THE ASYMPTOTIC VARIANCE OF SEMIPARAMETRIC ESTIMATORS

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 pres ence of nonparametric estimates of nuisance functions. The general for m of an adjustment factor for nonparametric estimates is derived and a nalyzed. The usefulness of the formula is illustrated by deriving prop ositions on invariance of the limiting distribution with respect to th e nonparametric estimator, conditions for nonparametric estimation to have no effect on the asymptotic distribution, and the form of a corre ction term for the presence of nonparametric projection and density es timators. Examples discussed are quasi-maximum likelihood estimation o f index models, panel probit with semiparametric individual effects, a verage derivatives, and inverse density weighted least squares. The pa per also develops a set of regularity conditions for the validity of t he asymptotic variance formula. Primitive regularity conditions are de rived for square-rootn-consistency and asymptotic normality for functi ons of series estimators of projections. Specific examples are polynom ial estimators of average derivative and semiparametric panel probit m odels.