, doi: http://dx.doi.org/10.1007/s10290-007-0124-x. un (2014), world investment report 2014: investing in the sdgs: an action plan, webdoc, http://unctad.org/en/publicationchapters/wir2014ch4_en.pdf (25.09.2015). the model of french development assistance – who gets the help? dynamic econometric models 15 (2015) 89–109 109 walker, m. (2008), indian ocean nexus, wilson quarterly, 32(2), 21–28. williams, j. h. (2014), us foreign aid, asian education and development studies, 3(1), 11–30. williamson, c. r. (2008), foreign aid and human development: the impact of foreign aid to the health sector, southern economic journal, 75(1), 188–207. appendix table 8. descriptive statistics of the explanatory variables mean median min max std.dev. within s.d. between s.d. migration 1.726 0.207 0 31.113 4.584 0.719 4.571 fdi 91.711 6.276 –7532.6 12918 789.420 827.64 144.22 import 5.12e+08 3.55e+07 1841.5 7.36e+09 1.17e+09 5.15e+08 1.06e+09 import fuel 7.01e+08 9.60e+07 19 7.05e+09 1.33e+09 7.70e+08 9.90e+08 import crude 2.06e+07 4.03e+06 3 2.68e+08 3.75e+07 1.59e+07 3.31e+07 export 5.51e+08 1.18e+08 37316 8.42e+09 1.21e+09 3.78e+08 1.16e+09 export fuel 3.29e+07 1.75e+06 185 8.61e+08 9.72e+07 7.4e+07 6.43e+07 export crude 6.16e+06 8.40e+05 1157.9 2.54e+08 2.15e+07 1.05e+07 1.83e+07 political stability –0.539 –0.36 –3.3 1.190 0.93 0.355 0.873 corruption control –0.607 –0.67 –1.92 1.260 0.58 0.193 0.555 girls out of ps 3.47e+05 1.57e+05 52 5.07e+06 7.32e+05 1.34e+05 7.63e+05 life expectancy 56.282 55.708 12.3 114.4 13.203 11.612 7.170 mortality rate 64.701 63.95 12.2 138.5 27.265 9 26.092 gdp per capita 2024.2 729.88 110.5 24355 3153.8 1434.8 2863.8 external debt 65.195 46.575 0.65 881.95 85.792 58.625 65.423 model francuskiej pomocy rozwojowej – kto dostaje pieniądze? z a r y s t r e ś c i. artykuł przedstawia analizę dystrybucji francuskiej pomocy rozwojowej wśród krajów afrykańskich, w latach 2001–2012. na podstawie wyników uzyskanych przy pomocy dynamicznych modeli panelowych autorki stwierdzają, że pomoc nie była kierowana do krajów najbardziej potrzebujących, ale do tych, z którymi łączą francję więzi polityczne i gospodarcze (m.in. byłe kolonie oraz kraje zasobne w ropę i gaz). s ł o w a k l u c z o w e: pomoc rozwojowa, deklaracja milenijna, ubóstwo, francja, dynamiczny model panelowy pusta strona dem_2015_129to156 © 2016 nicolaus copernicus university press. all rights reserved. http://www.dem.umk.pl/dem d y n a m i c e c o n o m e t r i c m o d e l s doi: http://dx.doi.org/10.12775/dem.2015.006 vol. 15 (2015) 129−156 submitted october 22, 2015 issn (online) 2450-7067 accepted december 15, 2015 issn (print) 1234-3862 ewa ratuszny* risk modeling of commodities using caviar models, the encompassing method and the combined forecasts a b s t r a c t. the aim of the research is to compare var methods/models for commodities. for risk measurement conditional autoregressive value at risk models (caviar), implied quantile model and encompassing method are used. the aim is to check whether simultaneous use of information both from historical time series and regarding markets' expectation can improve accuracy of forecasts. for this purpose four methods of combining forecasts are used: a simple average combining, an unrestricted linear combination, a weighted averaged combining and a weighted averaged combining using exponential weighting. in the case of the commodities neither the encompassing method nor the combining forecast method improve var forecasts. the method of choosing the most adequate model leads to simple caviar-sav model as the source of most optimal measure of risk forecasts. the kupiec test, the christoffersen and the dynamic quantile test indicate the model as an adequate to forecast var for gold and oil for short positions at the 0.01 and the 0.05 significance level, and for a long position at the 0.05 significance level. k e y w o r d s: caviar, var, encompassing method, combined forecasts, commodities j e l classification: c22, g17. introduction value at risk models should provide an adequate risk forecast both in stability period and in period with high volatility. accurate assessment of the risk is required for capital management purpose, limit settings and position * correspondence to: ewa ratuszny, warsaw school of economics, al. niepodległości 162, 02-554 warsaw, poland, e-mail: ewa.ratuszny@doktorant.sgh.waw.pl ewa ratuszny dynamic econometric models 15 (2015) 129–156 130 management. nowadays there exist many risk measurement methods but none of the models surpasses the others. this paper extends research proposed by jeon and taylor (2013) for commodities. moreover, we apply more complex way to compare var methods which helps to avoid overestimation and underestimation of risk. value at risk is defined as the maximum potential loss in portfolio value over a given time period due to adverse market movements (i.e. 500 days), with a given significance level of α (doman, doman, 2009; iwaniczdrozdowska, 2005). engle and manganelli (2004) have classified the existing var methods into three broad categories: parametric, semiparametric and nonparametric. parametric approach includes riskmetrics methodology and garch models (piontek, 2000; fiszeder, 2009; jajuga, 2011; mazur, pipień, 2012), but the weakness of those methods lies in possibility of incorrect specification both of variance model and the error distribution. an interesting parametric method, which becomes increasingly popular, is based on implied volatility. implied volatility is the expectation of volatility implied by the option market (chong, 2004; christoffersen, mazzotta, 2005; giot, 2005). the most common nonparametric approach is the historical simulation, used by about 73% of banks (pérignon, smith, 2010). the main advantage is that the historical var does not require an assumption about parametric form of the distribution of the risk factor returns. nevertheless the var forecast might be inaccurate due to inadequate rolling window of risk factors (boudoukh et al., 1998). a long data history will typically encompass several regimes with different behavior of market risk factors. boudoukh et al. (1998), mittnik and paolella (2000) and taylor (2008) propose to apply exponentially weighted approaches to var estimation to overcome those difficulties. in our research we apply semiparametric approach based on conditional autoregressive value at risk models (caviar). engle and manganelli (2004) have proposed models that derive a time-varying var directly via autoregression. the models are estimated using robust method, i.e. a nonlinear quantile regression proposed by koenker and bassett (1978). the robust approach is widely applied in risk measurement, hedging and portfolio allocation (taylor, 1999; umantsev, chernozhukov, 2001). this approach allows the shape of the conditional returns distributions to vary in time, and for the time-variation to differ for the different quantiles of distribution (jeon, taylor, 2013). the autoregressive structure is adequate in case of clustered time series. the previous researches by ratuszny (2013), risk modeling of commodities using caviar models… dynamic econometric models 15 (2015) 129–156 131 ratuszny (2015) indicated that caviar models successfuly compete with other var methods. jeon and taylor (2013) proposed to combine quantile forecasts – elaborated above 25 year ago by granger (1989) and granger, white and kamstra (1989). in their approach the quantile forecasts, obtained from caviar models and from method based on implied volatility, are combined. they applied their method not to economic indicators but to risk measurement of position in equity indices such as s&p500 and dax30. moreover, they included in the caviar models an additional regressor: a quantiles predictor based on implied volatility (encompassing method). the authors concluded that linear combining method and arithmetical method generate better forecast over the sample. the observation motivated us to apply their approach to commodities. the polish researches on combined forecast performed by grajek (2002), greszta, maciejewski (2005), piłatowska (2009) have also indicated the predominance of combined forecasts over the method based on single approach. in our research we apply caviar models, the encompassing method and four combining forecast methods: simple average combining, unrestricted linear combination, weighted averaged combining and weighted averaged combining optimized using exponential weighting. we try to verify the following hypothesis: the encompassing method or combining forecast methods based on caviar models and implied quantile model improve accuracy of var for commodities. the paper is organized as follows. firstly, we review value at risk methodology based on caviar models, implied quantile, encompassing method and combining forecast methodology. the part contains also the performance criteria. the second part contains empirical applications of the models. the last part contains concluding remarks. 