A Study on the Performance of Symmetric and Asymmetric GARCH Models in Estimating Stock Returns Volatility

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1 Vol., No. 4, 014, A Study on the Performance of Symmetric and Asymmetric GARCH Models in Estimating Stock Returns Volatility Mohd Aminul Islam 1 Abstract In this paper we aim to test the usefulness of two variants of Generalized Autoregressive Conditional Heteroscedasticity (GARCH) family-type models in estimating stock returns volatility for three Asian markets namely- Kuala Lumpur Composite Index (KLCI) of Malaysia, Straits Times Index (STI) of Singapore and the Bombay Stock Exchange Index (BSESN) of India. For this paper we have chosen the variants of the GARCH family models: the standard GARCH (1, 1) model represents as the symmetric model and the Threshold GARCH or TGARCH (1, 1) model represents as the asymmetric model. The study covers the period 0/01/007 31/1/013 comprising daily observations of 174 for KLCI, 1743 for Singapore and 175 for BSESN excluding the public holidays. Our results provide strong evidence that the daily stock returns can be characterized by these two models and they are better fit to capture the stylized facts about the index returns such as volatility clustering, leptokurtosis and the leverage effects. The results suggest that asymmetric GARCH performs relatively better for the case of Singapore while in the other two markets the standard GARCH performs better in explaining the data. Key words: Volatility; GARCH models; Leverage effects; Volatility Clustering JEL Classification Codes: C01, C13, C, C58, G15 1. Introduction Volatility plays very important role in many financial applications such as tracking the future movements of stock price change, calculating the value-at-risk, valuation of the derivative products and so on. The variation in the returns provided by the stocks due to changes in the daily price is generally termed as volatility which is measured by the standard deviation or the variance. In stock market perspective, usual up and down movement of the stock prices may not be bad but it turns out to be bad if the price swings are unusually very sharp or rapid over short time periods as it makes financial planning difficult. Sharp fluctuations in the prices increase the uncertainty about the future returns and hence increase the risk. In such an unstable market situation investors cannot reliably predict the future price movements. Uncertainty in prices in the future may prevent the investors to take risk and fund investment. Volatile market also causes difficulty for companies to raise funds in the capital markets as the investor may refrain themselves from taking risk and hence investing funds. Excess volatility may even lead to crashes or crisis in financial markets. Thus it is pertinent to estimate volatility accurately in managing the risk in fund investment. There has been observed a huge up and down shifts in the stock prices in many markets including developed and emerging markets worldwide. The fluctuations in the asset prices are widely believed to be the cause of changes in the economic factors such as interest rates, inflation, variability in speculative market prices, unexpected events (e.g., political unrest, natural calamities), and the instability of market performance. However, the biggest driver of the volatility in the financial market is a drop in the market 1 Department of Computational & Theoretical Sciences Faculty of Science, International Islamic University Malaysia Bandar Indera Mahkota, Kuantan 500, Pahang, Malaysia. 014 Research Academy of Social Sciences 18

2 performance. Volatility typically tends to decline as the stock market rises which in turn reduces the risk. In contrast, volatility tends to increase when the stock market falls and hence increases the risk. The stochastic nature of the financial market thus requires development of quantitative tools to explain and analyze the behavior of stock market returns and hence capable of dealing with such uncertainty in future price movements. A remarkable progress has already been made in developing sophisticated econometrics tools to explain and capture various characteristics of financial time series volatilities and hence to help managing the risks associated with them. Financial time series particularly stock/index prices often exhibit the phenomenon of volatility clustering (Stock et. al., 01 that is, the series exhibit sometimes high volatilities and sometimes low volatilities for an extended time periods. However, for a short period of time, there is a strong chance that a day of high volatility will be followed by another day of high volatility. In other words, if a high volatility is observed yesterday, it is more likely that a high volatility will also be observed today. This means that today s volatility is