Multivariate Modelling and Forecasting

Abstract

In this final chapter we return to the linear/normal framework to explore models for multivariate time series. Univariate DLMs can be extended in an obvious way to multivariate problems simply by taking the observations at each time as vectors rather than scalars. In fact such models have already been defined in Definition 4.1 of Chapter 4. This means that the observational errors become vectors too, the regression vectors become matrices, but the remaining model components are similar to the univariate case. The basic theory of the univariate DLM as developed in Chapter 4 extends directly to such models. The observational errors are now vectors, so that model specification requires observational variance matrices defining the joint stochastic structure of the observations conditional on state parameters. In DLMs having vector observations with observational errors following a multivariate normal distribution, the univariate theory extends easily only when it is assumed that the observational error variance matrices are known for all time. However, as soon as uncertainties about observational variance matrices are admitted, the tractability of analysis is lost. In general, there is no neat, conjugate analysis of multivariate DLMs whose observational errors are multivariate normal with unknown (constant or otherwise) variance matrices. The theory for models with known variance matrices is developed in Section 15.2.