Some Theoretical Results Concerning Time-varying Nonparametric Regression with Local Stationary Regressors and Error

With regard to a three-step estimation procedure, proposed without theoretical discussion by Li and You in Journal of Applied Statistics and Management, for a nonparametric regression model with time-varying regression function, local stationary regressors and time-varying AR(p) (tvAR(p)) error process , we established all necessary asymptotic properties for each of estimator. We derive the convergence rate and asymptotic normality of the preliminary estimation of nonparametric regression function, establish the asymptotic distribution of time-varying coefficient functions in the error term, and present the asymptotic property of the refined estimation of nonparametric regression function. In addition, with regard to the ULASSO method for variable selection and constant coefficient detection for error term structure, we show that the ULASSO estimator can identify the true error term structure consistently. We conduct two simulation studies to illustrate the finite sample performance of the estimators and validate our theoretical discussion on the properties of the estimators.