Estimation of partially linear regression models under the partial consistency property

Utilizing recent theoretical results in high dimensional statistical modeling, a flexible yet computationally simple approach is proposed to estimate the partially linear models. Motivated by the partial consistency phenomena, the nonparametric component in the partially linear model is modeled via incidental parameters and estimated by a simple local average over small partitions of the support of the nonparametric variables. The proposed least-squares based method seeks to strike a balance between computation burden and efficiency of the estimators while minimizing model bias. It is shown that given inconsistent estimators of the nonparametric component, square root-n consistent estimators of the parameters of the parametric component can be obtained with little loss in efficiency. Moreover, conditional on the parametric estimates, an optimal estimator of the nonparametric component can be obtained using classic nonparametric methods. The statistical inference problems regarding the parametric parameters and a two-population nonparametric testing problem regarding the nonparametric component are considered. The results show that the behavior of the test statistics is satisfactory. To assess the performance of the new method in comparison with other methods, three simulation studies are conducted and a real data set about risk factors of birth weights is analyzed.