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

Mikihito Nishi

TL;DR

This paper develops a unified kernel-regression framework for estimating time-varying parameters in linear models across a wide spectrum of TVP paths, including smooth deterministic trajectories, rescaled random walks, structural breaks, and thresholds. It establishes consistency and asymptotic normality for a local-constant estimator, showing that the admissible bandwidth rate depends on a smoothness parameter α via the set Γ(α), thereby linking convergence rates to the nature of the TVP. A data-driven bandwidth selection procedure, combining cross-validation and bootstrap, adapts to latent smoothness and is complemented by consistent estimators for the asymptotic variance-covariance matrices, enabling valid inference. Monte Carlo simulations and an empirical CAPM application demonstrate the method’s robustness and its ability to provide a unified approach across diverse TVP models, with practical guidance that challenges the conventional T^{-1/5} bandwidth in favor of rates aligned with the underlying TVP smoothness. The proposed framework thus offers a principled, flexible tool for inference in models with time-varying parameters subject to smooth changes, abrupt breaks, or mixtures thereof.

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.

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

TL;DR

This paper develops a unified kernel-regression framework for estimating time-varying parameters in linear models across a wide spectrum of TVP paths, including smooth deterministic trajectories, rescaled random walks, structural breaks, and thresholds. It establishes consistency and asymptotic normality for a local-constant estimator, showing that the admissible bandwidth rate depends on a smoothness parameter α via the set Γ(α), thereby linking convergence rates to the nature of the TVP. A data-driven bandwidth selection procedure, combining cross-validation and bootstrap, adapts to latent smoothness and is complemented by consistent estimators for the asymptotic variance-covariance matrices, enabling valid inference. Monte Carlo simulations and an empirical CAPM application demonstrate the method’s robustness and its ability to provide a unified approach across diverse TVP models, with practical guidance that challenges the conventional T^{-1/5} bandwidth in favor of rates aligned with the underlying TVP smoothness. The proposed framework thus offers a principled, flexible tool for inference in models with time-varying parameters subject to smooth changes, abrupt breaks, or mixtures thereof.

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 rate is valid only for sufficiently smooth TVPs, and the bandwidth should be proportional to for random-walk TVPs, where 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.
Paper Structure (24 sections, 12 theorems, 93 equations, 7 figures, 12 tables)

This paper contains 24 sections, 12 theorems, 93 equations, 7 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Smoothness of Time-Varying Parameters
  3. Asymptotics
  4. Assumptions
  5. Asymptotic properties of $\hat{\beta}_t$
  6. Estimation of variance-covariance matrices
  7. On Bandwidth Selection: Implications and a Guide
  8. When random-walk $\beta_{T,t}$ is assumed to be continuously differentiable
  9. The effect of neglected breaks
  10. A guide for bandwidth selection
  11. Monte Carlo Simulation
  12. Simulation for Section \ref{['subsec:rw_misspecify']}
  13. Simulation for Section \ref{['subsec:neglect_break']}
  14. Balance between robustness and efficiency
  15. Empirical Application
  16. ...and 9 more sections

Key Result

Theorem 1

Suppose Assumptions asm:kernel and asm:dgp hold. Then, for $t=\lfloor Tr \rfloor, \ r\in (0,1)$, we have where In particular, for $h =cT^{\gamma}, \ c>0, \ \gamma\in(-1,0)$, we have for $\gamma \in \Gamma(\alpha)$, where If $\beta_{T,t}$ belongs to $\mathrm{TVP}(\alpha)$ on the boundary, $\gamma$ close to the right endpoint of $\Gamma(\alpha)$ gives asymptotic normality and the fastest possibl

Figures (7)

  • Figure 1: Cross-validation criteria calculated using leave-$(2m+1)$-out estimators with $h=T^{\gamma}$, for V-G
  • Figure 2: Estimates and 95% confidence band from the kernel-based method ($h=\hat{c}T^{-1/3}$) for V-G
  • Figure 3: Estimates from the kernel method with $h=\hat{c}T^{-1/2}$ and Bayesian method (posterior means) for V-G
  • Figure E.1: Cross-validation criteria calculated using leave-$(2m+1)$-out estimators with $h=T^{\gamma}$, for G
  • Figure E.2: Estimates and 95% confidence band from the kernel-based method ($h=\hat{c}T^{-1/3}$) for G
  • ...and 2 more figures

Theorems & Definitions (28)

  • Definition 1
  • Remark 1
  • Example 1: Continuously differentiable functions
  • Example 2: The random walk
  • Remark 2
  • Example 3: Structural breaks
  • Example 4: Threshold models
  • Example 5: Mixed model
  • Theorem 1
  • Remark 3
  • ...and 18 more