positively correlated with yesterday s volatility. This fascinates to estimate volatility conditionally on the past volatility. Volatility can either be historical volatility which is a measure based on past data, or implied volatility which is derived from the market price of a market traded derivative particularly an option. The historical volatility can be calculated in three ways namely; (1) simple volatility, () Exponentially Weighted Moving Average (EWMA), and (3) GARCH. In this study, we will apply the most commonly used stochastic volatility model GARCH as it is theoretically superior to and more appealing than the other two approaches. Furthermore, GARCH is also said to be a preferred method for finance professionals as it provides a more real life estimate while forecasting parameters such as volatility, prices and returns. The GARCH is the extension of the Autoregressive Conditional Heteroscedasticity (ARCH) model. These two models are said to be volatility clustering models and are importantly applied to measuring and forecasting the time-varying volatility of high frequency financial data like daily stock or stock index returns. Since the introduction of these two models into the literature, they become very popular and most common predominantly in financial market research as they enable the financial analysts to estimate the variance of a series at a particular point in time (Enders, 004) more accurately. A large number of empirical studies utilized ARCH and its variations in many markets and their applicability in capturing the dynamic characteristics of stock index returns has been demonstrated successful. Some of the studies who, along with other asymmetric GARCH models, have applied the standard/ basic GARCH models across different countries are Floros (008); Elsheikh and Zakaria (011); Shamiri and Zaidi (009); Islam (013) to name a few. A lot of empirical studies also used the different extensions of the basic GARCH such as the Exponential GARCH (EGARCH) developed by Nelson (1991), the Threshold GARCH (TGARCH or ZGARCH) introduced by Zakoian (1994), the GJR-GARCH by Glosten, Jaganathan, and Runkle (1993), the Power GARCH (PGARCH) proposed by Ding, Granger and Engle (1993) and so on. These are called asymmetric GARCH as they are capable of modeling asymmetric response and leverage effect. Floros (008) applied GARCH-type models to model volatility and to explain financial market risk of two middle-east stock indices: Egypt (CMA General index) and Israeli (TASE-100 index) over the period Using daily data of these two markets, the study concludes that the GARCH models are capable of characterizing the dynamics of daily stock returns including volatility clustering. By utilizing GARCH-in-Mean model, the study found positive but insignificant relationship between increased expected risk and increased expected return leading to the conclusion that higher expected risk does not necessarily produce higher expected return in these two markets. In modeling and forecasting of the Malaysian stock market proxied by KLCI over the period 1/1/1998 to 31/1/008, Shamiri and Zaidi (009) used standard GARCH, EGARCH and non-linear asymmetric GARCH (NAGARCH). The study compared the performance of these three models with six different error distributions. The study found existence of standard leverage effects in the KLCI index returns. The study concludes that successful volatility model much more depends on the choice of error distribution than the choice of GARCH models. In other words, performance of the GARCH models depends much on the error 183

3 M. A. Islam distribution. Elsheikh and Zakaria (011) used GARCH-type models that include both symmetric and asymmetric models to estimate volatility in the daily returns of the Khartoum (Sudan) Stock Exchange over the period from January 006 to November 010. They found evidence that the GARCH models are fit to characterize the daily returns for the case of Sudan. With respect to risk-return relationship, this study found risk premium coefficient positive and statistically significant implying that increased risk leads to higher return as predicted in financial theory. Islam (013) applied the GARCH-type models including symmetric and asymmetric models to test their applicability in analyzing the stylized facts (e.g., volatility clustering, leptokurtosis and leverage effects) commonly observed in high frequency financial time series such as stock/stock indices for the cases of 4-asian stock indices. The study found strong evidence that the models can characterize the dynamics of daily stock returns in all four markets in the sample. With respect to the risk-return relationship, the study found positive correlation in all cases which is in consistent with the financial theory. Apart from these, there are many studies also who have investigated the relationship between