PANEL GARCH MODEL WITH

5. CROSS-SECTIONAL DEPENDENCE
BETWEEN CEE EMERGING MARKETS IN
TRADING DAY EFFECTS ANALYSIS

Josip ARNERIĆ1
Blanka ŠKRABIĆ PERIĆ2

Abstract
The presence of the weekday effects of 10 emerging CEE stock markets is explored.
Simultaneously, the cross-sectional dependence between daily returns of national stocks is
controlled. Most of the previous studies neglect cross-sectional dependence in case of
univariate analysis. Rare studies of the weekday effects include multivariate GARCH,
considering only a few markets as it suffers from high dimensionality. Thus, we specify and
estimate a panel GARCH with a relatively small number of parameters. Results indicate a
strong presence of the Monday effect in both mean and variance equations, while the
Tuesday effect is present only in the mean equation. Empirical findings also confirm the
existence of the cross-sectional dependence, particularly dependence of Poland with
Hungary, Czech and Croatia.

Keywords: panel GARCH, time-varying covariance, market anomalies, emerging CEE

markets, maximum likelihood estimates
JEL Classification: C32, C33, G1

1. Introduction
It is believed that anomalies in trading with stocks can influence stock market returns as well
as volatility. Finding a pattern of these anomalies can be helpful for investors to predict future
market movements in speculative purposes, hedging or portfolio management. These
anomalies are usually referred to as calendar effects, i.e. weekday effects (WDE). According
to French (1980), the average return on Monday should be significantly lower than the
average return over the other days of the week because of the accumulated information
during non-trading days when financial markets are closed. Therefore, accumulated
information over the weekend is reflected in prices on Monday. Among others, Fortune

1 University of Zagreb, Faculty of Economics and Business, Zagreb, Croatia, Email:

jarneric@efzg.hr.
2 Corresponding author: University of Split, Faculty of Economics, Split, Croatia, Email:
bskrabic@efst.hr.

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(1991) explains that firms and governments release good news during the trading time when
they can be absorbed and store up bad news after the close on Friday, when investors
cannot react until the Monday opening. Another explanation by behavioral finance is that
Monday is the worst day of the week for investors because it is the first working day and
consequently investors tend to be more pessimistic. On the other hand, investors feel
optimistic on Fridays and they are more inclined to sell when prices are rising due to the
increased demand.
Empirical results vary regarding different markets and methodology. While the most of
empirical findings have focused on developed stock markets as well as emerging markets
in Asia and Latin America, only a few studies were concentrating on the Central and Eastern
European (CEE) emerging markets. From previous research, there is no general conclusion
for WDE in CEE emerging markets. Also, it should be noted that most previous studies
focused mainly on OLS regression with dummy variables (Ajayi et al., 2004; Gajdošová et
al., 2011; Karanovic and Karanovic, 2018; Tonchev and Kim, 2004) and/or univariate
GARCH models, i.e. estimation of GARCH model for each market individually (Bubák and
Zikes, 2006; Patev et al., 2003; Stavarek and Heryan, 2012; Tonchev and Kim, 2004;
Andrieş et al., 2017). These studies neglect cross-sectional dependence in case of
univariate analysis, implying unreliable and imprecise conclusions. On the other hand,
numerous empirical studies confirmed cross-sectional dependence between CEE countries
(Babetskii et al., 2007; Harrison et al., 2010; Ferreira, 2018). Dependence of time series
returns across stock markets is opposite to the weak-form efficiency. Weak-form market
efficiency means there is no any kind of dependence of time series or anomalies that can
explain or predict market movements (Zhang et al., 2017)). Deviations from weak-form
market efficiency should not be ignored. Therefore, panel GARCH methodology is used in
this paper to explore not only market anomalies, i.e. weekday effect, but also the
dependence of time series returns across CEE stock markets. Moreover, dependence
across markets should be considered due to financial integration and increasing contagion
effect indicating that markets move more closely together.
Panel GARCH estimator is extremely valuable for testing panel regressions of financial data
for GARCH effects and has a more efficient panel estimator available if the error term is
found to be conditionally heteroscedastic.
The main contribution of this paper to the existing literature is employing panel GARCH
methodology to investigate WDE. Simultaneously, this methodology upgrades previous
studies of WDE by controlling cross-sectional dependence between CEE countries. In the
other words, it enables to estimate WDE by assuming already proven “inefficiency” of CEE
emerging markets. To the best of the authors’ knowledge, none of the previous studies
investigated the trading day effects in Central and Eastern European emerging stock
markets using panel GARCH.
Second, this methodology enables to find a common day of the week effect in both
conditional mean and conditional variance equations for all markets under consideration.
However, omitting WDE from both equations can cause spurious conclusion about cross-
sectional dependence intensity. Thus, the proposed panel GARCH model gives more
accurate cross-market correlations after including WDE.
Third, the existence of cross-sectional dependence, empirically confirmed in this paper,
should be considered in all similar studies, which explores any other anomalies in emerging
stock markets.
The remainder of the paper is organized as follows. Section 2 presents previous empirical
studies. Section 3 describes the data and methodology. Section 4 presents the estimation

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results. Finally, some conclusions and directions for future research are provided in
Section 5.

