Asymptotics of the Non-parametric Function for B-splines-based Estimation in Partially Linear Models

We consider least squares method for partially linear models based on polynomial splines. We derive the asymptotic property for the estimator, focusing on the estimation of the non-parametric function, in particular whether and how the estimation of the linear part will affect the non-parametric part (the converse relation, that is, how the linear part will be affected by the non-parametric part is much better known, which we will also review). One important goal along the way is to clarify the role of projection in semiparametric models, which was nevertheless a classical trick for proving the asymptotic normality of the linear part. A crucial question we try to answer is whether projection plays any role in the estimation of the non-parametric function. The answer is both positive and negative depending on the direction along which to assess asymptotic normality. The style of writing of the paper is somewhat expository, and it also contains several new results not found in the current literature. Finally, we demonstrate in our numerical studies that construction of the pointwise confidence intervals for the non-parametric function motivated by our theory improves upon those constructed by pretending the linear part is known.