Dynamic Matrix-Variate Graphical Models - A Synopsis 1 -

Bayesian dynamic linear models (DLMs) (West and Harrison, 1997) are used for analysis and prediction of time series of increasing dimension and complexity in many applied fields. The time-varying regression structure, or state-space structure, and the sequential nature of DLM analysis flexibly allows for the creation and routine use of interpretable models of increasingly realistic complexity. The inherent Bayesian framework naturally allows and encourages the integration of data, expert information and systematic interventions in model fitting and assessment, and thus in forecasting and decision making. The current work responds to the increasing prevalence of high-dimensional multivariate time series and the consequent needs to scale and more highly structure analysis methods. Contexts of high-dimensional and rapidly sampled financial time series are central examples, though similar needs are emerging in many areas of science, social science and engineering. This paper introduces a broad new class of time series models to address this: the framework synthesises multi- and matrix-variate DLMs with graphical modelling to induce sparsity and structure in the covariance matrices of such models, including time-varying matrices in multivariate time series. The presentation outlines the framework of matrix-variate DLMs and Gaussian graphical models for structured, parameter constrained covariance matrices based on the use of the family of hyper-inverse Wishart distributions. We then discuss formal model specification and details of the resulting methodology for both constant and, of more practical relevance, time-varying structured covariance matrices in the new models. We summarise the theory that extends DLM sequential updating, forecasting and retrospective analysis to this new model class. Our applied examples combine these flexible new Bayesian models with Bayesian decision analysis in financial portfolio prediction studies. We discuss theoretical and empirical findings in the context of an initial example using 11 exchange rate time series, and then a more extensive and practical study of 346 securities from the S&P Index. This latter application also develops and applies graphical model search and selection ideas, based on existing MCMC and stochastic search methods now translated to the DLM context, as well as illustrating the real practical utility, and benefits over existing models, of the new methodology.