Our general subject is model determination methods and their use in the prediction of economic time series. The methods suggested are Bayesian in spirit but they can be justified by classical as well as Bayesian arguments. The main part of the paper is concerned with 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. The paper continues some recent work on Bayesian asymptotics by the author and Werner Ploberger (1995), develops embedding techniques for vector martingales that justify the role of a 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 reset the initial conditions of a model.

John C. Ham, Robert J. Lalonde, The Effect of Sample Selection and Initial Conditions in Duration Models: Evidence from Experimental Data on Training, Econometrica, Vol. 64, No. 1 (Jan., 1996), pp. 175-205

This paper develops an asymptotic theory of Bayesian inference for time series. A limiting representation of the Bayesian data density is obtained and shown to be of the same general exponential form for a wide class of likelihoods and prior distributions. Continuous time and discrete time cases are studied. In discrete time, an embedding theorem is given which shows how to embed the exponential density in a continuous time process. From the embedding we obtain a large sample approximation to the model of the data that corresponds to the exponential density. This has the form of discrete observations drawn from a nonlinear stochastic differential equation driven by Brownian motion. No assumptions concerning stationarity or rates of convergence are required in the asymptotics. Some implications for statistical testing are explored and we suggest tests that are based on likelihood ratios (or Bayes factors) of the exponential densities for discriminating between models.