






































DEM_2015_49to69


D Y N A M I C  E C O N O M E T R I C  M O D E L S  
DOI: http://dx.doi.org/10.12775/DEM.2015.003  Vol. 15 (2015) 49−69 

Submitted September 4, 2015  ISSN (online) 2450-7067 
Accepted December 15, 2015 ISSN (print) 1234-3862 

Sabina Nowak, Joanna Olbryś* 

Day-of-the-Week Effects in Liquidity on the Warsaw 
Stock Exchange 

A b s t r a c t. The purpose of this study is to explore the day-of-the-week patterns in liquidity 
on the Warsaw Stock Exchange (WSE) using daily turnover as a liquidity measure. The exist-
ence of an inverted U-shape in the stock turnover across the trading days is examined. The 
research sample covers 2502 daily observations in the period January 2005 – December 2014. 
53 WSE-listed companies divided into three size groups are investigated. In the study the 
OLS method with the HAC covariance matrix estimation and the GARCH-type models are 
employed. The results indicate that liquidity on the WSE tends to be significantly lower on 
Mondays and higher on Wednesdays in comparison with the other days of the week. Howev-
er, the inverted U-shape in daily turnover occurs only among the companies with the largest 
market capitalization. 

K e y w o r d s: market microstructure, day-of-the-week effect, liquidity, turnover, HAC, 
GARCH, Warsaw Stock Exchange.  

J E L Classification: C10, C58, G10, G12. 

Introduction 

 The day-of-the-week patterns in returns and volatility on stocks and 
stock market indices rank among the most common seasonality anomalies. 
There is a growing body of empirical literature on that issue, also for the 
Polish capital market (see e.g. Fiszeder, 2009 and the references therein). On 

                                                 
* Correspondence to: Sabina Nowak, University of Gdansk, Faculty of Management, 

Armii Krajowej 101, 81-824 Sopot, Poland, e-mail: sabina.nowak@ug.edu.pl;  Joanna Olbryś, 
Bialystok University of Technology, Faculty of Computer Science, e-mail: 
j.olbrys@pb.edu.pl. 



Sabina Nowak, Joanna Olbryś 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 49–69 

50 

the contrary, relatively little empirical research has been conducted on the 
day-of-the-week effects in liquidity on equity markets (e.g. Jain, Joh, 1988; 
Foster, Viswanathan, 1993; Chordia et al., 2001; Chordia et al., 2005; 
Hameed et al., 2010; Alrabadi, 2012; Karolyi et al., 2012). 

The goal of this study is to examine day-of-the-week patterns in liquidity 
on the Warsaw Stock Exchange (WSE) using daily turnover as a liquidity 
measure. To address this issue, the OLS method with HAC covariance ma-
trix estimator (Newey, West, 1987) and the GARCH-type models are em-
ployed. The research covers the sample period January 2005  – December 
2014, during which 53 WSE-listed companies divided into three size groups 
are investigated. An inverted U-shape in the stock turnover across trading 
days is examined. This effect means that the trading volume tends to be at its 
lowest on Monday and Friday, while the most active periods are in the mid-
dle of the week (Jain, Joh, 1988). 

To the best of the authors’ knowledge, the empirical results concerning 
day-of-the-week effects in liquidity on the WSE are novel and have not been 
presented in the literature thus far. 

The remainder of the study is organized as follows. Section 1 specifies 
a methodological background and a brief literature review. In Section 2, we 
present and discuss the empirical results of the day-of-the-week patterns in 
liquidity on the WSE. Section 3 recalls the main findings and concludes. 

1. Methodological Background 

1.1. Brief Literature Review 

The presence of calendar anomalies has been investigated extensively 
since the nineteen seventies. The existence of seasonal behavior in returns 
and volatility has been widely documented in the finance literature. Some of 
the fundamental and most broadly citied papers are e.g. (Fama, 1965; 
French, 1980; Gibbons, Hess, 1981; Rogalski, 1984; French, Roll, 1986, 
etc). As the aim of this research is to investigate day-of-the-week patterns in 
liquidity and trading activity, we focus our analysis of previous literature on 
the studies related mostly to that issue. 

Among others, Jain and Joh (1988) studied the trading volume and re-
turns on the New York Stock Exchange (NYSE). They showed significant 
differences in the trading volume within and across days. The authors pro-
vided evidence of an inverted U-shape in volume across days, i.e. Monday 
and Friday had the lowest volume, and the most active periods were in the 
middle of the week. 



Day-of-the-Week Effects in Liquidity on the Warsaw Stock Exchange 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 49–69 

51

In their theoretical research, Foster and Viswanathan (1990) proposed 
a model to explain time-dependent patterns in securities trading. They ana-
lysed inter-day trading where an informed trader and a subset of the liquidity 
traders act strategically. In their model, the informed trader receives infor-
mation each day, but this information becomes less valuable through time, 
because there is a public announcement of some portion of the private in-
formation. The authors predicted a weekend effect in the trading volume and 
return volatility. They showed that the trading volume should be lower on 
Monday than Tuesday, and the trading costs are highest on Monday. In an-
other study, Foster and Viswanathan (1993) introduced empirical tests to 
document changes in the trading volume within and between days. They 
tested the null hypothesis that the trading volume is uniform through time. 
For the interday case, the authors used daily turnover as a measure of trading 
activity. They found that the Monday trading volume is significantly lower 
than the Tuesday and Wednesday trading volume for the most actively trad-
ed firms. These findings were consistent with the predictions of the Foster-
Viswanathan (1990) theoretical model. 

