We introduce and explore a new class of stationary time series models for variance matrices based on a constructive definition exploiting inverse Wishart distribution theory. The main class of models explored is a novel class of stationary, first-order autoregressive (AR) processes on the cone of positive semi-definite matrices. Aspects of the theory and structure of these new models for multivariate ``volatility'' processes are described in detail and exemplified. We then develop approaches to model fitting via Bayesian simulation-based computations, creating a custom filtering method that relies on an efficient innovations sampler. An example is then provided in analysis of a multivariate electroencephalogram (EEG) time series in neurological studies. We conclude by discussing potential further developments of higher-order AR models and a number of connections with prior approaches.
Keywords: Autoregressive models, Bayesian forecasting, Innovations sampling, Matrix-Variate Autoregressions, Multiple time series analysis, Multivariate stochastic volatility, Time-varying variance matrix
This work was supported in part by the National Science Foundation via a Mathematical Sciences Postdoctoral Research Fellowship and grants DMS-0342172 and DMS-1106516. Any opinions, findings and conclusions or recommendations expressed in this work are those of the authors and do not necessarily reflect the views of the NSF.