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The Asymptotic Variance of Semiparametric Estimators

Whitney K. Newey

Abstract

The purpose of this paper is the presentation of a general formula for the asymptotic variance of a semiparametric estimator. A particularly important feature of this formula is a way of accounting for the presence of nonparametric estimates of nuisance functions. The general form of an adjustment factor for nonparametric estimates is derived and analyzed. The usefulness of the formula is illustrated by deriving propositions on invariance of the limiting distribution with respect to the nonparametric estimator, conditions for nonparametric estimation to have no effect on the asymptotic distribution, and the form of a correction term for the presence of nonparametric projection and density estimators. Examples discussed are quasi-maximum likelihood estimation of index models, panel probit with semiparametric individual effects, average derivatives, and inverse density weighted least squares. The paper also develops a set of regularity conditions for the validity of the asymptotic variance formula. Primitive regularity conditions are derived for $\sqrt n$-consistency and asymptotic normality for functions of series estimators of projections. Specific examples are polynomial estimators of average derivative and semiparametric panel probit models.