Introduction

The Efficient Markets Hypothesis (EMH) postulates that securities’ prices in financial markets are highly sensitive to the arrival of new information which is rapidly reflected in prices through many buy and sell decisions. The speed and volume of the transactions triggered by new information arrival eliminate any distortions in prices, and prevents investors earn extraordinary returns (^{Fama, 1970}). Thus, the EMH implies that the information contained in historical asset prices is not useful to predict future returns since such prices are independent of each other, and follow unpredictable paths, known as random walks.

However, there is increasing evidence of the existence of market anomalies (^{Gayaker, Yalcin, & Berument, 2020}; ^{Plastun, Sibande, Gupta, & Wohar, 2020}; ^{Rossi, 2015}). Among these anomalies are the so-called calendar (or seasonal¹) effects such as the Day-of-the-Week (DOW) Effect (^{Anwar, Okot, & Suhendra, 2021}; ^{Santillán Salgado, Fonseca Ramírez, & Nelson Romero, 2019}; ^{Tadepalli & Jain, 2018}; ^{Vogelaar, Pimenta, Lima, & Gaio, 2014}; ^{Yardımcı & Erdem, 2020}; ^{Zhang, Lai, & Lin, 2017}) and the Holiday Effect (HE) (^{Caporale & Plastun, 2017}; ^{Dumitriu & Stefanescu, 2020}; ^{Eidinejad & Dahlem, 2021}; ^{Marques, 2014}; ^{Mazviona, Mah, & Choga, 2021}). The DOW and HE anomalies studied in the context of the MSE are the subject of interest in this paper. As in the case of other reported market anomalies (^{Bohl, Ehrmann, & Wellenreuther, 2020}; ^{Niroomand, Metghalchi, & Hajilee, 2020}), these seasonal effects are at odds with the EMH and require a careful documentation and analysis.

The DOW and the HE anomalies for the Mexican market have been the subject of several studies, but their findings are frequently contradictory (e.g., ^{Duarte, Sierra, & Garcés, 2013}; ^{Rojas & Kristjanpoller, 2014}; ^{Winkelried & Iberico, 2018}). The present research contributes to the literature on calendar anomalies of the Mexican Stock Exchange (MSE) by examining that market’s capitalization subindices besides the aggregate stock market index, the Índice de Precios y Cotizaciones (IPC), to contrast both seasonal effects in different subsamples. Another innovation consists on the estimation of GARCH-family models with dummy variables for the analysis.

The main findings were that the DOW Effect was consistently present in both the returns and the volatility of the MSE index and different subindices. Similarly, the Holiday Effect was present in the volatility of the four series examined. However, it was absent from the return’s series, except in the case of the Medium Capitalization subindex.

The second part of this paper briefly reviews the literature on stock market anomalies and highlights the contributions of DOW and HE studies that have focused on the Mexican market. The third part describes the estimation methodology; the fourth part reports some relevant descriptive statistics of the data, and the estimation results; the fifth part presents the main conclusions.

Brief literature review

Methodological aspects

This work studies the presence of the DOW and the HE on the MSE’s IPC (S&P/BMV IPC), as well as on the large-capitalization (S&P/BMV IPC LargeCap), medium-capitalization (S&P/BMV IPC MidCap) and small-capitalization (S&P/BMV IPC SmallCap) stock indices. The modeling approach is based on an Autoregressive Moving Average with GARCH effects (ARMA-GARCH) model with dummy variables, represented as equation (1), below:

where r_t corresponds to the daily return of the IPC (or one of the subindices) on period t, α₀ is the intercept that captures the average Monday return, that is, the reference date (c’) for this model is Monday. Coefficients α_1, α_2, α₃ and α₄ are the differences between the average return on Monday and the average returns on Tuesday, Wednesday, Thursday, and Friday, respectively, D_i,t is the dummy variable corresponding to the i-th day of the week from Tuesday (i = 1) to Friday (i = 4), and εt is the random disturbance.

Two additional dummy variables were added to equation (1) to examine the pre-holiday and post-holiday effects in the Mexican stock market, as in equation (2), below:

D_PRH,t is equal to one for the days before a holiday (i.e., pre-holidays) and zero otherwise, while D_POH,t is equal to one for the days after a holiday (i.e., post-holiday) and zero otherwise.

Considering the multiple market turbulence episodes that occurred during the sample period, an ARMA-GARCH model, capable to estimate the parameters of changing volatility, was selected. Anytime ARCH-LM tests did not reject the null hypothesis of constant variance of εt, OLS regression models were used to estimate equations (1) and (2). On the other hand, when the ARCH-LM test rejected the constant variance null, GARCH (1, 1) models were used to determine the conditional variance of the error term (^{Caporale & Zakirova, 2017}; ^{Seif et al., 2017}). Furthermore, ^{Engle (2001)} argues that this specification is the simplest and most robust of all volatility models and that it can be adapted and broadened in numerous ways.

