




































© 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 observa-

tions. 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 Eco-

nomic 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 autore-

gressive models. The aim of the study is to evaluate and compare the effi-

ciency 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 Ex-

change (PPE), commodity futures instruments for gas appeared, and on De-

cember 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 Janu-

ary 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 begin-

ning 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 cycli-

cality. 

 
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
at

a 

20
13

-0
4-

21
 

20
13

-0
6-

03
 

20
13

-0
7-

13
 

20
13

-0
8-

22
 

20
13

-1
0-

01
 

20
13

-1
1-

10
 

20
13

-1
2-

20
 

20
14

-0
1-

30
 

20
14

-0
4-

01
 

20
14

-0
5-

11
 

20
14

-0
6-

20
 

20
14

-0
7-

30
 

20
14

-0
9-

08
 

20
14

-1
0-

18
 

20
14

-1
1-

27
 

20
15

-0
1-

06
 

20
15

-0
2-

15
 

20
15

-0
3-

27
 

20
15

-0
5-

06
 

20
15

-0
6-

15
 

20
15

-0
7-

25
 

20
15

-0
9-

03
 

20
15

-1
0-

13
 

20
15

-1
1-

22
 

20
16

-0
1-

01
 

20
16

-0
2-

10
 

20
16

-0
3-

21
 

20
16

-0
4-

30
 

20
16

-0
6-

09
 

20
16

-0
7-

19
 

20
16

-0
8-

28
 

20
16

-1
0-

07
 

20
16

-1
1-

16
 

20
16

-1
2-

26
 

20
17

-0
2-

04
 

20
17

-0
3-

16
 

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 char-

acteristics as asymmetry, thick tails and leptokurticity. 

2. Risk Measurement – VaR 

 The formal definition of VaR does not take into account the process na-

ture 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 

20
14

-0
1-

02
 

20
14

-0
2-

18
 

20
14

-0
4-

13
 

20
14

-0
5-

23
 

20
14

-0
7-

02
 

20
14

-0
8-

11
 

20
14

-0
9-

20
 

20
14

-1
0-

30
 

20
14

-1
2-

09
 

20
15

-0
1-

18
 

20
15

-0
2-

27
 

20
15

-0
4-

08
 

20
15

-0
5-

18
 

20
15

-0
6-

27
 

20
15

-0
8-

06
 

20
15

-0
9-

15
 

20
15

-1
0-

25
 

20
15

-1
2-

04
 

20
16

-0
1-

13
 

20
16

-0
2-

22
 

20
16

-0
4-

02
 

20
16

-0
5-

12
 

20
16

-0
6-

21
 

20
16

-0
7-

31
 

20
16

-0
9-

09
 

20
16

-1
0-

19
 

20
16

-1
1-

28
 

20
17

-0
1-

07
 

20
17

-0
2-

16
 

20
17

-0
3-

28
 



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 develop-

ment of the VaR measurement theory. At present, in empirical financial 

studies of time series, which in most cases behave as non-stationary stochas-

tic 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 con-

centration 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 distribu-

tion of the analysed random variable tY  over time.  

Markov chain },{ NtYt   with phase space }...,,2,1{ rS   is called a ho-

mogeneous 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: 







 



otherwise0

state in the wasmoment at the and

state in the was1moment at theobject   when the1

)( jt

it

tnij
 






otherwise0

state in themoment at the object was when the1
)(

it
tni  

This estimator has desirable consistency properties, asymptotic unbiased-

ness, 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 decomposi-

tion 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 pre-

sented 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 recog-

nized 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 

2S  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, ac-

cording to which the interval 1S  is changed empirically and thus the interval 

2S , 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 con-

structed 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 indi-

cating 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 sim-

ple tool for determining VaR on a current basis. For the purposes of this arti-

cle, 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 esti-

mated 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 observa-

tions 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 esti-

mated 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 10
 

19
 

28
 

37
 

46
 

55
 

64
 

73
 

82
 

91
 

10
0 

10
9 

11
8 

12
7 

13
6 

14
5 

15
4 

16
3 

17
2 

18
1 

19
0 

19
9 

20
8 

21
7 

22
6 

23
5 

24
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 condi-

tions of white noise and display a significant ARCH effect, therefore model 

(14) is complemented by a model allowing for heteroscedasticity of vari-

ance: 

 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 esti-

mation 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 re-

turns 

Figure 6 presents a selected part of a time series of returns (zt) and the esti-

mated 05.0VaR  for the one-day investment horizon using the theory of sto-

chastic 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 10
 

19
 

28
 

37
 

46
 

55
 

64
 

73
 

82
 

91
 

10
0 

10
9 

11
8 

12
7 

13
6 

14
5 

15
4 

16
3 

17
2 

18
1 

19
0 

19
9 

20
8 

21
7 

22
6 

23
5 

24
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

ttI


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 

VaRw – the participation of VaR  exceedances in the process of the con-

sidered returns. 

T

K
wVaR 




 – the participation of VaR  exceedances (K – the number of 

exceedances), in the series of the considered returns (T- the 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: 
















11100100

11011000

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


 ;  

VaRw
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 ex-

ceedances, 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 

POFLR  0.3014 0.9736 
p-value 0.5830 0.3238 

INDLR  x 2.6466 
p-value x 0.1038 

 VaR0.05 estimated using the SARIMA-GJR model is slightly underesti-

mated, 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 consid-

ered 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. Neverthe-

less, 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, Wydaw-

nictwo Naukowe UMK, Toruń. 

Ganczarek, A. (2006), Wykorzystanie modeli zmienności wariancji GARCH w analizie ryzy-

ka 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 finanso-

wej, ujęcie bajesowskie, Wydawnictwo Uniwersytetu Ekonomicznego w Krakowie, 

Kraków. 

Piontek, K. (2002), Pomiar ryzyka metodą VaR a modele AR-GARCH ze składnikiem  lo-

sowym o warunkowym rozkładzie z „grubymi ogonami”, Rynek Kapitałowy. Skutecz-

ne 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 Fi-

nancial 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 oszaco-

wanej 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. Zgod-

ność 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. 

 


