Use of Restricted Residuals in SUR Systems: Some Finite Sample Results

Abstract

Consider a system of two seemingly unrelated regression equations: , where X₂ is a proper subset of X₁, i.e., X₁≡(X₂: L₂), and Σ = (σ_ij) is the disturbance covariance matrix. Let be the -estimators based, respectively, on the restricted estimate of Σ and on the unrestricted estimate S of Σ. This paper derives finite sample variances of the , and examines their efficiency with respect to the are identical to the OLS estimator obtained directly from , and are efficient. On the other hand, is more (less) efficient at the lower (upper) end of the scale of . In practical applications, it may be preferable to use when | ρ | ≤ .5 (approximately) and df ≥ 5, and otherwise.