Volatility Modeling with a Generalized t Distribution

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