1. review of value at risk methodology 1.1. caviar models the conditional autoregressive value at risk model has been introduced by engle and manganelli (2004). the basic intuition is to model directly the evolution of the quantile over time, rather than the whole distribution of portfolio returns. the general form of caviar models is defined by (engle, manganelli, 2004; doman, doman, 2009): ewa ratuszny dynamic econometric models 15 (2015) 129–156 132 ,)|,,(),(=),(=),( 11 1= 0 −++− ++ ∑ tqppiti p i tt lvarfvar fβββαβββα α kβx (1) where 1−tf is the information set available at time 1−t , and ),(= 0 ′+ qpββα kβ is the vector of parameters which is estimated using nonlinear regression quantile techniques. in most practical cases the above formulation is reduced to a first order model: ,)),(,,(),(=),( 112110 βαββαβββα −−− ++ tttt varylvarvar (2) where )|(= 1−− tttt rery f , tr is rate of return, )|( 1−ttre f is the expected value of rate of returns. the autoregressive term ),(11 βαβ −tvar ensures that the var changes smoothly over time. the role of )),(,,( 112 βαβ −− tt varyl is the linking the level of explained variable )(αtvar to the level of y at the moment 1−t . that is, it measures the impact of new information in y on the level of var. the following caviar models are analysed in our research both for long position (l) and short position (s) (engle, manganelli, 2004; doman, doman, 2009): 1. symmetric absolute value – sav ,),(=),( 12110 −− ++ t l t l t yvarvar ββαβββα (3) .),(=),( 12110 −− ++ t s t s t yvarvar ββαβββα (4) current var depends on the past value 1−tvar and absolute value of past rate of return. the model symmetrically responds to both negative and positive past returns. 2. asymmetric slope – as ,0)<(0)(),(=),( 1131111 −−−−− +≥+ tttt l t l t yiyyiyvarvar βββαβα (5) ,0)<(0)(),(=),( 1131111 −−−−− +≥+ tttt s t s t yiyyiyvarvar βββαβα (6) where )(⋅i is the indicator function. current var depends on its past value 1−tvar and on positive and negative returns that are treated in different way. 3. indirect garch for both short and long position: [ ] .),(=),( 1/22 132 121 −− ++ ttt yvarvar ββαβββα (7) current var is described as garch process. the model is correctly specified for rate of returns from garch(1,1) model. risk modeling of commodities using caviar models… dynamic econometric models 15 (2015) 129–156 133 4. adaptive – ad ( )[ ]{ },)],([exp1),(=),( 11111 αβαββαβα −−++ −−−− lttltlt varygvarvar (8) ( )[ ]{ },)],([exp1),(=),( 11111 αβαββαβα −−++ −−−− sttstlt varygvarvar (9) where g is some positive finite number. if ∞→g , the second term converges to ])),( ([ 11 αβαβ −− −− tt varyi for long position and to ])),(([ 11 αβαβ −≥ −− tt varyi for short position, where )(⋅i is the indicator function. in case of a var breach the var forecast should be increased, otherwise should be slightly decreased. the model aims to reduce the probability of sequences of var breaches and will also make unlikely that the var has never been reached. the disadvantage of this type of caviar models is lack of rate of return in explanatory variables set so that the information about extremal market movements is not effectivelly included in model (doman, doman, 2009). estimation of caviar models is performed on the basis of koenker and basset (1978) regression quantile methodology, which minimalises the regression quantile objective function of the following form for long (l) and short (s) position, respectively: ,),()(1 ),(min ),(> ),(      −−+      +− ∑ ∑ − − βαα βαα βα βα β l tt l tvartyt l tt l tvartyt vary vary (10) .),()(1 ),(min ),(< ),(      −−      +− ∑ ∑ ≥ βαα βαα βα βα β s tt s tvartyt s tt s tvartyt vary vary (11) ewa ratuszny dynamic econometric models 15 (2015) 129–156 134 1.2. implied volatility implied volatility reflects market’s expectations regarding future volatility. implied volatility is the key variable in financial investment decision, risk management, derivative pricing, market making, market timing and portfolio selection. in spite of huge volume of research, no consensus has been reached on usefulness of implied volatility as a predictor for future volatility in comparison with predictions from time series models. there are many empirical studies in which implied volatility overcomes the historical volatility (szakmary et al. (2003) for futures on equity indices, interest rates, currencies, commodities and crude oil; pong et al. (2004) for fx; corredor and santamaria (2004) for ibex; giot and laurent (2007) for stock indices such as the s&p100 and s&p500). noh and kim (2006) conclude that both implied volatility and historical volatility using high-frequency returns can outperform each other in forecasting volatility. in their empirical test, historical volatility from high frequency returns performed better in the ftse100 futures, which tend to be relatively close to normally distributed, while the result of implied volatility was better in the s&p500 futures, which displays excess skewness even with volatilities from high frequency returns. implied volatility is also considered as useful variable for estimating quantile of the returns distribution (giot, 2005; chong, 2004). jeon and taylor (2013) applied implied volatility to var for equity indices s&p500 and dax30. they construct an implied quantile ( iq ) estimator as the product of the implied volatility recorded in the previous period impliedt 1−σ , and the empirical distribution quantile )(αempq of ( ty ) standardised by the implied volatility. the iq estimator for long and short position is expressed in the following form (jeon, taylor, 2013): ,)(=)( 1 )()( implied t lempliq t qvar −σαα (12) .)(=)( 1 )()( implied t sempsiq t qvar −σαα (13) the iq approach captures the market’s expectation of future risk. another advantage is that the method does not assume a particular distribution for the asset returns, and it involves no parameter estimation. jeon and taylor (2013) note that this simple approach to capture an ‘implied quantile’ assumes returns standardised with implied volatility are i.i.d. risk modeling of commodities using caviar models… dynamic econometric models 15 (2015) 129–156 135 1.3. encompassing method jeon and taylor (2013) propose to construct one model encompassing competitive forecast models. such model should generate better forecast than every model separately. this approach is called encompassing (chong, hendry, 1986; diebold, 1989; grajek, 2002) or plug-in (jeon, taylor, 2013). day and lewis (1992) in their research show that the implied volatility and models based on historical volatility (egarch or garch models) does not reflect whole information about volatility. blair et al. (2001) received completely different results. they shows that implied volatility vix, is a significant explanatory variable for volatility forecast of s&p100. claessen and mittnik (2002) and giot (2005) performed research over the sample. claessen and mittnik (2002) included the implied volatility vdax to garch model, and show that implied volatility reflect market expectation about future volatility of dax. similarly giot (2005) for nasdaq and s&p500 shows that implied volatility vic and vxn included in garch models improves a volatility forecast. jeon and taylor (2013) analyzing the results of previous research of encompassing method, decided to check the possibility to receive a better estimate of var if information from historical time series and information regarding risk expected by market are combined. the rationale for their research was that if the implied volatility forecasts the future well, it should be useful in estimating future quantile of returns distribution. they include the implied quantile expressed by equation (12) or (13) to the caviar models as an explanatory variable. the impact of implied volatility on var can be determined by coefficient iqβ . the following models are analysed in our research (jeon, taylor, 2013): 1. symmetric absolute value plugin: (sav-plugin): ,)(),(=),( )(12110 αβββαβββα liq tiqt l t l t varyvarvar +++ −− (14) .)