conditional variance and risk premium using GARCH model. Bac et. al. (007) examined the relationship between volatility and risk premium for the case of the New York Stock Exchange (NYSE) index US over the period The study found evidence of positive correlation between volatility and risk premium. Appiah and Menyah (003) investigated the relationship between volatility and risk premium of 1-African stock markets for the period They found evidence of time-varying risk premium in five most volatile markets. Similarly, Algidede and Panagiotidis (009) from their study on the largest 7- African stock markets found evidence of positive association between high volatility and high risk premium. The present paper focuses on 3-Asian markets comprising of Malaysia, Singapore and India. Out of these three, Malaysia and India are considered as the emerging markets while Singapore is categorized as the developed market. To the best of our knowledge there are no empirical studies particularly applied to these three markets to compare the performance of symmetric and asymmetric models in explaining the data. In this study, we aim to test whether or not the symmetric and asymmetric GARCH models are capable of explaining the dynamics of stock returns behavior in these three countries and their relative performances. This paper is structured as follows: following the introduction, section briefly discusses the basic statistics about the data. Section 3 outlines methodological framework. Section 4 presents the results. Conclusions are provided in section 5 followed by a list of references used in this study.. Materials and Methodology We used the daily closing prices of stock index of each market collected from online database (yahoo.finance.com and cross-checked by marketwatch.com) over the period from January 007 to December 013. The daily index returns are expressed in the continuously compounded returns calculated as r t = log(p t ) log(p t-1 ) where p t and p t-1 are the index prices on day t and t-1 respectively. Before proceeding further for formal statistical tests, we plot the percentage changes in daily index prices along with bands of plus or minus one conditional standard deviation ( t ) based on the GARCH (1, 1) models over the study period as exhibited in figure 1 through figure 3 in order to get an initial clue about the likely nature of the series. Figure 1 KLCI: Percentage changes in the log of daily index prices and GARCH (1, 1) Bands Percent per annum Observations 184

4 Figure STI: Percentage changes in the log of daily index prices and GARCH (1, 1) Bands Percent per annum Observations Figure 3 BSESN: Percentage changes in the log of daily index prices and GARCH (1, 1) Bands Percent per annum Observations Figures (1 to 3) show the patterns of returns of the series leading up to the terminal value. They exhibit considerable swings or volatility in the return series over the sample period. The bulges in the return plots are the graphical evidence suggesting that the volatility is time varying. Putting differently, the bulges in the return plots indicate the presence of volatility clustering effect in the series whereby the series exhibit some periods of high volatility and some periods of relatively low volatility. Presence of volatility clustering also implies that there is autocorrelation in the squared returns. The narrow and wide conditional standard deviation bands indicate the periods of smaller and larger daily stock price volatilities. Tight bands indicate lower levels of risk and wide bands indicate higher levels of risk for investors holding the indices. The conditional volatility of stock price changes varies considerably over the period with higher volatility from the beginning of 007 till mid-009 followed by a relatively smaller volatility up to the terminal period as shown by the tight standard deviation bands. In Table 1, we present some illustrative statistics for each of the return series separately. The results show that during the sample period, Malaysian market observed the highest mean daily return of 0.098% followed by Indian market 0.04% and the Singapore market 0.004%. From the daily standard deviations, Indian equity market appears to be the most volatile followed by the Singaporean equity market. The Malaysian equity market is the least volatile. All the return series show evidence of fat tails, since the kurtosis exceeds 3 (the normal value). For KLCI and STI show negative skewness suggesting that the distributions have long left tail. Indian market is positively skewed meaning to say the distribution has a long 185

