Two-stage non Gaussian QML estimation of GARCH models and testing the efficiency of the Gaussian QMLE

In generalized autoregressive conditional heteroskedastic (GARCH) models, the standard identifiability assumption that the variance of the iid process is equal to 1 can be replaced by an alternative moment assumption. We show that, for estimating the original specification based on the standard identifiability assumption, efficiency gains can be expected from using a quasi-maximum likelihood (QML) estimator based on a non Gaussian density and a reparameterization based on an alternative identifiability assumption. A test allowing to determine whether a reparameterization is needed, that is, whether the more efficient QMLE is obtained with a non Gaussian density, is proposed.