Forecasting Volatility of Stock Indices with ARCH Model

The main motive of this study is to investigate the use of ARCH model for forecasting volatility of the DSE20 and DSE general indices by using the daily data. GARCH, EGARCH, PARCH, and TARCH models are used as benchmark models for the study purpose. This study covers from December 1, 2001 to August 14, 2008 and from August 18, 2008 to September 10, 2011 as in-sample and out-of-sample set sets respectively. The study finds the past volatility of both the DSE20 and DSE general indices returns series are significantly, influenced current volatility. Based on in-sample statistical performance, both the ARCH and PARCH models are considered as the best performing model jointly for DSE20 index returns, whereas for DSE general index returns series, ARCH model outperforms other models. According to the out – ofsample statistical performance, all models except GARCH and TARCH models are regarded as the best model jointly for DSE20 index returns series, while for DSE general index returns series, no model is nominated as the best model individually. Based on the in-sample trading performance, all models except GARCH are considered as the best model jointly for DSE20 index returns series, while ARCH model is selected as the best model for DSE general index returns series. A per outputs of out-of-sample trading performance, the EGARCH model is the best performing model for DSE20 index returns series, whereas the GARCH and ARCH models are considered as the best performing model jointly for DSE general index returns series.


Introduction
Forecasting of the stock exchange index is a motivating and tricky issue for both for investors and academics. The stock market is an extremely nonlinear vibrant system whose performance is manipulated by a number of factors, namely inflation rates, interest rates, economic atmosphere, political issues, and so on (Sutheebanjard and Premchaiswadi, 2010). In generic sense, financial markets and in particular sense, stock markets are characterized by uncertainty. The prices of financial securities, which are traded in the financial markets as well as interest rate and foreign exchange rates, are horizontal to constant inconsistency. For this type of changeability, their returns over the various periods of time are notably volatile and complicated to forecast. Volatility is an important variable for appraising the status of a financial market as well as for taking decision by its participants, like investors, investment managers, speculators and the financial supervisory body (Panait and Slăvescu, 2012). Forecasting volatility is a crucial and exigent financial matter, which have attained much concentration. It is broadly consented that though returns of financial securities prices are more or less unpredictable on daily as well as monthly basis, return volatility is forecastable, phenomenon along with vital inference for financial economics and risk management (Torben et al. 2009). Precise volatility forecasts are essential to traders, investors, financial analyst and researchers who are interested in realizing stock market dynamics (Ederington and Guan, 2005). Trading in stock market indices has achieved unparalleled attractiveness all over the world. The increasing diversity of financial index related instruments, along with economic growth enjoyed in the last few years, has broaden the dimension of global investment opportunity to both the individual and institutional investors. Index trading vehicles give an effectual way for the investors for hedging against prospective market risks as well as they generate new return making opportunities for market arbitragers and speculators. Therefore, being able to appropriately forecasting of stock index has thoughtful inference and important to researchers and practitioners identical (Leung et al. 2000). To conduct a study on stock market is a severe and challenging monetary activity. Enormous return may be earned through extremely accurate predictions by using a suitable forecasting model, however aggressive fluctuations in the stock market activity make forecasting as a tricky issue. So, forecasting accuracy is a most important concern of numerous investors, highlighting the magnitude of structuring a more appropriate forecasting model (Chang et al. 2009). The stock market is a set of connections that gives a platform for about each economic transaction in the business world at a dynamic rate entitled the stock value that is based on the market equilibrium. Forecasting this stock value offers huge arbitrage profit opportunities, which are the main motivation for doing research in this field (Gupta and Dhingra, n.d.). Volatility is a vital factor for determining price of financial instruments like stocks, options, and futures, is a measure of trade-off between risk and return on an investment. The volatility of stock market has a significant influence on financial rules and regulations, monetary and fiscal policies as well. The realistic significance of modelling and forecasting volatility in various finance applications represent that the accomplishment or failure of volatility models depend upon the features of experimental data which they attempt to capture and forecast. Volatility of share market is a crucial issue for the government's policy makers, market analysts, corporate and financial managers, since a remarkable volatility in a share market leads to an adverse impact for a country's economy (Islam et al. 2012).
The key motivation of the study is to forecast volatility of the stock indices with ARCH class model.