5 M. A. Islam right tail. The Jarque-Bera (JB) test of normality clearly rejects the null hypothesis of normality in all cases. The tests suggest that the distributions of the return series are non-normal. Table 1: Summary Statistics of Index Returns Series KLCI STI BSESN Mean (%) Maximum (%) Minimum (%) Std. Dev. (%) Skewness Kurtosis Jarque-Bera (J-B) Probability of J-.B No. of Observations Data Stationarity Test (Unit Root Test) In order to check whether the financial time series (returns) are stationary or not, we have applied the standard Augmented Dickey-Fuller (ADF) test (Dickey and Fuller,1979), Phillips- Perron (PP, 1988) test, and Kwiatkowski, et. al. (KPSS, 1991) test. All the test statistics suggested that the series at level are nonstationary but at first level (returns) they are stationary at 1% significance level. This ensures that we can use the time series stochastic models to examine the dynamic behavior of volatility of the returns over time. The results are presented in table as below: Table : Univariate Unit Root Tests for Stock Return Series. Level Ist Difference ADF PP KPSS ADF PP KPSS Malaysia [KLCI] (1) () 3.673[3] (0)* (7)* 0.15[]* Singapore [STI] (0) (9) 0.63[33]** (0)* (8)* 0.10[8]* India [BSESN] (1) -1.65(11).13[3] (0)* (14)* 0.06[1]* Notes: *and **indicate significance at the 1%, and 5% levels respectively. Figures in parenthesis refer to the lag order selected based on SIC for ADF and Newey-West Bandwidth for PP. The numbers in the brackets refer to lag order selected based on Newey-West Bandwidth using Bartlett Kernel. Testing For Arch-Effect The linear structural model assumes that the variance of the errors is constant over time. But this assumption is not applicable for many financial data particularly the stock prices or stock indices in which the errors exhibit time-varying heteroskedasticity. Before proceeding to applying GARCH models, it is necessary to ascertain the existence of ARCH effects in the residuals. To test for ARCH effects in the conditional variance of u t ( t = Var(u t t-1 )), we followed two steps: First we consider the AR(1) model for the returns series of each individual index as: rt 0 1 rt 1 ut (1) and run the linear regression on it to obtain the residuals u t. Secondly, we run a regression of squared OLS residuals (u t ) obtained from equation (1) on q lags of squared residuals to test for ARCH of order q. The ARCH (q) specification for t is denoted as- t 0 1u u... qu () The null hypothesis of no ARCH effect t tq 186

6 H0 : 1... q 0 is tested against the alternative hypothesis that, H1 : 1 0, 0,... q If the value of the LM version of test statistic is greater than the critical value from the (q) distribution, or the coefficient of the lagged term is statistically significant, then the null hypothesis is rejected that there is no ARCH effect in equation (1). The same conclusion can be achieved if the F-version of the test is considered. We carried out the test for a lag order of q=3. The test results are presented in Table 3. Table 3: ARCH-LM Test for Residuals of Returns Series KLCI STI BSESN ARCH-LM test statistic (n*r ) Prob. Chi-square {3} [0.000] [0.0000] [0.000] F-statistic Prob. (F-statistic) [0.000] Notes: Figure in {.} refers to the order of lag and in [.] refer to p-values. The LM version of the test statistic is defined as n*r (where n is the number of observations and R is the coefficient of correlation). The GARCH (P, Q) Models The ARCH model introduced by Engle (198) is one of the particular non-linear models that has proved very useful in the application to many economic time series especially to financial time series analysis. In the ARCH, the conditional variance of the error term u t is modeled as being normally distributed with mean zero and variance t, where the t is expressed as a function of past squared error values u t as stated in equation (). In estimating an ARCH model, it is required that the unknown coefficients ( 0, 1,. q ) are nonnegative since the variance cannot be negative. If these coefficients are positive and the recent squared residuals are large, the ARCH model predicts that the current squared error will be large in magnitude in the sense that its variance t is large (Stock, 01). An extension of ARCH model is the generalized ARCH or GARCH model developed by Bollerslev (1986). In GARCH model, the variance t is allowed to be dependent upon its own past values as well as lags of the squared error terms. The general form of a GARCH (p, q) model is defined as Mean equation: rt t t (3) p q Variance equation: t p t p q tq p1 q1 (4) Where, > 0, and, > 0, and t = residual returns defined as t = t e t, where e t N(0, 1) r t = index returns at time t t = average return p is the order of the moving average ARCH terms and q is the order of the autoregressive GARCH terms. The simplest and most commonly used GARCH model is the GARCH (1, 1) which is reasonably a good model for analyzing financial time series as well as to estimating and forecasting the time-varying volatility of returns of financial assets, especially the high frequency financial assets such as daily stock index returns. The Symmetric GARCH (1, 1) Model In financial markets, volatility is known as a measure of uncertainty about the return provided by the stocks or stock indices. The volatility of many economic time series, especially financial time series changes 187 0

