Semiparametric Regression with Kernel Error Model

We propose and study a class of regression models, in which the mean function is specified parametrically as in the existing regression methods, but the residual distribution is modeled nonparametrically 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 (as compared to the possibly pseudo consistency of the parameter estimation under the existing parametric regression model), and is asymptotically normal with rate sqrt{n} and efficient. The nonparametric 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.