Calendar Effects in the Stock Markets of Central European Countries

The efficient market hypothesis suggests that there are no  opportunities to  gain above‐normal profits using available information, because it is all reflected in the prices. However, calendar anomalies are found to contradict the efficient market hypothesis and enable investors to predict prices during specific days. Based on a review of papers on market efficiency and mar‐ ket anomalies, this paper examines and compares calendar effects known as ‘the month‐of‐the year effect’ and ‘the day‐of‐the‐week effect’ between the stock markets of three Central European countries: Poland, Hungary and the Czech Republic. The study has revealed the presence of calen‐ dar anomalies in the indexes representing small‐cap stocks listed on the Polish stock market and, to some extent, in the indexes used in the Hungarian and Czech stock markets.


Introduction
Market efficiency is one of the founding concepts of modern financial theory. According to the efficient market hypothesis (EMH), stock returns in an efficient market reflect all information available to investors, so in the long run they cannot beat the market to earn above-average returns (Fama, 1970: 383-417). However, some studies, for example, Gultkein (1983), Reinganum, Shapiro (1987: 281-295), have shown irregularities in the behaviour of stock prices which the EMH fails to explain and which the economic literature calls "anomalies". The probability of such anomalies occurring and their relative unpredictability largely determine the efficiency of investors' strategies, so it seems important to find the answer to the following question about the nature of calendar anomalies: Are they temporary irregularities or, perhaps, a sign of market inefficiencies?
In this paper, calendar anomalies known as the month-of-the-year effect and the day-of-the week effect are analysed and compared among the stock markets of three Central European countries: Poland, Hungary and the Czech Republic. One of the purposes of statistical analysis is to establish whether introducing control variables, such as: S&P500 index quotes, prices of gold, yields on 10-year Treasury bonds (YTM) and USD/PLN, USD/CZK and USD/HUF exchange rates, into regression models will offset or amplify the effect of the identified anomalies.
There are several features of our research that distinguish it from other studies. Firstly, we compare calendar anomalies among different emerging market countries. Most studies on market efficiency focus on developed economies, some emerging markets have also been tested for calendar anomalies, however, very few papers compare calendar anomalies among different emerging market countries. Secondly, we investigate how the financial crisis in 2007 impacted the occurrence of calendar anomalies. Thirdly, we introduce control variables to the testing of calendar effects. Last but not least, we explore whether calendar anomalies are still present in the markets, because some literature suggests that in recent years calendar effects have become less prominent and may be disappearing.
The paper is organised as follows. Section one gives an overview of the selected calendar anomalies and explains the efficient market hypothesis. Section two describes the research sample. Section three presents the research methodology, and in section four (last) the research results and conclusions are discussed. thereby tax liabilities, by selling the underperforming stocks. Then, in January, investors readily seize opportunities to purchase low-priced stocks and so their prices go up (Szyszka, 2009: 166).
Research shows, however, that the January effect is also present in countries without income tax, such as Japan (Kato, Schallheim, 1985: 243-260), and in countries where the end of the tax year and the calendar year fall on different dates coincide, for instance, in the UK and Australia (Reinganum, Shapiro, 1987: 281-295). This casts serious doubts on whether the hypothesis contributing the January effect to investors selling off in December for tax reasons holds for the stock markets of these countries. Globalisation processes and strong linkages between capital markets seem to be a more probable explanation.
Another interesting explanation of the January effect refers to the fact that the portfolio managers' compensation depends on the performance of their portfolios measured against a suitably selected benchmark index. Because better performance means higher fees, managers shed the worst-performing stocks at the year-end to maximise them. Then, in January next year, they replenish their portfolios with the stocks of risky companies, offering higher expected rates of return.
The explanation of the January effect that Kinney and Rozeff came up with attributes it to the effect of new information disclosed by companies in late December on investors' buying and selling decisions in January, and consequently on the prices of stocks (Kinney, Rozeff, 1976: 379-402).
Another interesting calendar anomaly has to do with the observation that on some trading days stock returns are consistently and repetitively different than on other days. Early on it was called the day-of-the-week effect but then the term the weekend effect became more popular. Studies of the US stock market revealed that Monday stock returns were, on average, lower than on other days (Higgins, Howton, Perfect, 2000: 19). Accordingly, the regularity was called the Monday effect, but new studies showed that Monday was not unique in that respect.
