Estimating the Spectral Density at Frequencies Near Zero

Estimating the spectral density function $f(w)$ for some $w\in [-π, π]$ 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 $2π$, 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 $w=\pm π$. 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 $\pm π$. 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.

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