2. Previous Empirical Studies

Empirical studies of WDE for CEE emerging markets appeared later than in other countries,
after their reforms and stabilization with increased domestic and foreign institutional trading
at the beginning of the 21st century. The first serious attempt to investigate WDE in eight
CEE emerging stock markets is Patev et al. (2003) using univariate GARCH-M model. They
have concluded that Monday returns were negative and significant for the Czech and
Romanian stock markets. Wednesday returns were significantly positive for the Slovenian
market. The WDE was not found in the Polish and Slovak stock markets.
Ajayi et al. (2004) have focused on the larger group of 11 emerging stock markets in Central
and Eastern Europe (OLS regression was used). Their results indicate negative Monday
returns in six markets but only returns in Estonia and Lithuania are significantly negative.
According to Tonchev and Kim (2004), various calendar anomalies in the Czech Republic,
Slovakia and Slovenia are examined in mean and variance equations. They found only a
weak WDE in mean return for Slovenia, but in opposite direction than theory suggests.
Bubák and Zikes (2006) have found significant day of the week effects in the mean of returns
on the Czech PX and the Polish WIG indices, and significant seasonality in the volatility of
the Hungarian BUX index within the framework of a periodic autoregressive model PAR-
PGARCH for both the mean and the variance of stock returns. Authors’ empirical results
indicated the presence of the non-trading effect in the mean of WIG stock returns.
During the crisis period studies of WDE in CEE emerging markets were not in the focus of
interest, while after the crisis several studies are provided. The evidence of WDE before and
during the financial crisis is compared by Gajdošová et al. (2011). Application of regression
models with dummy variables leads to the conclusion that WDE was present only in the
Czech market (decreasing Monday effect) and the Hungarian market (increasing Friday
effect) during the crisis. Results of Guidi et al. (2011) also confirm rather sporadic evidence
of WDE in CEE stock markets. They have investigated the random walk hypothesis as well
as the day-of-the-week effect for CEE stock indices by using parametric and non-parametric
tests, as well as OLS and conditional variance methodology. Using the variance ratio test,
they found that after the accession to the EU the random walk hypothesis is rejected for two
indices, SAX and SOFIX respectively, out of seven. Friday effect features predominantly
among indices in the full sample. When the GARCH-M model is employed in the full sample,
the day-of-the-week effect is present in both volatility and the returns: particularly Mondays
and Tuesdays show a significant effect in the volatility equation of four out of seven indices.
Splitting the sample into the pre-accession and post-accession period, results indicate that
the volatility Monday effect tends to be present in more indices in the post-accession than in
the pre-accession EU period.
Stavarek and Heryan (2012) divided the period 2006-2012 into six sub-periods capturing
individual phases of the financial and economic crisis and separately estimated a modified
GARCH-M(1,1) model for each country and each sub-period using daily returns of the major
national stock market indices. The results clearly indicate a little evidence of the day of the
week effect. Daily calendar anomalies are rather sporadic, isolated, and unstable over time.
They conclude that the day of the week effect is not typical for the Central European stock
markets and the recent financial crisis seems to have no impact on the existence of the WDE
phenomenon.

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In the recent time, WDE started again to be in the focus of empirical research. Andrieş et al.
(2017) investigated three seasonal anomalies in return and volatility. Results confirmed the
existence of WDE in most countries with exception of (Karanovic and Karanovic, 2018).
The results of recent empirical studies do not provide the general conclusion of WDE in CEE
countries. Moreover, all studies in these countries have neglected cross-sectional
dependence, which is confirmed in empirical studies (Babetskii et al., 2007; Harrison et al.,
2010; Ferreira, 2018). Based on this fact, it is necessary to investigate the existence of WDE
in CEE countries in detail. Therefore, our paper uses panel GARCH methodology focused
on finding the common WDE in 10 emerging markets that are EU members.

3. Data and Methodology

The dataset of daily prices of the main stock indices provided by Thomson Reuters service,
for 10 emerging European markets (Romania - BETI, Hungary - BUX, Croatia - CROBEX,
Latvia - OMXRGI, Estonia - OMXTGI, Czech -PX, Slovenia - SBITOP, Bulgaria - SOFIX,
Poland – WIG20 and Slovakia - SAX) covers the period from January 4, 2007 to May 13,
2015. These countries are considered because they have not been at the center of the
researchers’ interest within a recent period after the crisis. A full sample includes 2177
observations for each market. Table 1 presents descriptive statistics of returns according to
trading days to get a better insight of weekday effects for all emerging markets together. On
average, negative returns are observed on Monday, Tuesday and Thursday, while positive
returns are observed on Wednesday and Friday. The highest range of returns can be noticed
on Friday but with the smallest standard deviation and positive average return. These
findings are informative only but in line with pessimistic Monday and optimistic Friday.