Literature Review
Islam et al. (2012) conduct a study on forecasting volatility of Dhaka stock exchange by using linear as well as non-linear models and find that among linear model, the moving average model occupies first position according to root mean square error, mean absolute error, Theil-U and linex loss function criteria. They also find that non-linear models do not outperform linear models based on various error measurement criteria and moving average model nominates as the best model. Sutheebanjard and Premchaiswadi (2010) reveal that the projected prediction function not only yields the lowest MAPE for short-term periods but also yields a MAPE lesser than 1% for long-term periods. Dunis, Laws and Karathanasopoulos (2011) state that the mixed -HONNs and the mixed RNNs models carry out outstandingly as well and appear to have an capability in giving superior forecasts when autoregressive series are only applied as inputs. Louzis, Sisinis, and Refenes (2010) mention that compared with recognized HAR and Autoregressive Fractionally Integrated Moving Average (ARFIMA) realized volatility models, the proposed model shows superior in-sample fitting, as well as , out-of-sample volatility forecasting performance. According to Panait and Slăvescu (2012), the GARCH-in-mean model is unsuccessful to validate the theoretical hypothesis that there is a positive relationship between volatility and future returns, principally due to the variance coefficient from the mean equation of the model is not statistically significant for the majority of the time series analyzed and on most of the frequencies. MCMillan, Speight and Apgwilym (2000) reveal that the random walk model gives immensely better monthly volatility forecasts, whereas random walk, moving average and recursive smoothing models present moderately better weekly volatility forecast, and GARCH, moving average and smoothing models produce marginally better daily forecasts. Lee, Chi, Yoo and Jin (2008) find that among Back Propagation Neural Network (BPNN), Bayesian Chiao's (BC), and SARIMA models, the SARIMA model is nominated as the best model for mid-term and long-term forecasting, whereas BC model is selected as the best model for short-term forecasting. Chen (2011) reveals that the total index in terms of percentage is ten times that of the buy-and-hold method and two times that of Wang and Chan's (2007) model. Al-Zeaud (2011) conducts a study on modelling and forecasting volatility using ARIMA model and reveals that ARIMA (2,0,2) is the best model for banking sector, since this model provides the lowest mean square error followed by ARIMA (1,1,1). Leung, Daouk and Chen (2000) find that the classification model beat the level estimation model in the light of forecasting the direction of the stock market movement and maximizing returns from investment trading. Chang, Wei and Cheng (2009) demonstrate that the proposed model is better than the listing methods in respect of root mean square error. Yalama and Sevil (2008) reveal that the asymmetric volatility class models outperform the historical model for forecasting stock market volatility. According to Mehrara, Moeini, Ahrari and Ghafari (2010), the exponential moving average model beat the simple moving average model as well as the Group Method of Data Handling do better than Multi-Layered Feed Forward network model for forecasting stock price index. Tang, Yang and Zhou (2009) reveal that the proposed algorithm can assist to get better the performance of normal time series analysis in stock price forecasting significantly.

Data
Only the time series data is used in this study consists of the Dhaka Stock Exchange (DSE) indices, namely DSE20 Index and DSE General Index. The requisite data is obtained from the DSE library for the study purpose. The study period covers from December 1, 2001 to September 10, 2011 which contains 2600 trading days. The total data set is divided into in-sample and out-of-sample data set. The in-sample data set covers from December 1, 2001 to August 14, 2008 and includes 1733 observations, whereas out-of-sample covers from August 18, 2008 to September 10, 2011 and incorporates 867 observations.

Jarque-Bera Statistics
Jarque-Bera statistics is applied to examine the non-normality of the DSE20 and DSE general stock indices. Figure 1. DSE20 index summary statistics Figure 1 reveals that a positive skewness, 0.976402, and a high positive kurtosis, 3.145290. As per the Jarque-Bera statistics, DSE20 index is non-normal at the confidence interval of 99%, since probability is 0.0000 which is less than 0.01. So, it is mandated to convert the DSE20 index series into the return series.  Figure 2 demonstrates that a positive skewness, 1.201375 as well as a positive kurtosis, 3.600467. As per Jarque-Bera statistics, the DSE general index is non-normal at the confidence interval of 99%, since probability is 0.0000 which is less than 0.01. So, it is also needed to convert the DSE general index series into the return series.

