Symmetric Normal Mixture GARCH

Normal mixture (NM) GARCH models are better able to account for leptokurtosis in financial data and 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 NM-GARCH(1,1) models, derive the analytic derivatives for the maximum likelihood estimation of the model parameters and their standard errors and compute the moments of the error term. 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 mixing law. We show that there is a serious bias on parameter estimates for volatility 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 moment tests, a simple mixture of two normal densities is sufficient to capture the conditional excess kurtosis in the data. According to our chosen criteria, and given our simulation results, we conclude that a two regime symmetric NM-GARCH model, which quantifies volatility corresponding to 'normal' and 'exceptional' market circumstances, is optimal for these exchange rate data.