Table 1
Descriptive Statistics of Returns according to Trading Days
Monday Tuesday Wednesday Thursday Friday
Mean -0.0007 -0.0007 0.0002 -7.4e-06 0.0001
Std. Dev 0.0145 0.0136 0.0137 0.0132 0.0135
Minimum -0.0933 -0.1136 -0.1312 -0.1007 -0.1619
Maximum 0.1478 0.1188 0.1318 0.0915 0.1109

To examine the WDE in returns of emerging markets under consideration, the following
panel model is used:
rit   i  Mon  D1  Tue  D2  Thu  D3  Fri  D 4   it , i  1, , N , t  1, ,T
(1)
where: N is the number of cross-sectional units (10 national stock indices) and T is the
number of time periods (2177 daily observations per unit), r it is return of stock index i in the
period t calculated as continuously compounded return or log return, i.e. rit  ln pit  ln pit 1
with pit being the price of stock index of country i in period t,  i is a constant term for each
stock market, D1  D 4 are dummy variables of weekdays and Mon,Tue,Thu, Fri are
coefficients to be estimated. The dummy variable for Wednesday is omitted to avoid the
dummy variable trap. It seems reasonable to omit the dummy variable for Wednesday as
the middle day of the week. The omitted dummy variable becomes the reference category.

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Panel GARCH Model with Cross-Sectional Dependence

Equation (1) is, in fact, the conditional mean equation of returns written in panel data
notation. It is necessary to find an adequate estimator for unknowns in the panel model.
The frequently used dynamic panel data estimator difference GMM and system GMM
(Arellano and Bond, 1991; Arellano and Bover, 1995; Blundell and Bond, 1998) are proposed
for dataset with large N and small T. In case of large T as in this case ( T  2177 ), they can
produce inconsistent and potentially very misleading estimates and thus not adequate
estimators for an equation (1). On the other hand, there are several estimators for large T.
Mean group (MG) estimator and pooled mean group (PMG) estimator (Pesaran, Shin, and
Smith, 1999) are suitable for large N and T while Least Squares Dummy Variable (LSDV)
estimator is suitable for a large T regardless of the size of N. All these estimators assume
that independently distributed error terms across time and cross-sections, i.e. they assume
cross-sectional independence and constant variance. Finally, seemingly unrelated
regression equation (SURE) proposed by Zellner (1962) is suitable for dataset when N is
small relative to T and it assumes time-invariant correlation between error terms of different
cross-sections. To perform any of the given estimators, several tests must be provided.
Some of the considered estimators for a large T, such as LSDV and SURE, assume
stationarity. In the first step stationarity of returns is tested, i.e. panel unit root tests that have
shown good properties when T is large are computed using STATA software: Im, Pesaran
and ShinW-stat, ADF-Fisher Chi-square, PP-Fisher Chi-square and Pesaran’s CADF3 test.
Results in Table 2 indicate that returns are stationary (the unit root null hypothesis is
rejected).
Table 2
Panel Unit Root Tests for Returns
Test Statistic p-value
Im, Pesaran and Shin W-stat -671.34 0.0000
ADF - Fisher Chi-square 2027.90 0.0000
PP - Fisher Chi-square 268.99 0.0000
Pesaran's CADF test t-stat -15.46 0.0000

Also, it is necessary to test the presence and significance of the individual effects. Wald test
is used to test individual effects in equation (1). Wald test statistics is (9)
2
 1.45 , which is
not statistically significant (p-value=0.9975). Therefore, the null hypothesis that all cross-
sectional units have a common intercept is not rejected, i.e. all  i in equation (1) are equal.
Since all estimators except SURE assume cross-sectional independence, Breusch and
Pagan LM test for cross-sectional dependence is computed. This test is appropriate for data
sets with fixed N and T   (De Hoyos and Sarafidis, 2006). The result of LM test indicates
cross-sectional dependence according to chi-square value (45)
2
 10072.38 (p-value=0.000),
which is statistically significant at 1% level. Therefore, it is necessary to use an estimator
that enables cross-sectional dependence. Finally, test for the groupwise heteroscedasticity
(unconditional) and test of conditional variance are performed. Modified Wald test statistic
for groupwise heteroscedasticity in the residuals from a fixed-effect panel model obtains the
value (10)
2
 1116.04 (p-value=0.000). This test statistic is significant at the 1% level. It
indicates that variance is not constant between markets under consideration. Coefficients of

