A framework for normal mean variance mixture innovations with application to Garth modelling

Abstract

GARCH models are useful to estimate the volatility of financial return series. Historically the innovation distribution of a GARCH model was assumed to be standard normal but recent research emphasizes the need for more general distributions allowing both asymmetry (skewness) and kurtosis in the innovation distribution to obtain better fitting models. A number of authors have proposed models which are special cases of the class of normal mean variance mixtures. We introduce a general framework within which this class of innovation distributions may be discussed. This entails writing the innovation term as a standardised combination of two variables, namely a normally distributed term and a mixing variable, each with its own interpretation. We list the existing models that fit into this framework and compare the corresponding innovation distributions, finding that they tend to be quite similar. This is confirmed by an empirical illustration which fits the models to the monthly excess returns series of the US stocks. The illustration finds further support for the ICAPM model of Merton, thus supporting recent results of Lanne and Saikonnen (2006)