Some Exact Finite Results of Zellner’s Estimators in m Seemingly Unrelated Regressions

For m（≥3） seemingly unrleated regression equations: y_i = X_iβ_i+εi,（i=1,…, m）, with P₁ =… = P_k, P_k+1 =… = P_s, P_s+1 =… = P_m and P₁ =P_s+P_m, where P_i = X_i（X'_iX_i）^-1X'_i,the exact finite sample covariance matrix of Zellner’s two-step Aitken estimator ~β_i of β_i（i =1…，m）, based on the unrestricted estimate S of ∑ = （σ_ij ） are obtained.When m = 3, all the three ~β_i （i = 1, 2, 3） are shown to be more efficient then the OLS estimators of β_i, for moderate departures of \rho_i j^2=\sigma_i j^2 / \sigma_i i \sigma_j j from zero, and the efficiencies are shown to be increased with the sample size n. These results can be compared with those obtained by Revankar（1974）for a system of 2SUR with X₁=（X₂,L）