Estimation of Partially Linear Regression Model under Partial Consistency Property

Abstract: In this paper, utilizing recent theoretical results in high dimensional statistical modeling, we propose a model-free yet computationally simple approach to estimate the partially linear model $Y=X\beta+g(Z)+\varepsilon$. Motivated by the partial consistency phenomena, we propose to model $g(Z)$ via incidental parameters. Based on partitioning the support of $Z$, a simple local average is used to estimate the response surface. The proposed method seeks to strike a balance between computation burden and efficiency of the estimators while minimizing model bias. Computationally this approach only involves least squares. We show that given the inconsistent estimator of $g(Z)$, a root $n$ consistent estimator of parametric component $\beta$ of the partially linear model can be obtained with little cost in efficiency. Moreover, conditional on the $\beta$ estimates, an optimal estimator of $g(Z)$ can then be obtained using classic nonparametric methods. The statistical inference problem regarding $\beta$ and a two-population nonparametric testing problem regarding $g(Z)$ are considered. Our results show that the behavior of test statistics are satisfactory. To assess the performance of our method in comparison with other methods, three simulation studies are conducted and a real dataset about risk factors of birth weights is analyzed.