French, who compared the daily stock returns for companies comprising the S&P500 index in the years 1953 through 1977 (French, 1980: 57), found them to be higher on Mondays and Fridays. The regularity was correspondingly called 'the effect of the weekend'. Having studied the S&P500 and DJIA companies, Smirlock and Starks (1986: 197) concluded that the reason for negative rates of return on Mondays was related to falls in stock prices between the afternoon trading session on Friday and the commencement of trading on a Monday morning, but they failed to reach consensus over what was the cause of the phenomenon. Abnormal stock returns have been observed not only on Mondays and Fridays, but also on other days of the week.
Calendar anomalies have been studied also in the Polish capital market. Keller (2015: 69-79) studied the effect-of-the-day in the Polish stock market. In the analysis, he verified negatively the occurrence of the effect of weekdays, although small market-cap companies show certain tendencies in terms of the effect-of-the-week. His analysis showed large inconsistencies of the results, both in the case of the correlation analysis and the regression analysis. Grotowski (2008: 55-75) studied four calendar anomalies in the Polish stock market. He did not find evidence for the holiday effect and end-of-month effect, however, his analysis shows that there is the Thursday effect and Friday effect. He also found the January effect to be present in small-and medium-cap companies. The January rate of return was about 4% higher than the rates of return for the remaining months. Ślepaczuk (2006: 1-12) presented basic anomalies of the capital market, described both in the Polish and world literature. Fiszeder and Kożuchowska (2013: 217-229) used permutational tests and GARCH models. The results indicated the occurrence of the turn-of-the-month effect, no seasonal fluctuations and very weak weekly fluctuations. Although significantly positive rates of return were observed on Mondays, and significantly lower on Wednesdays, it was only for the WIG20 index, and only at the significance level of 0.1. However, none of these authors has convincingly answered the following question: "Are there calendar anomalies on the Warsaw Stock Exchange?".
Among authors who have investigated calendar anomalies in emerging stock markets are Tonchev and Kim (2006: 1035-1043. They studied the Czech Republic, Slovakia and Slovenia to investigate whether calendar effects are present in the newly developing financial markets. Out of the five calendar effects examined (the day of the week effect, the January effect, the half-month effect, the turn of the month effect, and the holiday effect), very weak evidence has been found for these calendar effects in the three countries, and the effects have different characteristics in different stock markets.
Calendar anomalies were tested after the 2007 crisis. Jayaraman, Muruganandan and Santhi (2017: 26-30) tested anomalies in Brazil, Russia, India and China in three sub-periods: the pre-financial crisis period (2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007), the financial crisis period (2008)(2009) and the post-crisis period (2010)(2011)(2012)(2013)(2014)(2015)(2016). Regression results show that after the crisis BRIC capital markets reached the efficient stage where day of the week trading rules lose the ground to earn the abnormal return. This could be attributed to changes in the capital markets regulations and vigilance of the stock market. Gajdošová, Heryán and Tufan (2011) analysed the day of the week effect in the European emerging markets (Czech, Hungarian, Polish, Slovak and Turkish stock markets) in the period from 2005 to 2010. The results show that anomalies appeared only during the financial crisis. Moreover, Marquering, Nisser and Valla (2006) point to strong evidence that the weekend effect and the January effect disappeared after the information about the occurrence of those anomalies had been published.
Calendar effects have attracted a large number of scholars, but results have often been mixed. The EMH, particularly in its semi-strong and strong forms, is often strongly criticised (Rossi, 2015). Despite the existence of great empirical and theoretical research papers, there is no clear picture whether calendar anomalies exist on a given stock exchange, if they are constant in time, and what circumstances influence them. The bulk of studies on market efficiency focus on the US market and markets in developed countries. Some of the emerging markets have also been tested for the presence of calendar anomalies. However, very few papers compare calendar anomalies among different emerging market countries. It is also worth investigating how the financial crisis in 2007 impacted the occurrence of calendar anomalies. The crisis had a different impact on emerging markets and developed countries. Even among emerging countries there were differences. For example, in the Czech Republic and Hungary there was a negative economic growth rate, whereas in Poland economic growth remained positive. Some literature, for example, Marquering, Nisser and Valla (2006), suggests that in recent years calendar effects have become less prominent and may be disappearing. The reason for this may be found in the crisis itself or perhaps in new regulations introduced to the economies after the crisis. The main research question in our paper is whether the best know anomalies (the month-of-the-year effect and the day-of-the week effect) still occur in markets after the crisis of 2007 or if they have become less prominent or have disappeared. In our research, we also have introduced control variables to the testing of calendar effects to check whether the detected anomalies disappear or gain significance if control variables are included.

