THE FLEXIBLE LEAST SQUARES APPROACH TO TIME-VARYING LINEAR REGRESSION* Robert KALABA and Leigh TESFATSION

Consider an investigator who obtains noisy observations on a process and who wishes to learn about the actual sequence of states through which the process has passed. Suppose the investigator believes the process is adequately described by non-linear dynamic and measurement equations. However, he sees no way to justify the assignment of specific probabilistic properties to residual error terms. In earlier studies [Kalaba and Tesfatsion (1980,1981,1985,1986a)] a solution is developed for this state estimation problem assuming the non-linear dynamic and measurement equations are known up to a parameterization. Currently we are considering what might be done when the dynamic equations are unknown but the process state evolves only slowly over time. A smoothness prior is introduced in place of an explicit specification for the unknown dynamic equations governing the evolution of the process state. In Kalaba and Tesfatsion (1986b) a special case of the latter problem is considered, namely, a variant of the well-known time-varying linear regression problem [Chow (1984)]. An investigator obtains noisy observations on a process which he believes can be adequately described by a linear regression model with a slowly evolving coefficient (state) vector. The actual dynamic equations governing the evolution of the coefficient vector are unknown and are proxied by a smoothness prior. Residual dynamic and measurement errors are anticipated to be small, but are otherwise unrestricted. The investigator wishes to estimate the sequence of time-varying coefficient vectors. The ‘flexible least squares (FLS) solution’ proposed for this time-varying linear regression problem consists of all coefficient sequence estimates which attain the ‘residual efficiency frontier’ i.e., which yield minimal pairs (r-6, r&) of squared residual dynamic error and measurement error sums, conditional on the given observations. A conceptually and computationally straightforward algorithm is developed which permits the exact sequential updating of