Instrumental regression in partially linear models

We consider the semi-parametric regression model Y = X-t beta + phi(Z) where beta and phi(center dot) are unknown slope coefficient vector and function, and where the variables (X, Z) are endogenous. We propose necessary and sufficient conditions for the identification of the parameters in the presence of instrumental variables. We also focus on the estimation of beta. It is known that the presence of phi may lead to a slow rate of convergence for the estimator of beta. An additional complication in the fully endogenous model is that the solution of the equation necessitates the inversion of a compact operator that has to be estimated non-parametrically. In general this inversion is not stable, thus the estimation of beta is ill-posed. In this paper, a root n-consistent estimator for beta is derived in this setting under mild assumptions. One of these assumptions is given by the so-called source condition that is explicitly interpreted in the paper. Monte Carlo simulations demonstrate the reasonable performance of the estimation procedure on finite samples.