Volatility Modeling with a Generalized t Distribution

Abstract

Exponential generalized autoregressive conditional heteroscedasticity models in which the dynamics of the logarithm of scaleare driven by the conditional score are known to exhibit attractive theoretical properties for the t distribution and general errordistribution. A model based on the generalized t includes both as special cases. We derive the information matrix for thegeneralized t and show that, when parameterized with the inverse of the tail index, it remains positive de?nite in the limit asthe distribution goes to a general error distribution. We generalize further by allowing the distribution of the observations tobe skewed and asymmetric. Our method for introducing asymmetry ensures that the information matrix reverts to the usualcase under symmetry. We are able to derive analytic expressions for the conditional moments of our exponential generalizedautoregressive conditional heteroscedasticity model as well as the information matrix of the dynamic parameters. The practicalvalue of the model is illustrated with commodity and stock return data. Overall, the approach offers a uni?ed, ?exible, robust,and effective treatment of volatility