Properties Of Estimators In Partially Linear Models Under Restrictions

Engle,Granger,Rice and Weiss(1986) were among the first to consider the partially linear model,when they analyzed the relationship between temperature and electricity usage . This kind of model includes not only a parametric component but also a nonparametric component .So it has the advantages of the parametric regression model and nonparametric regression model and more implements and stronger explanations than the pure parametric or nonparametric regression model.We discuss the properties of estimators of parametric β when y_i = x_i'β + g(t_i) + e_i, 1≤ i ≤ n , (â… )with Restrictions. Where the (x_i, t_i) are fixed and norandom design points. x_i = (x_i1,..., x_ip)' , β = (β_i,...,β_p)'(p≥1). g(·) is an unknow function over [0,1] ,β is an unknow parameter to be estimated ,0 ≤ t_i ≤ 1, e_i are i.i.d. random errors with Ee_i = 0,Ee_i² = σ² < ∞.There have been many results about model [1] ,and we always get the estimations of parametric β and nonparametric g(·) by classifing and discussing the data of predetemined variables.In the fact ,we always have auxiliary information of regression parametric β .This article discuss two aspects in the ideas:1. Under linear restrictions r = Rβ ,based on g estimated by the family of of nonparametric estimates including kernel and nearest neighbor estimates,the consistency and asymptotic normality of least-square of β are dicussed.2. Under stochastic linear restrictions r = Rβ+φ ,empirical likelihood ratio statistics are constructed for parametric β and the asymptotic distribution is discussed .Then the empirical likelihood ratio confidence is constructed.