Transformation of the DSE20 Index and DSE General Index Series
In general, the movements of the stock indices series are non-stationary, quite random and not appropriate for the study purpose. The series of DSE20 index and DSE general index are transformed into returns by using the following equation: - R t = the rate of return at time t P t = the stock index at time t P t -1 = the stock index just prior to the time t

Augmented Dickey-Fuller (ADF) Test and Phillips-Perron (PP) Test on DSE20 Index and DSE General Index Returns Series
ADF test as well as PP test are used to get confirmation regarding whether BDT/USD exchange rates return series is stationary or not.  -47.23678, is less than its test critical value, -2.862452, at 5%, level of significance which implies that the DSE20 index return series is stationary. An outcome of ADF test confirms that the DSE general index return series is stationary, because the values of ADF test statistic is less than its test critical value.  Table 2 illustrates the results of the PP test and proves that the DSE20 index returns series is stationary, because the values of PP test statistic, -47.43146, is less than its test critical value, 2.862452, at the level of significance of 5%. The findings of the PP test also confirms that the DSE general index returns series is stationary, since the values of PP test statistic is less than its test critical value.  Jarque-Bera 299961.9 Probability 0.000000 Figure 3 reveals a negative skewness, -2.494639, and a positive kurtosis, 55.38397. As per the Jarque-Bera statistics, the DSE20 index returns series is non-normal at 95% confidence level, since probability is 0.0000 which is less than 0.05. Based on the Jarque-Bera statistics, the DSE general index returns series is non-normal at 5% level of significance, because the probability, 0.0000, is less than 0.05.

Benchmark Model
ARCH model is benchmarked with GARCH, EGARCH, PARCH, and TARCH models in this study.
3.6.1.1 GARCH Model GARCH model is developed by Bollerslev (1986) & Taylor (1986 independently and according to this model the conditional variance to be dependent upon previous own lags. The form of this model is given below: 3.6.1.2 EGARCH Model EGARCH) model is developed by Nelson (1991). The conditional variance equation can be presented in the following form: The PARCH model is an extension of the GARCH model with an additional term added to account for possible asymmetries (Brooks, 2008). The conditional variance is now given by (4) 3.6.1.4 TARCH Model Zakoïan (1994) & Glosten et al. (1993 use the TARCH model with an intention of independence than for the asymmetric effect of the "news" (Brooks, 2008). Form of this model is as follows:

ARCH Model
It is a non-linear model which does not assume that the variance is constant, and it describes how the variance of the errors evolves. Many series of financial asset returns that provides a motivation for the ARCH class of models, is known as 'volatility clustering' or 'volatility pooling'. Volatility clustering describes the tendency of large changes in asset prices (of either sign) to follow large changes and small changes (of either sign) to follow small changes.  Under the ARCH model, the 'autocorrelation in volatility' is modelled by allowing the conditional variance of the error term, , to depend on the immediately previous value of the squared error and ARCH(1) model takes the following form (Brooks, 2008): The form of ARCH (q) model is as follows where error variance depends on q lags of squared errors: The statistical performance measures, like mean absolute error (MAE); mean absolute percentage error (MAPE); root mean squared error (RMSE); and theil-u, are applied to pick the best performing model both in the in-sample data set and the out-of-sample data set independently in this study. There is a negative association between the forecasting volatility accuracy of the model and the output of RMSE, MAE, MAPE and theil-U.

Measures of the Trading Performance of the Model
The trading performance measures, namely annualized return ( ); annualized volatility ; Sharpe ratio (SR); and maximum drawdown (MD), are applied to pick the best model. The values of annualized return and Sharpe ratio are positively associated with the forecasting volatility accuracy of a given model, whereas annualized volatility and maximum drawdown are inversely associated. The outputs of ARCH model on DSE20 index and DSE general index show that the constant, C, is not statistically significant both in the mean and variance equations, since the probability of C is greater than 0.00. The variance equation illustrates that RESID(-1)^2 term is also statistically significant at 1% level of significance which implies that the volatility of risk is influenced by past square residual terms. Therefore, it can be mentioned that the past volatility of both the DSE20 index and DSE general index is significantly, influencing the current volatility. The outputs of the GARCH model on DSE20 index and DSE general index illustrate that the constant, C, is not statistically significant both in the mean and variance equations. The variance equation describes that the RESID(-1)^2 term is statistically significant at both the DSE20 and DSE general indices returns which imply that the volatility of risk is influenced by past square residual terms. The GARCH (-1) term is also statistically significant in the both indices. So, it can be mentioned that the past volatility of both the DSE20 index and DSE general index is significantly, influencing the current volatility. Outcomes of the EGARCH model demonstrate that the term, C, is not statistically significant in the mean. The variance equation describes that the C(4), C(5), and C(6) terms are statistically significant which imply that past volatility of stock indices are significantly, influencing current volatility. The EGARCH variance equation also signifies that there exists the asymmetric behavior in volatility which means that positive shocks are effecting, differently, than the negative on volatility. Outputs of the PARCH model show that the term, C, is not statistically significant in the mean equation. The variance equation describes that the terms, C(4), C(5), C(6), C(7), and C (7) are statistically significant which imply that past volatility of DSE20 and DSE general indices are significantly, influencing current volatility. Results of the TARCH model represent that the terms, C, are not statistically significant in both the mean as well as variance equations. The variance equation describes that the RESID(-1)^2, RESID(-1)^2*(RESID(-1)<0), and GARCH(-1) terms are statistically significant which imply that past volatility of DSE20 and DSE general indices are significantly, influencing current volatility.

