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 finan- cial 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 defini- tions 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 nar- row definition there is an assumption that contagion is a shocks transmission to other countries or the relations between economies except the fundamental con- nections 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 fundamen- tally. These transmissions are real financial, economic and political connec- tions. The most often cause of the contagion in narrow sense is the herd inves- tors 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 transmis- sion 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 con- ditional 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 follow- ing 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 distri- bution. In example, the model that can generate the skewness and the leptokur- tosis 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 autocorrela- tion 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 analy- sis 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 alterna- tive 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 11.95- 12.00 10.95- 04.01 01.96- 02.96 04.97- 05.02 06.98- 05.99 01.96- 04.96 01.97- 05.03 07.97- 12.97 09.97- 11.97 03.02 09.98- 06.00 09.01- 02.02 11.01- 01.02 11.96- 02.97 07.07- 08.09 09.08- 08.09 12.96- 03.97 07.07- 08.09 07.98- 03.00 10.98- 02.99 10.08- 05.09 02.01- 12.01 07.03- 01.04 10.03- 12.03 12.97- 01.99 10.97- 01.98 02.01- 05.01 08.01- 11.01 05.02- 04.03 04.06- 09.06 04.99- 01.00 05.98- 01.00 08.01- 05.03 06.02- 09.00 01.08- 08.09 03.07- 03.08 03.00- 12.00 09.00- 01.01 09.03- 11.03 01.03- 05.03 09.08- 08.09 07.08- 08.09 03.01- 07.01 11.05- 12-05 10.08- 08.09 09.08- 05.09 05.02-07.02 05.06- 09.06 11.02-05.03 10.08- 08.09 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 Rus- sian 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 anal- ysis 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 funda- mental 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 pat- terns 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. 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(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 zmien- ności dla poszczególnych szeregów oraz przeprowadzono analizę ich występowania w odniesie- niu do globalnych kryzysów finansowych. Stwierdzono występowanie wspólnych okresów zmienności dla badanych indeksy. Okresy te wskazują na przenoszenie szoków pomiędzy rynka- mi, 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.