Capturing the month of the year e ﬀ ect in the Indian stock market using GARCH models

Purpose – Inthe research of stock market ef ﬁ ciency, it is arguedthatthe stock marketmoves randomly and absorbs all the available information. As a result, it is quite impossible to make predictions about the possible future movement by the investors. But literatures have detected certain calendar anomalies where a day(s) in a week or month(s) in a year or a particular event in a year becomes conducive for investors to earn more than the normal. Hence, the purpose of this study is to ﬁ nd out the month of the year effect in the Indian stock market. Design/methodology/approach – In this study, daily time series data of Sensex and Nifty from 1996 to 2021 is used. The study uses month dummies to capture the effect. Different variants of generalised autoregressive conditional heteroskedasticity (GARCH) models, both symmetric and asymmetric, are used in thestudytomodel the conditional volatilityin thepresencemontheffect. Findings – This study found the September effect in the return series of both the stock market. Apart from that, asymmetric GARCHmodels arefoundto be thebest ﬁ t modelto estimate conditionalvolatility. Originality/value – This study is an endeavour to study month of the year effect in the Indian context. This researchwill provide valuable insightforstudying thedifferentcalendaranomalies.


Introduction
Studying the behaviour of the stock market is one of the complex subjects in the field of finance. Different people observe, think differently and act accordingly. Researchers have carried out studies from different contexts to capture the behaviour of the stock market. Return and volatility are two interrelated and inherent parts of the stock market behaviour. The efficient market theory is one of the pillars of portfolio management. Developing the Efficient Market Hypothesis, Fama (1965Fama ( , 1970 laid the foundation stone for further research in understanding the stock market behaviour. It proposes that share price does not follow any pattern means random which reduces the chance to earn an abnormal return. But a number of studies have documented the presence of different patterns in return and volatility. Investors use these approaches to generate more than the normal return from the market. One such pattern is calendar effects. This calendar effect includes the day of the week effect, the month of the year effect, January effect, daylight savings effect, holiday effect, turn of the month effect, etc. This paper is an attempt in investigating the existence of the month of the year effect using generalised autoregressive conditional heteroskedasticity (GARCH) models.

Review of literature
The research evidence on the influence of the month of the year effect is relatively less observed comparing to the day effect as far as the emerging markets like India are concerned. The month of the year effect is a pattern in the returns of the financial assets where the return of a particular month of a year is significantly higher comparing to other months. This effect is documented in several studies in the world. One of the earlier studies by Berges et al. (1984) reported the higher average return in the month of January comparing to other months. Ignatius (1992) also found this effect in the case of India as well as US markets. The average return of December emerges as the highest return generating month. Analysing the return data of nine developed and emerging countries, Boudreaux (1995) also documented the presence of month of the year effect. Using a non-linear GARCH model in the Germany, US and UK stock markets, Choudhary (2001) found the January effect as well as the month of the year effect. Later on, the study of Parikh (2008) found a December effect in the Indian Stock market. His study supports the earlier study of Ignatius (1992). The study of Chaouachi and Douagi (2014) was on the Tunisian stock market from 1998 to 2011. The study found the presence of this effect. The study revealed the significant effect of January, March, April, August and September on the Tunisian stock market. Similarly, Ciccone and Etebari (2008) also captured the January and September effect in four international markets such as France, UK, Japan and Germany out of five considered for the study. Kaur (2004) studied this effect in the Indian context using the symmetric as well as asymmetric GARCH models to estimate the conditional volatility by capturing the month of the year effect and found the presence of the same effect and reported that the GARCH models that are asymmetric outperform symmetric. However, Floros (2008) reported the significance of June and October in the Greece stock market. Analysing for the period 1980-2009, Ke et al. (2014) found the February effect in the Taiwan stock exchange. Similarly, Munir and Ching (2018) disclosed the presence of month effect on the share value of the selected banking and non-banking companies in Malaysia. In a recent study by Bajaj et al. (2019), September effect was detected in Indian stock market.
The above discussed studies have used proper econometric methods in the estimation of the month effect. But some other studies are also there which uses only descriptive statistics or some simple estimation tools such as t-test, analysis of variance (ANOVA) to explain this behaviour of the stock market. The study of Verma and Vijay Kumar (2008) is one of them where they have used the descriptive statistics, ANOVA and regression analysis in Bombay stock exchange. The mean returns of the months are found to be positive except the month of October. The regression result finds the insignificance of the month coefficients which indicate the disappearance of month effect in the Bombay stock market. Similarly, Mehta and Chander (2009) studied by using the same techniques like descriptive statistics, Kruskal-Wallis H test and regression analysis revealed that November and December months can be considered to be important for investors to attain abnormal returns. Another study on the Ukrainian stock market by Caporale and Plastun (2017) discovered that while calendar irregularities are not evident in the spot market, they are present in the future market. After reviewing the above literatures, one thing is found that there are more or less certain trends or patterns in the stock markets across the world that affects the return generating process of the investors. But the lesser focus is on the world's fastest expanding economy, India. Another thing is the short study period that covers five to ten years. Hence, this study takes into account 25 years of data to analyse and generalise this phenomenon of the Indian stock market.

