Spectral Subsampling MCMC for Stationary Multivariate Time Series with Applications to Vector ARTFIMA Processes

Abstract

A multivariate generalisation of the Whittle likelihood is used to extend spectral subsampling MCMC to stationary multivariate time series by subsampling matrix-valued periodogram observations in the frequency domain. To assess the performance of the methodology in challenging problems, a multivariate generalisation of the autoregressive tempered fractionally integrated moving average model (ARTFIMA) is introduced and some of its properties derived. Bayesian inference based on the Whittle likelihood is demonstrated to be a fast and accurate alternative to the exact time domain likelihood. Spectral subsampling is shown to provide up to two orders of magnitude additional speed-up, while retaining MCMC sampling efficiency and accuracy, compared to spectral methods using the full dataset.