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Modeling stock market return volatility with reasonable accuracy is essential in asset allocation, asset pricing, and risk management. In this context, the primary aim of the present study is to investigate the volatility pattern of the Indian stock market based on the time-series data of the daily closing value of the Nifty-50 Index for the period from 1st April 2016 to 31st October 2021. The empirical investigation of the issue has been carried out by applying the Generalized Autoregressive Conditional Heteroscedastic (GARCH) Model. Both GARCH (1,1) and GARCH-M (1,1) were estimated. As per the Akaike Information Criterion (AIC), Schwarz Information Criterion (SIC), and model diagnostics, it is found that the GARCH-M (1,1) estimation is more appropriate to capture the symmetric volatility in the Indian stock market return.

The primary objective of the present study is to investigate the volatility pattern of the Indian stock market by applying appropriate Time Series Econometric techniques. For the empirical analysis of the issue related to volatility modeling, GARCH models will be estimated.

Several studies were made in modeling the stock market volatility bothin developed and developing countries.Many researchers investigatedthe performance of GARCH models in explaining the volatility of emergingstock markets (French, Schwert, and Stambaugh 1987; Chou 1988; BaillieandDeGennaro 1990; Bekaert andWu 2000;Chand,Kamal, andAli 2012;Kenneth 2013). Besides, few studies were attempted on the Egyptian markettoo. Zakaria and Winker (2012) examined the return volatility usingdaily prices of the Khartoum Stock Exchange (KSE) and Cairo and AlexandriaStock Exchange (CASE) and found that the GARH-M model describedconditional variance with statistically significant for both the markets;there existed a leverage effect in the returns of KSE and case with a positivesign.

Further, Floros (2008) investigated the volatility using daily data fromtwo Middle East stock indices viz., the Egyptian CMA index and the Israelitase-100 index, and used GARCH, EGARCH, TGARCH, Component GARCH (CGARCH), Asymmetric Component GARCH (AGARCH) and Power GARCH (PGARCH). The study found that the coefficient of the EGARCH model showed a negative and significant value for both indices, indicating the existence of the leverage effect. The ARCH modelshowed weak transitory leverage effects in the conditional variances andthe study showed that increased risk would not necessarily lead to an increasein returns. 

Ahmed andAal (2011) examined Egyptian stockmarketreturn volatility from1998 to 2009 and their study showed that EGARCH isthe best fit model among the other models for measuring volatility. Thestudy showed that there is no significant asymmetry in the conditionalvolatility of returns captured by GARCH (1,1) and GARCH (1,1) and it was found to be the appropriate model for volatility forecasting in the Nepalesestock market (Bahadur 2008).Although many research studies were undertaken on modeling thevolatility of the developed stock markets, only a few studies have been donein the Indian context. Recently, few studies have been done onmodeling thestock market volatility of the Indian market but most of the studies are limitedto only the symmetric model of the market. Karmakar (2005) estimatedthe volatility model to capture the feature of stock market volatility in India.The study also investigated the presence of the leverage effect in the Indian stockmarket and the study showed that the GARCH (1,1) model provided reasonablygood forecasts ofmarket volatility.Whereas, in another study,he (Karmakar 2007) found that the conditional variance was asymmetricduring the study period and the EGARCH-M was found to be an adequatemodel that reveals a positive relationship between risk and return.

Goudarzi and Ramanarayanan (2010) examined the volatility of the Indianstockmarket using the BSE 500 stock index as the proxy for ten years. And GARCH models were estimated and the best model was selected usingthemodel selection criterion viz., Akaike Information Criterion (AIC)and Schwarz Information Criterion (SIC). The study found that GARCH (1,1) was the most appropriate model for explaining volatility clusteringand mean-reverting in the series for the study period. Further, intheir (Goudarzi and Ramanarayanan 2011) study, they investigated the volatility of the BSE 500 stock index and modeled two non-linearasymmetricmodels viz., EGARCH (1,1) and TGARCH (1,1) and found that TGARCH (1,1) model was found to be the best-preferred model as per AIC, Schwarz Information Criterion (SIC) and Log-Likelihood (ll)criteria.

