Excerpt

## Abstract

*In this paper we examine the calendar anomalies in the stock market index of Athens. Specifically we examine the day of the week and the month of the year effects, where we expect negative or lower returns on Monday and the highest average returns on Friday for the day of the week effect and the higher average returns in January, concerning the January effect. For the period we examine we found insignificant returns on Monday, but significant positive and higher average returns on Friday. Also our results are consistent with the literature for the month of the year effect, where we find the highest average returns in January. Furthermore we estimate with ordinary least squares (OLS) and symmetric and asymmetric Generalized Autoregressive Conditional Heteroskedasticity (GARCH) rolling regressions and we conclude that the week day returns are not constant through the time period we examine but are changed. Specifically, while in the first half-period of the rolling regression there are negative returns on Mondays so we observe the day of the week effecting, in the last half-period of the rolling regression Friday presents the highest returns, but the lowest returns are reported on Tuesday and not on Monday, indicating a change shift in the pattern of the day of the week effect. Full programming routines of rolling regressions in EVIEWS and MATLAB software are described.*

## 1. Introduction

Many studies and researches have been made in calendar effects. One of them is the study of Aggarwal and Tandon (1994) test the day-of-the week found that Monday returns are negative in thirteen countries, but are significant only in seven countries. Also they found that Friday returns are significantly positive in almost all countries. Agathee (2008) examined the day of the week effect, who finds positive and significant ordinary least squares regression coefficients on Mondays, Wednesdays, Thursdays and Fridays, but however Fridays returns are the highest. Mills *et al.* (2000) haven’t found Monday effect , but a Tuesday effect similar to other papers is presented. Aggarwal and Rivoli (1989) find that Monday and Tuesday returns are lower than the overall average, while the Friday returns are higher, as also the volatility measured by the standard deviation is highest on Mondays. So in addition to the Monday effect, Aggarwal and Rivoli (1989) find a Tuesday effect in four Asian markets, which examined. Draper and Paudyal (2002) FT-All Share index and FTSE 100 Index from the beginning of 1988 until December 1997, and they found that Monday returns are negative and generally the returns of the other four days of the week are significantly higher. Floros (2008) rejects January effect for all the three indices which examined and he finds higher returns over other months rather January, but estimating coefficients are statistically insignificant, except significant negative returns in June for all indices. Mills *et al.* (2000) examine the month effect and found significant higher average returns on January and February. Choudhry (2001) reports significant negative returns in March and July for UK, while significant positive returns in February, August, September and December and significant negative returns in June and October were found for Germany. Aggarwal and Rivoli (1989) investigate the month –of-the-year effect and they find that January effects exist. The organization of the paper has as follows. In the section 2 we present the methodology which is followed. In section 3 we describe the nature of the data and we present the sources of them. In section 4 we present the results.

## 2. Methodology

### 2.1 The estimated model

In this section we present two simple models to examine the day-week effect and the second model is referred at the month of the year effect. The theoretical models have been derived by Panagiotidis and Alagidede (2006) which are

Abbildung in dieser Leseprobe nicht enthalten (1)

for the day of the week effect and

**Abbildung in dieser Leseprobe nicht enthalten ** (2)

for the month of the year effect

, where Abbildung in dieser Leseprobe nicht enthaltenrepresent the returns of the General Index of Athens Stock Market Abbildung in dieser Leseprobe nicht enthaltenrepresent the general index with one lag. Also In model (1) the variables Abbildung in dieser Leseprobe nicht enthaltento Abbildung in dieser Leseprobe nicht enthaltenare dummy variables and represent the days Monday to Friday and the dummy variables in model (2) express the months of the year. So for example in model (1) dummy variable Abbildung in dieser Leseprobe nicht enthaltenobtains value 1 for Mondays and 0 otherwise and so on where finally dummy variable Abbildung in dieser Leseprobe nicht enthaltentakes value 1 for returns on Fridays and 0 otherwise. Equations (1) and (2) are initially estimated with ordinary lest squares (OLS) method.

