Comparison of Forecasting Volatility in the Czech Republic
Stock Market
EleftheriosThalassinos1Yusuf Muratoglu2ErginbayUgurlu3
Abstract
This paper examined three GARCH models with three different distributions for
comparing their forecasting power for volatility of the return of Czech stock market, PX index
for the period 08.01.20001 -20.07.2012. We employ GARCH, GJR-GARCH and EGARCH
modelsagainst Normal, Student-t and Generalized Error distributions. Then we forecast stock
market volatility for Czech Republic by its return using GARCH, GJR-GARCH and
EGARCH model and compare their forecasting performance. The results show that return
volatility can be characterized by significant persistence and asymmetric effects. We estimate
estimated variance for all models for full sample period using static forecast. After comparing
the forecast performance of all nine models, it was found that the EGARCH model has the
best forecast performance comparing other models.
1. Introduction
Financial data mostly have some features which are leptokurtosis, volatility clustering
or volatility pooling and leverage effects.Linear structural (and time series) models are unable
to explain a number of the important features.
Leptokurtosis, volatility clustering or volatility pooling and leverage effects are
tendency for financial asset returns. The tendencies of these features are defined as; to have
distributions that exhibit fat tails and excess peakedness at the mean, to have volatility in
financial markets to appear in bunches which means large returns (of either sign) are expected
to follow large returns, and small returns (of either sign) to follow small returns and to have
volatility tends to rise more following a large price fall than following a price rise of the same
magnitude respectively (Brooks, 2008:380)
The most popular non-linear financial models are the Autoregressive Conditional
Heteroscedastic (ARCH) or Generalized Autoregressive Conditional Heteroscedastic
(GARCH) models used for modeling and forecasting volatility that are proposed by Engle
(1982) and Bollerslev (1986) .
In this paper, we investigate stock market volatility for Czech Republic by its return
using GARCH, GJR-GARCH and EGARCH model and compare their forecasting
performance.
The authors would like to thank Prof. Ph.D. BurcUlengin for useful comments. All remaining errors are solely
ours
1Professor, Department of Maritime Studies, University of Piraeus,Chair Jean Monnet,
e-mail:thalassic@unipi.gr
2 Research Assistant, Gazi University FEAS, Department of Economics,email:yusufmuratoglu@hitit.edu.tr
3 Instructor Ph.D., Hitit University FEAS, Department of Economics, e-mail:erginbayugurlu@hitit.edu.tr
1
The rest of the paper is organized as follows. The next section is literature review of
related subject of GARCH models and stock market volatility. The third section gives brief
information about ARCH/GARCH models and the estimation results are presented in the
fourth section. The five section summarizes and concludes the paper.
2. Literature Review
Andersen et al.(1999) stated that the expected future volatility of financial market
returns is the main ingredient in assessing asset or portfolio risk and plays a key role in
derivatives pricing models.
Emerson et al. (1996) investigate Bulgarian stock market and Scheicher (1999) studies
Polish stock returns, Shields (1997) modeling returns for the Warsaw and Budapest stock
exchanges returns. Also Scheicher(2001) analyses the movements of the short rates of
emerging markets in Central and Eastern Europe.Scheicher (2001) calledHungary, Poland and
Czech markets as a principal emerging stock markets in Europe. The author estimate a VEC
model and modeling its volatility with a Multivariate GARCH (M-GARCH) model. The
findings shows that countries which are investigated have limited interaction and their
volatility have a regional character.
Vošvrda and Žıkeš (2004) study the behavior of volatility and the distributional
properties of the Czech, Hungarian and Polish stock markets data for the period 1996- 2002
period using weekly data. They use PX-50 index for Czech Republic and find that
statistically significant results for GARCH (1,1) model and conclude that the volatility of the
returns on PX-50 is very persistent.
Syriopoulos (2007) investigates the relationships between Czech Republic, Hungary,
Poland, Slovakia as a emerging stock markets and Germany, US as a developed stock markets
over the period 1997-2003.Haroutounian and Price (2010) investigate Czech Republic,
Hungary, Poland and Slovakia using both univariate and multivariate GARCH models that are
GARCH, NGARCH, EGARCH, GJR-GARCH, AGARCH, NAGARCH and VGARCH.
Paper has two parts which are univariate models part and multivariate models part. Based on
the univariate models they conclude that strong GARCH effects are apparent for all four
markets with exception of Czech Republic where the coefficient of the lagged squared returns
is not significant in three out of seven specifications of conditional volatility.
