Research Article | Open Access
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.
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. , Roozbeh et al. , and Singh et al. .
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  and Srivastava and Giles  have discussed this problem. Wang  uses the covariance-improve method to estimate the parameter in two seemingly unrelated regressions. With the prior information, Wang  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  introduce the ridge estimator, Liu  proposes the principal regression component estimator, Qiu  proposes the classic type estimator, and Roozbeh et al.  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
In this section, we use the prior information to obtain the Liu-type estimator. Firstly, we give a lemma.
Lemma 1 (see ). Let be an estimator of , the prior information with , and
Then in the estimator , the estimator has minimum mean square error and
Now we establish the Liu-type estimator in SUR model. Set satisfy and has the max column full rank; then and
By Lemma 1, when is known, we establish a new estimator for : where is the estimator for the single model in (1), , , , , and . We call this estimator as Liu-type estimator in SUR model. By the definition of the Liu-type estimator, it is easy to know that it is a general estimator which includes the covariance-improve estimator proposed by Wang  and the ridge estimator proposed by Liu  as special cases.
(1) When , then
(2) When , then
In next section, we will give the properties of the new estimator.
3. MSE: The Superiority of the New Estimator over the Covariance-Improve Estimator
In this section, we discuss the statistical properties of the new estimator . Firstly, we give the definition of the mean square error (MSE).
Let be an estimator of ; the MSE of is defined as where .
In the following theorem, we give the superiority of the new estimator over the covariance-improve estimator in the mean square error.
Theorem 2. For the SUR model (1), when is known,
(a) when fixed,
(1) if , then MSEMSE;
(2) if , when /, then ;
(b) when fixed,
(1) if , then ;
(2) if , when , then .
Proof. By the definition of the covariance-improve estimator , we obtain
Then by the definition of the mean square error, we obtain
Since , there exists an orthogonal matrix such that , where is the ordered eigenvalues of ; thus we obtain
where . Since so we get that , .
Now we compute the MSE of the Liu-type estimator () for SUR model (1). Firstly, we compute that Thus we have where .
In order to compare and , we consider the following difference: When fixed, then we write (17) as follows: So
(1) If , then ;
(2) if , when , then .
When fixed, then we write (17) as follows:
(1) , then ;
(2) , when , then .
4. The Admissible of the New Estimator
As we all know, the admissible of an estimator is an important problem in SUR model. In this section, we discuss the admissible of the new estimator . Firstly, we give the definition of the admissible.
Definition 3. Let and be two estimators of , for arbitrarily risk loss function , (1) for arbitrarily . (2) There at least exist a such that the inequality is satisfied, Then we see that is uniformly better than for risk loss function . If, in some class estimators, there is not exist an estimator that is uniformly better than , we can say that is admissible estimator of for risk loss function in that class estimators.
Lemma 4 (see ). For linear model where is a known matrix and is full column rank. Let , ; then in the linear estimators of , is admissible estimator of if and only if
Now we discuss the admissibility of the new estimator. In this section we consider the quadratic loss (mean square error).
Theorem 5. For the SUR model (1), when is know, if , then is admissible estimator of in the class of linear estimators .
Proof. If , then
For SUR model (1), let , , , and , Then (1) can be written as
and (24) satisfies the following equation:
Then (25) can be written as follows:
When , (25) can be written as follows:
Then by (28), we may get
multiplication in (28); we have
Let and ; by Lemma 4, if we want to prove that is admissible estimator of , we only need to prove . Since , we may conclude that Then It is obvious that . The proof is completed.
Theorem 6. For the SUR model (1), when is known, if , then is admissible estimator of in the class of linear estimators .
Proof. For SUR model (1), when , by Lin , we obtain ; thus we obtain Similar to Theorem 5, we only need to prove ; and are defined as Theorem 5: Denote ; by Rao et al. , we have Then we have Obviously . The proof is completed.
5. Conclusion Remarks
In this paper we consider the parameter estimation in seemingly unrelated regression system. A Liu-type estimator is proposed to overcome the multicollinearity in seemingly unrelated regression system. The superiority of the new estimator has been also discussed and the admissibility of the new estimator is also discussed.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author is grateful to the editor and the anonymous referee for the valuable comments which improved the quality of the paper. This work was supported by the Scientific Research Foundation of Chongqing University of Arts and Sciences (Grant no. R2013SC12), Program for Innovation Team Building at Institutions of Higher Education in Chongqing (Grant no. KJTD201321), and the National Natural Science Foundation of China (nos. 11201505, 71271227).
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