Kernel-Based Semiparametric Estimators: Small Bandwidth Asymptotics and Bootstrap Consistency

This paper develops asymptotic approximations for kernel-based semiparametric estimators under assumptions accommodating slower-than-usual rates of convergence of their nonparametric ingredients. Our first main result is a distributional approximation for semiparametric estimators that differs from existing approximations by accounting for a bias. This bias is nonnegligible in general, and therefore poses a challenge for inference. Our second main result shows that some (but not all) nonparametric bootstrap distributional approximations provide an automatic method of correcting for the bias. Our general theory is illustrated by means of examples and its main finite sample implications are corroborated in a simulation study.