Time-varying additive model with autoregressive errors for locally stationary time series

In this article, we study the time-varying additive model with time-varying autoregression (tvAR) error in the locally stationary context, and propose the two-step estimation for it. B-spline method, which is computation efficient, is adopted to obtain the initial estimator of trend function and additive components. And then the structure of autoregression error is estimated by ULASSO, the consistency and asymptotical normality are proved. At last, with the initial estimator and the estimated error structure, the improved estimator of trend function and additive components is derived by local linear smoothing, and its asymptotic normality and oracle property are proved. Simulation studies validate the properties of the proposed estimators. A real data application illustrates the proposed model is applicable and more appropriate than the classical additive model in the presence of locally stationary regressors.