Feasible optimum Godambe scores for a semi-parametric GARCH time series

This paper concerns a semi-parametric GARCH time series for which the error distribution is unspecified. Godambe scores (GS) including quasi-likelihood scores are considered to estimate parameters of interest. Allowing the Godambe innovation to contain nuisance parameters associated with moments of the unknown error distribution, an optimum GS (oGS, for short) is obtained for each fixed nuisance parameters, and in turn the nuisance parameters are replaced by the quasi maximum likelihood (QML) residuals so that one can obtain computationally feasible zero of the oGS. It is verified under certain conditions that the solution of the feasible oGS continues to be asymptotically optimum, while extending the family of error distributions under consideration.