The Mean Variance Mixing Garch (1,1) Process a New Approach to Quantify Conditional Skewness

We present a general framework for a GARCH (1,1) type of process with innovations using a probability law of the mean-variance mixing type. We call the process the mean variance mixing GARCH (1,1) or MVM GARCH (1,1). One implication of this particular specification is a GARCH process with skewed innovations and constant mean dynamics. This is achieved without using a location parameter to compensate for time dependence that affects the mean dynamics. From a probabilistic viewpoint the idea is straightforward. We construct our stochastic process from the desired behavior of the cumulants. Having done this we provide explicit expressions for the unconditional second to fourth cumulants for the process in question. We present a specification of the MVM-GARCH process where the mixing variable follow an inverse Gaussian distribution. On the basis on this assumption we can formulate a conditional maximum likelihood approach for estimating the process. This approach is closely related to the approach used to estimate an ordinary GARCH (1,1). Under the distributional assumption that the mixing random process is an inverse Gaussian i.i.d. process, the MVM-GARCH process is then estimated on log return data from the Standard and Poor 500 index. An analysis for the conditional skewness and kurtosis implied by the process is also presented in the paper. Date: First draft: November 2003 This version: April 27, 2005.