A nonparametrically corrected likelihood for Bayesian spectral analysis of multivariate time series

Abstract

A novel approach to Bayesian nonparametric spectral analysis of stationary multivariate time series is presented. 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. A proof of mutual contiguity of the nonparametrically corrected likelihood, the multivariate Whittle likelihood approximation and the exact likelihood for Gaussian time series is given. 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, a generalisation 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 is presented. It is demonstrated 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. Its practical advantages are illustrated by a spectral analysis of two environmental time series data sets: a bivariate time series of the Southern Oscillation Index and fish recruitment and a multivariate time series of windspeed data at six locations in California.