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

TL;DR

This paper establishes the asymptotic properties of a three-step estimation procedure for nonparametric regression with time-varying functions, local stationary regressors, and time-varying AR errors, including variable selection via ULASSO.

Contribution

It provides the first theoretical analysis of a three-step estimator for such models, including convergence rates, asymptotic normality, and variable selection consistency.

Findings

The preliminary estimator converges at a specific rate and is asymptotically normal.

The ULASSO method can consistently identify the true error structure.

Simulation studies confirm the theoretical properties and finite sample performance.

Abstract

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.