Estimation and Model Identification of Locally Stationary Varying-Coefficient Additive Models

Nonparametric regression models with locally stationary covariates have received increasing interest in recent years. As a nice relief of "curse of dimensionality" induced by large dimension of covariates, additive regression model is commonly used. However, in locally stationary context, to catch the dynamic nature of regression function, we adopt a flexible varying-coefficient additive model where the regression function has the form $\alpha_{0}\left(u\right)+\sum_{k=1}^{p}\alpha_{k}\left(u\right)\beta_{k}\left(x_{k}\right).$ For this model, we propose a three-step spline estimation method for each univariate nonparametric function, and show its consistency and $L_{2}$ rate of convergence. Furthermore, based upon the three-step estimators, we develop a two-stage penalty procedure to identify pure additive terms and varying-coefficient terms in varying-coefficient additive model. As expected, we demonstrate that the proposed identification procedure is consistent, and the penalized estimators achieve the same $L_{2}$ rate of convergence as the polynomial spline estimators. Simulation studies are presented to illustrate the finite sample performance of the proposed three-step spline estimation method and two-stage model selection procedure.