7 M. A. Islam over time. In some periods the daily stock returns exhibit high volatility while in other periods they exhibit low volatility, a commonly observed phenomenon in financial time series which is referred to as volatility clustering. That is volatility comes in cluster. It is assumed that a day of high volatility most likely to be followed by another day of high volatility within each state or over a short period of time. As such, linear models which assume homoscedasticity (constant variance) are inappropriate to explain such unique behavior of financial time series data. It is preferable to use models that examine behavior of financial time series allowing the variance to depend upon its history. GARCH (1, 1) model is capable of capturing the volatility clustering effects in the financial time series data. The GARCH models are especially suitable for financial market data as the GARCH processes are fat-tailed compared to the normal distribution. The standard GARCH (1, 1) model is defined as t V L u (5) Where, V L is the long-run average variance rate, is the weight assigned to the V L, is the weight assigned to u t-1 and is the weight assigned to t-1. Weights must be equal to unity as, + + = 1. Equation (5) can be written by setting = V L as, t u, where, > 0,, > 0, > (6) A stable GARCH (1, 1) process requires + < 1. Once the parameters of the GARCH model are estimated, the long-term variance, V L and can be calculated as / and respectively. The GARCH (1, 1) model in equation (6) estimates the current volatility of assets returns based on a linear combination of the last period s squared returns and the last period s volatility. Since the GARCH model is no longer of the usual linear form, the parameters in GARCH (1, 1) model cannot be estimated by the usual OLS method. As such to estimate GARCH parameters, alternative technique is used. The most common method to estimate the GARCH parameter is to take the log likelihood which is the logarithm of the Maximum Likelihood (ML) method. ML employs trials and errors to determine the optimal values for the parameters that maximize the likelihood of the data occurring. t u t 1 t 1 (7) The parameter in the conditional mean return λ is the risk premium parameter. The time- varying risk premium is estimated by the significance of the λ coefficient of t in the conditional mean equation. If the coefficient of λ is positive and significant, then the increased expected return is said to be caused by the increased expected risk or conditional variance/standard deviation. GARCH (1, 1) and GARCH-M are considered to be symmetric models which imply that the positive and negative shocks of equal size elicit an equal response from the market. The Asymmetric Threshold GARCH (TGARCH) Model Threshold GARCH is another variant of GARCH models proposed by Zakoian (1994) which is capable of modeling the asymmetric effects of the past negative and positive innovations (bad or good news) on the conditional variance. The TGARCH model specification for the conditional variance is expressed as: t u u St 1 (8) t 1 Where, St 1 1if u t 1 0,(negative shocks) and S t 1 0 if u t 1 0, (positive shocks). In this model, u can have different effects on the conditional variance t depending on whether u is above or below the threshold value of zero. When u 0, the impact on the conditional variance is u t 1 (+), and when u t 1 0, the impact is u. The coefficient is known as the asymmetry parameter and is expected to be positive and significant so that bad news can have proportionally larger impact on the volatility of the returns than the positive shocks. This asymmetry is also known as a leverage effect in the 188

8 literature. The basic idea behind the TGARCH model is closely related to that of the GJR-GARCH model developed by Glosten, Jaganathan and Runkle (1993). 3. Results and Discussions The results of the ML estimates of the GARCH parameters are presented in Table 4. In the GARCH estimation, maximum likelihood method is used assuming student s t-distribution for the conditional distribution of the errors, u t. The results show that the estimate of the standard GARCH parameters and are positive and statistically highly significant for all specifications. The values of coefficient are found to be very high ranging between 83% - 91% which implies persistent volatility clustering. The statistical significance of and indicates that the news on volatility from the past periods have impact on the current volatility. It can be seen from the results that the sum of the two estimated coefficients ( and ) are very high but less than one ( + < 