### 2.2 Symmetric and Asymmetric Generalized Autoregressive Conditional Heteroskedasticity-GARCH

Because the data we use in order to examine the day of the week effect are daily we expect that OLS method will present ARCH effects and autocorrelation. These problems can be eliminated by applying Generalized Autoregressive Conditional Heteroskedasticity-GARCH models. The first model we estimate is the symmetric GARCH(1,1) model proposed by Bollerslev (1987) and is defined as:

*Abbildung in dieser Leseprobe nicht enthalten* (3)

Abbildung in dieser Leseprobe nicht enthalten (4)

, where *ε t* is the disturbance term or residuals of equation (1) and follows the distribution in (3). GARCH (1,1) equation is presented in relation (4), where *ω* denotes the constant of variance equation GARCH and coefficients *α* and *β* express the ARCH and GARCH effects respectively. The problem with symmetric GARCH is that only squared residuals with lags enter the conditional variance equation, and then shocks have no effect on conditional volatility. With the symmetric GARCH we can’t estimate the leverage effects. Leverage effects refer to the fact that “bad news” or negative shocks tend to have a larger impact on volatility than “good news” or positive shocks have. The first asymmetric GARCH we estimate is Exponential Generalized Autoregressive Conditional Heteroskedasticity-EGARCH (1,1) which was proposed by Nelson (1991) and is defined as:

Abbildung in dieser Leseprobe nicht enthalten

(5)

, where coefficient *γ* indicates the leverage effects.

The second asymmetric GARCH is Glosten-Jagannathan-Runkle Generalized Autoregressive Conditional Heteroskedasticity- GJR-GARCH (1,1) model proposed by Glosten *et al.* (1993). The variance equation is presented in (6).

Abbildung in dieser Leseprobe nicht enthalten

(6)

Abbildung in dieser Leseprobe nicht enthaltenis a dummy variable , where Abbildung in dieser Leseprobe nicht enthalten=1 if Abbildung in dieser Leseprobe nicht enthalten<0 and Abbildung in dieser Leseprobe nicht enthalten=0 otherwise. Also for a leverage effect we expect that *γ >0* and we require *α 1 + γ Abbildung in dieser Leseprobe nicht enthalten 0* and *α 1 Abbildung in dieser Leseprobe nicht enthalten 0* for non-negativity condition. As in the case of EGARCH coefficient *γ* in GJR-GARCH equation indicates the leverage effects.

## 3. Data

The data for the day of the week effect are daily and for the month of the year are in monthly frequency. As we mentioned we examine the General Exchange Stock Market Index of Athens. For both calendar anomalies we examine the period 2nd t January, 2002 to 30th October, 2007. The data are available at *www.enet.gr.*

## 4. Results

First of all we estimate the above models with the OLS method but we will test about heteroskedasticity and we will conclude that there are ARCH effects in the estimations for the day of the week effect. In an effort to eliminate this problem we estimate also with GARCH(1,1), GJR-GARCH (1,1) and EGARCH (1,1). The results are presented in appendix A . In table 1 the OLS estimations are reported. We conclude from the table 2 and the ARCH-LM test that the OLS is not the appropriate method to estimate this model. In tables 3, 4 and 5 the results of GARCH (1,1), GJR-GARCH (1,1) and EGARCH (1,1) respectively are presented.

For the GARCH(1,1) and the results of table 3 the sum of α1 + β is almost 0.96 , which is smaller than unit and the GARCH *β* term is positive and statistically significant , as it’s also in the GJR-GARCH (1,1) and EGARCH (1,1) estimations. The conclusion is that GARCH (1,1) is sufficient to capture the volatility clustering in the data and that shocks will be highly persistent in the conditional variance. Also in all models, including OLS, the AR(1) is positive and statistically significant. In GJR-GARCH (1,1) the parameter *γ* has the expected sign, which is positive and statistically significant and in EGARCH (1,1) parameter *γ* has also the correct and expected sign, which is negative and statistically significant indicating significant leverage effect in both models.