Research sample
Indexes: WIG, mWIG40, sWIG80 (Warsaw), PX (Prague) and BUX (Budapest) were analysed for the presence of the day-of-the-week effect and the month-of-the-year effect. WIG and BUX are total return indexes, which means they take into consideration the price fluctuations of the components of the index as well as dividends that companies pay. PX, mWIG40 and sWIG80 are price indexes, which means they calculate only the changes in the price of the index components. We used daily returns (around 2600 observations) and monthly returns (126 observations) from a period spanning from 1 January 2008 to 30 June 2018. The historical WIG, mWIG40 and sWIG80 prices were sourced from http://www.biznesradar.pl and https://www.gpw.pl (accessed on 1 July 2018), and PX and BUX prices from https:// stooq.pl (accessed on 1 July 2018). Descriptive statistics and the number of observations for each index are reported in Table 1. For control variables, we also used daily returns (around 2600 observations) and monthly returns (126 observations) from a period spanning from 1 January 2008 to 30 June 2018. The data were derived from https://stooq.pl (accessed on 1 July 2018).

Research method
In the study, we analysed the Polish, Czech and Hungarian capital markets and compared for the presence of two calendar anomalies: the month-of-the-year effect and the day-of-the-week effect. The focus of the analysis was on the following stock indexes: 1) the Warsaw Stock Exchange (WSE) -WIG -Total Return index, mWIG40 and sWIG80 -both Price Return indexes, 2) the main index of the Prague Stock Exchange -PX -a Price Return index, 3) the main index of the Budapest Stock Exchange -BUX -a Total Return index.
The statistical analysis of both anomalies was carried out using mainly linear regression models, and additionally non-parametric testing.
For the linear regression, we calculated logarithmic rates of return, then assessed the probability of calendar effects being present by estimating parameters of regression models. Our approach is similar to that of French (1980), Junkus (1986), Grotowski (2008), Gajdošová, Heryán and Tufan (2011). The index prices were converted into logarithmic rates of return using the following formula: 1 ln , where r t -the logarithmic rate of return, P t -the index price at the end of the trading day t, P t-1 -the index price at the end of the trading day t -1. The testing for the day-of-the-week effect involved the use of the following model: where r t-1 is the logarithmic rate of return on the previous day, i t d is a dummy variable, representing consecutive trading days (Monday through Friday) and taking the value of "1" when t is Monday and "0" for the other four days. Hence, 1 t d stands for Monday, 2 t d for Tuesday, etc. The model was estimated using the ordinary least squares method (OLS). The model employed to determine whether the month-of-the-year effect was present in the selected stock markets was as follows: d is a dummy variable representing consecutive months of the year (January through December) and taking the value of "1" when t is January and "0" for each consecutive month. Hence, 1 t d denotes January, 2 t d February, etc. This model, too, was estimated by the OLS.
Testing for the presence of a calendar effect basically comes down to estimating the statistical significance of the β i coefficient for a given dummy variable.
To be estimable by the OLS method, a model has to meet a number of restrictive assumptions about the distribution of random term ε t , including the absence of autocorrelation. Because the autocorrelation of the time series of daily stock returns is a frequent problem in statistical analysis, an independent variable represented by a one period lagged-return was introduced as a precaution.
Both the day-of-the-week effect and the month-of-the-year effect were subjected to the Kruskal-Wallis test, a non-parametric alternative to one-way ANO-VA enabling the comparison of three or more samples. The Kruskal-Wallis test is applied when ANOVA assumptions are not met or when the nature of variables prevents its use.
In modelling economic phenomena, the model's robustness is a critical piece of information, because models that are insufficiently robust may misrepresent the mechanisms underlying the phenomenon under consideration and lead to incorrect conclusions and decisions. The robustness of both models used in the study was checked against four control variables: 1) X 1 -the US stock index S&P500, 2) X 2 -the exchange rate between the domestic currency and the US dollar, the exchange rates are: USD/PLN, USD/CZK and USD/HUF, 3) X 3 -the yield on 10-year Treasury bonds in each country, 4) X 4 -gold futures price (New York Mercantile Exchange and Commodity Exchange -Comex).