Foerster and Keim (1993) explored the frequency of non-trading for 
NYSE and AMEX stocks, in the sample period 1973–1990. Among other 
results, they found an interesting day-of-the-week pattern: non-trading in-
creased monotonically through the week. 

Chordia et al. (2001) documented strong day-of-the-week effects in trad-
ing activity for the U.S. stock market. They found that Fridays accompanied  
a significant decrease in trading activity and liquidity, while Tuesdays dis-
played the opposite pattern. In another paper, Chordia et al. (2005) investi-
gated the U.S. stock and bond markets and they found distinct seasonal pat-
terns in stock and bond liquidities. Both stock and bond market liquidities 
were higher at the beginning of the week compared with Friday. 

Hameed et al. (2010) used bid/ask spread as a measure of liquidity. They 
regressed the quoted spread of the stocks on a set of variables known to cap-
ture seasonal variation in liquidity. The estimated parameters showed calen-
dar effects in the liquidity measure. 

Alrabadi (2012) investigated day-of-the-week regularities on the Amman 
Stock Exchange (ASE). The author confirmed the significant seasonal pat-
terns in aggregate market liquidity on the ASE but, in contrast to the U.S. 
evidence of Chordia et al. (2001), trading activity reached its minimum in 
the middle of the week and was significantly higher on Thursdays. Probably, 
the contradictory results arose from the nature of the ASE as an emerging 
market. 



Sabina Nowak, Joanna Olbryś 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 49–69 

52 

Examining commonality in liquidity around the world, Karolyi et al. 
(2012) followed the approach taken by Hameed et al. (2010) and they meas-
ured whether fluctuations in liquidity of individual stocks are correlated 
within a country. The authors employed the model with day-of-the-week 
dummies and ran regressions for each stock using two different liquidity 
measures. The residuals of these regressions were treated as the daily inno-
vations in liquidity and they were subsequently used as the endogenous vari-
ables in the monthly regressions. Their coefficients of determination were 
regarded as the measures of commonality in liquidity of individual stocks. 

1.2. Measuring of Liquidity 

 As Lesmond (2005) emphasized, liquidity, by its very nature, is difficult 
to define and even more difficult to estimate. Given the uncertainty sur-
rounding liquidity estimation, some measures are especially often advocated 
in the literature to provide empirical research in liquidity/illiquidity effects 
(e.g. Olbryś, 2014a; 2014b). The popular measures of trading activity, i.e. 
volume, dollar trading volume, and share or market turnover are among 
them. The raw trading volume is the number of shares traded. The stock 
turnover is defined as the ratio of the number of shares traded in a day to the 
number of shares outstanding at the end of the day. It is worthwhile to note 
that using turnover disentangles the effect of firm size from the trading vol-
ume. 

In this research, we compute daily turnover as a measure of liquidity for 
stock i on day d: 

  ,
,

,
,

di

di
di NSO

V
T =  (1) 

where  

diT ,  is the turnover of stock i on day d, 

diV ,  is the trading volume of stock i on day d, 

diNSO,  is the number of shares outstanding at the beginning of the quarter 

for stock i on day d. 

1.3. Econometric Analysis of Day-of-the-Week Effects 

 Many studies investigating day-of-the-week effects employ the standard 
OLS methodology by regressing an endogenous variable on daily dummy 
variables. However, using that methodology has two disadvantages. Firstly, 
errors in the model may be autocorrelated resulting in misleading inferences. 



Day-of-the-Week Effects in Liquidity on the Warsaw Stock Exchange 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 49–69 

53

Secondly, error variances may not be constant over time (Kiymaz, 
Berument, 2003). Hamilton (2008) stresses that even if the researcher’s pri-
mary interest is in estimating the conditional mean, having a correct descrip-
tion of the conditional variance can still be quite important. By incorporating 
the observed features of the heteroskedasticity into the estimation of the 
conditional mean, substantially more efficient estimates of the conditional 
mean can be obtained. 
 To account for daily seasonality, dummy variables are incorporated into 
the model. In order to avoid the dummy variable trap (Foster, Viswanathan, 
1993), one selected dummy is always excluded from the regression. As the 
main aim of the research is to examine an inverted U-shape in the stock 
turnover on the WSE, three different versions of the model are employed. 
The first version excludes the dummy variable for Monday, the second for 
Wednesday, and the third for Friday.  
 Finally, for each stock, daily turnover (1) on day t is composed of a fixed 
effect for Monday (Wednesday, Friday) (0a ), an inter-day adjustment for 
days other than Monday (Wednesday, Friday), and an idiosyncratic error 
term with zero expected value (tε ).  
 The model describing the effect for Monday is given as follows:  

  ∑
=

++=
5

2
,0

j
ttjjt dbaT ε  (2) 

where  
5,4,3,2,1,, =jd tj  are the dummy variables for Monday (j=1), Tuesday 

(j=2), Wednesday (j=3), Thursday (j=4) and Friday (j=5) at time t, 
respectively,  

5,4,3,2,1, =jb j  are the corresponding coefficients.  

 The respective models for Wednesday and Friday effects are also con-
sidered1. 
 The values of the coefficients of the dummy variables are central to test 
for inter-day variations in daily turnover. We initially estimate day-of-the-
                                                 

1 In the first version of the research in order to address the autocorrelation problem the 
lagged values of the variable tT  were included as the explanatory variables in the model (2). 

