Gauss-Markoff估计的协方差改进形式及其在SUR模型中的应用

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

摘要: 本文采用并推广Rao¹的协方差改进原理，证明了线性模型（1．1）的Gauaa－Markoff估计（1．2）具有协方差改进形式\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，其中N＝I－X（X'X）^-1X'．这一结果用于SUR系统y_i＝X_iβ_i＋ε_i（i＝1，2，…，m），容易得到Zellner两步估计的有限样本性质．本文得到了一类系统的有限样本方差结果，从而完善了一些已有结果。

Abstract: 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