SUMMARYSome recent specifications for GARCH error processes explicitly assume a conditional variance that is generated by a mixture of normal components, albeit with some parameter restrictions. This paper analyses the general normal mixture GARCH(1,1) model which can capture time variation in both conditional skewness and kurtosis. A main focus of the paper is to provide evidence that, for modelling exchange rates, generalized two-component normal mixture GARCH(1,1) models perform better than those with three or more components, and better than symmetric and skewed Student's t-GARCH models. In addition to the extensive empirical results based on simulation and on historical data on three US dollar foreign exchange rates (British pound, euro and Japanese yen), we derive: expressions for the conditional and unconditional moments of all models; parameter conditions to ensure that the second and fourth conditional and unconditional moments are positive and finite; and analytic derivatives for the maximum likelihood estimation of the model parameters and standard errors of the estimates. Copyright 2006 John Wiley & Sons, Ltd.
INTRODUCTIONThe classic econometric approach to volatility modelling is the generalized autoregressive conditional heteroscedasticity (GARCH) framework that was pioneered by Engle (1982) and Bollerslev (1986). Whilst some degree of leptokurtosis in the unconditional returns distribution can be captured by the normal GARCH(1,1) model, Bollerslev (1987), Baillie and Bollerslev (1989), Hsieh (1989a,b), Nelson (1996), Johnston and Scott (2000) and many others have all concluded that, in daily or higher frequency data, the observed non-normalities in both conditional and unconditional returns is higher than can be predicted by normal GARCH(1,1) models. Consequently, a number of non-normal conditional densities have been considered in the GARCH framework. Notably, Bollerslev (1987) introduced the Student's t-GARCH and, more recently, non-normalities have also been captured by GARCH models with a flexible parametric error distribution such as those based on the exponential generalized beta (Wang et al., 2001).1 Alternatively, the inclusion of a trend for the long-term volatility can increase the unconditional leptokurtosis, as in the component model of Engle and Lee (1999). Non-distributional models, such as the semi-parametric ARCH model of Engle and Gonzalez-Rivera (1991), have also been considered.These developments in GARCH models are clearly important for modelling exchange rate returns where non-normalities are highly significant (Boothe and Glassman, 1987;de Vries, 1994;Huisman et al., 2002). Amongst the earliest applications of GARCH models to exchange rates were Engle and Bollerslev (1986) and Hsieh (1988), who rejected the hypothesis that these data have a heavy-tailed distribution with fixed parameters over time. The normal GARCH model
