Investigation of the structural properties of Kalman filter models for forecasting non-stationary time series

Abstract

This work examines structural properties of dynamic linear models (DLMs) and considers their implications for predictors and minimum-variance estimators of these models. The techniques employed are those of statistical time series analysis and modern control theory, and particular use is made of the concept of observability. Structural properties are derived and considered for the constant forecast model, the polynomial model of arbitrary degree, both with and without a forward shifted forecast function, and for the more general non-seasonal models with an asymptotic forecast function. Several equivalence theorems are established for these DLMs. It is shown that the invertibility of the time series model is equivalent to the stability of the estimation scheme for the DLM in the equilibrium state. It follows that all predictors of a stable DLM are identical in the steady state to those of the Box-Jenkins forecasting schemes. Furthermore, the observability requirement yields an upper bound on the dimension of the state vector, and a lower bound is necessary to avoid specified restrictions on the equivalences. Examples are considered which show that the practical requirement that the system error covariance matrix be diagonal can further restrict the equivalence. In addition, the Cramer-Rao bound is considered for estimators of the state vector. It is shown that the information matrix is invertible, and there is a unique estimator which achieves the Cramer-Rao bound if and only if the DLM is observable. This result is also discussed in varying degrees of generality.