Dynamic matrix-variate graphical models

Abstract

This paper introduces a novel class of Bayesian models for multivariate time series analysis based on a synthesis of dynamic linear models and graphical models. The synthesis uses sparse graphical modelling ideas to introduce structured, conditional independence relationships in the time-varying, cross-sectional covariance matrices of multiple time series. We define this new class of models and their theoretical structure involving novel matrix-normal/hyper-inverse Wishart distributions. We then describe the resulting Bayesian methodology and computational strategies for model fitting and prediction. This includes novel stochastic evolution theory for time-varying, structured variance matrices, and the full sequential and conjugate updating, filtering and forecasting analysis. The models are then applied in the context of financial time series for predictive portfolio analysis. The improvements defined in optimal Bayesian decision analysis in this example context vividly illustrate the practical benefits of the parsimony induced via appropriate graphical model structuring in multivariate dynamic modelling. We discuss theoretical and empirical aspects of the conditional independence structures in such models, issues of model uncertainty and search, and the relevance of this new framework as a key step towards scaling multivariate dynamic Bayesian modelling methodology to time series of increasing dimension and complexity.