  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.
