Efficient semiparametric estimator for heteroscedastic partially linear models

We study the heteroscedastic partially linear model with an unspecified partial baseline component and a nonparametric variance function. An interesting finding is that the performance of a naive weighted version of the existing estimator could deteriorate when the smooth baseline component is badly estimated. To avoid this, we propose a family of consistent estimators and investigate their asymptotic properties. We show that the optimal semiparametric efficiency bound can be reached by a semiparametric kernel estimator in this family. Building upon our theoretical findings and heuristic arguments about the equivalence between kernel and spline smoothing, we conjecture that a weighted partial-spline estimator could also be semiparametric efficient. Properties of the proposed estimators are presented through theoretical illustration and numerical simulations.