Hajek (2007) tests the Efficient Market Hypothesis on the Czech capital market for
1995–2005 period for monthly, weekly and daily data.. The author analyses efficiency and
linear dependency of several index closing values and stock closing prices on the Prague
Stock Exchange and concludes that both daily stock returns and daily index returns are
significantly linearly dependent and the heteroskedasticity-consistent methodology must be
therefore applied to avoid significant biases.
Rockinger and Urga (2012) investigate two groups of country which are transition
economies and established economies. Transition economies are; CzechRepublic, Poland,
Hungary and Russia, established economies are USA, Germany and UK. Although they
focus on a sample of Central and Eastern European Financial Markets (CEEFM)4 they only
use these four countries. The model results are very similar for the Czech Republic, Hungary,
and Polandand for all countries investigated there exist significant GARCH effects.
3. Methodology
We use three different types of GARCH models which are GARCH, GJR-GARCH
and EGARCH model.
4 Czech Republic , Poland, Hungary, Russia, Bulgaria, Slovenia, Romania, Croatia and Estonia
2
ARCH Model
ARCH models based on the variance of the error term at time t depends on the realized
values of the squared error terms in previous time periods. The model is specified as:
yt  u t
(1)
u t ~ N0, h t 
q
h t   0    j u 2t i
(2)
t 1
This model is referred to as ARCH(q), where q refers to the order of the lagged
squared returns included in the model. If we use ARCH(1) model it becomes
h t   0  1 u 2t 1
(3)
Since h t is a conditional variance, its value must always be strictly positive; a negative
variance at any point in time would be meaningless. To have positive conditional variance
estimates, all of the coefficients in the conditional variance are usually required to be nonnegative. Thus coefficients must be satisfy
and
.
GARCH Model
Bollerslev (1986) and Taylor (1986) developed the GARCH(p,q) model . The model
allows the conditional variance of variable to be dependent upon previous lags; first lag of the
squared residual from the mean equation and present news about the volatility from the
previous period which is as follows:
q
p
i 1
i 1
h t   0    i u 2t i   i h t i
(4)
In the literature most used and simple model is the GARCH(1,1) process, for which
the conditional variance can be written as follows:
h t   0  1 u 2t 1  1 h t 1
(5)
Under the hypothesis of covariance stationarity, the unconditional variance h t can be
found by taking the unconditional expectation of equation 5.
We find that
h   0  1h  1h
(6)
Solving the equation 5 we have
h
0
1  1  1
(7)
For this unconditional variance to exist, it must be the case that 1  1  1 and for it to
be positive, we require that  0  0 .
GJR GARCH
The GJR model is a simple extension of GARCH with an additional term added to
account for possible asymmetries (Brooks, 2008:405).Glosten, Jagananthan and Runkle
3
(1993) develop the GARCH model which allows the conditional variance has a different
response to past negative and positive innovations.
q
p
i 1
i 1
ht   0   i ut2i   i ut2i d t 1    j ht  j
where
(8)
is a dummy variable that is:
 1 if u t 1  0, bad news
d t 1  
0 if u t 1  0, good news
In the model, effect of good news shows their impact by  i , while bad news shows
their impact by    . In addition if   0 news impact is asymmetric and   0 leverage
effect exists. To satisfy non-negativity condition coefficients would be  0  0 ,  i  0 ,   0
and  i   i  0 . That is the model is still acceptable, even if  i  0 , provided that
 i   i  0 (Brooks, 2008:406).
Exponential GARCH
Exponential GARCH (EGARCH) proposed by Nelson (1991) which has form of
leverage effects in its equation. In the EGARCH model the specification for the conditional
covariance is given by the following form:
log ht    0    j log ht  j     i
q
p
j 1
i 1
u t i
ht i
r
  k
k 1
ut k
ht k
(9)
Two advantages stated in Brooks (2008) for the pure GARCH specification; by using
log ht  even if the parameters are negative, will be positive and asymmetries are allowed for
under the EGARCH formulation.
In the equation  k represent leverage effects which accounts for the asymmetry of
the model. While the basic GARCH model requires the restrictions the EGARCH model
allows unrestricted estimation of the variance (Thomas and Mitchell2005:16).
If  k  0 it indicates leverage effect exist and if  k  0 impact is asymmetric.
meaning of leverage effect bad news increase volatility.
