VOLATILITY MODELING OF THE JSE ALL SHARE INDEX AND RISK ESTIMATION USING THE BAYESIAN AND FREQUENTIST APPROACHES

(ProQuest: denotes formulae omitted.)1.IntroductionFinancial time series data is generally characterized by volatility clustering, leptokurtosis and heavy tailed distributions. In this paper we carry out a comparative analysis of Bayesian and Maximum Likelihood (ML) parameter estimates of the All Share Index (ALSI) at the Johannesburg Stock Exchange (JSE) Volatility modeling of financial time series data was pioneered by Engle (1982) who developed Autoregressive Conditional Heteroskedasticity (ARCH) models. The developed ARCH models were aimed at capturing the high volatility in the log returns of financial time series data. The ARCH models were extended to Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models by Bollerslev (1986) To date there are several variants of the GARCH type models which include integrated GARCH model (see Engle and Bollerslev (1986) exponential GARCH (Nelson, 1991) GJR GARCH model of Glosten et al. (1993) asymmetric power ARCH model of Ding et al. (1993) family GARCH model of Hentschel (1995) Component GARCH model by Lee and Engle (1999) among others.Modeling volatility using heavy tailed distributions such as the Student t distribution is discussed extensively in literature. Cifter (2012) investigates the performance of an asymmetric normal mixture generalized autoregressive conditional heteroskedasticity (NM GARCH) model with benchmark GARCH type models. Results from this study show that the NM GARCH model captures the time variation in kurtosis and skewness. Cifter (2012) shows that the predictive power of volatility models can be improved by using the NMGARCH (1,1) with skewed Student t distribution.ARMA GARCH type models for modeling volatility and financial market risk of shares on the Johannesburg Stock Exchange (JSE) are developed in Makhwiting et al. (2012) Results from this study show that ARMA(0,1)GARCH(1,1) model achieves the most accurate volatility forecast. Mzamane (2013)investigates volatility in the JSE index using univariate and multivariate GARCH models. Empirical results from this study show the presence of the leverage effect in the log returns of the JSE index. Niyitegeka and Tewari (2013) investigate the volatility of stock return on the JSE using GARCH type models. Empirical results show that stock returns volatility is persistent and that the leverage effects do not exist. Harvey and Sucarrat (2014) show how the Beta t EGARCH model generalizes to a Skew t model. The authors show that the Beta t EGARCH model provides a better fit compared to the GARCH GJR which is used as a benchmark model. Xia et al. (2015) develop autoregressive models with exogeneous variables and power transformed and threshold GARCH errors (ARX PPTGARCH) Parameter estimation is done using the Bayesian approach. A comparative analysis is done with an ARX GARCH model. Results from this study show that the ARX PPTGARCH model captures well the important financial characteristics such as asymmetry and heteroscedasticity among others.This paper focuses on volatility modeling of the JSE all share index and risk estimation using the Bayesian and frequentist approaches. The Bayesian approach is more attractive than the frequentist approach since it combines prior information with data, enables analysis with small samples, and the estimation is robust (Ardia and Hoogerheide, 2010) Bayesian analysis provides a coherent modeling framework for solving problems under conditions of uncertainty (Marin et al. 2003) Similar studies on a comparative analysis of the frequentist and Bayesian approaches are discussed in literature. Nakatsuma and Tsurumi (1996) compare ARMA GARCH type models on small samples using both the frequentist and Bayesian approaches. Empirical results show that Bayes estimation outperforms the frequentist approach (maximum likelihood estimation) Hoogerheide et al. (2012) compare the performance of some GARCH type models using stock index returns.