(),(=),( )(12110 αβββαβββα siq tiqt s t s t varyvarvar +++ −− (15) 2. asymmetric slope plugin: (as-plugin): ,)( 0)<(0)(),(=),( )( 1131111 αβ βββαβα liq tiq tttt l t l t var yiyyiyvarvar + ++≥+ −−−−− (16) ewa ratuszny dynamic econometric models 15 (2015) 129–156 136 .)( 0)<(0)(),(=),( )( 1131111 αβ βββαβα siq tiq tttt s t s t var yiyyiyvarvar + ++≥+ −−−−− (17) 3. indirect garch(1,1) plugin (igarch-plugin): [ ] [ ]{ }212)(2 132121 )(),(=),( αβββαβββα liqtiqtltlt varyvarvar +++ −− (18) [ ] [ ]{ } .)(),(=),( 2 1 2)(2 13 2 121 αβββαβββα siq tiqt s t s t varyvarvar +++ −− (19) 4. adaptive plugin (ad-plugin): ( )[ ]{ } ,)()],([exp1 ),(=),( )(1 113 121 αβαβαβ βαβββα liq tiq l tt l t ll t varvaryg varvar +−−+ ++ − −− − (20) ( )[ ]{ } .)()],([exp1 ),(=),( )(1 113 121 αβαβαβ βαβββα siq tiq s tt s t s t varvaryg varvar +−−++ ++ − −− − (21) notations in the models are the same as in part 1.1. and 1.2. 1.3. 1.4. combining method to var forecast if it is not clear which of two forecasts performs better, a combination can be the best option (bates, granger, 1969). combining methods include information contained in each of individual forecast. according to armstrong (2001) the combined forecasts should be applied if several different models can be combined to obtain better forecast, there is no certainty about the future state of the object forecast, and where large forecasting error involves a high cost. by combining forecasters should able to reduce inconsistency in estimates and to cancel out biases to some extent. the work by bates and granger (1969) often is considered to be the seminal article on combining forecasts. they combined two separate sets of forecasts of airline passenger data to form a composite set of forecasts. they concluded that the composite set of forecasts can yield lower mean-square error than either of the original forecasts. past errors of each of the original forecasts are used to determine the weights to attach to these two original forecasts in forming the combined forecasts. they also examined different methods of deriving these weights. combined forecasts for economy risk modeling of commodities using caviar models… dynamic econometric models 15 (2015) 129–156 137 indicators are subject of research of crane and crotty (1967), zarnowitz (1967), nelson (1972, 1984). despite the criticism of combined forecasts (e.g. diebold (1989) showed that it is better to improve one of the single models, rather than relying on combining methods of forecast derived from models with incorrect specifications), this approach has become the subject of further research. combining forecasts for the variation is subject of research of doidge and wei (1998), armendola and storti (2008), donaldson and kamstra (2005). there are very few studies about combining quantile forecasts. granger (1989) and granger et al. (1989) introduce the idea of using quantile regression to combine quantile forecasts. taylor and bunn (1998) assess the usefulness of different restrictions on the parameters of the quantile regression combination. giacomini and komunjer (2005) describe how encompassing tests can be performed for two quantile predictors using the quantile regression combining framework. they apply their proposal to var estimates of the s&p500 based on two time series volatility forecasting methods. jeon and taylor (2013) in their research applied the following four combined mehods: simple average combining (simpavg), unrestricted linear combination (linearcomb), weighted averaged combining (wtdavg) and weighted averaged combining optimized using exponential weighting (wtdavgexp). 2. simple average combining (simpavg) the simplest and most widely used forecast combining method is to take the simple arithmetic mean of the individual forecasts. we consider the simple average of the quantile forecasts from the iq method and one caviar model, as in expression (3–9) (jeon and taylor, 2013): ,),( 2 1 )( 2 1 =)( )()( βααα lcaviart liq t l t varqvar + (22) .),( 2 1 )( 2 1 =)( )()( βααα scaviart siq t s t varqvar + (23) the method will be here denoted as simpavg according to jeon and taylor (2013) nomenclature. the aim of this approach is to determine the combination of forecasts with lower error variance than in case of individual forecasts. ewa ratuszny dynamic econometric models 15 (2015) 129–156 138 3. unrestricted linear combination (linearcomb) a traditional approach to combining is to compute linear combinations of forecasts, called also regression method (jeon and taylor, 2013). the method will be dnoted as linearcom according to jeon and taylor (2013) nomenclature. forecast is formed on the basis of an iq forecast and one of caviar models (jeon and taylor, 2013): ,),()(=)( )(3 )( 21 βαγαγγα lcaviar t liq t l t varqvar ++ (24) .),()(=)( )(3 )( 21 βαγαγγα scaviar t siq t s t varqvar ++ (25) the parameters 2γ and 3γ inform about the dynamics of forecasted variable. if the sum of the parameters 2γ and 3γ is less than unity, the individual predictions are more volatile than the risk measure var. if the sum of the parameters is greater than one, then the individual forecasts are of less dynamic than var. there are several difficulties with the combination method. the first is related to collinearity of individual forecasts. if the individual predictions are quite good, they would not differ significantly and this entails the phenomenon of collinearity. consequently, the low-significance and high randomness of estimated weights are obtained. another issue is the autocorrelation of the random component, caused by autocorrelation of dependent variable. in order to solve this problem diebold (1988) proposed to estimate the arch model. the third issue is related with the inability to impose zero restrictions for correlation between the errors of individual forecasts, when examining the behavior of individual forecasts in the past. in addition, regression method requires a large data sets, which in case of time series is fullfilled. the advantage of this method is the lack of restrictions on the parameters and lack of assumptions about unbiasedness of individual forecasts. 4. weighted averaged combining (wtdavg) the weighted averaged combining method is based on the relation between forecast error in the past. in this approach the unbiasedness of quantile forecast is assumed (granger (1989)). error variance of combined forecast will be equal or smaller than of the individual forecasts. the method in our research will be noted as wtdavg according to jeon and taylor (2013) nomenclature. the resultant quantile forecast is of the form (26–27), without risk modeling of commodities using caviar models… dynamic econometric models 15 (2015) 129–156 139 constant, where combining weights are constrained to be between zero and one. ,),()(1)(=),( )()( βαωαωωα lcaviart liq t l t varqvar −+ (26) .),()(1)(=),( )()( βαωαωωα scaviart siq t s t varqvar −+ (27) clemen (1986) advocates the use of the weighted average even if the forecasts are biased, arguing that gains in efficiency can be made at the cost of some bias. bunn (1989) noted greater robustness of the method compared with regression method. taylor and bunn (1998) pointed out that the value of the weight indicates the relative explanatory powers of the two quantile predictors. 