Covariance-Improved Estimator Of Seemingly Unrelated Regression Equation System Under Linear Restriction

In the late 30 or 40 years, many scholars have a lot of studies on a Seemingly Unrelated Regression (SDR) system with two linear regression models, and some important results are obtained: Zellner (1962) put forward two-stage estimator (TSE); Based on Zellner' s, Lin chun-shi (1984) Obtained the sufficient and necessary condition of two-stage estimator; Chen chang-hua (1986) discussed the TSE and its optimalities without any condition for designed-matrix X; Ulteriorly, Wang song-gui and Van li-qing (1997) obtained an iteration sequence of estimator by using the covariance-improved approach; Liu jin-shan (1994), Li wen and Lin ju-gan (1997) generalized the covariance-improved estimator respectively. In. the practical application, the second linear regression equation of this SUR system is usually regarded as auxiliary information of the first one, and. all of the results of unknown regression coefficient β2 are parallel to that of β1. Based on [4], the author adds a linear restriction r=Rβ to this SUR system, and discusses the estimator and the optimalities of unknown regression coefficient ft in the SUR system under the linear restriction. The main results are calculated as the following:1.The covariance-improved estimator sequence bk(1)(E) (k=1, 2......) of unknown regression coefficient ft under thelinear restriction r=Rβ and its convergence, that is:2. The optimalities of covariance-improved estimator sequence under three MDE crlterions;3. Showed the unbiasedness of two-stage estimator under normal distribution assumption on the random errors when the covariance matrixes of errors is unknown: and the monotonousness of covariance matrixes of the two-stage covariance-improved estimator sequence bk(1)(S) :4. Solved the convergence among covariance matrixes of β1*, covariance-improved estimator sequence and two-suage covariance-improved estimator sequence, that is: (1): For any and regular k:(2): For regular n: limcov (bk(1)(S)) = cov (β1*(S))In this paper the author extended and improved the covariance-improved estimator introduced by Wang song-gui, the results show clearly the power of the covariance improved approach.