1) indicating that the changes in the conditional variances are persistent. This also implies the long periods of volatility clustering as seen in figure 1 through 3. The ML estimates of the asymmetry parameter () in TGARCH (1, 1) appeared to be significant and with correct sign suggesting the existence of standard leverage effect in all three markets. In other words, negative shocks have higher impact on conditional variance (volatility) than the positive shocks of the same magnitude. Table 4: Estimation Results of GARCH (1, 1) and TGARCH (1, 1) Models GARCH (1, 1): Variance equation KLCI STI BSESN 1.33e-06* 8.84e-07**.16e-06** * * * * * * ARCH-LM test statistic (n*r ) Prob. Chi-Square[3] (0.7404) (0.4637) (0.3010) Long-term variance rate, V L Long-term volatility: Per-day 1.184% 1.794% 1.968% Per year 18.79% 8.47% 31.4% TGARCH (1, 1): Variance equation 1.4e-06* 8.64e-07*.98e-06* * * * ** * * * * ARCH-LM test statistic (n^r ) Prob. CHSQ[] (0.785) (0.596) (0.3044) Notes: *, and ** refer to significance levels at 1%, and 5% respectively. Figures in (.) refer to p-values. The LM version of the test statistic is defined as n^r (where n is the number of observations and R is the coefficient of correlation). 189

9 Performance Comparison: M. A. Islam In order to see the performance of the two GARCH models considered in this study we examine the autocorrelation structure of the squared residuals. According to Hull (01, p. 508), if a GARCH model is working well, it should remove the autocorrelation. To see which model has succeeded in removing the autocorrelations and to what extent, we performed a Ljung-Box (LB) test for the first 15 lags at 99% confidence interval. This test calculates the Ljung-Box statistic defined ask LB= ( m ) k m n ( m k) k 1 Where k is the autocorrelation for a lag of k and m is the observations in the series. LB statistics follow the chi-square distribution with n df. For K = 15, zero autocorrelation can be rejected with 99% confidence when the LB statistic is greater than The LB test results are presented in table 5. From the results it can be seen that before the use of GARCH, the LB statistics for the series were (KLCI), 05.0 (STI) and (BSESN) respectively. After the use of GARCH models, the LB statistics for all cases reduced to much smaller than suggesting that zero autocorrelation cannot be rejected at the chosen 99% confidence level. In other words, autocorrelation has been largely removed by the GARCH models. However, in the case of Malaysia & India, GARCH (1, 1) model appears to have done slightly better job than the TGARCH. For Singapore market, TGARCH seems to have performed better job in removing the autocorrelation in the series. Table 5: Autocorrelations Before and After the Implementation of GARCH (1, 1) and TGARCH (1, 1) Models: Ljung-Box (LB) test (at a lag of 15) KLCI % STI % BSESN % LB statistic critical value LB statistic (autocorrelation) LB statistic GARCH(1, 1) LB statistic TGARCH(1, 1) Conclusions In this paper, we have utilized two of the GARCH family models with imposing names GARCH (1, 1) and the TGARCH (1, 1). The objectives of this paper are to test the usefulness of these models in explaining the unique characteristics of financial time series and to assess the relative performance in estimating the volatility of stock index returns for the 3-Asian markets- Malaysia, Singapore and India. As a benchmark to compare the performance, we look at the extent of the models ability to remove the autocorrelation in the residual series. The key results of this study are as follows: both models are found to be sufficiently capable of capturing the dynamics of the financial time series particularly with respect to volatility clustering, the leptokurtic characteristic of the distribution of the daily return series and the asymmetric effects. Secondly, in terms of performance comparison in removing autocorrelation, GARCH (1, 1) appears to be a better fit model for Malaysia and India, while TGARCH is found to have performed better for Singapore market. Finally, we found Indian market as more volatile or riskier for investors as compared to Singapore and Malaysia markets. However, Malaysian market appeared to have the lowest volatility suggesting that this market is relatively safer for the investors who avoid taking risk. The results can be beneficial for traders and investors in making investment decisions. We have utilized only two types of GARCH models in this paper. There are a number of variations of GARCH type-models that have been developed since Engle (198) and Bollerslev (1986) such as- Power GARCH, Exponential GARCH, Component GARCH to name a few that can be considered well worth for further study in assessing the extent of these models abilities to explain the characteristics of financial time series. 190

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