State space modeling of multiple time series

Time series methods offer the possibility of making accurate forecasts even when the underlying structural model is unknown, by replacing the structural restrictions needed to reduce sampling error and improve forecasts with restrictions determined from the data. While there has been considerable success with relatively simple univariate time series modeling procedures, the complex interrela- tionships possible with multiple series requite more powerful techniques.Based on the insights of linear systems theory, a multivariate state space methos for both stationary and nonstationary problems is described and related to ARMA models. The states or dynamic factors of the procedure are chosen to be robust in the presence of model misspecification, in constrast to ARMA models which lack this property. In addition, by treating th emidel choice as a formal approximation problem certain new optimal properties of the procedure with respect to specification are established; in particular, it is shown that no other model of equal or smaller order fits the observed autocovariance sequence any better in the sense of a Hankel norm. Finally, in the treatment of nonstationary series, a natural decomposition into long run and short run dynamics results in easily implemented two step procedures that use characteristics of the data to identify and model trend and cycle components that correspond to cointegration and error correction models. Applications include annualo U.S. GNP and money stock growth rates, monthly California beef prices and inventories, and monthly stock prices for large retailers.