Improved Liu-Type Estimator of Parameters in Two Seemingly Unrelated Regressions

We consider the parameter estimation in two seemingly unrelated regression systems. To overcome the multicollinearity, we propose a Liu-type estimator in seemingly unrelated regression systems. The superiority of the new estimator over the classic estimator in the mean square error is discussed and we also discuss the admissibility of the Liu-type estimator. 1. Introduction Consider the following two seemingly unrelated regressions (SUR): where are vectors of observations, are matrices with full column rank, are vectors of unknown regression parameters, are vectors of error variables, and where is a positive definite matrix. This system has been used in many fields, such as social biological sciences and econometrics. Zellner [1, 2] firstly defined this system and later it was discussed by many researchers, such as Wang et al. [3], Roozbeh et al. [4], and Singh et al. [5]. For system (1), when , we can get the estimator of as follows: When the error vector is related, the estimator may not fully use the information of the parameter. How to use the information to improve the is a problem. Revankar [6] and Srivastava and Giles [7] have discussed this problem. Wang [8] uses the covariance-improve method to estimate the parameter in two seemingly unrelated regressions. With the prior information, Wang [8] proposes the covariance-improve estimator of and : where , , . When the design matrix is ill-conditioned, the estimator , is no longer a good estimator. Many researchers have discussed this problem. One method to overcome this problem is to consider biased estimator, such as Liu and Wang [9] introduce the ridge estimator, Liu [10] proposes the principal regression component estimator, Qiu [11] proposes the classic type estimator, and Roozbeh et al. [12] introduce the ridge estimator. In this paper, we propose a Liu-type estimator to overcome the multicollinearity in the two seemingly unrelated regressions. Then we discuss the superiority of the Liu-type estimator over the covariance-improve estimator in the mean square error criterion, and we also discuss the selection of the parameter in the proposed estimator. Since the estimator of is similar to the estimator of , so in this paper, we only discuss the estimator of . The rest of the paper is organized as follows. The Liu-type estimator is given in Section 2 and the properties of the new estimator under the mean square error criterion are discussed in Section 3. The admissibility of the proposed estimator is studied in Section 4 and some conclusion remarks are given in Section 5. 2. The Proposed Estimator