On the Expectation of the Local-to-Zero Cross-Validated Log Likelihood Criterion for Bandwidth Selection in Kernel Spectral Estimation

2026-07-17

AI Summary

This study addresses the challenge of data-driven bandwidth selection for spectral density estimation near zero frequency by proposing a local-to-zero cross-validated log-likelihood criterion, denoted CVLL_c, which selects the bandwidth by summing over the frequency interval $[1, n^c]$ with $0 < c < 1$. It is shown for the first time that when $4/5 < c < 1$, the expectation of the key term in CVLL_c converges to the asymptotic mean squared error of the zero-frequency spectral estimator, thereby overcoming limitations inherent in conventional global CVLL approaches. The theoretical foundation for applying this criterion to heteroskedasticity- and autocorrelation-consistent (HAC) standard error estimation is established through Fourier frequency truncation, Taylor expansion, and asymptotic analysis.

Abstract

We consider data-driven bandwidth selection for a kernel spectral estimator at zero frequency based on a local-to-zero version of the cross validated log-likelihood (CVLL) criterion. The modified version is $\mbox{CVLL}_c$, based on a sum over Fourier frequencies from $1$ to $n^c$ with $0<c<1$, where $n$ is the sample size. We focus on the expectation of a key term in a Taylor series expansion for $\mbox{CVLL}_c$ and show that in the case $4/5 < c < 1$ it converges to the corresponding asymptotic mean squared error of the spectral estimator at zero frequency. This provides some justification for the use of the local CVLL criterion for Heteroskedasticity and Autocorrelation Consistent (HAC) standard error estimation. Our theoretical results do not follow from existing literature on CVLL because those results exploit the fact that CVLL is global, summing over all frequencies in $(0,π)$ rather than local-to-zero frequency, as is the case for $\mbox{CVLL}_c$.

Problem

• bandwidth selection
• kernel spectral estimation
• cross-validated log likelihood
• local-to-zero frequency
• HAC standard errors

Innovation

• local-to-zero CVLL
• bandwidth selection
• kernel spectral estimation
• HAC standard errors
• asymptotic mean squared error