3 Pesaran CADF test (Pesaran, 2007) assumes cross-sectional dependence.

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the partial autocorrelation of squared residuals from a fixed-effect panel model are also
calculated as in Cermeño and Grier (2006). Results from Table 3 show that all coefficients
of partial autocorrelation up to lag 5 are statistically significant at 1% level. The number of
lags included in testing corresponds to the number of trading days in a week. This is the
evidence of existing time-varying variance, i.e. conditional heteroscedasticity. Therefore, it
is necessary to introduce conditional variance equation that follows the GARCH (1,1)
process.
Additionally, Hosking’s multivariate Portmanteau autocorrelation test on squared
standardized residuals is performed for each time lag (this test is a multivariate extension of
the univariate Ljung-Box Q test and requires unstacked data of N separated time series)
indicating the presence of ARCH effects in residuals for all markets under consideration.
Table 3
Partial Autocorrelation Coefficients of Squared Residuals from Fixed-
effect Panel Model and Hosking’s Multivariate Portmanteau Test
Lag PAC p-value Q p-value
1 -0.4647 0.000 2447.32 0.000
2 -0.4719 0.000 4668.53 0.000
3 -0.4677 0.000 6615.85 0.000
4 -0.4611 0.000 8242.38 0.000
5 0.5469 0.000 9531.81 0.000

Considering all pre-estimation results, it can be concluded that the above-mentioned panel
data estimators are not suitable for modeling of WDE in emerging markets. The results in
Table 3 indicate that ARCH effects are present, so it is necessary to employ the estimator
which is a combination of panel data and GARCH methodology. Also, the results of cross-
sectional dependence test indicate that range of correlation coefficient across countries is
wide (from -0.035 to 0.5710) and thus covariance equation, among mean and variance
equations, should be included additionally. Namely, except mean equation, it is necessary
to include the equation of variance and the equation of covariance simultaneously in the
panel model due to the cross-sectional dependence across markets and time-varying
variance. Therefore, the panel GARCH model with cross-sectional dependence includes
three equations in the context of panel data. This model was firstly proposed by Cermeño
and Grier (2006) and it is adopted here to investigate the presence of the day of the week
effects in both mean and variance equations. This methodology is relatively new in financial
applications, and it is the only adequate estimation method for this research.
Conditional mean equation (1) can be expressed as a dynamic panel model with fixed
effects:
k
rit  i  Mon  D1  Tue  D2  Thu  D3  Fri  D4    j ri ,t  j   it , i  1,, N, t  1,,T (2)
j 1

where: ri ,t  j are lagged values of returns and j are additional coefficients that should be
estimated. In the finance literature, it is known that returns are serially uncorrelated or with
minor lower order serial correlation4, but they are not independent. Significant serial

4 Estimation process was carried out by comparing the results before and after the inclusion of
lagged returns up to 5-time lags, but they were not significant. Thus, only one-time lag ( k  1 )

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Panel GARCH Model with Cross-Sectional Dependence

correlation of the squared returns and squared residuals respectively indicate the
dependence of stock returns in the context of ARCH effects.
Further, the disturbance  it is assumed to have zero mean and normal distribution with the
following conditional moments:
( i ) E  it  js    it2 , i  j and t  s
( ii ) E  it  js    ij ,t , i  j and t  s
(3)
( iii ) E  it  js   0, i  j and t  s
( iv ) E  it  js   0, i  j and t  s
Assumptions (i) and (ii) define a very general conditional variance and covariance process,
condition (iii) assumes no autocorrelation and condition (iv) assumes non-contemporaneous
cross-sectional correlation. The conditional variance and conditional covariance equations
are adopted according to Bollerslev et al. (1988) specification of GARCH(1,1) model due to
its simplicity and sufficiency to capture the ARCH effects:

 it2  i   i2,t 1   i2,t 1 i  1,2,...,N, t  1,...,T (4)

 ij ,t  ij   ij ,t 1   i ,t 1 j ,t 1 i j (5)

The model defined by equations (2), (4) and (5) is a dynamic panel data model with
conditional covariance (DPD-CCV). It is convenient that both equations (4) and (5) follow a
common dynamic according to GARCH (1,1) specification, i.e. it captures the dynamic
properties of disturbances  it and their cross-products  it   jt . In this way, the number of
parameters is considerably reduced (there are 4  N  N  1 / 2 parameters in the covariance
matrix). Given the “excessive” number of parameters estimated in DPD-CCV model, it still
includes fewer parameters compared to some MGARCH type specifications5. Furthermore,
the conditional variance equation (4) assumes  i  0 ,   0 ,   0 and      1 as
sufficient restrictions to ensure the positivity and convergence of  it2 . Equation (5) is
conditional covariance equation, assuming       1 to ensure the covariance processes
to converge to some fixed value. A different individual effect for each market in conditional
variance equation is supported from the modified Wald test for heteroscedasticity. Some
authors have also tested the day of the week effect in volatility using univariate approach
(Choudhry, 2000; Berument and Kiymaz, 2001; Tonchev and Kim, 2004; Bubák and Zikes,
2006). Therefore, conditional variance equation (4) is extended in the following way:
 it2  i  Mon  D1  Tue  D2  Thu  D3  Fri  D4   i2,t 1   i2,t 1, i ,1..., N, t  1,,T (6
)

