Generalized Least Squares Applied to Time Varying Parameter Models

This paper shows the formal equivalence of Kalman filtering and smoothing techniques to generalized least squares. Smoothing and filtering equations are presented for the case where some of the parameters are constant. The paper further shows that generalized least squares will produce consistent estimates of those parameters that are not time varying. When linear models have been used to model economic problems, it has been useful many times to allow for parameter variation across observa-tions. Various statistical procedures have been developed to estimate and test this hypothesis of nonstable regression coefficients. ' Recently, it has been recognized that a technique known as the Kalman filter has useful applications in estimating economic models with nonconstant coeffi-cients.2 The purpose of this paper is to show the formal equivalence of Kalman filtering and smoothing techniques with generalized least squares, to derive the Kalman filter and smoother. without assuming all of the parameters are subject to stochastic variation, and to show that gen-eralized least squares produces consistent estimates of those parameter