In order to capture a one week delay, the lag equal to 5 was considered. However, employing 
AR part in formula (2) did not substantially improve the values of the Lagrange Multiplier 
statistics of the Breusch-Godfrey test for autocorrelation of order 5 – the hypothesis of no 
autocorrelation was continually rejected. The OLS estimator was therefore inconsistent. 

 



Sabina Nowak, Joanna Olbryś 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 49–69 

54 

week effects in Eq. (2) by using the OLS method and the robust HAC esti-
mates2. However, the Newey-West corrections may not fully correct for the 
influence problems introduced by the ARCH effect. For this reason, the es-
timation of the day-of-the-week effect model as a GARCH-type model is 
appropriate for this study. To test for the ARCH effect, the test of Engle 
(1982) with the Lagrange Multiplier (LM) statistic is employed. 
 In order to test the presence of seasonality anomalies in stock returns, 
volatility or liquidity, various versions of GARCH-type models (Bollerslev, 
1986) have been applied in the literature (e.g. Choudhry, 2000; Franses, 
Paap, 2000; Berument, Kiymaz, 2001; Kiymaz, Berument, 2003; Apolinario 
et al., 2006; Žikeš, Bubák, 2006; Alrabadi, 2012). In this research, the 
GARCH(p, q) model is utilised. According to the literature, the lower order 
GARCH(p, q), p, q = 1, 2, models are used in most applications (Tsay, 
2010). The GARCH(p, q) models are usually compared and selected by the 
Akaike (AIC) and Schwarz (SC) information criteria3. 

The GARCH(p, q) model, with the excluded Monday dummy variable, is 
given by Eq. (3): 

  

,0,,,1,0,0,,,1,0,0

,

),1,0(~,

0

11

2
0

5

2
,0

≥=≥>=≥>

⋅+⋅+=

=

++=

∑∑

∑

=
−

=
−

=

pplqqk

hh

Nzhz

dbaT

lk

p

l
ltl

q

k
ktkt

tttt

j
ttjjt

KK βαα

βεαα

ε

ε

 (3) 

where  

tε  is the innovation in a linear regression with ,)( 2σε =V   

th  is the variance function,  
and remaining notation like in Eq. (2).  

 Similarly, the GARCH(p, q) model with excluded dummy variable for 
either Wednesday or Friday could be written respectively.  

                                                 
2 HAC – heteroscedasticity and autocorrelation consistent covariance matrix estimation 

(Newey, West, 1987). 
3 When the values of the information criteria AIC or SC for different variants of the 

GARCH(p, q) models are almost equal, the statistical significance of the parameters in the 
conditional mean and conditional variance equation of the GARCH(p, q) model could be 
analysed to choose the appropriate model. 



Day-of-the-Week Effects in Liquidity on the Warsaw Stock Exchange 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 49–69 

55

 The parameters of GARCH(p, q) models are almost invariably estimated 
via Maximum Likelihood (ML) or Quasi-Maximum Likelihood (QML) 
(Bollerslev, Wooldridge, 1992) methods4. 

2. Empirical Results on the Warsaw Stock Exchange 

In this research, a database containing data for the WSE-listed stocks for 
the period from January 2, 2005 to December 30, 2014 was utilised. When 
forming the database, we included only those securities which existed on the 
WSE for the whole sample period since December 31, 2004, and were not 
suspended. The stock daily trading volumes (in items) were obtained from 
the website http://www.gpwinfostrefa.pl. The data of the number of shares 
outstanding are coming from the Notoria Serwis. All companies entered into 
the database (147) were sorted according to their market capitalization at the 
end of each year. Next, the stocks were divided into three size groups based 
on the breakpoints for the bottom 30% (SMALL – 44 companies), middle 
40% (MEDIUM – 59 companies) and top 30% (BIG – 44 companies) (e.g. 
Fama, French, 1993). The companies that remained in the same group during 
the period investigated were selected. Finally, the 53 WSE companies were 
entered into separate, representative groups, specifically: 8 firms into the 
SMALL group, 18 firms into the MEDIUM group and 27 firms into the BIG 
group (Nowak, Olbryś, 2015). We computed daily turnover diT ,  given by 

Eq. (1), providing 2,502 observations for each company. To avoid numerical 
problems, the data were rescaled by multiplying by 104 (Lucchetti, Balietti, 
2014). All calculations were done using Gretl 1.10.1 software (Adkins, 
2014).  

First we detected stationarity of the analysed daily turnover series for 53 
stocks included in the size groups. We employed the ADF-GLS test (Elliott 
et al., 1996) and we proved that the unit-root hypothesis can be rejected for 
all series at 5 per cent significance level. 
 Second, in order to carry out an initial assessment of the existence of an 
inverted U-shape in the stock turnover on the WSE, the graphs showing the 
average level of the stock turnover on each day of the week were created. On 
such basis we noticed the occurrence of an inverted U-shape in the turnover 
across the majority of the analysed big companies. Their level of turnover 
turned out to be the lowest on Mondays, increasing on Tuesdays and the 

                                                 
4 To choose the conditional distribution of innovations, various variants of the model (3) 

were estimated. Unfortunately, they did not yield satisfactory results and even estimation 
failed in many cases. Therefore, the distribution for the innovations is supposed to be normal. 



