Semiparametric Regression with Kernel Error Model

We propose and study a class of regression models, in which the mean function is specified parametricaUy as in the existing regression methods, but the residual distribution is modelled non-parametrically by a kernel estimator, without imposing any assumption on its distribution. This specification is different from the existing semiparametric regression models. The asymptotic properties of such likelihood and the maximum likelihood estimate (MLE) under this semiparametric model are studied. We show that under some regularity conditions, the MLE under this model is consistent (when compared with the possibly pseudo-consistency of the parameter estimation under the existing parametric regression model), is asymptotically normal with rate $\sqrt n $ and efficient. The non-parametric pseudo-likelihood ratio has the Wilks property as the true likelihood ratio does. Simulated examples are presented to evaluate the accuracy of the proposed semiparametric MLE method.