was considered due to robustness check of the weekday effects in mean, assuming that AR
parameter satisfies the stability condition.
5 For example DVEC(1,1) model, proposed by Bollerslev et al. (1988), would include 165
parameters, while it’s vector-diagonal version proposed by Bollerslev et al. (1994), would
include 75 parameters. Due to dimensionality problem, many of the estimated parameters would
not be statistically significant, while additional computational problems arise in the numerical
optimization procedure.

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Equation (6) follows the similar specification of the day of the week effect as in equation (2)
and it is considered in this paper.
In matrix notation equation (2) with AR(1) parameter can be written as:
rt    Z t    t , t  1,...,T (7)

where: rt and t are vectors of dimension N 1, matrix Zt  rt 1, D1, D2 , D3 , D4  has dimension
N 5 ,  is a N 1 vector of individual specific effect, while   1, Mon, Tue, Thu, Fri  ' is a
column vector of unknown coefficients. Considering previous assumptions, the N-
dimensional vector of disturbances  t N  0, t  . The covariance matrix  t is time-
dependent; its diagonal and off-diagonal elements are given by equations (4) and (5) or (6)
respectively. The vector of returns rt is therefore conditionally normally distributed with
mean    Zt  and variance-covariance matrix  t with its conditional density:
N 1 1
 rt    Zt   ' t1  rt    Zt  
f  rt Zt ,  , ,     2 


2 t

2 e 2 (8)
where:  includes the parameters in equations (4) and (5) or (6) respectively. The log-
likelihood function of the complete panel GARCH can be written as:
NT 1 T 1 T
l  ln  2    ln t    rt    Zt  ' t1  rt    Zt  (9)
2 2 t 1 2 t 1
Equation (9) is maximized with respect to ,  and  using the numerical optimization
algorithm BFGS. It is well known that under regularity conditions the MLE estimators are
consistent, asymptotically efficient and asymptotically normally distributed with mean equal
to the true parameter vector and a covariance matrix equal to the inverse of the
corresponding information matrix. It is important to note that these asymptotic properties
would hold for N fixed and T approaching infinity since we are modeling the N-dimensional
vector of disturbances of the panel as a multivariate time series process.

4. Estimation Results
To examine the day of the week effects in both mean and variance equations using daily
returns of 10 national stock indices from CEE emerging markets, four panel GARCH models
are estimated in the first step, while in the second step two additional panel GARCH are
estimated for robustness check. In theory, it is expected that returns are serially uncorrelated
but in practice, many researchers found positive and statistically significant autocorrelation
of returns for some countries. Therefore, the basic model, labeled as Model 1, is specified
without the lagged dependent variable. To examine the possibility that returns follow
autoregression process of the first order lagged dependent variable is introduced in Model 2
and Model 4. Moreover, some empirical researches have provided evidence that WDE has
the influence on a variance of returns thus dummy variables of weekdays are also included
in two specifications (Model 3 and Model 4). All results are obtained using RATS software
using MLE within BFGS algorithm (Table 4).

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Table 4
Estimation Results of Different Panel GARCH Specifications using MLE
Parameter Model 1 Model 2 Model 3 Model 4 Model 5 Model 6
1 0.0132 0.0132
(0.0092) (0.0088)
 0.0003** 0.0003 0.0002 0.0003 0.0001 0.0004**
(0.0001) (0.0002) (0.0002) (0.0002) (0.0003) (0.0002)
-0.0012** -0.0012*** -0.0012*** -0.0013*** -0.0006** -0.0014***
Mon
(0.0003) (0.0004) (0.0003) (0.0003) (0.0003) (0.0001)
-0.0005** -0.0005 -0.0006*** -0.0005** -0.0004* -0.0009***
Tue
(0.0002) (0.0004) (0.0003) (0.0003) (0.0002) (0.0001)
-0.0002 -0.0002 -0.0002 -0.0002 -0.0001 -0.0005***
Thu
(0.0003) (0.0004) (0.0003) (0.0003) (0.0003) (0.0001)
0.0002 0.0002 0.0002 0.0002 0.0002 0.0001
Fri
(0.0003) (0.0004) (0.0003) (0.0003) (0.0003) (0.0001)
0.8830*** 0.8829*** 0.8830*** 0.8830*** 0.8555*** 0.8789***
 (0.0067) (0.0068) (0.0065) (0.0066) (0.0049) (0.0077)
0.9175*** 0.9176*** 0.9174*** 0.9176*** 0.8976*** 0.9145***
 (0.0067) (0.0070) (0.0071) (0.0068) (0.0071) (0.0077)
 0.0809*** 0.0809*** 0.0809*** 0.0809*** 0.0872*** 0.0820***
(0.0043) (0.0046) (0.0043) (0.0042) (0.0062) (0.0049)
 0.0297*** 0.0296*** 0.0297*** 0.0296*** 0.0356*** 0.0304***
(0.0024) (0.0025) (0.0023) (0.0024) (0.0025) (0.0027)
-0.0012*** -0.0012*** -0.0002 -0.0014***
Mon
(0.0004) (0.0004) (0.0005) (0.0003)
-0.0006 -0.0005 -0.0004 -0.0009***
Tue
(0.0004) (0.0003) (0.0004) (0.0003)
-0.0002 -0.0002 -0.0001 -0.0005
Thu
(0.0004) (0.0004) (0.0005) (0.0003)
0.0002 0.0002 0.0001 0.000007
Fri
(0.0004) (0.0004) (0.0005) (0.0003)
Note: *, **, *** represent significance at the 1%, 5% and 10% level, while standard
errors are in parentheses.

