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

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.