5. weighted averaged combining optimized using exponential weighting (wtdavgexp). the method is similar to weighted averaged combining but additionally the exponential weighting factor for the optimisation of the combining weight is applied. the factor gives greater weight to the more recent observations in the quantile regression optimisation (taylor (2008)). in this way the nonstationarity problem of weights is solved. this is particularly important when the time series exhibits time-varing and cyclical volatility. boudoukh et al. (1998) insist that such an approach is a reasonable compromise between statistical precision and adaptation to the latest information. exponentially weighted quantile regression (ewqr) method solves the following minimizing problem (jeon, taylor, 2013):      −−+      +− − − − −≤ ∑ ∑ ),()(1 ),(min ),(>| ),( | ωααλ ωααλ ωα ωα ω l tt tt l tvartyt l tt tt l tvartyt vary vary (28)      −−+      +− − − ≥ ∑ ∑ ),()(1 ),(min ),(<| ),(| ωααλ ωααλ ωα ωα ω s tt tt s tvartyt s tt tt s tvartyt vary vary (29) ewa ratuszny dynamic econometric models 15 (2015) 129–156 140 where )(αltvar and )(α s tvar are expressed in equations (26)–(27). a lower value of the decay parameter λ implies faster exponential decay, and hence more weight is given to the recent observations and less historical information is captured. this method is noted as wtdavgexp according to jeon and taylor (2013). 5.1. out-of-sample diagnostics regulators can apply backtest for evaluating the accuracy of the var models, but this method misclassifies forecasts from inaccurate models as acceptably accurate. in our research the out-of-sample diagnostic of var is performed on the basis tests and measures, i.e. backtests, tests based on bernoulli trials model, the dynamic quantile test, regulatory loss, binary loss, firm’s loss. the lr test of unconditional coverage (kupiec test) evaluates the model, taking into account both too much and too few exceedances. its disadvantage, however, is that it does not take into account the distribution of exceedances in the sample. a well-functioning var model should be characterized by the absence of autocorrelation in the indicator function what can be done by performing the dynamic quantile test. the model is considered adequate if the number of exceedances corresponds to the assumptions and there is no autocorrelation. in case of no exceedances we consider that the model is inadequate, because of overestimation of var. recall the construction and interpretation of these tests/measures. kupiec (1995), christoffersen (1998), rachev and mittnik (2002) proposed the indicator variable ( tξ ) for time t, made at time t–1, which is defined as: , )(<1 )(0 =    − −≥ + + α αξ l tnt l tntl t varyfi varyfi (30) . )(>1 )( 0 =    + + α αξ s tnt s tnts t varyif varyfi (31) on basis of the variable we perform backtesting. in the backtest we check ex post if observed loss ( nty + ) breaches the forecast var at the time t. if the var is indicated on the significance level of α , an appriopriate model should also indicate fraction of exceedances of the realised loss at the level of α . if the fraction of breaches is much greater than assumed, it means that risk modeling of commodities using caviar models… dynamic econometric models 15 (2015) 129–156 141 the model underestimates the size of var. lower number of exceedances means in turn that the model overestimates the value at risk (doman, doman, 2009). kupiec (1995) proposes to treat tttttpozt ′++ ,1,,=: kξ , where poz denotes instrument position, i.e. l – long or s – short, as sequence of bernoulli trials of independent variables with the same probability of success: , )(<1 1)(0 =    − −−≥ + + αα ααξ ithwvaryif ithwvaryfi l tnt l tntl t (30) . )(>1 1)( 0 =    − + + αα ααξ .ithwvaryif ithwvaryif s tnt s tnts t (31) the null hipotesis of kupiec test is: αα =:0h . the likelihood ratio test statistic uclr (ang. the lr test of unconditional coverage) is given by an equation (pipień, 2006): ( ){ } ( ){ }[ ],1lnˆ1ln2= 11 sstsstuclr αααα −+′−−′ −−− (32) where pozt tt tt s ξ∑ ′+ = = means the total number of exceedances and poz t tt ttt ξα ∑ ′+ +′ =1 1 =ˆ is the assessment of the likelihood of success. with a true null hypothesis, test statistic has asymptotic distribution 21χ . the null hypothesis is rejected if the statistical value is above a critical value, i.e. 84.3>uclr . christoffersen (1998) combines the above tests for unconditional coverage and independence. in effect, the null hypothesis of the unconditional coverage test will be tested against the alternative of the independence test. the statistics of the joint test of coverage and independence is expressed by the following equation (christoffersen, 1998): .= induccc lrlrlr + (33) where: ( ) ( ){ }[ −−− 1111101101010001 ˆˆ1ˆˆ1ln2= ttttindlr ππππ ( ) ( ) ( ){ }].ˆˆ1ln 1101210002 tttt ++−− ππ (34) ewa ratuszny dynamic econometric models 15 (2015) 129–156 142 where )/(=ˆ 10 iiijij ttt +π ; ttt )/(=ˆ 1101 +π for 0,1=, ij , ijt – number of points at time { }ttt ≤≤;2 for which the ii t = follows ji t =1+ . the test statistic has asymptotic distribution 22χ . the null hypothesis is rejected if the statistical value is above a critical value, i.e. 99.5>cclr . the dynamic quantile test was constructed to check the absence of autocorrelation in sequence },1,,=:{ tttttpozt ′++ kξ , where poz tξ is binary variable expressed by equations (30)–(31). define for long position ααα −− ))(<(=)( tt l t varyihit and for short position: )())(>(=)( ααα −tt s t varyihit , where )(⋅i is indication function. the dynamic quantile test verifies two hypothesis simultaneously: • ,0=))((:01 α poz thiteh • :02h variable )(α poz thit is uncorrelated with the variables included into information set. engle and manganelli (2004) jointly verify the above hypothesis by the regression of the following form: ,=)( t pozhit ελα +x (35) where x is the matrix ][= , jtxx of dimension kt × , where in the first column are the ones, then p columns contains variables ptt hithit −− k,1 , and k–p–1 remaining columns – an additional independent variables (including tvar )cthe dynamic quantile test statistic is expressed by the equation: , )(1 ˆˆ αα λλ − ′xx' (38) where ( ) yxxx ′′ −1=λ̂ is the ols estimate of parameters λ . the test statistic is asymptotically distributed 2kχ . detailed test results are available from the author on the request. in table (12)–(13) bolded value indicate models which are adequate under assumed criterion. lopez (1998) proposed the loss functions evaluation method not based on hypothesis testing framework, but rather on assigning to the var estimates a numerical score that reflects specific regulatory or firm’s risk modeling of commodities using caviar models… dynamic econometric models 15 (2015) 129–156 143 concern. in our research we take into account binary loss, regulatory loss and firm’s loss. let loss function will be implied by the binomial method which takes value 0 or 1 related with observed var breach:    − −≥ + + + )(<1 )(0 =))(,( α αα l tnt l tntl tnt l t varyfor varyfor varyf , (39) . )(1 )(<0 =))(,(    ≥+ + + α αα s tnt s tnts tnt s t varyfor varyfor varyf (36) binary loss (bl) is described by the number of exceptions observed in period from ntt += to nttt +′+= (lopez, 1998; pipien, 2006): pozt tt tt poz fbl ∑ ′+ = = . (37) the smaller is the number of exceedances, the better rating for the models will be assigned. this criterion favors models which overestimate the var and assign low score for the models that generate liberal var forecasts. in the assessment of var forecasts an important issue is to take into account the size of the losses that are associated with exceptions of var by observation nty + . function (41) includes only the fact of exceptions, and does not take into account the size of the losses arising from an extremal market movement. the second loss function proposed by lopez (1999) contains both the magnitude and the number of exceptions:    −++ −≥ ++ + + )(<))((1 )(0 =))(,( 2 αα αα l tnt l tnt l tntl tnt l t varyforvary varyfor varyf , (38) . )())((1 )(<0 =))(,( 2    ≥−+ ++ + + αα αα s tnt s tnt s tnts tnt s t varyforvary varyfor varyf (39) regulatory loss (rl) is expressed as follows (sarma et al., 2003): .= = poz t tt tt poz frl ∑ ′+ (40) thus, as before, a score of one is imposed when an exception occurs, but also, an additional term based on its magnitude is included. the numerical score increases with magnitude of the exception and can provide additional ewa ratuszny dynamic econometric models 15 (2015) 129–156 144 information on how the underlying var model forecasts the lower tail of the inderlying distribution. sarma et al. (2003) pointed out that in financial institutions exists conflict between profit maximization and the duty of protection against market risk. the duty is related with basel iii which imposes an obligation to maintain the capital requirements to cover potential losses. sarma et al. (2003) propose to incorporate in the loss function the additional costs arising from the capital adequacy. the loss function is:    −++ −≥ ++ + + )(<))((1 )()( =))(,( 2 αα ααα l tnt l tnt l tnt l tl tnt l t varyforvary varyforcvar varyf , (45) . )())((1 )(<)( =))(,( 2    ≥−+ ++ + + αα ααα s tnt s tnt s tnt s ts tnt s t varyforvary varyforcvar varyf (46) where the parameter 0>c specifies the opportunity cost associated with non-use of the capital which the institution must hold in order to hedge against the risk predicted by var (sarma et al.,2003; pipień, 2006). in our research we assume 1=c . the cumulated value of pozf is expressed as follows (pipień, 2006): .