In both models with lagged dependent variable, Model 2 and Model 4, respectively, the AR
parameter is not statistically significant even positive. This finding confirms theoretical
assumption about serially uncorrelated returns. Moreover, the introduction of lagged
dependent variable didn’t change other parameter estimates. Comparing the first and the
second model, statistically significant common parameters are Mon,  , ,  and  , while in
Model 1  and Tue are additionally statistically significant. Model 3 and Model 4 are
augmented by including the day of the week effect in conditional variance equation. Results
of both models are almost identical. Statistically significant parameters are
Mon, Tue,  , ,  ,  and Mon , while  1 is not significant in Model 4. From all results, it can
be concluded that Model 3 is the most appropriate thus results from this model are
considered in economic interpretation. Moreover, the results from Model 1, 2 and 4 are
almost equal as in the Model 3.

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To additionally ensure the robustness of the results, the entire sample was split into two
subsamples. The first subsample of 800 observations covers pre-crisis period while the
second subsample includes 1400 observations. Therefore, two additional models were
estimated, i.e. Model 5 and Model 6, with results in the last two columns of Table 4.
Robustness is confirmed in the context of parameter estimates but minor differences in
statistical significance can be noticed. Namely, Monday and Tuesday effects in mean are
statistically significant in both subsamples but Thursday effect is significant in crisis and post-
crisis period only, with the expected negative sign. Even though the Friday effect is found to
be positive, supported by a theoretical explanation, it is not statistically significant in both
equations and both subsamples. In the variance equation, Monday and Tuesday effects are
significant only in the second subsample period. In the recent paper of Andrieş et al., (2017),
authors found significant Monday effect in the more than half of the examined countries.
Therefore, the significance of the Monday effect in our research is expected. However,
results of the Tuesday effect are in the line with research of Karanović and Karanović (2018).
It should be noted that parameters  , ,  and  , are statistically significant in all estimated
models and their values are very similar. In particular, the GARCH (1,1) model usually
indicates high persistence in the conditional variance when     1 , i.e. almost integrated
behavior of the conditional variance. High volatility persistence means that a long time is
needed for shocks in volatility to die out. Based on estimated results from Model 3, as most
appropriate, the sum σ+γ=0.9639 indicates high volatility persistence at observed emerging
markets. In other words, the conditional variance converges to unconditional long-run
variance very slowly (3.61% daily). We find that all individual specific effects in the variance
equation are statistically significant at the 1% level (see Appendix A). It is expected that
different levels of unconditional variance within each market can be computed due to
different constant terms  i and hence it can be easily noted that unconditional variance
depends straightforward on individual specific effects. Therefore, our results suggest that
Hungary, Romania, Poland Slovakia and the Czech Republic are expected to be riskier in
the future, as they have a high level of unconditional volatility, while Estonia, Latvia, Slovenia,
Croatia, and Bulgaria have a low level of unconditional volatility (see Appendix B).
From the constant terms ij , i  j in covariance equation (5), implied conditional cross-
sectional dependence in pairs of markets can be computed. As the results for the covariance
equation also indicate a quite persistent although stationary GARCH (1,1) process
(σ+λ=0.9471), the unconditional covariance can be computed as ij /      . Moreover, from
unconditional covariance and unconditional variance, the long-run cross-correlation
coefficient in pairs can be obtained. Long-run cross-correlation coefficients matrix is
presented in Appendix C. Results indicate that in the long-run high positive cross-correlation
is expected between Poland and the Czech Republic, Poland and Hungary and Poland and
Croatia. These results are expected, i.e. Harrison et al. (2010) confirmed that the Czech
Republic, Hungary and Poland display higher correlations among them and with the other
EU markets. Additionally, Šikić and Šagovac, (2017), confirm that the Croatian stock market
is more correlated with the Polish stock market than the US. These results confirm the
dominance of Poland stock market in the CEE region. The cross-correlation is negative but
close to zero, between Slovakia and most other countries. These results are in the line with
Babetskii et al. (2007). They found that Slovak returns are not correlated with the Czech
Republic, Hungary and Poland. They d escribed the Slovak stock market as moderate
development and lower than others. Therefore, it can be concluded that during the ten years
Slovak stock market stays isolated from other new EU countries.

