Convergence Rates for Parametric Components in a Partly Linear Model

Abstract

Consider the regression model $Y_i = X'_i\beta + g(t_i) + e_i$ for $i = 1, \cdots, n$. Here $g$ is an unknown Holder continuous function of known order $p$ in $R, \beta$ is a $k \times 1$ parameter vector to be estimated and $e_i$ is an unobserved disturbance. Such a model is often encountered in situations in which there is little real knowledge about the nature of $g$. A piecewise polynomial $g_n$ is proposed to approximate $g$. The least-squares estimator $\hat\beta$ is obtained based on the model $Y_i = X'_i\beta + g_n(t_i) + e_i$. It is shown that $\hat\beta$ can achieve the usual parametric rates $n^{-1/2}$ with the smallest possible asymptotic variance for the case that $X$ and $T$ are correlated.