  Spectral subsampling MCMC was recently proposed to speed up Markov chain
Monte Carlo (MCMC) for long stationary univariate time series by subsampling
periodogram observations in the frequency domain. This article extends the
approach to stationary multivariate time series. It also proposes a
multivariate generalisation of the autoregressive tempered fractionally
differentiated moving average model (ARTFIMA) and establishes some of its
properties. The new model is shown to provide a better fit compared to
multivariate autoregressive moving average models for three real world
examples. We demonstrate that spectral subsampling may provide up to two orders
of magnitude faster estimation, while retaining MCMC sampling efficiency and
accuracy, compared to spectral methods using the full dataset.