The general situation in all models is that there isn’t day of the week effect, except Friday effect which coefficient is positive and statistically significant indicating 2,5-3,0% positive returns. In figures 1-3 presented the rolling regressions of OLS, GJR-GARCH and GARCH models separately are presented with 95% confidence intervals. We observe that coefficients are not stable and they present significant deviations and fluctuations around the mean. To be specific while in the first half-period in rolling regressions negative returns are reported only on Monday in the second half-period Monday presents positive average returns, but the lowest average returns are reported on Tuesday, indicating that there is a change shift in the pattern of the day of the week effect. Generally there is only the Friday effect which this fact it is possible to explain that the Athens Stock market is not been characterized by market efficiency. Based on the period we examine as also the methodology we apply. On the other hand Friday effect depends on the ARCH effects, heteroskedasticity and autocorrelation, which after correcting this effect might be disappeared (Alagidede and Panagiotidis, 2006). But we observe that in GARCH and GJR-GARCH estimations only the coefficient representing Friday is a statistically significant, as also the coefficient of *Rt-1 ,* indicating that there significant average returns only on Friday and these are positive. In EGARCH estimations the results are quite different. Besides the two previous coefficients one additional coefficient is statistically significant, and it’s the coefficient expressing Tuesday. Furthermore in α=0.10 the highest and positive average returns are presented on Tuesday and not on Friday. But if we accept only in α=0.05 and α=0.01, then we reject the above statement, as Tuesday and Friday returns are statistically insignificant. However, we accept the estimations of GJR-GARCH, based on the Log-Likelihood statistic, which is the maximum among the various estimations, as also based on information criteria, Schwarz and Akaike, which are the minimum among all estimations.

For the month of the year effect the results are presented in table 6 with OLS method. Also in table 7 we apply the ARCH-LM test and in table 8 the correlogram of residuals squared are reported and we conclude that there aren’t ARCH effects neither autocorrelation. We expected this phenomenon as the data we obtained to investigate about the month of the year effect are monthly. In the case we would had been taken daily data it would be very possible to found ARCH effects. So there isn’t any need to estimate with GARCH models. In figure 4 the rolling estimates of the dummy variables with the OLS method with 95% confidence intervals are presented From figure 4 and the rolling OLS regressions for the month of the year effect we observe that in the first half-period the highest returns are not presented on January, but are reported on May, while in the second-half period January presents the highest returns. So we conclude that there is a change shift in the pattern of January or the month of the year effect. Moreover, in whole period March presents negative returns, while the average returns in November and December remain constant and always positive. Changes in the shift in the remained months are observed too.

The general situation is that there is the January effect, which is the only coefficient which is statistically significant in α=0.01 and α=0.05, as also November returns are statistically significant if we take α=0.10, but the average returns in January are higher than those of November. Also in appendix B we present the programming routine for the rolling regressions in EVIEWS software.

*Conclusions*

We examined the day of the week ad the month of the year effects for the Athens Stock market index and we found significant positive returns only on Friday. On the other hand we found significant highest positive returns in January and the returns in November are followed. The conclusion is that Athens exchange stock market might not be characterized by market efficiency, as calendar anomalies are presented. Another explanation of the presence of the calendar anomalies might be the methodology approach we follow. Additionally based on the rolling regressions we observe that in the half-period Monday presents negative returns, while positive returns are reported in the second half-period and the lowest returns are presented on Tuesday. Additionally, as we discussed in the main part of the study we concluded that there is a shift in the patter of the month of the year effect, where initially January presents positive average returns, but not the highest, while in the last months the highest and positive average returns are reported in January.

**[...]**

- Quote paper
- Eleftherios Giovanis (Author), 2008, The Day of the Week and the Month of the Year Effects: Applications of Rolling Regressions in EVIEWS and MATLAB, Munich, GRIN Verlag, https://www.grin.com/document/146637

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