Asymptotics for nonparametric and semiparametric fixed effects panel models

In this paper, we investigate the problem of estimating nonparametric and semiparametric panel data models with fixed effects. We focus on establishing the asymptotic results for estimators using smooth backfitting methods. We consider two estimators for the smooth unknown function in nonparametric panel regressions. One is a local linear estimator constructed similar as the one in Mammen et al. (2009) which was proposed for the additive nonparametric panel model. The other is the local profile likelihood based estimator proposed by Henderson et al. (2008) (HCL hereafter). We build the link and compare the difference between these two estimators which are constructed under different sets of conditions. We put both of these estimators in the smooth backfitting algorithm framework discussed in Mammen et al. (1999). Following the recently developed theories on backfitting kernel estimates in Mammen et al. (2009), we establish the asymptotic normality of these estimators, and hence verify the conjectures made by HCL and complement their paper. Further, we consider a partially linear fixed effects panel data model with the nonparametric component estimated using the methods discussed in the first part of the paper. We give the asymptotic result for the estimators of finite dimensional parameters, which shows that the first-step plug-in estimators will not affect the asymptotic variance in the second-step estimation.