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Panel GARCH Model with Cross-Sectional Dependence

5. Conclusion
In this paper, we employ the panel GARCH model with cross-sectional dependence to
investigate the presence of the WDE in daily returns on 10 European emerging markets
in both conditional mean and conditional variance equations. Most of the previous
empirical studies neglect possible cross-country dependence using the univariate
approach which reports only country-by-country results. Empirical evidence of time-
varying conditional variance and covariance of these markets imposed the need to
include cross-sectional dependence in a panel model. This finding makes panel GARCH
methodology unique and appropriate one. Therefore, the main contribution of this paper
is the extension of traditional GARCH model to a panel context, i.e. novel methodology
enables to explore not only market a nomalies but also the dependence of time
series returns across CEE stock markets.
The empirical findings indicate the strong presence of the common Monday effects in
both mean and variance equations, while the Tuesday effect is significant only in the
mean equation. High volatility persistence at observed emerging markets is present.
Our results implicate that Hungary, Romania, Poland Slovakia and the Czech Republic
are expected to be riskier in the future, while Estonia, Latvia, Slovenia, Croatia and
Bulgaria have the low level of unconditional volatility. In the long-run high positive cross-
correlation is expected between Poland and the Czech Republic, Poland and Hungary,
Poland and Croatia. The cross-correlation is negative but close to zero, between
Slovakia and most of the other countries. Trading anomalies findings on Mondays and
Tuesdays can be helpful to investors to predict future market movements in speculative
purposes, hedging or portfolio adjustment. Moreover, information about cross-markets
dependence is extremely valuable for international portfolio diversification.
Recommendation for risk-averse international investors is to invest in stocks of Estonia,
Latvia, Slovenia, Croatia and Bulgaria. Moreover, investors who are interested in
emerging markets should include Poland stocks in their international portfolios for
diversification purpose, even though it has a relatively high risk position among
neighboring countries.
As the purpose was to find common trading anomalies across emerging markets, i.e.
weekday effects, developed ones are not included in the analysis. This is a limitation of the
research. It would be interesting to include not only other European markets but also US
markets, as we can assume that emerging markets depend on developed markets. In that
case, a panel GARCH model should be adapted to control local and global cross-sectional
dependence. Moreover, high-frequency data (thick-by-thick) should be explored to check the
number of transactions and trading volumes on Mondays, Tuesdays, Wednesdays,
Thursdays and Fridays for emerging markets under consideration. Such data can be used
to identify significant jumps within trading days of the week. Finally, our results can motivate
researchers to deeply investigate cross-sectional dependence between emerging markets
in the other economics behaviors and motivate them to perform panel GARCH in their
studies.

Acknowledgement
This work has been fully supported by Croatian Science Foundation under the project
“Volatility measurement, modeling and forecasting” [number 5199].

