  This paper presents a novel approach to Bayesian nonparametric spectral
analysis of stationary multivariate time series. 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 mutual contiguity of the nonparametrically
corrected likelihood, the multivariate Whittle likelihood approximation and the
exact likelihood for Gaussian time series. A multivariate extension of the
nonparametric Bernstein-Dirichlet process prior for univariate spectral
densities to the space of Hermitian positive definite spectral density matrices
is specified directly on the correction 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. We demonstrate that the nonparametrically corrected
likelihood combines the efficiencies of a parametric with the robustness of a
nonparametric model. Its numerical accuracy is illustrated in a comprehensive
simulation study. We illustrate its practical advantages by a spectral analysis
of two environmental time series data sets: a bivariate time series of the
Southern Oscillation Index and fish recruitment and time series of windspeed
data at six locations in California.

Methodology