Filters for Linear Stochastic Discrete-Time Systems

Abstract

Observed variables in the real economy will be accompanied by observation errors, and any assumed linear dynamic economic system must contain a random disturbance term unexplainable by the system scheme. “Filtering” here is the problem of estimating all state variables from hitherto available input and output data in such a system. Filtering is, of course, a prerequisite for providing optimal feedback control values. Following Rhodes (1971), we verify the Kalman recursive formulas for the predictor and filter for our dynamic system in a tutorial manner (Section 4.2). (Section 4.1 is devoted to a preliminary least-squares estimation.) In Sections 4.3 and 4.4 we are concerned with the problem of finding a minimal-order filter of observer type, and we rely exclusively on Tse and Athans (1970). Their idea is to minimize the covariance matrix of estimation errors and to establish a dynamic minimal-order observer-estimator. The estimator is also applied to a general distributed-lag system. Finally, in Section 4.5, related economic applications are presented and examined.