  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.
