The Asymptotic Variance of Semiparametric Estimators

Abstract

Knowledge of the asymptotic variance of an estimator is important for large sample inference, efficiency, and as a guide to the specification of regularity conditions.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 asymptotic equivalence for different nonparametric estimators of the same function, conditions for estimation of the nuisance functions to have no effect on the asymptotic variance, and the form of a correction term for the presence of linear function of a conditional expectation estimator, or other projection estimator (e.g.partially linear and/or additive nonparametric projections), and for a function of a density.Specific results cover a semiparametric random effects model for binary panel data, nonparametric consumer surplus, nonparametric prediction, and average derivatives.Regularity conditions are given for many of the propositions.These include primitive conditions for v'n-consistency, asymptotic normality, and consistency of an asymptotic variance estimator with series estimators of conditional expectations (or projections), in each of the examples.