Following ^{Caporale & Zakirova (2017)}, ^{Kristjanpoller & Arenas (2015)}, and ^{Kristjanpoller (2012a)}, an ARMA-GARCH (i.e., AR(m)-GARCH (p, q)) specification was used:

In expression (4), which represents the mean equation of the GARCH (1, 1) model, r_t-j denotes the j^th lag of the returns. The number of lags to be used was determined based on the results of the AR(p) model described by equation (3), and only those lags that turned out to be significant and contributed to minimizing the Akaike information criterion (AIC) were included in the final version of the model.

In equation (5) α₁ and β₁ represent the variance equation’s coefficients for the lags of the error term and the variance, respectively. ^{Connolly (1989)} asserts that it is feasible to expand the GARCH model to include other effects in the conditional variance. Hence, the δi, the dummy variable coefficients, capture the average impact of the i-th calendar effect on volatility, both on direction and size, relative to a certain calendar reference date (c’) to circumvent the dummy variable trap.

Besides using the conventional GARCH (1,1) model with dummy variables, this research also estimated the Exponential GARCH (EGARCH) model proposed by ^{Nelson, (1991)} and the Threshold GARCH (TGARCH) specification developed independently by ^{Glosten, Jagannathan, & Runkle (1993)} and ^{Zakoïan (1994)}, revised to include exogenous variables (dummy variables in this case), following an approach similar to the procedure described for GARCH parameterizations (^{Begiazi & Asteriou, 2015}). Equations (6) and (7) represent the TGARCH (1, 1) and EGARCH (1, 1) models used to examine asymmetrically clustered volatility, respectively³:

Coefficient Ι-t-1 in equation (6) is an indicator variable that takes the value of 1 when εt-1<0 and zero otherwise, so that bad news have an impact of α1+γ1 and good news have an impact of α1 only. If γ1 is statistically different from zero, the data have a threshold effect; and when γ1>0 negative shocks will have larger effects on volatility than positive shocks. Equation (6) is subject to the following restrictions: ω≥0, α1≥0, β1≥0, and α1+γ1≥0.

The γ1 coefficient in equation (7) is related to asymmetry since such a leverage coefficient should be significantly negative⁴ if there is greater volatility after negative shocks. Thus, the existence of leverage effects is tested by the hypothesis that γ1<0. Also, there is an asymmetric impact whenever γ1 is statistically different from zero. Moreover, given that the logarithm of the conditional variance, rather than the conditional variance itself, is parameterized, the implied value of σt2 cannot be negative, which prevents the need to include artificial nonnegativity constraints on the model parameters.

GARCH family models fail to capture the non-normality of the data, so we use a non-parametric Kruskal-Wallis (K-W) test to examine the differences between the returns on a specific day of the week and other days. Equation (8) represents the K-W model:

where N is the total number of observations,Rj2 is the average rank of observations in the j-th group, k is the number of groups, and n_j is the total number of observations in the j-th group.

Data and results

Daily observations on the closing level of the IPC and the three subindices were retrieved from Economática. The complete sample period for the IPC comprised from January 3, 2000 to September 28, 2018. However, due to data availability limitations the series for the LargeCap, MidCap, and SmallCap indices only include observations from October 26, 2006, to September 28, 2018⁵.

^{Ortiz Bolaños, Hernández Henao, & Quintanilla Dieck (2015)}, ^{Russo & Katzel (2012)}, and ^{Santillán-Salgado (2015)}, among others, have stressed the extensive influence of the Subprime Mortgages Crisis and its consequence, the Great Recession (2007-2009), on the social, economic, and financial arenas around the globe. The European Sovereign Debt Crisis started during the autumn of 2009 also had important economic and financial consequences, in several regions of the world (^{Cabello, Moncarz, & Ortiz, 2015}). Given the extraordinary nature of the period, we considered it appropriate to determine the performance of the calendar anomalies in the MSE before, during, and after the two financial crises episodes, an issue rarely examined in the literature.

The complete period was divided in four subperiods: a) before the Great Recession, from January 2000 to March 2007; b) during the Great Recession, from April 2007 to September 2009; c) during the Eurozone Sovereign Debt Crisis, from October 2009 to December 2012; and, d) after the Eurozone crisis, from January 2013 to September 2018 (see Figure 1).

Source: Authors’ own elaboration with data retrieved from Economática.

Figure 1 Evolution of the IPC 2007-2018. 

The subperiods for the IPC were determined based on previous studies (^{Cabello et al., 2015}; ^{Do, Powell, Singh, & Yong, 2018}; ^{Russo & Katzel, 2012}; ^{Santillán-Salgado, 2015}). In the case of the subindices, the full period was divided into two subperiods: a) from October 2006 to September 2009 and b) from October 2009 to September 2018, due to data availability limitations (see Figure 2).