The
Applying process of GARCH models to return series, it is often found that GARCH
residuals still tend to be heavy tailed. To accommodate this, rather than to use normal
distribution the Student’s t and GED distribution used to employ ARCH/GARCH type models
(Mittnik et al. 2002:98).
4. Empirical Application
We use daily data in stock exchanges of Czech Republic PX5 for the period
08.01.20001 -20.07.2012. Data collected from Reuters.We use return, which is in Yu (2002),
defined as follows:
5 See: http://www.pse.cz/dokument.aspx?k=Exchange-Indices for description of PX
4
 x 
r  log  t 
 x t 1 
where
as a RPX.
(10)
is capital index. Figure 1 shows graph of PX and its return which is defined
.15
2,000
.10
1,600
.05
1,200
.00
-.05
800
-.10
400
-.15
1/2/12
1/3/11
1/1/10
1/1/09
1/1/08
1/1/07
1/2/06
1/3/05
1/1/04
1/1/03
1/2/12
1/3/11
1/1/10
1/1/09
1/1/08
1/1/07
1/2/06
1/3/05
1/1/04
1/1/03
1/1/02
1/1/02
-.20
0
RPX
PX
Table 1: Descriptive Statistics
RPX
Mean
0.000210
Median
0.000721
Maximum
0.123641
Minimum
-0.161855
Std. Dev.
0.015431
Skewness
-0.524060
Kurtosis
15.43870
Jarque-Bera
18821.75
Probability
0.000000
Sum
0.609240
Sum Sq. Dev.
0.690062
Observations
2899
Table 1 summarizes descriptive statistics of PX, RPX have negative skewness and
high positive kurtosis. These values signify that the distributions of the series have a long left
tail and leptokurtic. Jarque-Bera (JB) statistics reject the null hypothesis of normal
distribution at the 1% level of significance for the variable. Table 2 shows Augmented
Dickey-Fuller (ADF) test results and concludes that RPX is stationary.
5
Table 2: ADF Test Results
Without Trend
ADF stat
p
-39.7972***
0.0000
Variable
RPX
With Trend
ADF stat
p
-39.8306***
0.0000
Note: *** denotes significant at the 1% level
We test mean model for the ARCH effect by ARCH-LM Test. Table 3 shows ARCHLM test results. If the value of the test statistic is greater than the critical value from the
distribution, the null hypothesis is rejected.
Table 3 : ARCH (1) Test Results
Dependent Variable of Model
RPX
ARCH(1)LM Stat
429,7907***
P
0.0000
Note: *** denotes significant at the 1% level. The hull hypothesis, there is no
ARCH effect is rejected.
It is stated in We employ GARCH, GJR-GARCH and EGARCH model using both
Student-t and Generalized Error distributions additional to Normal distribution. Results show
that (Appendix: Table1, Table 2, Table 3) that strong GARCH and GJR-GARCH effects are
apparent for returns. The sum of coefficient of
and
less than one in GARCH and GJRGARCH model for all distribution. Nevertheless the estimate of
is smaller than the
estimate of
in both cases that is to show negative shocks haven’t a larger effect on
conditional volatility than positive shocks of the same magnitude. In GJR-GARCH model
  0 the news impact is asymmetric on the other words bad news increase volatility. In the
E-GARCH model negative and significant leverage effect parameter indicating the existence
of the leverage effect in returns. If GED parameter (r)equals two it means normal distribution
if it is less than two it means leptokurtic distribution. In all models r is less than two and
statistically significant which indicate that RPX is leptokurtic. This result is consistent with
the skewness values shown in Table 1 .
After all model are estimated ARCH effect is tested and except GJR-GARCH model
in normal distributionis rejected in %10 level, the null hypothesis that there is no ARCH
effect cannot be rejected for all models (see: Appendix) .
The in-sample evidence provides the history performance of models. We estimate
estimated variance for all models for full sample period using static forecast. Then we
compare forecasting performance of models which are used in this paper. We consider four
statistics as the forecasting error criterions which are employed in Wang and Wu(2012 ). Four
measures are used to evaluate the forecast accuracy for full samplenamely, Mean Square
Error (MSE) andMean Absolute Error (MAE):
n

MSE1  n 1   2t  ˆ 2t

2
(11)
t 1
6
MSE 1  n 1   t  ˆ t 
n
2
(12)
t 1
n
MAE 1  n 1   2t  ˆ 2t
(13)
t 1
n
MAE 2  n 1   t  ˆ t
(14)
t 1
2
ˆ2
where, n is the number of forecasts,  t is the actual volatility and  t is the volatility
forecast at day t.
