Some Exact Finite Sample Results of Estimators Based on Restricted Residuals in A Class of Seemingly Unrelated Regressions

Consider a system of two seemingly unrelated regression equautionsy_i=X_iB_i+u_i(i=1,2）, Where X_i（i=1,2） is the repression matrices for whichP₁P₂=P₂P₁（P_i=X_i(X'_iX_i)^-1X'_i,i=1,2）. This system includes some special special models considered in some paperes 1-4.Let β_iand ∧β_i（i= l,2） be the β-estimatiors based,respectively, on the restricted estimate S of Σ and on tile unrestricted estimate S of Σ. This paper derives the finite sample variances of the β_i（i = 1, 2）, and examines their efficiency with respect to the ∧β_iand with respect tile OLS estimator b_i obtained directly from the ith equation.It is shown that for large values of the sample size n or the correlation coefficient|ρ|the estimatorβ_imay be more efficient than b_i,and for samll values of|ρ|the restricted estimator β_i can work better than the unrestricted estimator ∧β_i（i = 1, 2）.