Recursive differencing: Bias reduction with regular kernels

It is well known that it is important to control the bias in estimating conditional expectations in order to obtain asymptotic normality for quantities of interest (e.g. a finite dimensional parameter vector in semiparametric models or averages of marginal effects in the nonparametric case). For this purposes, higher order kernel methods are often employed in developing the theory. However such methods typically do not perform well at moderate sample sizes. Moreover, and perhaps related to their performance, non-optimal windows are selected with undersmoothing needed to ensure the appropriate bias order. We propose a recursive differencing approach to bias reduction for a nonparametric estimator of a conditional expectation, where the order of the bias depending on the stage of the recursion. It performs much better at moderate sample sizes than regular or higher order kernels while retaining a bias of any desired order and a convergence rate the same as that of higher order kernels. We also propose an approach to implement this estimator under optimal windows, which ensures asymptotic normality in semiparametric multiple index models of arbitrary dimension. This mechanism further contributes to its very good finite sample performance.