Note---On the Relationship of Adaptive Filtering Forecasting Models to Simple Brown Smoothing

This Note establishes two linkages between adaptive estimation procedures used in the analysis and forecasting of multivariate time series and the simple Brown smoothing technique. First, for a simple model it is shown that AEP and LMS are identical to the Brown technique. Thus they are accorded the theoretical properties possessed by the Brown estimator for the simple model. This linkage establishes that the Brown technique has been extended in two complementary directions: to higher order smoothing models which use explicit functions of time and implicitly include effects of other variables and to adaptive time-varying parameter models which explicitly include the influences of other variables and implicitly incorporate time. Second, for the same simple model, Stochastic Approximation, Kalman Filtering, and Bayesian Analysis reduce to a form similar to the Brown technique except that they have a time-varying smoothing constant which approaches zero as the sample size grows large. Hence, while they provide optimal estimates which converge in mean square error, they are not responsive to change.