TIME-VARYING LINEAR REGRESSION VIA FLEXIBLE LEAST SQUARESt

-Suppose noisy observations obtained on a process are assumed to have been generated by a linear regression model with coefficients which evolve only slowly over time, if at all. Do the estimated time-paths for the coefficients display any systematic time-variation, or is time-constancy a reasonably satisfactory approximation? A "flexible least squares" (FLS) solution is proposed for this problem, consisting of all coefficient sequence estimates which yield vector-minimal sums of squared residual measurement and dynamic errors conditional on the given observations. A procedure with FORTRAN implementation is developed for the exact sequential updating of the FLS estimates as the process length increases and new observations are obtained. Simulation experiments demonstrating the ability of FLS to track linear, quadratic, sinusoidal, and regime shift motions in the true coefficients, despite noisy observations, are reported. An empirical money demand application is also summarized. 1. I N T R O D U C T I O N 1.1. Overview Suppose an investigator undertaking a time-series linear regression study suspects that the regression coefficients might have changed over the period of time during which observations were obtained. The present paper proposes a conceptually and computationally straightforward way to guard against such a possibility. The dynamic equations governing the motion of the coefficients will often not be known. Nevertheless, for many linear regression applications in the natural and social sciences, an assumption that the coefficients evolve only slowly over time seems reasonable. In this case two kinds of model specification error can be associated with each choice of an estimate b = (b, . . . . . bs) for the sequence of coefficient vectors bn: residual measurement error given by the discrepancy between the observed dependent variable Yn and the estimated linear regression model x~bn at each time n; and residual dynamic error given by the discrepancy lb,+, b , ] between coefficient vector estimates for each successive pair of times n and n + 1. Suppose a vector of "incompatibility costs" is assigned to each possible coefficient sequence estimate b based on the specification errors which b would entail. For example, suppose the cost assigned to b for measurement error is given by the sum of squared residual measurement errors, and the cost assigned to b for dynamic error is given by the sum of squared residual dynamic errors. The "flexible least squares" (FLS) solution is defined to be the collection of all coefficient sequence estimates b which yield vector-minimal sums of squared residual measurement and dynamic errors for the given observations--i .e, which attain the "residual efficiency frontier". The frontier characterizes the efficient attainable trade-offs between residual measurement error and residual dynamic error. In particular; the frontier reveals the cost in terms of residual measurement error that must be paid in order to attain the zero residual dynamic error (time-constant coefficients) required by ordinary least squares estimation. Coefficient sequence estimates b which attain the residual efficiency frontier are referred to as "FLS estimates". Each FLS estimate has a basic efficiency property: no other coefficient sequence tThe present paper is a revised version of Ref. [1], presented at the 1987Ninth Annual Conference of the Society for Economic Dynamics and Control, in an April 1987 seminar at UC Berkeley, and in a January 1988 seminar at the University of Arizona. The authors are grateful to conference and seminar participants for numerous helpful suggestions. :~Author for correspondence.