Efficient estimation and model selection for single-index varying-coefficient models

The single-index varying-coefficient models include many types of popular semiparametric models, i.e. single-index models, partially linear models, varying-coefficient models, and so on. In this paper, we first establish the semiparametric efficiency bound for the single-index varying-coefficient model, and develop an estimation method based on the efficient estimating equations. Although our main focus is more on homoscedastic models for simplicity, the calculated efficiency bound and efficient estimating equations are for the more general heteroscedastic models. It shows that the estimator of the finite dimensional parameter is √ n consistent and asymptotically normal and attains the semiparametric efficiency bound. Moreover, for the homoscedastic model, a two-stage variable selection procedure is proposed to select the important nonparametric components and parametric components. We also find that the proposed procedures can divide the predictors into varying-coefficient predictors and constant-coefficient predictors automatically. Some simulation studies are conducted to evaluate and illustrate the proposed methods.