Data and methodology
The analysis was carried out with the time-series data of the index values collected from two active Indian stock exchanges. They are the Bombay stock exchange and the National stock exchange. The daily Sensex and Nifty values from 1996 to 2021 were collected and further processed to find out the return for the purpose of the analysis. A total of 6,203 and 6,223 observations in the case of Sensex and Nifty, respectively, were used in this study. The difference in the total number of observations is primarily attributable to the trading halt, may be due to the activation of circuit breaker, in a trading day or trading holidays declared in the stock exchanges. The main intention of the current research is to estimate the conditional volatility by capturing the month of the year effect. For this purpose, GARCH, exponential generalised autoregressive conditional heteroskedasticity (EGARCH), threshold generalised autoregressive conditional heteroskedasticity (TGARCH) and power generalised autoregressive conditional heteroskedasticity (PGARCH) models are used. The autoregressive conditional heteroskedasticity (ARCH) and GARCH terms used in the equations are of order one. GARCH (1,1) model is the basic model that explains the conditional variance in the time series. Tim Bollerslev (1986) introduced this concept in, which is an expansion of Engel's ARCH model. This model is based on the premise that today's conditional variance is influenced by past conditional variance. The GARCH models conveniently account for the clustering of the volatility in the time series data of any financial assets. But the basic GARCH model has a major limitation because of the fact that it is symmetric. It means that the signs of the error term are ignored as this term is squared in the variance equation. However, the financial market exhibits positive and negative shocks. To capture these shocks in the time series data, Nelson (1991), Glosten et al. (1993), Zakoi±an (1994) and Ding et al.(1993) introduced certain extensions to the basic model. They are TGARCH, EGARCH and PGARCH models. The mean equation and variance equation of different GARCH models are as follows.
3.1 GARCH model where R t is the return of Sensex and Nifty returns which is considered to be linearly related to the month dummy variables and past return. M 1 to M 12 are the months starting from January to December. In equation (2), h t is the conditional variance. Thus equation (1) is the mean equation, whereas equation (2) is the variance equation. In the variance equation, a i is the ARCH coefficient and b j is the GARCH coefficient.

EGARCH model
Here, the conditional variance of the return is expressed in logarithmic form, resulting in an exponential rather than quadratic leverage effect. As a consequence, the conditional variance would be greater than zero. Here, g j is the leverage parameter and if g 1 = g 2 = . . .. . .. . .. = 0, then the model is asymmetric. When g j < 0, then it can be said that the volatility because of positive news is lower than that of bad news.

TGARCH model
Here a dummy variable "d t " is used in equation [4] which takes the value of 1 in the case when u t < 0, otherwise it is 0. Thus, in TGARCH model, the impact of good news is captured by the g i coefficient, whereas the impact of the bad news can be captured by adding the coefficients of the residual term with the coefficient of the multiplicative dummy variable. If the value of the coefficient g i, is greater than 0, then it indicates the dominant role of bad news in increasing the volatility.
3.4 PGARCH model where d > 0, jgj < = 1 for i = 1, 2, . . .. . ..,r, i = 0 for all i > r, and r # p. All i are set to 0 in this symmetric model. A PGARCH model is just a regular GARCH specification if d = 2 and i = 0 for every i. If the value of g is less than 1, the asymmetric effect is evident. In the current study, d is kept as 1 so that standard deviation or volatility can be directly calculated.