DYNAMIC ECONOMETRIC 

56 

highest on Wednesdays. However, it subsequently decreased on Thursdays
and finally achieved on Fridays the level close to the level of Mondays. It is 
pertinent to mention that the inverted U
rarely for the medium and small companies. 
average daily turnover 
BOS, BPH, KGH, KTY, LPP, NET, OPL, PKN.
 

Figure 1. An inverted U

 In the next step, 
ters of three versions of 
Monday, Wednesday and Friday,
estimated, comprising
SMALL companies. Due to the existence of five or

Sabina Nowak, Joanna Olbryś 

CONOMETRIC MODELS 15 (2015) 49–69 

highest on Wednesdays. However, it subsequently decreased on Thursdays
and finally achieved on Fridays the level close to the level of Mondays. It is 
pertinent to mention that the inverted U-shape in the turnover occurred very 
rarely for the medium and small companies. In Figure 1, one can observe the 
average daily turnover of the nine selected big companies, namely BDX, 
BOS, BPH, KGH, KTY, LPP, NET, OPL, PKN. 

 

 
 

  
An inverted U-shape in the average daily turnover of selected WSE

step, we employed the OLS method to estimate the 
ersions of the model (2), excluding the dummy variable f

Monday, Wednesday and Friday, respectively. In total, 159 models were 
comprising 81 models for BIG, 54 for MEDIUM and 

companies. Due to the existence of five order serial autocorrelation 

highest on Wednesdays. However, it subsequently decreased on Thursdays 
and finally achieved on Fridays the level close to the level of Mondays. It is 

shape in the turnover occurred very 
one can observe the 

of the nine selected big companies, namely BDX, 

 

 

 
turnover of selected WSE-stocks  

employed the OLS method to estimate the parame-
the dummy variable for 

respectively. In total, 159 models were 
and 24 for 

der serial autocorrelation 



Day-of-the-Week Effects in Liquidity on the Warsaw Stock Exchange 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 49–69 

57

and – in some cases – also the heteroscedasticity of residuals, the  Newey-
West covariance matrix estimator (1987) was employed.  
 The results of the model (2) estimation with the Monday dummy varia-
ble excluded are presented in Tables 2 –3 in Appendix5. In the case of 35 
‘without Monday’ models (comprising 16 models for big, 11 for medium 
and 8 for small companies), the ARCH effect in residuals was detected. 
Therefore, for those 35 companies the GARCH (p, q), p, q = 1, 2, models 
were estimated. The number of lags p, q was selected on the basis of the 
Akaike (AIC) and Schwarz (SC) information criteria. The results of the es-
timation of the ‘without Monday’ model (3) are reported in Tables 4 –6 in 
Appendix6.  

Table 1 presents a brief summary of the major day-of-the-week effects in 
daily turnover on the WSE. 

Table 1. Summarized day-of-the-week effects in daily turnover on the WSE 

Effect in daily 
turnover 

BIG group MEDIUM 
group 

SMALL 
group  

Monday effect BPH, BOS, BDX, BHW, BZW, ECH, GTC, GTN, ING, 
KGH, KTY, LPP, MBK, MIL, MOL, NET, OPL, ORB, 

PEO, PKN, PKO, SNS, STP, TVN 

CNG, MCI, 
MNI, STX 

VST 

MZA 

Wednesday 
effect 

BPH, BOS, BDX, BHW, BZW, GTC, GTN, ING, KGH, 
KTY, LPP, MBK, MIL, NET, OPL, PEO, PKN, PKO, TVN 

MNI, STF, 
STX 

MZA 

Partial inverted 
U-shape 

BPH, BOS, BDX, BHW, BZW, GTC, GTN, ING, KGH, 
KTY, LPP, MBK, MIL, NET, OPL, PEO, PKN, PKO, TVN 

MNI, STX MZA 

Full inverted 
U-shape 

BPH, MIL – – 

Note: The Monday effect means that the level of daily turnover is statistically significantly lower on 
Mondays than on the other days of the week; the Wednesday effect means that the level of daily turnover 
is statistically significantly higher on Wednesdays in comparison with Mondays and/or the other days of 
the week; the partial inverted U-shape in turnover means the presence of the Monday and Wednesday 
effects, but the absence of the Friday effect; the full inverted U-shape in turnover means the presence of 
the Monday, Wednesday and Friday effects at the same time. 

 The obtained estimation results of the models (2) and (3) led to the con-
clusion that the ‘Monday effect’ was the most frequently present in the stock 
turnover time series. In the case of 24 big and 3 medium firms, the level of 
daily turnover was statistically significantly lower on Mondays than on the 
other days of the week7. The ‘Wednesday effect’ (recognized in those cases 
when the level of daily turnover was statistically significantly higher on 

                                                 
5 The estimation results of the models with Wednesday and Friday dummies excluded are 

available upon request. 
6 See footnote 3. 
7 At the 5 per cent level.  



