Root Root Estimator Of Seemingly Unrelated Regression System

SUR (Seemingly Unrelated Regression) System is very useful in metrology econometries, life sciences, industry, metrology geography and so on. Therefore, the study in SUR Equations is highly regarded.C.R.Rao introduced the method of Covariance Adjustment Approach in 1967. It was first applied to SUR System by Wang Songgui to get better estimators, which are Covariance Adjustment Estimator and Two-Stage Covariance Adjustment Estimator. 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 coefficientβ2 are parallel to that ofβ1. When the designed matrix is ill-conditioned, Covariance Adjustment Estimatorβ~ 1 is not accurate, so we propose a class of new estimators—root root estimator, which is based on the mean square error criterion.Firstly, we give the expression of root root estimator and proved under some condition root root estimator is more efficient than the Covariance Adjusted Approach, and the properties are also discussed for the two-stage estimator.Besides, the relative efficiencies of root root estimator are given and the higher and lower bound are also obtained.Secondly, we introduced a biased estimator-generalized root root estimator, which is based on the ordinary root root estimator, and the properties are also discussed, we obtain the condition that estimator-generalized root root estimator is more efficient than the Covariance Adjusted Approach. By studying a special expression of estimator-generalized root root estimator we proved that estimator- generalized root root estimator more greatly reduces the Mean Square Error of the estimated coefficients than ordinary root root estimator. And the properties are also discussed for the two-stage estimator.Besides, the relative efficiencies of root root estimator are given.Finally, we add a linear restriction Rβ=0 to this SUR System, and first give the expression of restricted Covariance Adjusted Approach. Furthermore, we obtain the expressions of restricted the expression of root root estimator and estimator- generalized root root estimator and discussed the estimator and optimality of unknown regression coefficientβ1 in the SUR system under the linear restriction.