= = poz t tt tt poz ffl ∑ ′+ (47) the function is called firm’s loss (fl) and enables to compare the var forecasts generated by different models in scope of market risk hedging. the model that generates too conservative var predictions will – unlike to (44) – be penalized by the (47) due to inefficient maintenance of excess capital in order to hedge against market risks. 6. empirical study 6.1. descriptive statistics we analysed close price ( tp ) from august 1 st, 2008 to october 10th, 2014 in case of gold (1593 observation), and from may 10th, 2008 to october 10th, 2014 (1851 observation) for oil. as an implied volatility we used cboe gold volatility index and cboe crude oil volatility index ( )impliedtσ . the indices measure the market's expectation of volatility implicit in the prices of options. the indices are leading barometers of investor risk modeling of commodities using caviar models… dynamic econometric models 15 (2015) 129–156 145 sentiment and market volatility relating to listed options on an instrument with different strike prices, at the money (atm) and out of the money (otm), which are then averaged to provide hypothetical price of atm options with a maturity of one month (22 days business). daily volatility is calculated using scaling rule: .252= 1252 impliedimplied σσ table 1. descriptive statistics instrument median mean std minimum maximum skewness kurtosis deviation gold 0.0005 0.0002 0.0127 –0.0888 0.1044 –0.2497 10.0226 oil 0.0008 0.0002 0.0239 –0.1274 0.1503 0.0379 9.0206 table 2. the lomnicki-jarque-bera test results instrument statistics p-value gold 24.5934 0.2174 oil 83.5494 0.0000 table 3. the ljung-box test results lags 10 15 20 instrument statistics p-value statistics p-value statistics p-value gold 13.7835 0.1831 22.9158 0.0859 24.5934 0.2174 oil 46.6134 0.0000 74.5049 0.0000 83.5494 0.0000 table 4. the engle test results lags 10 15 20 instrument statistics p-value statistics p-value statistics p-value gold 89.3389 0.0000 105.6773 0.0000 148.0272 0.0000 oil 413.6957 0.0000 488.4326 0.0000 522.1252 0.0000 table 5. the mcleod-li test results lags 10 15 20 instrument statistics p-value statistics p-value statistics p-value gold 150.8873 0.0000 207.7756 0.0000 303.2220 0.0000 oil 1 300.5209 0.0000 1 908.8351 0.0000 2 553.6536 0.0000 the time series of the quotations, prices and rates of return were checked for the presence of the following features: fatter tails than in the normal distribution (identified on the basis of the quantile-quantile plots, histograms and the lomnicki-jarque-bera test); stationarity; autocorrelation of the rates of returns (checked with the ljung-box test); skewness, kurtosis of rates of return. the rate of returns have high degrees of kurtosis, a negative skewness is evident in case of gold. the oil time series is characterised by positive skewness. the lomnicki-jarque-bera test rejects normality at the 5%-level ewa ratuszny dynamic econometric models 15 (2015) 129–156 146 in case of oil. the standard deviation of the rate of returns is the highest in the case of oil. the q test ljung-box in case of oil indicates autocorrelation. the engle and mcleod-li test confirms the existence of a strong and permanent nonlinear dependence. table 6. gold. estimated parameters of caviar and caviar-plugin models model parametr caviar plugin long position short position long position short position 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 sav 1β 0.00041 0.00008 0.00037 0.00025 –0.00223 –0.00414 –0.00156 –0.00271 (0.0004) (0.0002) (0.0004) (0.0002) (0.0073) (0.0050) (0.0043) (0.0027) 2β 0.93857 0.94362 0.95419 0.94818 –0.35921 –0.28538 0.53545 0.33501 (0.0222) (0.0244) (0.0277) (0.0163) (0.5209) (1.1716) (0.4215) (0.4223) 3β 0.18587 0.11934 0.10555 0.07401 0.24365 0.08086 –0.27554 –0.14731 (0.0698) (0.0537) (0.0612) (0.0189) (0.2447) (0.1983) (0.1014) (0.0757) iqβ 1.47378 1.55136 0.64168 0.95140 (0.6732) (1.4107) (0.3212) (0.5991) as 1β 0.00068 0.00005 0.00032 0.00024 –0.00194 –0.00365 –0.00995 –0.00163 (0.0007) (0.0002) (0.0003) (0.0002) (0.0063) (0.0021) (0.0099) (0.0021) 2β 0.94322 0.94893 0.96706 0.94679 –0.14498 0.34562 0.07883 0.40237 (0.0373) (0.0216) (0.0141) (0.0179) (0.6099) (0.2039) (0.3584) (0.2845) 3β 0.25113 0.13757 0.10653 0.07312 –0.32960 –0.24975 –0.21780 –0.10120 (0.1068) (0.0492) (0.0474) (0.0333) (0.3141) (0.1194) (0.1142) (0.0753) 4β –0.03450 –0.08393 –0.02525 –0.08316 –0.23637 –0.15212 0.55140 0.22534 (0.0785) (0.0491) (0.0316) (0.0208) (0.3469) (0.1318) (0.3182) (0.1333) iqβ 1.25475 0.87372 1.52731 0.80735 (0.6264) (0.3152) (0.8703) (0.3249) indirect garch 1β 0.00002 0.00000 0.00001 0.00000 0.00004 0.00015 0.00023 0.00003 (0.0004) (0.0002) (0.0004) (0.0001) (0.0036) (0.0011) (0.0006) (0.0002) 2β 0.91453 0.95049 0.95819 0.95292 0.32477 0.37988 0.03281 0.40679 (0.4147) (0.4265) (0.7352) (0.3599) (4.6846) (2.7296) (0.6178) (0.8225) 3β 0.47486 0.15482 0.16744 0.07178 0.71078 0.03784 0.22551 0.04628 (0.5825) (0.6343) (1.0813) (0.1408) (7.8410) (2.0967) (3.0804) (0.5717) iqβ 1.50097 1.98863 1.59270 0.79010 (9.4800) (6.6522) (1.6841) (0.9189) ad 1β –0.00003 –0.00001 –0.00001 –0.00001 –0.05703 –0.04389 0.05029 0.00480 (0.0000) (0.0000) (0.0000) (0.0000) (0.0546) (0.1450) (0.0470) (0.0470) 2β 0.01650 0.56209 –0.34653 0.42903 (0.2477) (1.0987) (0.3483) (0.9800) 3β 0.11134 0.08988 –0.12186 –0.01272 (0.0232) (0.0000) (0.0018) (0.0000) iqβ 1.36557 0.84537 1.53685 0.61808 (0.0546) (0.1540) (0.0677) (0.1400) note: 0.01; 0.05 – α-significance level of var; standard errors in bracktes; bolded values indicate significant parameters according to t statistics. risk modeling of commodities using caviar models… dynamic econometric models 15 (2015) 129–156 147 6.2. empirical research we perform estimation of caviar models (equations (3)–(9)), implied quantile model (equations (12)–(13)), encompassing method (equations (14)–(21)) and combining method of forecast (equations (22)–(27)). we use 1093 periods for gold and 1350 periods for oil to estimate parameters (insample) and 500 periods for post-sample evaluation of day-ahead quantile estimates using rolling window. for both instruments we indicate rate of returns tr and and an average returns in the sample ( µ ). we perform estimation for residuals µ−tt ry = . we estimate the parameters using, as engle and manganelli (2004), doman, doman (2009), jeon and taylor (2013), the differential evolution algorithm in c++ and matlab. the algorithm was presented by price and storn (1997). parameters of estimated models are contained in tables (6) and (7). for gold in the sav models for long and short positions at the 0.05 level of probability and in the case of the model indirect garch both long and short positions explanatory variable as empirical quantile turns out to be significant. for oil we observe a different situation. only in the case of models sav and the as for a short position at the 0.05 significance level and for the ad model for long and short positions at the 0.01 significance level attached explanatory variable of empirical quantile turned out to be irrelevant. to optimize the parameters of regression methods and the variancecovariance method we expressed the quantile regression minimization as a linear programme and applied the nelder-mead simplex algorithm. the estimated parameter for the combined forecasts are contained in the table (8). in case of gold the forecast based on implied volatility receives a higher weight than the predictions from the caviar models. in case of the oil the implied quantile receives significantly higher weight when forecast is combined on the basis of caviar-ad model and implied quantile for long and short positions at the 0.01 significance level, and for a long position at the 0.05 