Published online by Cambridge University Press: 25 April 2018
This article studies two-step sieve M estimation of general semi/nonparametric models, where the second step involves sieve estimation of unknown functions that may use the nonparametric estimates from the first step as inputs, and the parameters of interest are functionals of unknown functions estimated in both steps. We establish the asymptotic normality of the plug-in two-step sieve M estimate of a functional that could be root-n estimable. The asymptotic variance may not have a closed form expression, but can be approximated by a sieve variance that characterizes the effect of the first-step estimation on the second-step estimates. We provide a simple consistent estimate of the sieve variance, thereby facilitating Wald type inferences based on the Gaussian approximation. The finite sample performance of the two-step estimator and the proposed inference procedure are investigated in a simulation study.
We gratefully acknowledge insightful comments from Xiaohong Chen, who was a co-author of the initial version. We appreciate useful suggestions from Liangjun Su, the coeditor and three anonymous referees. The outstanding editorial input by the Editor, Professor Phillips, in our last version of the manuscript is greatly appreciated. All errors are the responsibility of the authors.