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DYNAMIC ECONOMETRIC MODELS 15 (2015) 49–69 

58 

Wednesdays in comparison with Mondays and/or the other days of the week) 
occurred in the case of 19 big companies. The findings of the ‘Friday effect’ 
existence were ambiguous.  
 The results mentioned above justify the preliminary conclusions from 
analysing the graphs, where a pronounced ‘Monday effect’ accompanied by 
a weaker ‘Wednesday effect’ were detected. Besides, both effects occurred 
more often in the case of big companies. 
 Continuing the analysis of the model (2) estimation results, we observed 
the presence of the so-called ‘partial inverted U-shape’ in daily turnover 
related to the presence of the Monday and Wednesday effect, but not the 
Friday effect, in the case of 19 out of 27 big firms (namely BPH, BOS, 
BDX, BHW, BZW, GTC, GTN, ING, KGH, KTY, LPP, MBK, MIL, NET, 
OPL, PEO, PKN, PKO, TVN). For these companies, the level of daily turn-
over turned out to be either statistically significantly lower on Mondays than 
on Wednesdays and/or the other days of the week, or statistically significant-
ly higher on Wednesdays than on Mondays. Moreover, on the basis of the 
model (3) estimation, we found the ‘partial inverted U-shape’ in daily turno-
ver for 14 companies, including 11 big (BPH, GTC, ING, KGH, MBK, MIL, 
NET, OPL, PEO, PKN, PKO), 2 medium (MNI, STX) and 1 small firm 
(MZA). 
 Furthermore, we did not confirm the occurrence of the ‘full inverted  
U-shape’ in daily turnover of the companies analysed, since in the majority 
of the cases the level on Friday turnover was not statistically significantly 
lower than on the other days of the week. For 17 big companies (BHW, 
BZW, ECH, GTC, GTN, ING, KGH, KTY, LPP, MBK, MIL, NET, OPL, 
PEO, PKN, PKO, TVN) the level of turnover on Fridays was statistically 
significantly higher compared with the level on Mondays. Only for 2 big 
companies (BPH and OPL) the daily turnover level turned out to be statisti-
cally significantly lower on Fridays in comparison with Wednesdays. 
 Barely for 2 companies (BPH and MIL) we can venture the conclusion 
of the existence of the ‘full inverted U-shape’ in daily turnover (involving 
Monday, Wednesday and Friday effect at the same time). In the case of the 
BPH company, on the basis of the model (2) estimation, we noted that the 
turnover was simultaneously: (i) statistically significantly lower on Mondays 
compared with Tuesdays, Wednesdays and Thursdays; (ii) statistically sig-
nificantly higher on Wednesdays compared with Mondays, Thursdays and 
Fridays; (iii) statistically significantly lower on Fridays compared with 
Tuesdays and Wednesdays. Those findings were only partially confirmed by 
the results of the estimation of the model (3), which showed rather the exist-
ence of the ‘partial inverted U-shape’ in daily turnover of the BPH. Con-



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DYNAMIC ECONOMETRIC MODELS 15 (2015) 49–69 

59

versely, in the case of the MIL company, the results of the model (2) estima-
tion proved the occurrence of the ‘partial inverted U-shape’, whereas the 
results of model (3) estimation – the existence of the ‘full inverted U-shape’ 
in the stock turnover.  

Conclusions 

 The main goal of this paper was to explore and to document day-of-the-
week effects in liquidity on the WSE, using daily turnover as a liquidity 
measure. To address this issue, we employed the OLS method with the HAC 
covariance matrix estimation and the GARCH-type models. To account for 
daily seasonality in the turnover, dummy variables were incorporated into 
the models. We investigated 53 WSE-listed stocks from three size groups. 
Our research provided evidence for pronounced Monday and Wednesday 
effects in daily turnover on the WSE, especially in the BIG group. Further-
more, we observed the so-called ‘partial inverted U-shape’ in daily turnover 
in the case of 22 out of 53 firms. Moreover, the graphs showing the average 
daily stock turnover on each day of the week were created and they revealed 
an inverted U-shape in some cases. 
 Although relatively little empirical research has been conducted on the 
day-of-the-week effects in liquidity on stock markets in the world, our find-
ings are rather consistent with the existing literature. In light of our empirical 
results, it seems that the trading volume on the WSE is usually the lowest on 
Mondays, but the most active periods are in the middle of the week. From an 
investor’s point of view it is important that these findings are also in accord-
ance with the investor’s intuition. 
 It is worth stressing that our study situates itself in the broad strand of 
literature concerning commonality in liquidity, which is nowadays the centre 
of attention of many empirical research papers (e.g. Olbryś, 2014a; 2014b; 
Karolyi et al., 2012). Given the importance of the topic, one of the possible 
directions for further investigation could be to examine day-of-the-week 
patterns in liquidity on the WSE following the methodology proposed by 
Franses and Paap (2000) or Žikeš and Bubák (2006). The former authors 
employ the PAR-PGARCH model to investigate the seasonality in the S&P 
500 index, while the latter use the same model to explore daily returns on the 
Central European stock markets. 
  



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DYNAMIC ECONOMETRIC MODELS 15 (2015) 49–69 

60 

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Analiza efektu dnia tygodnia w płynności spółek notowanych  
na Giełdzie Papierów Wartościowych w Warszawie S.A. 