Mittal, Arora, and Goyal (2012) examined the behavior of Indian stock prices and investigated to test whether volatility is asymmetric using daily returns from 2000 to 2010. The study reported that GARCH and PGARCH models were found to be the best-fitted models to capture symmetric and asymmetric effects respectively. Vijayalakshmi and Gaur (2013) used eight different models to forecast volatility in Indian and foreign stock markets. NSE and BSE index was considered as a proxy for the Indian stock market and the exchange rate data for the Indian rupee and foreign currency over the period from 2000 to 2013. Based on the forecast statistics the study found that TGARCH and PARCH models lead to better volatility forecast for BSE and NSE return series for the stock market evaluation and ARMA (1,1), ARCH (5), and EGARCH for the foreign exchange market. In the same line, Singh and Tripathi (2016) also modeled the stock market return volatility for the Indian economy. They used daily closing prices of the S&P CNX Nifty Index for fifteen years from 1st April 2001 to31st March 2016. A Series of GARCH models were estimated to capture symmetric and asymmetric volatility. The study concluded that the GARCH-M model is more suitable to capture symmetric volatility in data.

Most of the Indian studies that attempted on modeling volatility found that the GARCH (1,1) is considered the best model to capture the symmetric effect and for leverage effects, EGARCH-M and PGARCH models are appropriate to the previous studies. However, the choice of the best fitted and adequate model depends on the model that is included for the evaluation in the study. Hence, the present study used different GARCH models both to capture the facts of return and to study the volatility in the Indian stock market.

Research Methodology   

To investigate the volatility pattern of the Indian stock market, the time-series data of the daily closing value of the Nifty-50 Index for the period from 1^st April 2016 to 27^th April 2021 has been used. The source of the data is the Nifty database. The volatility has been estimated at the daily return (r_t). The daily return can be calculated as follow;

Where r_t is the logarithmic daily return on the Nifty-50 index for the t period, P_t is the closing price at time t, and P_t-1 is the corresponding price during the previous period.

Descriptive Statistics

To specify the distributional properties of the daily return series of the BSE market index during the study period, descriptive statistics such as mean, standard deviation, skewness, kurtosis, and Jarque-Bera statistics have been applied.
Unit Root Test

When time-series data are used for the analysis, it is an essential econometric process to check for the order of integration of the time series. If a time series is stationary (has no unitroot), its mean, variance, and autocovariance (at various legs) remain the same no matter at what point the measurement is done. In other words, the probability distribution of a stationary time series does not change over a period of time. If a series does not follow the said characteristics, the time series is understood as a nonstationary process. For GARCH analysis, the order of integration of a series is the essential aspect.  Therefore, the first task is to check for the stationarity properties in the time series. To analyze this behavior of time series, Augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) Tests are applied.
Augmented Dickey-Fuller Test (ADF)

The ADF test for the unit root tests the null hypothesis H0: µ=0 against the one-sided alternative H1: µ<0 in the regression which is described as follows;

Under the null hypothesis, the series X_t has a unit root which means the series is not stationary and under the alternative hypothesis, the series is said to be stationary. If the series is stationary around a deterministic linear trend, then the trend t must be added and the following equation is estimated.

The ADF tests whether µ=0, meaning the time series under consideration is nonstationary. The leg length p can be selected based on Akaike's information criteria.

