Prior Information on the Coefficients when the Disturbance Covariance Matrix is Unknown

Abstract

usefulness of prior information. Viewed broadly, this theme has encompassed such diverse topics as the estimation of a system of simultaneous structural equations, the specification and estimation of distributed lags, and the formal integration of stochastic prior and sample information through Bayes' theorem. Without exception, the results have encouraged the incorporation of further prior information into our statistical procedures, in the sense that the judicious use of such information has produced unambiguously better estimators. That this need not be the case in a typical linear regression application is thus somewhat surprising and constitutes the topic of this paper. Here, we develop the relationship between the specification of the deterministic and the stochastic components of a linear model and show that whenever the covariance structure of the disturbance process is effectively misspecified,2 one can no longer justify the use of prior information about the deterministic part of the model. If the error covariance matrix differs substantively from that required by the Gauss-Markov theorem, the imposition of correct linear restrictions on the regression coefficients leads to less efficient estimators of some estimable functions of the parameters.3 Prior information can hurt! We begin by introducing notation and examining the efficiency of least squares estimators in linear models with varying amounts of prior information. An application of the theory of regular pencils then produces the main results (Section 3), practical implications are drawn in Section 4, and we conclude with an application (Section 5).