Bayesian Nonparametric Spectral Analysis of Locally Stationary Processes

Stationarity plays a pivotal role in time series analysis. It is not only the basis for the derivation of general asymptotic theory but it also allows an efficient analysis in the frequency domain via the Whittle likelihood, based on the asymptotic independence of the Fourier coefficients. However, many regularly sampled data derived from the observation of physical or ecological processes, for instance, are only locally stationary. They exhibit slowly evolving spectra and asymptotically non-vanishing dependencies between the Fourier coefficients. In this article we construct a novel dynamic Whittle likelihood approximation for a locally stationary process and propose a Bayesian nonparametric approach to estimate its time-varying spectral density. This dynamic likelihood approximation in the frequency domain can capture the time-frequency evolution of the process by using moving periodograms previously introduced in the bootstrap literature. The posterior distribution is obtained by updating a bivariate extension of the Bernstein-Dirichlet process prior with the dynamic Whittle likelihood. Asymptotic properties such as sup-norm posterior consistency and L2-norm posterior contraction rates are presented. In addition, simulation studies and applications to real-life datasets demonstrate the competitive empirical performance compared to other state-of-the-art methods under finite-sample conditions. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.