significance level. but for a short position at the 0.05 significance level the implied quantile receives a significantly higher weight in the combination of implied quantile with the forecast based on sav or indirect garch model. ewa ratuszny dynamic econometric models 15 (2015) 129–156 148 table 7. oil. estimated parameters of caviar and caviar-plugin models model parametr caviar plugin long position short position long position short position 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 *sav 1β 0.00251 0.00011 0.00278 0.00006 –0.00062 –0.00196 –0.00074 –0.00288 (0.0007) (0.0002) (0.0015) (0.0002) (0.0032) (0.0029) (0.0037) (0.0013) 2β 0.87120 0.91757 0.76330 0.90003 0.68751 0.78916 0.49449 0.79478 (0.0211) (0.0370) (0.0683) (0.0265) (0.1239) (0.1580) (0.2313) (0.0520) 3β 0.27242 0.17969 0.55563 0.20181 0.24017 0.14208 0.51237 0.22536 (0.0466) (0.0882) (0.2154) (0.0561) (0.0486) (0.1666) (0.2829) (0.0638) iqβ 0.26605 0.20524 0.33959 0.18180 (0.1846) (0.1679) (0.2533) (0.0843) as 1β 0.00303 0.00010 0.00218 0.00048 –0.00396 –0.00053 0.00225 –0.00089 (0.0008) (0.0002) (0.0008) (0.0002) (0.0084) (0.0011) (0.0013) (0.0009) 2β 0.84521 0.93895 0.86735 0.91141 0.60910 0.91166 0.86953 0.85358 (0.0358) (0.0301) (0.0375) (0.0279) (0.2047) (0.0795) (0.0387) (0.0638) 3β 0.25223 0.09902 0.48895 0.24716 0.22894 0.07790 0.48150 0.26358 (0.0585) (0.0823 (0.1723) (0.0589) (0.0530) (0.1848) (0.1674) (0.1156) 4β –0.44066 –0.16163 –0.01570 –0.06286 –0.38298 –0.16073 –0.02043 –0.05617 (0.1968) (0.0653) (0.0790) (0.0564) (0.2921) (0.0923) (0.0785) (0.0831) iqβ 0.40265 0.05087 –0.00280 0.09615 (0.3612) (0.1275) (0.0396) (0.0533) indirect garch 1β 0.00009 0.00002 0.00027 0.00001 0.00010 0.00001 0.00020 0.00014 (0.0008) (0.0004) (0.0005) (0.0003) (0.0007) (0.0008) (0.0009) (0.0010) 2β 0.92011 0.89439 0.57487 0.88718 0.76380 0.85947 0.33369 0.69506 (0.2345) (0.4421) (0.1830) (0.3737) (0.5136) (0.5754) (1.6465) (0.9587) 3β 0.33190 0.30253 2.23002 0.26633 0.36438 0.30606 2.15737 0.27593 (0.0955) (0.2939) (1.6959) (0.2245) (0.0785) (0.5608) (1.8255) (0.5857) iqβ 0.16353 0.05555 0.41244 0.33313 (0.4457) (0.4795) (1.6413) (1.0470) ad 1β –0.00006 –0.00001 –0.00009 –0.00001 –0.01450 –0.01260 –0.00243 0.01063 (0.0000) (0.0000) (0.0000) (0.0000) (0.0327) (0.0000) (0.0587) (0.0000) 2β 0.46859 0.87643 0.13146 0.79313 (0.1660) (0.0000) (0.2360) (0.0000) 3β 0.01776 0.02359 –0.01067 –0.02933 (0.0000) (0.0000) (0.0000) (0.0000) iqβ 0.72811 0.24512 0.99871 0.20844 (0.0636) (0.0000) (0.0955) (0.0000) note: 0.01; 0.05 – α-significance level of var; standard errors in bracktes; bolded values indicate significant parameters according to t statistics. weights received on the basis of weighted averaged combining method are included in a table (9). for gold, we observe that the forecast based on implied volatility receives a higher weight than the forecast from caviar models, and in the case of oil implied quantile receives a higher weight only in combining forecast of the implied quantile and caviar-ad. risk modeling of commodities using caviar models… dynamic econometric models 15 (2015) 129–156 149 table 8. estimated parameters of linear combination method model long position short position 0.01 0.05 0.01 0.05 1γ 2γ 3γ 1γ 2γ 3γ 1γ 2γ 3γ 1γ 2γ 3γ gold sav –0.002 0.826 0.265 –0.005 1.153 0.104 –0.009 2.121 –0.584 –0.001 1.624 –0.403 as –0.001 1.091 –0.025 –0.005 1.187 0.090 –0.011 1.571 0.020 –0.001 1.598 –0.381 indirect garch –0.001 1.120 –0.057 –0.005 1.057 0.178 –0.008 2.253 –0.746 0.000 1.622 –0.468 ad 0.028 1.215 –0.956 0.005 1.294 –0.460 0.027 1.637 –1.287 0.005 1.188 –0.378 oil sav –0.010 0.650 0.572 –0.008 0.581 0.639 –0.011 0.593 0.601 –0.013 0.907 0.454 as –0.006 0.561 0.575 –0.006 0.296 0.863 –0.008 0.198 0.950 –0.007 0.447 0.758 indirect garch –0.014 0.617 0.686 –0.007 0.201 0.969 –0.010 0.492 0.704 –0.012 0.943 0.417 ad –0.016 1.338 0.035 –0.015 1.314 0.162 –0.029 1.235 0.236 –0.123 1.476 3.111 note: 0.01; 0.05 – α-significance level of var; bolded values indicate models with higher value of parameter for forecasts derived on the basis of implied quantile model for determining the coefficient λ the ewqr method is applied. ewqr estimation was carried out on the in-sample data with the last 500 observations excluded and considering a grid values for λ between 0.97 and 1 with a step size of 0.001. the smallest value of the function (28)–(29) was the criterion to determine the optimum values for λ for the analysed significance levels of var ( α ). the discount factor λ for long and short positions and considered significance levels are shown in the table (10). the lower values for long positions mean that the 0.01 and the 0.05 quantile change more dynamically over time than the 0.01 and the 0.05 quantile in the case of a short position. table 9. estimated weights of weighted averaged combining method instrument model long position short position α 0.01 0.05 0.01 0.05 gold sav 0.5996 0.7290 0.8339 0.9066 as 0.7771 0.6058 0.7016 0.9067 indirect garch 0.5198 0.7028 0.7731 0.9281 ad 0.9519 0.9696 0.9760 0.9762 oil sav 0.3193 0.3928 0.4083 0.2192 as 0.0822 0.2564 0.0439 0.0966 indirect garch 0.3288 0.0832 0.3052 0.2480 ad 0.9770 0.9280 0.9071 0.9295 note: 0.01; 0.05 – α-significance level of var; bolded values indicate models with higher weight ω assigned to forecasts from implied quantile model ewa ratuszny dynamic econometric models 15 (2015) 129–156 150 table 10. estimated exponential weight λ instrument model long position short position α-significance level of var 0.01 0.05 0.01 0.05 gold sav 0.996 0.996 1.000 1.000 as 1.000 0.982 1.000 1.000 indirect garch 1.000 0.991 0.993 1.000 ad 0.991 1.000 1.000 0.995 oil sav 0.970 0.998 1.000 0.999 as 1.000 0.998 1.000 1.000 indirect garch 1.000 0.998 1.000 0.998 ad 0.993 0.981 0.997 1.000 the estimated values of weights for weighted averaged combining optimized using exponential weighting is included in the table (11). we see the opposite situation than in the case of weighted averaged combining without discounting factor. in the considered combinations, the forecasts from the caviar models receive a significantly higher weight ( 93.0> ). table 11. estimated weights of weighted averaged combining optimized using exponential weighting instrument model long position short position α-significance level of var 0.01 0.05 0.01 0.05 gold sav 0.008 0.026 0.009 0.036 as 0.007 0.028 0.009 0.036 indirect garch 0.008 0.028 0.009 0.037 ad 0.007 0.023 0.009 0.036 oil sav 0.016 0.060 0.019 0.062 as 0.016 0.058 0.016 0.058 indirect garch 0.015 0.063 0.018 0.063 ad 0.015 0.062 0.025 0.066 results of measures are presented in tables (12) for gold and (13) for oil. conclusions in the present study the caviar models, the encompassing method and the combined forecasts methods are applied to determine the risk measure var. since none of the forecasts is dominant and there is no universally accepted ranking of the various methods, we decided to check if the encompassing method or the forecast combination methods may reduce the risk of a large forecast error compared to individual forecast. risk modeling of commodities using caviar models… dynamic econometric models 15 (2015) 129–156 151 table 12. gold. loss functions for post sample model/metoda bl fl rl long short long short long short position position position position position position α 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 iq 15 40 8 32 87.22 153.25 23.33 56.22 76.02 146.41 13.73 50.34 sav caviar 7 35 2 16 63.80 133.08 17.27 35.02 49.26 124.91 4.60 27.46 plugin 13 40 10 35 66.86 144.30 25.66 58.53 54.80 137.31 16.25 52.83 linearcomb 10 43 12 29 75.95 160.81 28.79 52.09 63.61 154.35 20.01 46.00 simpavg 10 35 3 20 73.65 139.98 17.96 41.08 60.78 132.44 6.79 34.31 wtdavg 10 38 5 32 75.23 146.53 20.08 56.27 62.67 139.32 9.93 50.40 wtdavgexp 7 35 2 16 63.90 133.41 17.26 35.14 49.39 125.27 4.62 27.63 as caviar 8 35 2 16 79.72 142.30 17.04 35.28 65.61 134.49 4.03 27.79 plugin 15 52 10 31 69.21 159.32 25.85 54.38 57.75 152.94 16.70 48.60 linearcomb 14 45 6 29 83.31 163.03 21.51 51.85 71.70 156.58 11.41 45.72 simpavg 11 38 3 20 82.60 147.74 17.77 41.23 69.94 140.42 6.43 34.50 wtdavg 12 38 4 32 83.83 148.39 18.84 56.27 71.96 141.16 8.17 50.40 wtdavgexp 8 35 2 16 79.72 142.41 17.04 35.40 65.62 134.63 4.05 27.95 indirect garch caviar 9 31 2 14 56.45 118.36 17.44 32.92 