Convergence Rates for Parametric Components in a Partly Linear Model

Consider the regression model Yᵢ = X'ᵢβ + g(tᵢ) + eᵢ for i = 1, ⋯, n. Here g is an unknown Holder continuous function of known order p in R, β is a k × 1 parameter vector to be estimated and eᵢ 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 gₙ is proposed to approximate g. The least-squares estimator β is obtained based on the model Yᵢ = X'ᵢβ + gₙ(tᵢ) + eᵢ. It is shown that β can achieve the usual parametric rates n-1/2 with the smallest possible asymptotic variance for the case that X and T are correlated.