Table 4 reports the forecasting performance of the GARCH, GJR-GARCH and
EGARCH models. Generally, we find that EGARCH model have the greatest forecasting
accuracy according to MSE2, MAE1 and MAE2, only MSE1 shows GJR-GARCH is better
performance.
Table 4:ComparisonForecasting Performance of GARCH Models
Normal Distribution
Student t Distribution
GED
GJRGJRGJRGARCH GARCH EGARCH GARCH GARCH EGARCH GARCH GARCH EGARCH
MSE1 6.30E-07 6.05E-07 6.25E-07 6.31E-07 6.08E-07 6.29E-07 6.31E-07 6.06E-07 6.27E-07
MSE2 0.000480 0.000478 0.000460 0.000478 0.000477 0.000459 0.000478 0.000476 0.000459
MAE1 0.000243 0.000239 0.000234 0.000242
MAE2 0.015476 0.015387 0.015342 0.015447
Note: The values in bold face refer to smallest loss.
0.000239
0.000234
0.000242 0.000239
0.000234
0.015398
0.015321
0.015439 0.015387
0.015315
5. Conclusion
This paper examined three GARCH models namely GARCH, GJR-GARCH and
EGARCH model with three differentdistributionswhich are Normal, Student-t and
Generalized Error distributions for comparing their forecasting power for volatility of the
return of Czech stock market, PX index.
All models are employed and their coefficients are interpreted. The results show that
significant ARCH and GARCH effects are present in the data indicates that return volatility
can be characterized by significant persistence and asymmetric effects consistent with papers
Rockinger and Urga (2012), Haroutounian and Price (2010), Vošvrda And Žikeš (2004)
which has GARCH type models to Czech stock market.
Finally we compare the insampleforecasting performance of all nine models. The
evidence shows that the EGARCH model has a best forecast performance under most of the
loss criterions
7
REFERENCES
Andersen T. G.,Bollerslev T., Lange S.(1999), “Forecastingfinancial market
volatility: Samplefrequencyvis-a-visforecasthorizon”, Journal of Empirical
Finance,Volume 6, Issue 5, 457-477.
Bollerslev, T., (1986) “GeneralizedAutoregressiveConditionalHeteroskedasticity”.
Journal of Econometrics, Vol. 31, issue 3, pages 307-327 .
Bollerslev, T., (1987). “A ConditionalHeteroskedastic Time Series Model
ForSpeculativePricesAndRates Of Return”. Review of EconomicsandStatistics 69,
542–547.
Brooks C.,(2008) “IntroductoryEconometricsFor Finance: Second Edıtıon ”
CambrıdgeUnıversıtyPress
Emerson, R.,Hall, S. andZelweska-Mitura, A. (1997) “Evolving Market
Efficiencywith an Application toSomeBulgarianShares” , Economics of Planning,
Volume 30, 75-90
Engle, R.F., (1982) “AutoregressiveConditionalHeteroskedasticityWithEstimates
of TheVariance Of UK İnflation”, Econometrica 50, 987–1008.
Glosten, L.R.,Jagannathan, R. AndRunkle, D. (1993) “On
TheRelationBetweenTheExpectedValuesAndTheVolatility Of The Nominal
Excess Return On Stocks”, Journal of Finance 48, 1779–1801.
Hájek J. (2007), “CzechCapital Market Weak-Form Efficiency, SelectedIssues” ,
PragueEconomicPapers, 4, 303-318
Haroutounıan M. K. andPrice S. (2001) “Volatility İn TheTransitionMarkets of
Central Europe”, Applied Financial Economics, 11, 93-105
Mittnik S.,Paolella M.S. Rachev S.T (2002) “Stationarity Of StablePowerGARCH Processes”, Journal Of Econometrics, 106, 97–107
Nelson, D. B. (1991). ConditionalHeteroscedasticity İn AssetReturns: A New
Approach. Econometrica, 59(2),
Rockinger, M.,Urga, G., (1999) Time VaryingParameters Model To Test
ForPredictabilityAndİntegration İn StockMarkets Of TransitionEconomies”.