In-Sample Statistical Performance
The following table presents the comparison of the in-sample statistical performance results of the selected models.   Table 14 demonstrates that both all models have the same and lowest MAE at 0.0074., whereas AR has the lowest MAPE at 67.43%. ARCH model has the lowest MAPE and RMSE 176.57% and at 0.0105, whereas the EGARCH model has the lowest theil's inequality coefficient at 0.7984. Therefore, the best is the ARCH model in case of DSE general index returns series.  Table 15 illustrates that all models have the same and the lowest MAE at 0.0109. PARCH model has the lowest MAPE at 142.64%, EGARCH model has the lowest RMSE at 0.0171 and ARCH model has the lowest theil's inequality coefficient at 0.8499. So, it can be mentioned that all models except GARCH and TARCH models are nominated as the best model once in case of DSE20 index return series.  Table 17 shows that the TARCH model has the highest annualized return at 39.43%, whereas EGARCH, PARCH, and TARCH models have the same and the lowest annualized volatility at 17.10%. Both the PARCH and TARCH models have the same and highest Sharpe ratio at 3.13. The ARCH model has the minimum downside risk s measured by maximum drawdown at -16.25%. All models except GARCH are selected as the best model once in case of DSE20 index returns series.  Table 19 demonstrates that the EGARCH model has the highest annualized return, lowest annualized volatility, and highest at Sharpe ratio at 48.06%, 40.15%%, and 1.20 respectively. ARCH model has the lowest maximum drawdown at -53.35%. Therefore, the EGARCH model is selected as the best performing model DSE20 index returns series.  Table 20 shows that the GARCH model has the highest annualized return, and Sharpe ratio at 40.64% and 0.96 respectively. On the other hand, ARCH model has the lowest annualized volatility and maximum drawdown at 41.63% and -61.02% accordingly. Therefore, the GARCH ARCH models are nominated as the best performing model twice, whereas other models are not nominated single time in case of DSE20 index returns series.

Conclusion
In this study, ARCH model is used to forecast volatility of the stock indices, namely DSE20 index and DSE general index. GARCH, EGARCH, PARCH, and TARCH models are applied as benchmark models for the study purpose. The daily data from December 1, 2001 to September 10, 2011 is used in this study out of which, in-sample data set covers from December 1, 2001 to August 14, 2008 and, whereas out-of-sample covers from August 18, 2008 to September 10. The results of ARCH models on both the DSE20 and DSE general indices series show that in the variance equation the terms, C and RESID(-1)^2 are statistically significant which imply that the volatility of risk is influenced by past square residual terms. The outcomes of the GARCH model on the selected stock indices returns also demonstrate that the RESID(-1)^2 term is statistically significant which imply that the volatility of risk is influenced by past square residual terms. The GARCH (-1) term is also statistically significant for the both indices. Outputs of EGARCH model on the sample stock indices show that the C(4), C(5), and C(6) terms are statistically significant which imply that past volatility of stock indices are significantly, influenced current volatility. Therefore, the EGARCH variance equation demonstrates that the asymmetric behavior are existed in volatility which means that positive shocks are affected, differently, than the negative shocks on volatility. The outputs of PARCH model on the selected stock indices returns series demonstrate that the terms, C(4), C(5), C(6), C(7), and C(7) are statistically significant which imply that past volatility of DSE20 and DSE general indices returns series are significantly, influenced present volatility. In addition, the results of TARCH model on the sample stock indices illustrate that in the variance equation the RESID(-1)^2, RESID(-1)^2*(RESID(-1)<0), and GARCH(-1) terms are statistically significant which mean that past volatility of DSE20 and DSE general indices returns series are significantly, influenced current volatility.