A further note on flexible least squares and Kalman filtering

Abstract

Tucci (1990) logically errs when he attempts to equate the flexible least squares (FLS) approach [Kalaba and Tesfatsion (KT) (1989)] with Kalman filtering. FLS addresses a multicriteria model specification problem which does not require probability assumptions either for its motivation or for its solution: the characterization of the set of all state sequence estimates which achieve vector-minimal incompatibility between imperfectly specified theoretical relations and process observations. Kalman filtering is a point estimation technique for determining the most probable state sequence estimate for a stochastic model assumed to be correctly and completely specified. To illustrate, consider the simple time-varying linear regression problem analyzed both in KT (1989) and in Tucci (1990). Scalar observations yi, . . . , yr have been obtained on an economic process which is not yet well understood. For simplicity, a linear relation is postulated between the observation y, and a vector X, of given regressor variables at each time t, of the form y, = x;bt, t=l 9 . , T. Although it is recognized that some systematic time variation in the coefficient vectors b, might have occurred over the observation period, it is anticipated that such evolution will have been gradual, so that the successive coefficient vectors satisfy b,+l = b,, t = 1,. . . , T - 1. Each possible estimate 5 = (ii,. . . ,6,) for the sequence of coefficient vectors thus entails two concep- tually distinct types of model specification errors: namely, measurement errors consisting of the discrepancies [v, - ,x:6!] between the observation and the estimated linear regression at each time t, and dynamic errors consisting of the discrepancies [it+ 1 - h,] between the estimated coefficient vectors at successive times t and t + 1. The FLS approach acknowledges the multicriteria nature of this estimation problem, and the minimal nature of the availableAinformation concerning the true underlying process. A measurement cost cM( b) and a dynamic cost cD( b) are separately assessed for the two disparate types of model specification errors entailed by the choice of a coefficientnsequen:e estimate 6. Letting C denote the collection of cost vectors (c,(b), c,,,(b)) corresponding to all