Multiple time series models with stochastic regressors are considered and primary attention is given to vector autoregressions (VAR’s) with trending mechanisms that may be stochastic, deterministic or both. In a Bayesian framework, the data density in such a system implies the existence of a time series “Bayes model” and “Bayes measure” of the data. These are predictive models and measures for the next period observation given the historical trajectory to the present. Issues of model selection, hypothesis testing and forecast evaluation are all studied within the context of these models and the measures are used to develop selection criteria, test statistics and encompassing tests within the compass of the same statistical methodology. Of particular interest in applications are lag order and trend degree, causal eﬀects, the presence and number of unit roots in the system, and for integrated series the presence of cointegration and the rank of the cointegration space, which can be interpreted as an order selection problem. In data where there is evidence of mildly explosive behavior we also wish to allow for the presence of co-motion among variables even though they are individually not modelled as integrated series. The paper develops a statistical framework for addressing these features of trending multiple time series and reports an extended empirical application of the methodology to a model of the US economy that sets out to explain the behavior of and to forecast interest rates, unemployment, money stock, prices and income. The performance of a data-based, evolving “Bayes model” of these series is evaluated against some rival ﬁxed format VAR’s, VAR’s with Minnesota priors (BVARM’s) and univariate models. The empirical results show that ﬁxed format VAR’s and BVARM’s all perform poorly in forecasting exercises in comparison with evolving “Bayes models” that explicitly adapt in form as new data becomes available.
Phillips, Peter C.B., "Bayes Methods for Trending Multiple Time Series with an Empirical Application to the US Economy" (1992). Cowles Foundation Discussion Papers. 1268.