Ruoyao Shi (University of California, Riverside)
Kalai Family Workshop in Econometrics
Sep 10 2020
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An Averaging Estimator for Two Step M Estimation in Semiparametric Models
Abstract: In this paper, we study the two step M estimation of a finite dimensional parameter which depends on a first step estimation of a potentially infinite dimensional nuisance parameter. We present 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 quadratic risk functions. This averaging estimator strikes a balance between the robust semiparametric estimator and the efficient parametric estimator, as we show that the averaging estimator uniformly dominates the semiparametric estimator in terms of asymptotic quadratic risk regardless of whether the first step parametric restrictions hold or not. In particular, we prove that under certain sufficient conditions, the asymptotic lower bound of the truncated quadratic risk differences between the averaging estimator and the semiparametric estimator is strictly less than zero under a class of data generating processes that includes both correct specification and misspecification of the first step parametric restrictions, and the asymptotic upper bound is weakly less than zero. We illustrate our estimator in a variety of applications using simulations and real data.