Z a r y s  t r e ś c i. Celem artykułu jest analiza występowania efektu dnia tygodnia w płynno-
ści spółek notowanych na Giełdzie Papierów Wartościowych w Warszawie S.A., z wykorzy-
staniem dziennych wartości względnego wolumenu jako miary płynności. Badaniu poddano 
w szczególności występowanie tzw. efektu odwróconego U w dziennym względnym wolu-
menie 53 spółek, z podziałem na grupy według wartości rynkowej, w okresie od stycznia 
2005 r. do grudnia 2014 r. W badaniu wykorzystano modele OLS-HAC oraz GARCH. Na 



Sabina Nowak, Joanna Olbryś 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 49–69 

62 

podstawie uzyskanych wyników stwierdzono, że dzienny względny wolumen na giełdzie 
warszawskiej jest generalnie istotnie niższy w poniedziałki (tzw. efekt poniedziałku) oraz 
istotnie wyższy w środy (tzw. efekt środy) w porównaniu do pozostałych dni tygodnia. Po-
nadto zaobserwowano, że częściowy efekt odwróconego U występuje głównie w dziennym 
względnym wolumenie spółek o największej kapitalizacji. Efekt ten oznacza jednoczesną 
obecność efektów poniedziałku i środy, przy braku tzw. efektu piątku, czyli spadku dziennego 
względnego wolumenu w piątek do poziomu z początku tygodnia. 

S ł o w a  k l u c z o w e: mikrostruktura rynku, efekt dnia tygodnia, płynność, względny 
wolumen, HAC, GARCH, Giełda Papierów Wartościowych w Warszawie S.A. 

 
 
 



Appendix 

Table 2. Estimation results of model (2), OLS-HAC, Monday dummy variable excluded, the BIG group 

BIG group 

 BPH BNP BOS BDX BZW DBC ECH GTN GTC BHW ING KTY KGH LPP 

2b
 1.196*** 0.093 0.346** 1.590 1.850*** 0.977 1.657*** 2.052*** 2.432*** 1.406* 0.152* 4.653*** 7.685*** 0.470 

3b
 1.802***  –0.010 0.606** 2.466** 2.176*** 1.270* 0.755* 3.415*** 4.006*** 1.306** 0.372** 4.991*** 11.346*** 2.403*** 

4b
 0.804**  –0.042 0.179 1.147 1.512** 1.589* 1.255*** 2.767*** 5.197*** 0.578 0.283** 5.359*** 12.181*** 1.809** 

5b
 0.265  –0.012 0.139 1.376 2.476*** 1.190 1.448*** 4.077*** 4.390*** 1.387** 0.184** 4.926*** 12.150*** 2.483** 

LM 47.640 
[0.000] 

874.406 
[0.000] 

3.037 
[0.694] 

11.683 
[0.039] 

5.826 
[0.324] 

0.489 
[0.993] 

4.562 
[0.472] 

240.258 
[0.000] 

13.985 
[0.016] 

11.028 
[0.051] 

23.055 
[0.000] 

0.051 
[0.999] 

279.072 
[0.000] 

0.748 
[0.980] 

2TR  605.480 
[0.000] 

606.167 
[0.000] 

248.992 
[0.000] 

198.401 
[0.000] 

580.037 
[0.000] 

128.216 
[0.000] 

231.766 
[0.000] 

764.311 
[0.000] 

589.677 
[0.000] 

206.991 
[0.000] 

391.061 
[0.000] 

108.312 
[0.000] 

661.087 
[0.000] 

85.203 
[0.000] 

W  13.352 
[0.010] 

2.752 
[0.600] 

5.947 
[0.203] 

3.715 
[0.446] 

5.378 
[0.251] 

3.969 
[0.410] 

6.425 
[0.170] 

6.298 
[0.178] 

4.444 
[0.349] 

2.386 
[0.665] 

3.189 
[0.527] 

4.435 
[0.350] 

5.903 
[0.207] 

5.023 
[0.285] 

 



 

 

Table 2 cont. Estimation results of model (2), OLS-HAC, Monday dummy variable excluded, the BIG group 

BIG group 

 MBK MIL MOL NET ORB PEO PKN PKO STP SNS OPL TVN ZWC 

2b
 1.664*** 1.979*** 0.034 5.087*** 0.409 3.884*** 4.587*** 3.230*** 0.553 1.529** 3.973*** 1.993*** 0.033 

3b
 2.834*** 2.565*** 0.101* 5.478*** 1.248 4.791*** 6.311*** 5.246*** 0.420 0.824* 6.573*** 2.052*** 0.013 

4b
 2.181*** 2.762*** 0.197*** 4.224*** 2.971** 4.651*** 7.472*** 5.610*** 1.000** 0.214 4.668*** 2.001*** 0.000 

5b
 2.308*** 2.018*** 0.168* 3.644*** 1.683 4.363*** 7.437*** 5.262*** 0.271 3.078 3.967*** 4.161*** 0.009 

LM 184.953 
[0.000] 

33.166 
[0.000] 

3.164 
[0.675] 

16.050 
[0.007] 

2.372 
[0.796] 

542.272 
[0.000] 

74.756 
[0.000] 

471.023 
[0.000] 

16.104 
[0.007] 

0.008 
[1.000] 

36.749 
[0.000] 

0.851 
[0.974] 

195.769 
[0.000] 

 727.798 
[0.000] 

277.598 
[0.000] 

298.113 
[0.000] 

537.436 
[0.000] 

46.544 
[0.000] 

616.530 
[0.000] 

580.538 
[0.000] 

922.911 
[0.000] 

147.017 
[0.000] 

79.600 
[0.000] 

390.676 
[0.000] 

484.002 
[0.000] 

461.129 
[0.000] 

W  9.385 
[0.052] 

5.842 
[0.211] 

6.865 
[0.143] 

4.434 
[0.350] 

