The subject of this paper is modelling, estimation, inference and prediction for economic time series. Bayesian and classical approaches are considered. The paper has three main parts. The first is concerned with Bayesian model determination, forecast evaluation and the construction of evolving sequences of models that can adapt in dimension and form (including the way in which any nonstationarity in the data is modelled) as new characteristics in the data become evident. This part of the paper continues some recent work on Bayesian asymptotics by the author and Werner Ploberger, develops embedding techniques for vector martingales that justify the role of a general class of exponential densities in model selection and forecast evaluation, and implements the modelling ideas in a multivariate regression framework that includes Bayesian vector autoregressions (BVAR's) and reduced rank regressions (RRR's). It is shown how the theory in the paper can be used: (i) to construct optimized BVAR's with data- determined hyperparameters; (ii) to compare models such as BVAR's, optimized BVAR's and RRR's; (iii) to perform joint order selection of cointegrating rank, lag length and trend degree in a VAR; and (iv) to discard data that may be irrelevant and thereby help determine the "lifetime" of an econometric model. Simulations are conducted to study the forecasting performance of these model determination procedures in some multiple time series models with cointegration. The final part of the paper reports an empirical application of these ideas and methods to US and UK macroeconomic data.
