Symmetric Normal Mixture GARCH

Normal mixture GARCH models offer a more intuitive and tractable framework for risk analysis and option pricing than student's t-GARCH models. We present a general, symmetric parameterisation for normal mixture GARCH(1,1) models, with analytic derivatives for the maximum likelihood estimation of the model parameters, deriving also the standard errors of the estimates and expressions for the unconditional moments. Also, we formulate specific conditions on the model parameters to ensure positive, finite conditional and unconditional second and fourth moments. Simulations quantify the potential bias and inefficiency of parameter estimates as a function of the weight of the variance component in the mixture. We show that there is a substantial bias of a definite sign on parameter estimates for variance components having very low weight in the mixing law. An empirical application uses moment specification tests and information criteria to determine the optimal number of normal densities in the mixture. For daily returns on three US dollar foreign exchange rates (British pound, Euro and Japanese yen) we find that, whilst normal GARCH(1,1) models fail the specification tests, a simple mixture of two normal densities is found sufficient to capture the conditional and unconditional excess kurtosis in the data. According to our chosen criteria, and given our simulation results, we conclude that a symmetric NM(2)-GARCH(1,1) model, which has the additional benefit of quantifying volatility corresponding to 'normal' and 'exceptional' market circumstances, is optimal for these exchange rate data