Modelling multivariate asymmetric financial volatility

The univariate Generalised Autoregressive Conditional Heterscedasticity (GARCH) model has successfully captured the symmetric conditional volatility in a wide range of time series financial returns. Although multivariate effects across assets can be captured through modelling the conditional correlations, the univariate GARCH model has two important restrictions in that it: (1) does not accommodate the asymmetric effects of positive and negative shocks; and (2) assumes independence between conditional volatilities across different assets and/or markets. In order to capture such asymmetric effects, Glosten et al. (1993) proposed a univariate asymmetric GARCH (or GJR) model. However, the univariate GJR model also assumes independence between conditional volatilities across different assets and/or markets. Several multivariate GARCH models have been proposed to capture such interdependencies, but none has been designed to capture asymmetry, apart from the constant correlation multivariate asymmetric GARCH (CC-MGJR) model of Hoti et al. (2002). The CC-MGJR model captures asymmetric effects and permits interdependencies between conditional volatilities across different assets and/or markets. In addition, the structural and statistical properties of the CC-MGJR model have been established, and the sufficient conditions for consistency and asymptotic normality can be verified in practice. The aim of this paper is to model the multivariate asymmetric conditional volatility of three different stock indexes, namely S&P 500, Nikkei and Hang Sang, using the CC-MGARCH model of Bollerslev (1990), the vector ARMA-GARCH model of Ling and McAleer (2003), and the CC-MGJR model of Hoti et al. (2002). Extensive empirical results support the presence of asymmetric effects across the stock indexes, as well as interdependencies in conditional volatilities across different markets.