An Averaging Estimator for Two Step M Estimation in Semiparametric Models

In a two step extremum estimation (M estimation) framework with a finite dimensional parameter of interest and a potentially infinite dimensional first step nuisance parameter, I propose an averaging estimator that combines a semiparametric estimator based on nonparametric first step and a parametric estimator which imposes parametric restrictions on the first step. The averaging weight is the sample analog of an infeasible optimal weight that minimizes the asymptotic quadratic risk. I show that under mild conditions, the asymptotic lower bound of the truncated quadratic risk difference between the averaging estimator and the semiparametric estimator is strictly less than zero for a class of data generating processes (DGPs) that includes both correct specification and varied degrees of misspecification of the parametric restrictions, and the asymptotic upper bound is weakly less than zero.