3.801 
[0.434] 

5.465 
[0.243] 

14.533 
[0.006] 

9.897 
[0.042] 

2.515 
[0.642] 

4.168 
[0.384] 

3.358 
[0.500] 

5.732 
[0.220] 

2.572 
[0.632] 

Note:  5432  , , , bbbb  – the estimates of the model (2) coefficients using the OLS method with the HAC covariance matrix estimator (Newey, West, 1987); ** 

(***, *) – indicates statistical significance at 5 per cent (1 per cent, 10 per cent) significance level; LM – the Lagrange Multiplier statistic of the Engle’s test 

(1982), order lag equal to 5; 2R  – the adjusted 2R , 2TR  – the Lagrange Multiplier statistic of the Breusch-Godfrey test for autocorrelation of order 5; W – 
the White statistic; relevant p-values in brackets under the estimates. 

2TR



 

 

Table 3. Estimation results of model (2), OLS-HAC, Monday dummy variable excluded, the MEDIUM group 

MEDIUM group 

 ALM AMC ATM ATG COL IPL IND LTX MCI MNI CNG PEK SKA STX 

2b
 0.683  –0.161  –0.038 0.787 0.310  –0.030 1.632 0.237 4.375** 0.949 3.768** 0.191 2.644* 0.003 

3b
 1.331 0.199 0.124 0.144 0.024 1.688 0.684  –0.348 4.873* 4.299 3.331* 1.162 0.689 0.301 

4b
 0.128 1.252 1.229  –0.098 0.272 1.647 2.898*  –1.140 7.143** 3.708 4.117** 1.011 1.906* 1.630 

5b
 0.672 0.379  –0.456 1.016 0.758 1.191 0.720  –1.106 2.718 2.797 2.487* 0.353 1.138 0.978 

LM 21.847 
[0.001] 

67.675 
[0.000] 

41.616 
[0.000] 

1.936 
[0.858] 

84.352 
[0.000] 

4.026 
[0.546] 

4.568 
[0.471] 

569.325 
[0.000] 

501.096 
[0.000] 

438.636 
[0.000] 

7.008 
[0.220] 

38.001 
[0.000] 

4.068 
[0.540] 

692.530 
[0.000] 

2TR  298.859 
[0.000] 

551.466 
[0.000] 

205.035 
[0.000] 

293.065 
[0.000] 

664.712 
[0.000] 

305.668 
[0.000] 

107.632 
[0.000] 

1245.440 
[0.000] 

1175.298 
[0.000] 

964.227 
[0.000] 

269.080 
[0.000] 

314.258 
[0.000] 

52.412 
[0.000] 

1189.473 
[0.000] 

W  2.351 
[0.671] 

0.975 
[0.914] 

2.566 
[0.633] 

2.921 
[0.571] 

3.700 
[0.448] 

3.811 
[0.432] 

6.533 
[0.163] 

0.903 
[0.924] 

2.354 
[0.671] 

1.316 
[0.859] 

3.668 
[0.453] 

2.186 
[0.702] 

5.264 
[0.261] 

1.610 
[0.807] 

 



 

 

Table 3 cont. Estimation results of model (2), OLS-HAC, Monday dummy variable excluded, the MEDIUM and SMALL 
groups 

MEDIUM group SMALL group 

 STF TIM VST PUE APL BDL EFK ENP KMP MZA PLA SME 

2b  2.262* 1.997 4.450** 1.426 17.500  –0.062 3.036*  –3.250 1.984 8.368*  –8.366  –0.459 

3b   –0.382 1.979* 2.782 3.657* 12.106 0.176 1.021 6.769 1.071 12.335  –3.698 0.978 

4b  0.718 2.126 5.249 3.567 9.275 0.033 2.840  –3.308  –2.865 11.356  –5.348 0.985 

5b   –1.033 0.104 0.689 1.011  –2.722 0.269 2.103  –2.614  –7.695* 5.771  –1.797  –0.155 

LM 426.069 
[0.000] 

0.153 
[0.999] 

613.836 
[0.000] 

0.173 
[0.999] 

279.249 
[0.000] 

1245.050 
[0.000] 

1029.52 
[0.000] 

380.345 
[0.000] 

665.594 
[0.000] 

619.648 
[0.000] 

565.978 
[0.000] 

929.439 
[0.000] 

2TR  777.058 
[0.000] 

58.392 
[0.000] 

1106.698 
[0.000] 

34.729 
[0.000] 

987.168 
[0.000] 

1159.090 
[0.000] 

1095.110 
[0.000] 

1135.479 
[0.000] 

1153.429 
[0.000] 

1224.741 
[0.000] 

1196.819 
[0.000] 

1370.149 
[0.000] 

W  3.111 
 [0.539] 

3.244 
[0.518] 

3.694 
[0.449] 

2.987 
[0.560] 

4.153 
[0.386] 

1.408 
[0.843] 

1.450 
[0.835] 

1.977 
[0.740] 

3.193 
[0.526] 

3.555 
[0.470] 

3.043 
[0.551] 

0.498 
[0.974] 

Note: See Table 2 for explanation. 



