Estimating the Spectral Density at Frequencies Near Zero

TL;DR

This paper introduces a local polynomial regression approach for estimating the spectral density at frequencies near zero and the boundaries, improving accuracy over traditional kernel smoothing methods.

Contribution

It proposes a boundary-aware local polynomial regression technique for spectral density estimation at critical boundary points, addressing limitations of kernel smoothing.

Findings

Improved estimation accuracy at boundary frequencies.

Enhanced inference capabilities for the mean based on spectral density.

Addresses boundary bias issues in spectral density estimation.

Abstract

Estimating the spectral density function $f(w)$ for some $w\in [-\pi, \pi]$ has been traditionally performed by kernel smoothing the periodogram and related techniques. Kernel smoothing is tantamount to local averaging, i.e., approximating $f(w)$ by a constant over a window of small width. Although $f(w)$ is uniformly continuous and periodic with period $2\pi$, in this paper we recognize the fact that $w=0$ effectively acts as a boundary point in the underlying kernel smoothing problem, and the same is true for $w=\pm \pi$. It is well-known that local averaging may be suboptimal in kernel regression at (or near) a boundary point. As an alternative, we propose a local polynomial regression of the periodogram or log-periodogram when $w$ is at (or near) the points 0 or $\pm \pi$. The case $w=0$ is of particular importance since $f(0)$ is the large-sample variance of the sample mean; hence, estimating $f(0)$ is crucial in order to conduct any sort of inference on the mean.