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References
Ajayi, R.A. Mehdian, S. and Perry, M.J., 2004. The Day-of-the-Week Effect in Stock Returns :
Further Evidence from Eastern European Emerging Markets. Emerging
Markets Finance and Trade, 40(4), pp. 53–62.
Andrieş, A.M. Ihnatov, I. and Sprincean, N., 2017. Do Seasonal Anomalies Still Exist In
Central And Eastern European Countries? A Conditional Variance
Approach. Romanian Journal of Economic Forecasting, 20(4), pp.60–83.
Arellano, M. and Bond, S., 1991. Some Tests of Specification for Panel Data: Monte Carlo
Evidence and an Application to Employment Equations. The Review of
Economic Studies, 58(2), pp.277–297.
Arellano, M. and Bover, O., 1995. Another look at the instrumental variable estimation of
error-components models. Journal of Econometrics, 68(1), pp. 29–51.
Babetskii, I. Komárek, L. and Komárková, Z., 2007. Financial Integration of Stock Markets
among New EU Member States and the Euro Area. Czech Journal of
Economics and Finance (Finance a Uver), 57(7-8), pp. 341–362.
Berument, H. and Kiymaz, H., 2001. The day of the week effect on stock market volatility.
Journal of Economics and Finance, 25(2), pp.181–193.
Blundell, R. and Bond, S., 1998. Initial conditions and moment restrictions in dynamic panel
data models. Journal of Econometrics, 87(1), pp. 115–143.
Bollerslev, T. Engle, R.F. and Nelson, D. B., 1994. ARCH models. Handbook of
Econometrics, 4, pp. 2959–3038.
Bollerslev, T. Engle, R.F. and Wooldridge, J.M., 1988. A Capital Asset Pricing Model with
Time-Varying Covariances. Journal of Political Economy, 96(1), pp.116–
131.
Bubák, V. and Zikes, F., 2006. Seasonality and the Nontrading Effect on Central-European
Stock Markets. The Czech Journal of Economics and Finance, 56, pp.1–7.
Cermeño, R., and Grier, K.B., 2006. Conditional Heteroscedasticity and Cross-Sectional
Dependence in Panel Data: An Empirical Study of Inflation Uncertainty in
the G7 countries. In : B. H. Baltagi, ed. 2006. Contributions to Economic
Analysis 274, Emerald Group Publishing Limited, Ch,10.
Choudhry, T., 2000. Day of the week effect in emerging Asian stock markets: evidence from
the GARCH model. Applied Financial Economics, 10(3), pp.235-242.
De Hoyos, R.E. and Sarafidis, V., 2006. Testing for cross-sectional dependence in panel-
data models. Stata Journal, 6(4), pp.482–496.
Ferreira, P., 2018. Long-range dependencies of Eastern European stock markets: A
dynamic detrended analysis. Physica A: Statistical Mechanics and Its
Applications, 505, pp.454–470.
French, K.R., 1980. Stock returns and the weekend effect. Journal of Financial Economics,
8(1), pp.55–69.
Fortune, P., 1991. Stock market efficiency: an autopsy?. New England Economic Review,
No. Mar, pp. 17–40.
Gajdošová, K. Heryán, T. and Tufan, E., 2011. Day of the week effect in the European
emerging stock markets: recent evidence from the financial crisis period.
Scientific Papers of the University of Pardubice, Series D, 15(19), pp.38–
51.
Guidi, F. Gupta, R. and Maheshwari, S., 2011. Weak-form market efficiency and calendar
anomalies for Eastern Europe equity markets. Journal of Emerging Market
Finance, 10 (3), pp.337–389.

82 Romanian Journal of Economic Forecasting –XXI (4) 2018

Panel GARCH Model with Cross-Sectional Dependence

Harrison, B. Lupu, R. and Lupu, I., 2010. Statistical Properties of the CEE Stock Market
Dynamics. A Panel Data Analysis 1. The Romanian Economic Journal,
13(37), pp.41–54.
Karanovic, G. and Karanovic, B., 2018. The Day-of-the-Week Effect: Evidence from
Selected Balkan Markets. Scientific Annals of Economics and Business,
65(1), pp.1–11.
Patev, P.G. Lyroudi, K. and Kanaryan, N.K., 2003. The Day of the Week Effect in the Central
European Transition Stock Markets. SSRN Scholarly Paper 434501
Rochester, NY: Social Science Research Network. [online] Avaliable at:<
http://papers.ssrn.com/abstract=434501> [Accessed on July 2018]
Pesaran, M.H., 2007. A simple panel unit root test in the presence of cross-section
dependence. Journal of Applied Econometrics, 22(2), pp.265–312.
Pesaran, M.H. Shin, Y. and Smith, R.P., 1999. Pooled Mean Group Estimation of Dynamic
Heterogeneous Panels. Journal of the American Statistical Association,
94(446), pp.621–634.
Šiki, L. and Šagovac, M., 2017. An international integration history of the Zagreb Stock
Exchange. Public Sector Economics, 41.(2), pp.227–257.
Stavarek, D. and Heryan, T., 2012. Day of the week effect in central European stock markets
MPRA Paper 38431. [online] Avaliable at:
<http://mpra.ub.unimuenchen.de/38431/> [Accessed on July 2018]
Tonche, D. and Kim, T.-H.K., 2004. Calendar effects in Eastern European financial markets:
evidence from the Czech Republic, Slovakia and Slovenia. Applied
Financial Economics, 14(14), pp.1035–1043.
Zellner, A., 1962. An Efficient Method of Estimating Seemingly Unrelated Regressions and
Tests for Aggregation Bias. Journal of the American Statistical Association,
57(298), pp.348–368.
Zhang, J. Lai, Y. and Lin, J., 2017. The day-of-the-Week effects of stock markets in different
countries. Finance Research Letters, 20, pp.47–62.

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Appendix
A. Matrix of Estimated Individual Specific Effects in Conditional Variance and
Conditional Covariance

Note: *, **, *** represent significance at the 1%, 5% and 10% level.

B. Matrix of Unconditional Variance and Covariance

C. Matrix of Long-run Cross-correlation Coefficients

84 Romanian Journal of Economic Forecasting –XXI (4) 2018