Phillips-Perron (PP) Test

PP test has an advantage over the ADF test as it gives a robust estimate when the series has serial correlation and time-dependent heteroscedasticity, and there is a structural break. The PP test can be carried out by estimating the following equation;

The null hypothesis that the series is non-stationary time series is rejected if π is less than zero and statistically significant.
GARCH Specification

As noticed above, the financial data always exhibit the phenomenon of 'volatility clustering', that is, the periods in which the data show wide swings for an extended period followed by a period in which there is relative calm. To analyze this kind of nature and behavior of the data, the GARCH model is useful.
To understand, the GARCH model, it is important to understand the first ARCH model. A simple way to measure the volatility is to run the following regression;

Where r_t is daily stock return, c is a constant and u_t represent the error term. The constant c in this case measures simply the mean value of the daily stock return. Notice that there is not an explanatory variable added to the equation. Equation 1 is also called a mean equation and proper specificationof the mean equation to capture the volatility in the series properly is an important condition of GARCH analysis. The validity of the mean equation can be examined using Correlogram – Q Statistics of the estimated residuals of the mean equations. If the residuals exhibit the presence of AR (Auto-Regressive) or MA (Moving Average) or a combination of both (ARMA), it is important to incorporate these components in the mean equation too for the correct identification of the mean equation. After the estimation of the mean equation, the estimated equation is tested for the ARCH effect in the equation using the ARCH-LM test. And the null hypothesis in the test is that there is no ARCH effect in the estimated equation.

In the presence of the ARCH effect, the following equation is estimated to capture the volatility.

Where σ²₁ is the error variance at time t, is equal to some constant plus a constant multiplied by the squared error term in the previous period that is obtained from equation 1 (mean equation). It is assumed that the coefficients in equation 2 are positive because the variance cannot be a negative number. Equation 2  is also called the conditional variance equation and it is also called the ARCH (1) model. The ARCH model of order (p) can be represented as follow;

From equation 2,  which is an ARCH model, the GARCH (1) can be represented as follow;

Notice that the conditional variance at time t  depends not only on the lagged squared error term but also on the lagged variance at a time (t-1). Equation 4 is known as the Generalized Autoregressive Conditional Heteroscedastic (GARCH -1,1) model. It is again important to note that the value of β₁ and β₂ will be positive. The size of parameters β₁ and β₂ determine the short-run dynamics ofthe volatility time series. If the sumof the coefficient is equal to one, thenany shock will lead to a permanent change in all future values. Hence,a shock to the conditional variance is ‘persistence’.

Further in the study, the GARCH-M model will also be estimated to model the return. It is will enable us to examine the presence of conditional variance in the mean equation. This implies that the return also depends on the volatility. For the simple GARCH-M model, the mean equation can be written as follow.

Where R_t is the return of the asset at time t. µ represents the average return (specified mean equation) and σ²₁ represents the GARCH term in the mean equation. In the presence of conditional variance in the mean equation, the variance equation for GARCH-M (1,1) can be written as follow.

The parameter V₁ in the mean equation can be identified as Risk Premium. The positive V₁ indicates that the return ispositively related to its volatility, i.e. a rise in mean return is caused by an increase in conditional variance as aproxy of increased risk.

Figure 1 depicts the movement of the closing stock index during the study period and it is quite evident that the mean over a period of time is changing which means that the series is non-stationary indicating the presence of unit root in the data. To analyze the phenomenon under investigation, the return has been calculated and the behavior of the series is shown in Figure2. It is observed that in the daily return series, there is the presence of volatility clustering which needs to be investigated further. From figure 2, it is inferredthat the period of low volatility tends to be followed by a period oflow volatility for a prolonged period and the period of high volatility isfollowed by a period of high volatility for a prolonged period, whichmeansthe volatility is clustering and the return series vary around the constantmean but the variance is changing with time.

Table 1 represents the descriptive statistics of the Daily Return series. The mean value of the return is positive, indicating the fact that the price has increased over a period of time. The descriptive statistics also indicate that the return series is negatively skewed. This implies that there is a high probability of earning greater than the average return. The Kurtosis is greater than 3 indicating that the series is not normally distributed which is also clear from the Jarque-Bera p-value which is < 0.05 and the null hypothesis is that the series is normally distributed.