Cahier De RechercheDuGroupeHec 635/1998.
Scheicher, M. (1999). ”ModelingPolishStockReturns.” in Helmenstein, C., ed.,
CapitalMarkets in TransitionEconomies. Cheltenham, UK : Edward Edgar, pp.
417–437.
8
Scheicher , M., (2001). “TheComovements Of StockMarketsInHungary, Poland
AndTheCzechRepublıc” InternatıonalJournal Of FınanceAndEconomıcsInt. J. Fin.
Econ. 6: 27–39
Shields K., ( 1997) “Stock Return Volatility On
EmergingEasternEuropeanMarkets”, The Manchester School Supplement: 118–
138.
Syriopoulos, T., (2007)
DynamicLinkagesBetweenEmergingEuropeanAndDevelopedStockMarkets: Has
TheEmuAnyİmpact? “, International Review Of Financial Analysis 16, 41-60.
Taylor, S. J. (1986), Modelling Financial Time Series, John Wiley&Sons,
Chichester.
Thomas S. andMitchell H. (2005) “GARCH Modeling of HighFrequencyVolatility in Australia’sNationalElectricity Market”, DiscussionPaper.
Melbourne Centrefor Financial Studies,
Vošvrda M. andŽıkeš F. (2004). “An Applıcatıon Of TheGarch-t Model On
Central EuropeanStockReturns” PragueEconomıcPapers, 1, 26-39
Wang, Y.,Wu, C.(2012). “ForecastingEnergy Market Volatility Using GARCH
Models: Can MultivariateModelsBeatUnivariateModels?”, EnergyEconomics.
Yu , J., (2002). “ForecastingVolatility in the New ZealandStock Market”, Applied
Financial Economics, 2002, 12, 193-202
9
APPENDIX
Table 1: Normal Distribution
GARCH(1,1)
GJR-GARCH
E GARCH
Value
p
Value
p
Value
p
0.0009
0.0000
0.0006
0.0023
0.0005
0.0055
0


4.69E-06
0.0000
6.07E-06
0.0000
-0.5147
0.0000
0.1318
0.0000
0.0727
0.0000
0.2531
0.0000
-
-
0.1036
0.0000
-0.0687
0.0000

0.8496
0.0000
0.8441
0.0000
0.9636
0.0000
AIC
-5.9192
-5.9291
-5.9282
SIC
-5.9110
-5.9188
-5.9179
DW-stat
1.8863
1.8888
1.8890
Meanequation
0
Variationequation
ARCH-LM (1) Test
p-value of ChiSq
Obs.
0.306626
0.0846
0.1493
2899
Table 2: Student’s T Distribution
GARCH(1,1)
GJR-GARCH
E GARCH
Value
p
Value
p
Value
p
0.0009
0.0000
0.0008
0.0000
0.0008
0.0001
4.34E-06
0.0000
5.36E-06
0.0000
-0.4700
0.0000
0.1192
0.0000
0.0696
0.0000
0.2354
0.0000
-
-
0.0895
0.0000
-0.613
0.0000
0.8617
0.0000
0.8563
0.0000
0.9673
0.0000
Meanequation
0
Variationequation
0



AIC
-5.9526
-5.9583
-5.9577
SIC
-5.9423
-5.9459
-5.9453
DW-stat
1.8855
1.8872
1.8873
0.3879
0.1387
0.2596
ARCH-LM(1) Test
p-value of ChiSq.
Obs.
2899
10
Table 3: GeneralizedError Distribution(GED)
GARCH(1,1)
GJR-GARCH
E GARCH
Value
p
Value
p
Value
p
0.0009
0.0000
0.0007
0.0001
0.0007
0.0002
4.56E-06
0.0000
5.72E-06
0.0000
-0.4947
0.0000
0.1256
0.0000
0.0712
0.0000
0.2446
0.0000
-
-
0.0960
0.0000
-0.0646
0.0000

0.8546
0.0000
0.8495
0.0000
0.9653
0.0000
r
1.5083
0.0000
1.534154
0.0504
1.5329
0.0498
Meanequation
0
Variationequation
0


AIC
-5.9427
-5.9493
-5.9484
SIC
-5.9324
-5.9369
-5.9360
DW-stat
1.8856
1.8876
1.8878
ARCH-LM (1) Test
p-value of ChiSq.
0.2896
0.074
0.1979
Obs.
2899
11