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

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 $CVL{L}_c$, 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 $CVL{L}_c$ and show that in the case $4\mathrm{/}5<c<1$ 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 $(0,\pi )$ rather than local-to-zero frequency, as is the case for $CVL{L}_c$.