Non‐Parametric Spectral Density Estimation Under Long‐Range Dependence

Abstract

One major aim of time series analysis, particularly in the physical and geo‐sciences, is the estimation of the spectral density function. With weakly dependent time processes, non‐parametric, kernel‐based methods are available for spectral density estimation, which involves smoothing the periodogram by a kernel function. However, a similar non‐parametric approach is presently unavailable for strongly, or long‐range, dependent processes. In particular, as the spectral density function under long‐range dependence commonly has a pole at the origin, kernel‐based methods developed for weakly dependent processes (i.e., with bounded spectral densities) do not apply readily for long‐range dependence without suitable modification. To address this, we propose a non‐parametric kernel‐based method for spectral density estimation, which is valid under both weak and strong dependence. Based on the initial or pilot estimator of the long‐memory parameter, the method involves a frequency domain transformation to dampen the dependence in periodogram ordinates and mimic kernel‐based estimation under weak dependence. Under mild assumptions, the proposed non‐parametric spectral density estimator is shown to be uniformly consistent, and general expressions are provided for rates of estimation error and optimal kernel bandwidths. The method is investigated through simulation and illustrated through data examples, which also consider bandwidth selection.