  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.