Research results
The linear regression results for the day-of-the-week effect are shown in Tables  2-6 consisting of five panels each. Tables 7-11 show the linear regression results for the month-of-the-year effect. In these tables, Panel I contains the values of β i coefficients for models without control variables and Panels II-V for models with X 1 , X 2 , X 3 and X 4 control variables, respectively. Table 12 presents the Kruskal-Wallis test statistics for both effects. The values of β i coefficients on particular days of the week calculated for the WIG and mWIG40 indexes are not statistically significant, but in the case of the sWIG80 they are significant for Monday and Friday (p-value = 0.012 and 0.027 respectively). Monday returns on this index are, on average, lower than on other trading days, and on Fridays they are higher. The significance of the β i coefficients for the sWIG80 does not change after the inclusion of the control variables (α = 0.05).
All three WSE indexes are statistically significantly influenced by the S&P500 index and the USD/PLN exchange rate. Their effect is positive (S&P500) and negative when the Polish currency is depreciating (USD/PLN). The analysis also shows that the WIG and mWIG40 indexes are sensitive to changes in T-bond yields.
The average return growth rate for the PX is lower on Tuesday compared to other days of the week. The β i coefficients for daily returns are not statistically significant, but Tuesday returns become significantly different from zero (α = 0.05) following the inclusion of the control variables. The S&P500 and the USD/CZK exchange rate have a significant effect on the PX, like in the case of the Polish indexes. The S&P500 increases returns on the PX and a depreciating Czech koruna decreases them.
The β i coefficients for variables representing individual daily returns on the Hungarian stock exchange are not statistically significant either, but the inclusion of the control variables causes the β i coefficient for 'Monday' to become statistically significant (α = 0.1). The Monday returns on the BUX are, on average, higher than those noted on other trading days. Interestingly, of all indexes studied, only the BUX is statistically significantly influenced by all control variables (α = 0.001).
Overall, the analysis does not provide grounds to conclude that the day of the week effect is present in the main indexes of the analysed stock markets, with the exception of the sWIG80, which generates statistically significantly lower returns on Mondays and higher on Fridays. Statistically different returns on the PX and the BUX (both made up of fewer companies than the Polish WIG) occur on Tuesday (PX) and Monday (BUX). The effect of the S&P500 and domestic currencies/ USD exchange rates on the analysed indexes is considerable.
The linear regression results for the month-of-the-year effect are shown in Tables 7-11, which also consist of five panels each. The β i coefficients in Panel I were calculated for each month's returns without the control variables. Their effect is accounted for in the statistics presented in Panels II-V. The monthly returns on the broadest-based Polish index, WIG, are generally not significantly different from zero, except for July when they are positive (α = 0.1), but only until the control variables are included. The S&P500 and the USD/PLN exchange rate's effect on monthly returns is statistically significant.
Monthly returns on the mWIG40 are statistically significantly lower (α = 0.1) in June than in other months, and this does not change after the inclusion of the control variables.
Monthly returns yielded by the sWIG80 (the small-cap companies) are significantly different from zero in January and June, and their β-coefficients remain statistically significant (α = 0.05), even after the inclusion of the control variables. Higher returns in January are typical of the January effect. In June, stock returns are lower than in other months. The sWIG80 is statistically significantly influenced by the S&P500.
Stock returns on the Czech stock market are, on average, lower in June and September implying the presence of the month-of-the year effect. The β-coefficients for the monthly returns are significantly different from zero even after the control variables are included. The July's β i is also statistically significant, but only until the inclusion of the control variables. As in the case of the Polish indexes, the S&P500 is the only control variable to have a statistically significant effect on the PX.
The β-coefficients on the monthly returns on the Hungarian BUX become statistically different from zero only under the influence of the control variables. January returns are, on average, higher than in other months (α = 0.05), implying the presence of the January effect, but the conclusion may be premature because the effect is not discernible until the control variables are included. As far as the control variables are concerned, a significant effect on the BUX is exerted by the US S&P500 index and the USD/HUF exchange rate.