The results of the unit root tests have been presented in Table 2. It is inferred from the ADF test and PP test, as the test value, for both with intercept and without intercept, is highly significant. The null hypothesis that the series is non-stationary is rejected and the alternative hypothesis is accepted i.e. that the daily return series is stationary.

As discussed above, the primary condition for the GARCH analysis is the stationarity of data and the Daily Return series is integrated of order zero which means the series is stationary and now further analysis can be carried out.

Equation 1  is estimated and the results are presented in Table 3. Further to check the validity of the mean equation the residual diagnosis is very important. The Correlogram – Q Statistics have been applied to check the validity of the estimated mean equation. The results of the Correlogram – Q Statistics are presented in Table 4. The probability value is increasing in the beginning and significant after four lags. This kind of behavior indicates that there is a presence of ARMA in the data.

The ARMA models were estimated with different orders of AR and MA. The best possible model has been presented in Table 5.The coefficients are highly significant as indicated by the Student’s T-Test. Further, the validity of the AR(1) is also checked by applying the Correlograme Q Statistics and the results are shown in Table  6. The probability value for all the lags is more than 5% meaning that the equation is correctly specified.

Now, it is important to check for the presence of Heteroskedasticity in the mean equation. The presence of Heteroskedasticity in the modal indicates that there is an ARCH effect in the model. The results of the Heteroskedasticity Test: ARCH are presented in Table 7. The null hypothesis in the test is that there is no ARCH effect. The results indicate that the probability values are less than 5% indicating the presence of Heteroskedasticity in the model. So it is important to apply the ARCH/GARCH model.

Table 7: Heteroskedasticity Test: ARCH

As it has been explained in section 3, both GARCH (1, 1) and GARCH-M (1,1) have been estimated. The results of the estimated models have been presented in Table 8. The parameters of ARCH and GARCH effects of both the models are non-negative and significant which is an important condition for the GARCH model. The total of ARCH and GARCH coefficients should be less than or equal to one. The total of these coefficients is 0.53 and 0.97 respectively for GARCH (1,1) and GARCH-M (1,1) models. However, it is important to note that the AIC value is minimum and the Log-Likelihood value is higherfor the model GARCH-M (1,1). This implies that GARCH-M (1,1) model should be selected out of these two models. It is also important to check for the further presence of the ARCH effect in the estimated models. The results of the ARCH-LM test have also been presented in Table 8. The results indicate that there is a presence of heteroscedasticity in the model GARCH (1,1) as the P-value is almost zero. On the other side, the ARCH-LM test accepts the null hypothesis that there is no presence of further ARCH effect in the estimated model. This also confirms that GARCH-M (1,1)is more appropriate to capture the symmetric volatility in the Indian stock market return. It can be noticed from the estimated model that the ARCH term is moving toward zero and the GARCH term is trending toward one. Thie implies that GARCH effects are stronger than ARCH effects. This suggests that the volatility impact is more persistent than the past sock impact. As the total of these two terms is 0.97 which is quite close to one indicates that there may be a long memory process in the volatility. The empirical results of the study are in line with the study conducted by Singh and Tripathi (2016).

The presented study has attempted to model the stock market volatility by using the daily return of Nifty-50 from the period from 1st April 2016 to 27th April 2021. It has been observed that the daily return series is not normally distributed as most of the financial data reveals. The daily return series is intergraded of order zero which means that the series is stationary and can be used further for the univariate analysis. ARMA (3,3) model fits well as the mean equation. To model the volatility both GARCH (1,1) and GARCH-M (1,1) have been estimated. The results indicated that the GARCH-M (1,1) model is more appropriate to capture the symmetric volatility in the Indian stock market return. This means that there is also the presence of GARCH in the mean equation and the return is positively related to the risk.The estimated model explains well the volatility clustering in the daily return and it the market has a memory longer than oneperiod and that volatility is more sensitive to its lagged values than it isto new surprises in the market values. The conditional variance does not only depend on the previous squared error term in the respective period tbut it also depends on the previous variance.

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