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Nonparametric regression models with locally stationary covariates havereceived increasing interest in recent years. As a nice relief of "curse ofdimensionality" induced by large dimension of covariates, additive regressionmodel is commonly used. However, in locally stationary context, to catch thedynamic nature of regression function, we adopt a flexible varying-coefficientadditive 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 eachunivariate nonparametric function, and show its consistency and $L_{2}$ rate ofconvergence. Furthermore, based upon the three-step estimators, we develop atwo-stage penalty procedure to identify pure additive terms andvarying-coefficient terms in varying-coefficient additive model. As expected,we demonstrate that the proposed identification procedure is consistent, andthe penalized estimators achieve the same $L_{2}$ rate of convergence as thepolynomial spline estimators. Simulation studies are presented to illustratethe finite sample performance of the proposed three-step spline estimationmethod and two-stage model selection procedure.

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