Introduction

The Autoregressive Conditional Heteroskedasticity (ARCH) model was introduced by Engle (1982). In this model the conditional variance of the errors is assumed to be a function of the squared past errors. Engle derives the Maximum Likelihood (ML) estimator for the ARCH model under the assumption that the conditional density of the error term is normal. Bollerslev (1986), suggested the Generalized Autoregressive Conditional Heteroskedasticity model (GARCH) in which the conditional variance of the errors is assumed to be a function of its lagged values and the squared past errors. Bollerslev derives the Maximum Likelihood (ML) estimator for the GARCH model under the assumption that the conditional density of the error term is normal. The ARCH and GARCH models are useful in modelling economic phenomena, mainly in the theory of finance (see e.g., Bollerslev et al. 1992 and Engle, 2002). In the above models the conditional density of the error term is assumed to be normal but in the applications with actual data, distributions other than the normal have been observed with fatter tails or with skewness significantly different from zero. For this reason, in particular applications with real data, other distributions have been used. Bollerslev (1987) used the Student's t distribution to model the monthly returns composite index. Baillie and Bollerslev (1989) also used the Student's t distribution while Hsieh (1989) chose the mixture Normal–Lognormal to model daily foreign-exchange rates.