Empirical results
The results indicated the presence of month effects in the return series of these stock exchanges.
The descriptive statistics of Sensex and Nifty returns are displayed in Tables 1 and 2. This includes the mean, median, standard deviation, skewness, kurtosis and Jarque-Bera test statistics along with the probability values. It can be observed from the table that, the mean returns of the selected index were positive except March and October. The highest return was in the month of December. Similarly, the standard deviation in both the markets was highest in the month of March and May followed by October. Kurtosis suggests that the index data is leptokurtic. The result of Jarque-Bera statistics rejects the normality assumption about the index data series. The unit root test outcomes are displayed in Table 3. Both the tests confirmed that there is no presence of unit root. Table 4 shows the results of the estimated mean equation as well as variance equation [i.e. equations (1) and (2)  October, November and December. All these months are statistically significant at 5%. ARCH and GARCH coefficients are so significant and the sum of the coefficients is less than one in the case of Sensex and Nifty. Thus, the conditional volatility seems to be a little bit explosive in nature, meaning that the volatility has increased gradually over the time in an exponential manner. The large GARCH coefficient indicates its persistence. All the months have been found to have a positive effect on both the return series. The diagnostic tests such as Ljung Box Q statistics represents the squared residuals did not show the presence of autocorrelation and in the same way, autoregressive conditional heteroskedasticity (ARCH) LM test shows the absence of the ARCH effect.
The estimated results of equation (3) are shown in Table 5. It is evident that there is the presence of asymmetries in both the stock markets while capturing the month of the year effect. Interestingly, Sensex and Nifty show the September effect. Apart from the above mentioned months, the Nifty return series shows some different effects. Here, April in case of Bombay Stock Exchange and March in the case of National Stock Exchange shows significance. Looking at the variance equation, it is found that there is the presence of leverage effect in both the return series. The leverage parameter is negative and significant in both cases. The negative leverage term shows that bad news plays a greater role in increasing the volatility more comparing the good news of a similar scale would do while capturing the month of the year effect. The significance of conditional volatility is also observed. Both are significant at 1%. Here, the GARCH coefficient in the above estimation is covariance stationary as X p i¼1 b i < 1 (Zivot, 2008). The diagnostic tests suggest the absence of autocorrelation and ARCH effect in the residuals.
Similarly, Table 6 displays the estimated results of the TGARCH equation. This also indicated some asymmetric evidence while capturing the month of the year effect in the return series. Like the EGARCH model, the study exhibits the September effect which is significant statistically at a 5% level. Interestingly, May and December also show their significance in both the return series. All the parameters in the conditional variance equation are found to be statistically significant in the presence of the month of the year effect. The diagnostic tests suggest that there is no presence of autocorrelation and ARCH effect in the residuals.
As indicated in Table 7, PGARCH results reveal the presence of asymmetries. Both indices' return series show that September has a considerable effect and, additionally, Nifty also captured the December effect. March in the case of Nifty and April in the case of Sensex are found to be significant. ARCH and GARCH coefficients are statistically significant including the asymmetry coefficient. Like in all other estimations, the diagnostic tests suggest that there is no presence of autocorrelation and ARCH effect in the residuals. Notes: Ljung Box Q statistics represents the squared residuals up to lag 5. *, ** and *** indicate statistically significant at 10%, 5% and 1%, respectively Notes: Ljung Box Q statistics represents the squared residuals up to lag 5. *, ** and *** indicate statistically significant at 10%, 5% and 1%, respectively Notes: Ljung Box Q statistics represents the squared residuals up to lag 5. *, ** and *** indicate statistically significant at 10%, 5% and 1%, respectively Notes: Ljung Box Q statistics represents the squared residuals up to lag 5. *, ** and *** indicate statistically significant at 10%, 5% and 1%, respectively Table 7. PGARCH (1,1) estimates for the month-of-the-year effect

Discussion and conclusion
In this article, an attempt was made to capture the month effect in the Indian stock market and further to estimate the conditional variance or volatility. By using both forms of the GARCH models, i.e. symmetric and asymmetric, it is found the presence of the above effects in the returns and volatility. The nature of the month of the year effect seems to be a little different across months. Both the series shows the same months to influence the return series except June in the case of Sensex return while March in the case of the Nifty series. This study did not find the presence of January effect as was detected by Berges et al. (1984), Ignatius (1992) and Choudhary (2001). However, the study detected the September effect consistently throughout the study period in all the estimations of return series. This present study supports the study of Ciccone and Etebari (2008), Ke et al. (2014) and Bajaj et al. (2019). The September effect as detected in the study has several plausible explanations. For Indians, the festive season begins in September. This season provides a significant stimulus to India's economy resulting in a favourable effect on the stock market. The quarterly results of the first quarter come in August which has also a certain effect on the stock market. A good monsoon has an impact on the stock market performance as well. Seasonal psychological bias has a beneficial effect on the market. The study's findings revealed the presence of the leverage effect. Therefore, the investors should be careful, particularly to good news and bad news, while deciding the investment. Except for September, the study did not find any strong signs of any other months. Hence, prediction by the investors becomes difficult. Integration of domestic markets with foreign markets may be responsible for this phenomenon. The most important implication that can be drawn from this study is that the past return plays a significant role in deciding today's return in the presence of month dummies. The ARCH and GARCH parameters are also found highly significant in the variance equation. Hence, one should take care of past volatility in his/her strategies for stock market investment. The asymmetric models outperform the symmetric models. Finally, except for few months, the study found no significant effect of other months on the return series which implies that the Indian stock market is becoming fairly efficient. To corroborate the findings of this study, a survey among the investors can be carried out in the future which can assess their perception towards these types of anomalies.