Volatility Modeling with a Generalized t-distribution

Beta-t-EGARCH 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 (GED). 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 as the tail index goes to infinity and the distribution becomes a GED. Hence it is possible to construct Lagrange multiplier tests of the null hypothesis of light tails against the alternative of fat tails. Analytic expressions may be obtained for the unconditional moments in the EGARCH model and the information matrix for the dynamic parameters obtained. The distribution may be extended by allowing for skewness and asymmetry in the shape parameters and the asymptotic theory for the associated EGARCH models may be correspondingly extended. For positive variables, the GB2 distribution may be parameterized so that it goes to the generalised gamma in the limit as the tail index goes to infinity. Again dynamic volatility may be introduced and properties of the model obtained. Overall the approach offers a unified, flexible, robust and practical treatment of dynamic scale.