 

 

Table 4.  Estimation results of model (3), GARCH(p, q), Monday dummy variable excluded, the BIG group 

BIG group 

 BPH BDX GTN GTC ING KGH MBK MIL NET PEO PKN PKO STP OPL ZWC 

Conditional mean equation 

2b
 

0.81 5.61** 1.19* 1.84 0.11 6.05*** 0.88 1.73** 1.88 3.60*** 4.10*** 2.53*** 0.81 5.86*** 0.07*** 

3b
 

1.30** 1.83 2.06*** 3.70*** 0.26*** 10.30*** 1.85*** 6.67*** 8.38*** 4.20*** 5.77*** 3.97***  –0.16 5.80*** 0.02** 

4b
 

1.46** 3.65 1.11 2.99*** 0.15 11.73*** 2.50**  –0.85 6.53*** 3.61*** 6.77*** 4.17*** 0.26 5.04*** 0.01 

5b
 

2.59  –0.55 2.83*** 3.57 0.16** 10.31*** 2.21*** 0.93 4.68** 4.05*** 6.80*** 3.73***  –0.16 5.17*** 0.10*** 

Conditional variance equation 

 
7.09** 12.99 9.19* 2.49 0.002 18.04*** 0.05 4.43* 6.04 1.03* 117.67*** 1.11 7.88 64.49 0.002* 

 
1.07 1.25*** 0.12*** 0.61*** 1.02*** 0.29*** 0.63*** 1.12*** 0.93 0.25*** 0.41*** 0.57*** 0.36 0.44** 0.15** 

 
–  –1.06*** –  –0.56***  –0.98***  –0.24***  –0.63***  –1.01***  –0.88*  –0.21*** –  –0.55*** – – 5.10*** 

 
0.44*** 0.87*** 0.86*** 1.22*** 0.98*** 0.93*** 1.11*** 0.92*** 1.08*** 1.02*** 0.21 0.98*** 0.02 0.55*** 0.04* 

 
– – –  –0.27* – –  –0.12* –  –0.11  –0.07 – – 0.60 – 0.10*** 

LL  –8207  –10416  –10081  –10413  –3702  –11568  –8918  –9601  –11323  –9286  –10357  –9822  –8045  –10613 206 

Note: The variance-covariance matrix of the estimated parameters of the model (3) is based on the QML algorithm (Bollerslev, Wooldridge, 1992); the distri-
bution for the innovations is supposed to be normal; ** (***, *) – indicates statistical significance at 5 per cent (1 per cent, 10 per cent) significance level.  

0α

1α

2α

1β

2β



 

 

Table 5.  Estimation results of model (3), GARCH(p, q), Monday dummy variable excluded, the MEDIUM group 

MEDIUM group 

 ALM AMC ATM COL LTX MCI MNI PEK STX STF VST 

Conditional mean equation 

2b
 

0.356 0.915 1.307 0.146 2.545 1.344 4.570 1.322  –1.637 2.092** 0.436 

3b
 

 –1.222 2.788 1.678*  –0.723 0.246 2.157 5.974 0.843 9.686***  –0.127  –0.013 

4b
 

0.202 3.672 1.873**  –0.022  –1.214 0.025 13.307 1.051  –0.607 2.837***  –0.067 

5b
 

0.293 0.984 0.182  –0.166  –0.887 0.825 1.569 0.772 1.870 1.392 0.755 

Conditional variance equation 

 
22.65*** 300.9*** 1.773 46.40** 3.142* 1.647 17.07*** 8.132*** 20.584 8.344 0.102 

 
0.663 0.760** 0.020* 0.488*** 1.033*** 0.575*** 1.002*** 0.058*** 2.405** 0.118* 0.120 

 
 –0.529 – 0.197** –  –0.919***  –0.548***  –0.930*** – – – 0.771 

 
0.854***  –0.006 0.019  –0.009*** 0.916*** 1.183*** 0.944***  –0.009**  –0.001 0.349*** 0.020 

 
– 0.278** 0.862*** 0.203 –  –0.208 – 0.929*** 0.389** 0.529*** 0.711 

LL  –10551  –11844  –9794  –8995  –9651  –12393  –12368  –10248  –11052  –10560  –11631 

Note: See Table 4 for explanation. 

0α

1α

2α

1β

2β



 

 

Table 6.  Estimation results of model (3), GARCH(p, q), Monday dummy variable excluded, the SMALL group 

SMALL group 

 APL BDL EFK ENP KMP MZA PLA SME 

Conditional mean equation 

2b
 

0.867  –0.104*  –0.471 0.370  –0.679 2.822**  –2.956  –1.948 

3b
 

1.454  –0.170**  –0.876 0.449 0.394 4.426***  –1.930 0.203 

4b
 

1.094 0.040  –0.365 3.465* 0.906 1.736  –8.793*** 8.155*** 

5b
 

 –0.097  –0.018  –0.267 2.289* 0.678 1.725  –4.354**  –1.315 

Conditional variance equation 

 
4.418 0.009 0.536* 2.309** 1.141* 1.806 9.578** 60.21 

 
2.724*** 1.906** 0.712** 0.879*** 0.993*** 1.021*** 1.544*** 0.877** 

 
 –0.810** –  –0.639**  –0.855***  –0.847** –  –1.063*** 1.814 

 
0.302** 0.017 0.893*** 0.979*** 0.916*** 0.042 0.815*** 0.029 

 
0.322*** 0.478*** 0.057 – – 0.639*** – 0.193 

LL  –14474  –3394  –10205  –12904  –12458  –12211  –12636  –12340 

 Note: See Table 4 for explanation. 

 

 

0α

1α

2α

1β

2β




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