Estimating the spectral density at frequencies near zero

Estimating the spectral density function <i>f</i>(<i>w</i>) for some w∈[−π,π] has been traditionally performed by kernel smoothing the periodogram and related techniques. Kernel smoothing is tantamount to local averaging, i.e., approximating <i>f</i>(<i>w</i>) by a constant over a window of small width. Although <i>f</i>(<i>w</i>) is uniformly continuous and periodic with period 2π, in this paper we recognize the fact that <i>w</i> = 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 <i>w</i> is at (or near) the points 0 or ±π. The case <i>w</i> = 0 is of particular importance since <i>f</i>(0) is the large-sample variance of the sample mean; hence, estimating <i>f</i>(0) is crucial in order to conduct any sort of inference on the mean.