Nonparametric estimation of fixed effects panel data varying coefficient models

In this paper, we consider the nonparametric estimation of a varying coefficient fixed effect panel data model. The estimator is based in a within (un-smoothed) transformation of the regression model and then a local linear regression is applied to estimate the unknown varying coefficient functions. It turns out that the standard use of this technique produces a non-negligible asymptotic bias. In order to avoid it, a high dimensional kernel weight is introduced in the estimation procedure. As a consequence, the asymptotic bias is removed but the variance is enlarged, and therefore the estimator shows a very slow rate of convergence. In order to achieve the optimal rate, we propose a one-step backfitting algorithm. The resulting two-step estimator is shown to be asymptotically normal and its rate of convergence is optimal within its class of smoothness functions. It is also oracle efficient. Further, this estimator is compared both theoretically and by Monte-Carlo simulation against other estimators that are based in a within (smoothed) transformation of the regression model. More precisely the profile least-squares estimator proposed in this context in Sun et al. (2009). It turns out that the smoothness in the transformation enlarges the bias and it makes the estimator more difficult to analyze from the statistical point of view. However, the first step estimator, as expected, shows a bad performance when compared against both the two step backfitting algorithm and the profile least-squares estimator.