The foregoing analysis offers the following conclusions. The Polish sWIG80 index and, to some extent, the Hungarian BUX index exhibit the January effect. Returns generated by the Czech PX index and the Polish mWIG40 and sWIG80 indexes are, on average, lower in January than in other months. The Czech stock market is the only one to generate statistically significantly different returns in September. In all the three countries, the S&P500 index and exchange rates against the US dollar influence stock index returns (excluding the PX). Table 12 contains the results of the Kruskal-Wallis test for the day-of-the-week and month-of-the-year effects. As it can be seen, statistically significant differences between returns were only obtained for the sWIG80 tested for the 'day-of-the-week-effect', namely between Monday and Friday, Tuesday and Friday, Wednesday and Friday. The validity of this result is indirectly supported by the regression analysis, which shows that the Monday and Friday rates of return for this index are significantly different from zero. For the other indexes, no statistically significant differences were found.
Of the three WSE indexes studied, positive returns on Fridays and negative on Mondays, pointing to the presence of the day-of-the week-effect, were only found for the sWIG80. The result was confirmed by both the linear regression analysis and non-parametric testing. A tendency for stock prices to drop between Friday and Monday was also reported by French (1980) for the US stock market. One of the explanations of this weekend effect is connected with the new information reaching investors during the weekend, however, it was not the subject of our study. The behaviour of the Polish stock market is also well explained by the results of earlier reports which attribute the presence of calendar effects to changes in small-cap returns such as Hull, Mazachek and Ockree (1998). However, the day-of-the-week-effect was not found to be present in the WIG and mWIG40, which confirms the research conducted by Patev (2003), who studied calendar anomalies for the main market indexes in the Central European stock markets. In his work, he indicates that the Polish indexes do not show effects related to the days of the week, but the significance of individual days in other markets can be indicated.
The PX yielding lower returns on Tuesdays and the BUX positive returns on Mondays partially confirmed weak presence of the day-of-the-week effect on the Prague and Budapest stock markets.
As far as the month-of-the-year effect is concerned, the sWIG80 returns were positive in January and negative in June. The presence of January effect on small market companies is consistent with Grotowski (2008). The Czech PX generated statistically significantly lower returns in February, June and September. The January effect was found in the Hungarian stock market, which generated the biggest above-normal returns in January among the studied markets.
The occurrence of the January effect in the Polish and Hungarian stock markets was reported also by Asteriou and Kavetos (2006). Their results supported the existence of seasonal effects, particularly the January effect, in Poland and Hungary, but not in the Czech Republic, stronger evidence (in terms of statistical significance) was seen for the cases of Hungary and Poland. The January effect may have two probable explanations. One assumes that investors sell off underperforming stocks in December to reduce their tax bill and buy stocks again in January. The other explanation holds that the effect may be caused by portfolio managers who shed 'loser stocks' at the year-end to maximise their fees (which depend on the performance of their portfolios) and in January readily buy riskier stocks with higher expected rates of return. The likely cause of the June anomaly (negative returns) is investors selling part of their portfolios for the summer to have more time for themselves or simply to have the money for summer holidays. Whatever the reason, the increasing supply of stock drives stock prices upwards.
The analysis of the effect of the control variables on the selected indexes shows that all of them are sensitive to the US S&P500 index and exchange rates against the US dollar, and partially to yields on 10-year Treasury bonds.
The obtained results indicate that some anomalies are still present in the stock markets of the three emerging countries in the period after the financial crisis.

Conclusions
This paper tested for the existence of calendar effects in the stock markets of Central European countries: Poland, Hungary and the Czech Republic. The stock markets of all the three countries analysed in the study exhibit calendar anomalies such as the day-of-the-week effect and the month-of-the-year effect in the period after the financial crisis. The day-of-the-week (Friday and Monday) was found for the sWIG80 in Poland, and to some extent for the BUX (Monday) in Hungary and the PX (Tuesday) in the Czech Republic. However, statistical significance was weaker for the Hungarian and Czech Republic stock markets. The month-of-the-year effect was found in all the three studied markets, mainly the January effect in the case of Poland and Hungary. Our conclusion requires several comments, though. Firstly, it only holds for the selected time period. Secondly, it would have been more convincing had the stocks of individual companies been used rather than stock market indexes. Lastly, the classical linear regression analysis requires a number of restrictive assumptions (e.g. concerning the presence of normal distributions, the absence of autocorrelation, etc.) to be fulfilled, therefore the authors in their upcoming paper plan to use the GARCH models.