41.77 109.45 4.74 25.15 plugin 3 4 2 12 36.75 61.98 17.73 29.92 18.47 46.66 2.80 21.95 linearcomb 15 41 14 33 86.04 155.96 32.44 57.16 74.58 149.32 24.03 51.25 simpavg 10 34 3 20 68.26 133.03 18.05 40.74 55.28 125.13 6.87 33.89 wtdavg 11 36 5 32 69.72 140.40 20.06 56.49 56.84 132.93 9.73 50.65 wtdavgexp 9 31 2 15 56.60 118.96 17.43 34.02 41.96 110.10 4.76 26.31 ad caviar 10 25 4 32 81.23 123.45 19.68 61.71 68.48 114.45 5.98 55.48 plugin 15 51 8 32 68.35 153.60 24.49 55.07 57.01 147.15 14.59 49.10 linearcomb 5 40 4 15 62.64 149.55 19.06 34.03 47.34 142.51 7.67 26.68 simpavg 12 28 4 30 83.04 132.13 18.93 56.13 71.06 124.17 7.24 50.06 wtdavg 14 42 7 32 86.03 159.76 22.29 56.27 74.75 153.26 12.57 50.38 wtdavgexp 10 26 4 32 81.22 124.64 19.66 61.38 68.48 115.70 6.00 55.16 note: 0.01; 0.05 – α-significance level of var; bolded value indicates models/methods that both tests based on bernoulli trials model and dynamic quantile test indicate as adequate, underlined value is the lowest value among the models/methods for the analysed position and at the significance level. the subject of the study were two assets: gold and oil. for crude oil, in the encompassing method we observe significant contribution of the implied volatility to the var. for gold, the implied quantile has been assigned a higher weight in the method of linear combination and weighted averaged combining method. conclusions made by jeon and taylor (2013), who analyzed the caviar models, the encompassing method and the combination methods for the capital market indices differ from the one presented in this study for the commodities. jeon and taylor (2013) using backtests and the dynamic quantile test pointed out the forecast determined by combining methods as more adequate than coming from caviar models or the implied quantile model. in their view, the encompassing method derives more accurate ewa ratuszny dynamic econometric models 15 (2015) 129–156 152 forecasts than the individual caviar models or the implied quantile model individually, but considering only the criterion of the smallest fraction of exceedances hit (table (5) in jeon and taylor (2013)). among the methods of combining forecasts jeon and taylor (2013) pointed out that the linear combination method and the arithmetic mean method generate the most adequate var. table 13. oil. loss functions for post sample method/model bl fl rl long short long short long short position position position position position position α 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 iq 8 24 4 28 53.53 91.59 19.90 49.10 38.92 81.89 5.19 39.78 sav caviar 6 24 2 30 27.18 75.88 18.94 50.55 7.74 65.51 2.02 41.36 plugin 9 51 10 65 44.49 132.40 25.07 112.84 29.46 124.69 11.22 106.70 linearcomb 8 42 10 59 40.44 115.63 24.97 105.85 24.87 107.44 11.76 99.59 simpavg 7 25 0 27 35.47 81.99 15.88 46.29 18.46 72.00 0.00 37.01 wtdavg 7 24 0 27 31.72 79.47 16.08 46.76 13.85 69.38 0.00 37.50 wtdavgexp 6 23 2 29 27.27 75.25 18.90 49.29 7.91 64.92 2.02 40.08 as caviar 5 23 1 20 26.86 77.04 18.58 35.75 6.62 66.64 1.10 25.07 plugin 11 38 1 34 55.47 108.30 18.73 60.07 41.76 99.63 1.07 51.86 linearcomb 7 35 4 27 36.01 101.16 19.97 48.09 19.04 92.17 4.60 39.09 simpavg 6 25 0 25 34.13 83.56 16.15 41.35 16.70 73.55 0.00 31.40 wtdavg 5 24 1 22 27.17 79.86 18.43 37.66 7.39 69.66 1.08 27.14 wtdavgexp 5 23 1 21 26.90 77.38 18.52 36.69 6.75 67.01 1.09 26.10 indirect garch caviar 3 18 0 23 25.45 58.71 18.65 39.65 4.52 47.11 0.00 29.57 plugin 6 20 0 0 33.06 63.24 18.08 16.03 15.50 52.08 0.00 0.00 linearcomb 6 26 5 53 37.19 78.81 20.19 92.12 20.71 68.92 5.10 85.37 simpavg 6 18 0 25 34.84 69.12 16.72 42.12 17.11 58.41 0.00 32.44 wtdavg 5 18 0 26 30.59 60.08 17.47 42.59 11.78 48.63 0.00 32.75 wtdavgexp 3 19 0 24 25.52 60.71 18.58 40.57 4.68 49.24 0.00 30.57 ad caviar 0 5 0 5 29.80 34.08 19.28 20.99 0.00 16.00 0.00 5.50 plugin 12 45 13 90 68.43 142.19 29.38 161.95 54.95 134.43 16.61 156.82 linearcomb 15 44 59 205 77.90 132.14 114.91 470.52 65.61 124.19 107.58 468.66 simpavg 0 8 0 11 34.45 49.03 18.96 25.55 0.00 35.05 0.00 13.10 wtdavg 10 34 4 55 59.29 113.07 19.77 101.50 45.54 104.52 5.42 94.84 wtdavgexp 0 5 0 5 30.73 35.10 19.21 20.69 0.00 17.51 0.00 5.57 note: 0.01; 0.05 – α-significance level of var; bolded value indicates models/methods that both tests based on bernoulli trials model and dynamic quantile test indicate as adequate, underlined value is the lowest value among the models/methods for the analysed position and at the significance level. in our research, the kupiec test is used in conjunction with the dynamic quantile test and the christoffersen test, so the models and methods that underestimate or overestimate var are rejected. we show that the criteria based on the kupiec, the christoffersen and the dynamic quantile test risk modeling of commodities using caviar models… dynamic econometric models 15 (2015) 129–156 153 together more precisely classify the var forecasting method than the one used by jeon and taylor (2013). proposals of evaluation var models formulated in this study may protect the institution against errors arising from the application of methods pointed out by jeon and taylor (2013), and inefficiently maintaining too high level of capital. in the case of the commodities neither the encompassing method nor the combining forecast method improve var forecasts. the method of choosing the most adequate model presented in this paper leads to the caviar-sav model selection as the source of most optimal measure of risk forecasts. the kupiec test, the christoffersen and the dynamic quantile test indicate the model as an adequate to forecast var for gold and oil for short positions at the 0.01 and the 0.05 significance level, and for a long position at the 0.05 significance level. the model generates also the lowest value of loss functions. in our study, none of the models and none of the methods have predicted adequately the var for a long position at the 0.01 significance level, leading to the conclusion that the choice of model should also depend on the financial institution's investment strategy and portfolio structure. references armendola, a., storti, g. (2008), a gmm procedure for combining volatility forecasts, computational statistics and data analysis, 52, 3047–3060. armstrong, j. s. (2001), principles of forecasting: a handbook for researchers and practitioners, dordrecht, kluwer academic publishers, doi: http://dx.doi.org/10.1007/978-0-306-47630-3. artzner, p., delbaen, f., ebe,r j.-m., heath, d. (1999), coherent measures of risk, mathematical finance, 9, 203–228. bates, j. m., granger, c. w. j. (1969), the combination of forecasts, operations research quarterly, 20, 451–468, doi: http://dx.doi.org/10.2307/3008764. blair, b. j, poon, s .h., taylor, s. j. (2001), forecasting s&p 100 volatility: the incremental information content of implied volatilities and high-frequency index returns, journal of econometrics, 105, 5–26, doi: http://dx.doi.org/10.1016/s0304-4076(01)00068-9. boudoukh, j., richardson, m., whitelaw, r. f. (1998), the best of both worlds, risk, 11, 64–67. bunn, d. w. (1989), forecasting with more than one model, journal of forecasting, 8, 161–166, doi: http://dx.doi.org/10.1002/for.3980080302. chong, j. (2004), value at risk from econometric models and implied from currency options, journal of forecasting, 23, 603–620, doi: http://dx.doi.org/10.1002/for.934. chong, y., henry, d. f. (1986), econometric evaluation of linear macro-economic models, review of economic studies, 53, 671–690, doi: http://dx.doi.org/10.2307/2297611. christoffersen, p. (1998), evaluating interval forecasts, international economics review, 39, 841–862, doi: http://dx.doi.org/10.2307/2527341. ewa ratuszny dynamic econometric models 15 (2015) 129–156 154 christoffersen, p .f., mazzotta, s. (2005), the accuracy of density forecasts from foreign exchange options, journal of financial econometrics, 3, 578–605, doi: http://dx.doi.org/10.1093/jjfinec/nbi021. claessen, h, mittnik, s. (2002), forecasting stock market volatility and the informational efficiency of the dax-index options market, the european journal of finance, 8, 302–321, doi: http://dx.doi.org/10.1080/13518470110074828. clemen, r. t. (1986), linear constraints and the efficiency of combined forecasts, journal of forecasting, 5, 31–38, doi: