A matrix Gamma process and applications to Bayesian analysis of multivariate time series

Abstract

While there is an increasing amount of literature about Bayesian time series analysis, only few Bayesian nonparametric approaches to multivariate time series exist. Most methods rely on Whittle's Likelihood, involving the second order structure of a stationary time series by means of its spectral density matrix. This is often modeled in terms of the Cholesky decomposition to ensure positive de niteness. However, with such nonlinear transformations, the modeling of certain prior knowledge aspects such as mean or variance often proves to be di cult. Furthermore, and most importantly, asymptotic properties such as posterior consistency or posterior contraction rates are not known. A different idea is to model the spectral density matrix by means of random measures. This is in line with existing approaches for the univariate case, where the normalized spectral density is modeled similar to a probability density, e.g. with a Dirichlet process mixture of Beta densities. In this work we present a related approach for multivariate time series, with matrix-valued mixture weights induced by a Hermitian positive de nite Gamma process. The process construction is inspired by Kingman's construction of the Gamma process and utilizes an in nitely divisible Hermitian positive de nite Gamma distribution. In conjunction with Whittle's Likelihood, our proposed Bayesian nonparametric procedure for spectral density inference is shown to perform well for both simulated and real data. Important theoretical properties such as posterior consistency and contraction rates are also established. As a preliminary result for these asymptotic considerations, we establish mutual contiguity of Whittle's Likelihood and the full Gaussian Likelihood for stationary multivariate Gaussian time series, a fact that has so far only been known in the univariate case. We also present a semiparametric model extension, accommodating a parametric linear model in which the nonparametric time series component constitutes the error term. This model is investigated with both numerical simulations and in terms of asymptotic properties of the joint posterior and the marginal posterior of the linear model coefficients.