Estimating the Spectral Density at Frequencies Near Zero

Abstract

Estimating the spectral density function f(w) for some w ∈ [ − π , π ] has been traditionally performed by kernel smoothing the periodogram and related techniques. Kernel smoothing is tantamount to local averaging, that is, approximating f(w) by a constant over a window of small width. Although f(w) is uniformly continuous and periodic with period 2 π , in this article 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 = ± 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 ± 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. Supplementary materials for this article are available online.