The Covariance Adjustment Version of Gauss-Markoff Estimator and Its Application to Seemingly Unrelated Regression Equations

A lemma of Rao’s covariance adjustment theory is extended, and the Gauss-Markoff estimator \hat\beta of β in linear model Y ＝ Xβ＋ε,ε～ （0, V） is given as \hat\beta=\left(X^\prime X\right)^-1 X^\prime Y-\left(X^\prime X\right)^-1 X^\prime V NN V N^+ N Y, where N＝I－X（X'X）^-1X'. The application of this version to the system of Seemingly Unrelated Regression （SUR） equations: y_i＝X_iβ_i＋ε_i（i＝1,…，m） is considered, and some exact finite sample results in a class of SUR system with P₁＝…＝ P_k, P_k+1 ＝…＝ P_m,P₁P_m ＝ P_m P₁ are obtained