Source: Authors’ own elaboration with data retrieved from Economática.

Figure 2 Evolution of the Capitalization Size Subindices 2006-2018. 

To analyze the returns of the Mexican stock market indices, daily data were used to compute the logarithmic returns of the four indices in the sample:

In equation (9) r_t is the close-to-close continuous return corresponding to day t, P_t is the index adjusted closing price on day t, P_t-1 is the index adjusted closing price on the previous day, t - 1.

Table 1 displays descriptive statistics for the returns of the IPC, along with the results of the Augmented Dickey-Fuller (ADF) and KPSS unit root tests for the whole period, as well as the four subperiods of interest. Table 2 shows the same information for the IPC LargeCap, the IPC MidCap, and the IPC SmallCap indices.

According to the Jarque-Bera test⁶ in Table 1 and Table 2, the indices’ returns did not follow a normal probability distribution. Regarding the subindices, the returns of the IPC SmallCap exhibited the highest daily mean and median, as well as the lowest standard deviation. The LargeCap index presented the highest standard deviation.

Conclusions

Market efficiency is a subject of great interest for policymakers and investors in charge of designing investment strategies. According to the EMH, investors cannot systematically beat the market through the inherent information of asset prices. However, the reported existence of market regularities, such as the Day-of-the-Week (DOW) and the Holiday Effect (HE), questions the absolute validity of this notion.

Although our results reveal that the DOW effect did not exist on the IPC MidCap, we find supporting evidence for the presence of the DOW effect on the returns of the IPC, large- and small-capitalization indices, and on the four indices’ volatility. Specifically, the results show evidence of Monday and Thursday effects in the returns of the IPC LargeCap, as well as a Friday anomaly in the IPC SmallCap. Furthermore, such effects have not disappeared from the size-related indices’ returns. The time- and size-related disaggregation of the series we performed allowed the identification of a recent significantly negative Thursday effect in the returns of the IPC and IPC LargeCap. This outcome contrasts with the findings reported in the literature regarding the vanishing of the DOW effect (^{Alt, Fortin, & Weinberger, 2011}; ^{Plastun, Sibande, Gupta, & Wohar, 2019}) and questions the increase of efficiency in the MSE.

The results obtained for the DOW IPC’s volatility agree with those reported by ^{Kristjanpoller (2012a)}, since the analysis found reduced volatility on Fridays; the small-capitalization index also showed that effect. Thus, the notion that Fridays are ‘good’ days for liquidity traders (^{Kiymaz & Berument, 2003}) was confirmed for the IPC and SmallCap indices. Also, the analyses indicate that the IPC had significantly lower volatility on Wednesdays. The same anomaly was also present in the large- and medium-capitalization indices throughout the examined subperiods.

The regression model tests rejected the hypothesis that the HE was present in the returns of the IPC, the IPC LargeCap, and the IPC SmallCap, a conclusion that was further supported by the Kruskal-Wallis test results. This result agrees with the findings of other studies that report the absence of the holiday effect in the MSE (^{Duarte et al., 2013}; ^{Winkelried & Iberico, 2018}). In contrast with the conclusion of nonexistence of the DOW effect in the IPC MidCap, the results denote a significant presence of the HE in the returns of that index from 2006 to 2018. This coincides with the conclusions reported in the literature about the existence of such an anomaly in the MSE (^{Duarte et al., 2013}; ^{Kristjanpoller, 2012b}; ^{Seif et al., 2017}).

A pattern of low pre-holiday volatility and high post-holiday volatility in the four indices was significant and consistent across all time intervals considered. Our results indicate no evidence that this phenomenon disappeared from the size-related indices, which also challenges the notion of an increase in the MSE efficiency. A plausible explanation for the DOW effect is that volatility is higher on Friday and lower on Monday, and so are the corresponding average returns for those days (^{Campbell & Hentschel, 1992}). However, that explanation does not hold in the case of the MSE’s indices. In any case, for example, significantly higher Monday average returns were generally accompanied by lower, although not significant, volatility.

On the other hand, the explanation proposed by ^{Campbell & Hentschel (1992)} for the Weekend Effect seems reasonable for the pre-holiday and post-holiday volatility and returns relationship found in this research. It has also been suggested that the HE may be explained by behavioral factors such as the change in the mood of investors before and after holidays (^{Dumitriu & Stefanescu, 2020}; ^{Gama & Vieira, 2013}). In the case of the MSE, the significantly lower pre-holiday and higher post-holiday volatility patterns may, respectively, may relate to higher and lower trading volumes since such variables do not always move in tandem (^{Bollerslev, Li, & Xue, 2018}). Moreover, liquidity traders are reluctant to trade in periods where the prices are more volatile (^{Foster & Viswanathan, 1990}; ^{Kiymaz & Berument, 2003}). However, the pre-holiday and post-holiday pattern in the MSE leaves the door open for further research.