http://dx.doi.org/10.1002/for.3980050104. corredor, p., santamaria, r. (2004), forecasting volatility in the spanish option market, applied financial econometrics, 14, 1–11, doi: http://dx.doi.org/10.1080/0960310042000164176. crane, d. b., crotty, j. r. (1967), a two-stage forecasting model: exponential smoothing and multiple regression, management science, 13, 501–507, doi: http://dx.doi.org/10.1016/b978-0-08-019605-3.50019-5. day, t. e., lewis, c. m. (1992), stock market volatility and the information content of stock index options, journal of econometrics, 52, 267–287, doi: http://dx.doi.org/10.1016/0304-4076(92)90073-z. diebold, f. x. (1988), serial correlation and combination of forecasts, journal of business and economic statistics, 6, 105–111, doi: http://dx.doi.org/10.1080/07350015.1988.10509642. diebold, f. x. (1989), forecast combination and ecompassing: reconciling two divergent literatures, international journal of forecasting, 2, 589–592. doidge, c., wei, j. z. (1998), volatility forecasting and the efficiency of the toronto 35 index options market, canadian journal of administrative sciences, 15, 28–38, doi: http://dx.doi.org/10.1111/j.1936-4490.1998.tb00150.x. doman, m., doman, r. (2009), volatility and risk modeling, oficyna wolters kluwer business, kraków. donaldson, r. g., kamstra, m. j. (2005), volatility forecasts, trading volume, and the arch versus option-implied volatility trade-off, the journal of financial research, 28, 519–538, doi: http://dx.doi.org/10.1111/j.1475-6803.2005.00137.x. capital requirements directive iv, crd iv, dz. urz. ue l176, june 27th, 2013. engle, r. f., manganelli, s. (2004), caviar: conditional autoregressive value at risk by regression quantiles, journal of business and economic statistics, 22, 367–381, doi: http://dx.doi.org/10.1198/073500104000000370. fiszeder, p. (2009), garch class models in empirical financial research, wydawnictwo naukowe umk, toruń. giacomini, r., komunjer, i. (2005), evaluation and combination of conditional quantile forecasts, journal of business and economic statistics, 23, 416–431, doi: http://dx.doi.org/10.1198/073500105000000018. giot, p. (2005), implied volatility indexes and daily value at risk models, journal of derivatives, 12, 54–64, doi: http://dx.doi.org/10.3905/jod.2005.517186. giot, p., laurent, s. (2007), the information content of implied volatility in light of the jump/continuous decomposition of realized volatility, journal of futures markets, 27, 337–359, doi: http://dx.doi.org/10.1002/fut.20251. grajek, m. (2002), prognozy łączone (combining forecasts), przegląd statystyczny, 2, 70–81. granger, c. w. j. (1989), invited review: combining forecasts–20 years later, journal of forecasting, 8, 167–173. risk modeling of commodities using caviar models… dynamic econometric models 15 (2015) 129–156 155 granger, c. w. j., white, h., kamstra, m. j. (1989), interval forecasting: an analysis based upon arch-quantile estimators, journal of econometrics, 40, 87–96, doi: http://dx.doi.org/10.1016/0304-4076(89)90031-6. greszta, m., maciejewski, w. (2005), kombinowanie prognoz gospodarki polski (combining forecasts of the polish economy), gospodarka narodowa, 5–6, 49–60. iwanicz-drozdowska, m. (2005), zarządzanie finansowe bankiem (financial management of bank), warsaw, pwe. jajuga, k., jajuga, t. (2011), instrumenty finansowe, aktywa niefinansowe, ryzyko finansowe, inżynieria finansowa (financial instruments, non-financial assets, financial risk, financial engineering), pwn, warsaw. jarque, c. m., bera, a. k. (1987), a test for normality of observations and regression residuals, international statistical review, 55, 163–172. doi: http://dx.doi.org/10.2307/1403192.jeon j., taylor j.w (2013), using caviar models with implied volatility for value at risk estimation, journal of forecasting, 32, 62–74, doi: http://dx.doi.org/10.1002/for.1251. koenker, r., bassett, g. s. (1978), regression quantiles, econometrica, 46, 33–50, doi: http://dx.doi.org/10.2307/1913643. kupiec, p. (1995), techniques for verifying the accuracy of risk measurement models, journal of derivatives, 3, 73–84, doi: http://dx.doi.org/10.3905/jod.1995.407942. lomnicki, z. a. (1961), tests for departure from normality in the case of linear stochastic processes, metrika, 4, 37–62, doi: http://dx.doi.org/10.1007/bf02613866. lopez, j. a. (1998), testing your risk test, the financial survey, 18–20. lopez, j. a. (1999), methods for evaluating value at risk estimates, frbny economic policy review, 2, 3–15, doi: http://dx.doi.org/10.2139/ssrn.1029673. mazur, b., pipień, m. (2012), on the empirical importance of periodicity in the volatility of financial returns – time varying garch as a second order apc(2) process, central european journal of economic modelling and econometrics, 4, 95–116. mittnik, s., paolella, m. s. (2000), conditional density and value-at-risk prediction of asian currency exchange rates, journal of forecasting, 19, 313–333, doi: http://dx.doi.org/10.1002/1099-131x(200007)19:4%3c313::aidfor776%3e3.0.co;2-e. nelson, c. r. (1972), the prediction performance of the f.r.b.-m.i.t.-penn model of the u.s. economy, american economic review, 62, 902–917. nelson, c. r. (1984), a benchmark for the accuracy of econometric forecasts of gnp, business economics, 19, 52–58. noh, j., kim, t. h. (2006), forecasting volatility of futures market: the s&p 500 and ftse 100 futures using high frequency returns and implied volatility, applied economics, 38, 395–413, doi: http://dx.doi.org/10.1080/00036840500391229. perignon, c., smith, d. r. (2010), the level and quality of value-at-risk disclosure by commercial banks, journal of banking and finance, 34, 362–377, doi: http://dx.doi.org/10.2139/ssrn.952595. piłatowska, m. (2009), prognozy kombinowane z wykorzystaniem wag akaike’a, w: ekonomia xxxix. dynamiczne modele ekonometryczne, wydawnictwo naukowe uniwersytetu mikołaja kopernika, toruń, 51–62. piontek, k. (2000), modelowanie finansowych szeregów czasowych warunkową wariancją, w: prace naukowe akademii ekonomicznej we wrocławiu, 890, 218–226. pipień, m. (2006), wnioskowanie bayesowskie w ekonometrii finansowej, w: zeszyty naukowe akademia ekonomiczna, 176, kraków. ewa ratuszny dynamic econometric models 15 (2015) 129–156 156 pong, s., shackleton, m. b., taylor, s. j., xu, x. (2004), forecasting currency volatility: a comparison of implied volatilities and ar (fi) ma models, journal of banking and finance, 28, 2541–2563, doi: http://dx.doi.org/10.2139/ssrn.301981. price, k., storn, r. (1997), differential evolution, dr. dobb’s journal, 18–24, doi: http://dx.doi.org/10.1007/978-3-642-30504-7_8. rachev, s., mittnik, s. (2002), stable paretian models in finance, john wiley, new york. ratuszny, e. (2013), robust estimation in var modelling – univariate approaches using bounded innovation propagation and regression quantiles methodology, central european journal of economic modelling and econometrics, 5, 35–63. ratuszny, e. (2015), influence of robust estimation on value at risk. bounded innovation propagation and regression quantiles method, journal of management and financial sciences, forthcoming. sarma, m., thomas, s., shah, a. (2003), selecting of var models, journal of forecasting, 22, 337–358. szakmary, a., ors, e., kim, j. k., davidson iii, w. n. (2003), the predictive power of implied volatility: evidence from 35 futures markets, journal of banking and finance, 27, 2151–2175, doi: http://dx.doi.org/10.1016/s0378-4266(02)00323-0. taylor, j. w. (2008), using exponentially weighted quantile regression to estimate value at risk and expected shortfall, journal of financial econometrics, 6, 382–406, doi: http://dx.doi.org/10.1093/jjfinec/nbn007. taylor, j. (1999), a quantile regression approach to estimating the distribution of multiperiod returns, journal of derivatives, 64–78, doi: http://dx.doi.org/10.3905/jod.1999.319106. taylor, j. w., bunn, d. w. (1998), combining forecast quantiles using quantile regression: investigating the derived weights, estimator bias and imposing constraints, journal of applied statistics, 25, 193–206, doi: http://dx.doi.org/10.1080/02664769823188. umantsev, l., chernozhukov, v. (2001), conditional value at risk: aspects of modeling and estimation, empirical economics, 26, 271–292. zarnowitz, v. (1967), an appraisal of short-term economic forecasts, national bureau of economic research, new york.