Abstract: In a time-varying coefficient model, the regression coefficient is allowed to change over time as a nonparametric function to capture the time-varying feature. Owing to its popularity in time series applications, where the assumption of independence typically does not hold, it is desirable to allow dependent and nonstationary observations. We consider the problem of semiparametric variable labeling and estimation for multi-output time-varying coefficient models in a time series setting, where a variable can be labeled as time-varying, time-constant, or irrelevant, in a nested structure. We first show that the natural approach of imposing separate penalties on the local linear estimator and its derivative do not work as intended for semiparametric labeling, owing to the lack of connection between the coefficient and the derivative estimators in the popular local linear method. We then propose a stratified fix that borrows information from the coefficient estimator and combines it with the derivative into the same stratum that simultaneously achieves successful labeling and estimation. We establish the theoretical properties of the proposed method, including its estimation and labeling consistency, for a general class of nonstationary processes. Numerical examples, including a Monte Carlo simulation study and a real-data application, are presented to illustrate the proposed method.

Key words and phrases: Kernel smoothing, local linear estimation, nonstationary time series, time-varying coefficient model, variable selection.