Abstract:
While there is an increasing amount of literature about Bayesian time series analysis, only few nonparametric approaches to multivariate time series exist. Most notably, [1] and [2] rely on Whittle's likelihood, involving the second order structure of the time series by means of the spectral density matrix. The latter is modeled with a smoothing splines prior for the components of the Cholesky decomposition. While these approaches are shown to perform well in many applications, their theoretical asymptotic behavior in terms of posterior consistency is not known. Consistency results are typically restricted to parameters outside a prior null set, which is unsatisfactory in in nite dimensions, or even fail entirely [3]. Having a posterior consistency result in mind, we investigate multivariate extensions of the Bernstein-Dirichlet prior from [4], for which consistency under the Gaussianity assumption has been shown in the univariate case. We also consider a multivariate extension of the corrected parametric likelihood, a generalization of Whittle's likelihood, which has recently been developed by Kirch et al [5].