*(English)*Zbl 0637.62067

Consider the regression model ${Y}_{i}={X}_{i}^{\text{'}}\beta +g\left({t}_{i}\right)+{e}_{i}$ for $i=1,\xb7\xb7\xb7,n$. Here g is an unknown Hölder 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 $\widehat{\beta}$ is obtained based on the model ${Y}_{i}={X}_{i}^{\text{'}}\beta +{g}_{n}\left({t}_{i}\right)+{e}_{i}$. It is shown that $\widehat{\beta}$ can achieve the usual parametric rates ${n}^{-1/2}$ with the smallest possible asymptotic variance for the case that X and T are correlated.

##### MSC:

62J05 | Linear regression |

62J10 | Analysis of variance and covariance |

62G99 | Nonparametric inference |

41A15 | Spline approximation |