Abstract: The analysis of multivariate time series can give important insights into periodicities and coherencies. We present a novel approach to Bayesian nonparametric spectral analysis of stationary multivariate time series which is based on Whittle’s likelihood. Starting with a parametric vector-autoregressive model, the parametric likelihood is nonparametrically adjusted in the frequency domain to account for potential deviations from parametric assumptions. We show contiguity of the nonparametrically corrected likelihood, the multivariate Whittle likelihood approximation and the exact likelihood for Gaussian time series. The nonparametric prior used is a multivariate extension of the nonparametric Bernstein-Dirichlet process prior for univariate spectral densities to the space of Hermitian positive definite spectral density matrices. An infinite series representation of this prior is then used to develop a Markov chain Monte Carlo algorithm to sample from the posterior distribution. The code is made publicly available for ease of use and reproducibility. With this novel approach we provide a generalization of the multivariate Whittle-likelihood-based method of Meier et al. (2020) as well as an extension of the nonparametrically corrected likelihood for univariate stationary time series of Kirch et al. (2019) to the multivariate case. In a simulation study, we demonstrate that the nonparametrically corrected likelihood combines the efficiencies of a parametric with the robustness of a nonparametric model. We illustrate the practical benefits using case studies of EEG and windspeed time series.
