Estimating Time-Varying Parameters of Various Smoothness in Linear Models via Kernel Regression

TL;DR

This paper develops a kernel-based method for estimating various types of time-varying parameters in linear models, establishing theoretical properties, optimal bandwidth selection, and demonstrating its effectiveness through simulations and real data.

Contribution

It introduces a unified kernel estimation framework for diverse TVPs, analyzes bandwidth choices based on smoothness, and proposes an adaptive data-driven bandwidth selection procedure.

Findings

The estimator is consistent and asymptotically normal for various TVPs.

Bandwidth must be chosen according to the TVP's smoothness for optimal performance.

The proposed adaptive bandwidth method performs well in simulations and real data applications.

Abstract

We study kernel-based estimation of nonparametric time-varying parameters (TVPs) in linear models. Our contributions are threefold. First, we establish consistency and asymptotic normality of the kernel-based estimator for a broad class of TVPs including deterministic smooth functions, the rescaled random walk, structural breaks, the threshold model and their mixtures. Our analysis exploits the smoothness of the TVP. Second, we show that the bandwidth rate must be determined according to the smoothness of the TVP. For example, the conventional $T^{-1/5}$ rate is valid only for sufficiently smooth TVPs, and the bandwidth should be proportional to $T^{-1/2}$ for random-walk TVPs, where $T$ is the sample size. We show this highlighting the overlooked fact that the bandwidth determines a trade-off between the convergence rate and the size of the class of TVPs that can be estimated. Third, we propose a data-driven procedure for bandwidth selection that is adaptive to the latent smoothness of the TVP. Simulations and an application to the capital asset pricing model suggest that the proposed method offers a unified approach to estimating a wide class of TVP models.