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

Abstract

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): 𝑌₁=𝑋₁𝛽₁+𝜀₁,𝑌₂=𝑋₂𝛽₂+𝜀₂, (1) where 𝑌_𝑖(𝑖=1,2) are 𝑛×1 vectors of observations, 𝑋_𝑖(𝑖=1,2) are 𝑛×𝑝 matrices with full column rank, 𝛽_𝑖(𝑖=1,2) are 𝑝×1 vectors of unknown regression parameters, 𝜀_𝑖(𝑖=1,2) are 𝑛×1 vectors of error variables, and 𝐸𝜀_𝑖=0,Cov𝜀_𝑖,𝜀_𝑗=𝜎_𝑖𝑗𝐼_𝑛,𝑖,𝑗=1,2, (2) where 𝑉=(𝜎_𝑖𝑗) is a 2×2 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 𝜎₁₂=0, we can get the estimator of 𝛽 as follows: 𝛽𝛽=¹𝛽₂=𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑌₁𝑋^′₂𝑋₂⁻¹𝑋^′₂𝑌₂. (3) When the error vector is related, the estimator 𝛽 may not fully use the information of the parameter. How to use the information to improve the 𝛽 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 𝛽₁ and 𝛽₂: 𝛽₁=𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑌₁−𝜎₁₂𝜎₂₂𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑁₂𝑌₂,𝛽₂=𝑋^′₂𝑋₂⁻¹𝑋^′₂𝑌₂−𝜎₁₂𝜎₁₁𝑋^′₂𝑋₂⁻¹𝑋^′₂𝑁₁𝑌₁, (4) where 𝑁_𝑗=𝐼−𝑃_𝑗, 𝑃_𝑗=𝑋_𝑗(𝑋_𝑗𝑋^′_𝑗)⁻¹𝑋^′_𝑗, 𝑗=1,2.

When the design matrix 𝑋_𝑖 is ill-conditioned, the estimator 𝛽_𝑖, 𝑖=1,2 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 𝑐−𝑘 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 𝛽₂ is similar to the estimator of 𝛽₁, so in this paper, we only discuss the estimator of 𝛽₁.

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 [10]). Let 𝜃 be an estimator of 𝜃, 𝜃 the prior information with 𝐸(𝜃)=0, and 𝜃𝜃=𝑉Cov¹¹𝑉₁₂𝑉₁₂𝑉₂₂. (5)
Then in the estimator 𝐴={𝜃^∗(𝑋)=𝜃+𝑋𝜃}, the estimator 𝜃^∗=𝜃−𝑉₁₂𝑉⁻¹₂₂𝜃 has minimum mean square error and MSE𝜃^∗=MSE𝜃−tr𝑉₁₂𝑉⁻¹₂₂𝑉₂₁. (6)

Now we establish the Liu-type estimator in SUR model. Set 𝑍₂ satisfy 𝑋^′₂𝑍₂=0 and 𝑋₂ has the max column full rank; then 𝐸(𝑍^′₂𝑌₂)=0 andCov𝛽₁()𝑍𝑘,𝑑_′2𝑌₂=𝜎₁₁𝑋^′₁𝑋₁+𝑘𝐼⁻²𝑋^′₁𝑋₁+𝑑𝐼²𝑋^′₁𝑋₁⁻¹𝜎₁₂𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′𝑋+𝑑𝐼𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑍₂𝜎₁₂𝑍^′₂𝑋₁𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′𝑋+𝑑𝐼𝜎₂₂𝑍^′₂𝑍₂. (7)

By Lemma 1, when 𝑉 is known, we establish a new estimator for 𝛽₁: 𝛽₁(𝑘,𝑑)=𝛽₁()−𝜎𝑘,𝑑¹²𝜎₂₂𝑋^′₁𝑋₁+𝑘𝐼⁻¹×𝑋^′₁𝑋₁+𝑑𝐼𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑍₂𝑍^′₂𝑍₂⁻¹𝑍^′₂𝑌₂=𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁+𝑑𝐼𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑌₁−𝜎₁₂𝜎₂₂𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁×𝑋+𝑑𝐼_′1𝑋₁⁻¹𝑋^′₁𝑁₂𝑌₂, (8) where 𝛽₁(𝑘,𝑑)=(𝑋^′₁𝑋₁+𝑘𝐼)⁻¹(𝑋^′𝑋+𝑑𝐼)(𝑋^′₁𝑋₁)⁻¹𝑋^′₁𝑌₁ is the estimator for the single model in (1), 𝑘>0, 0<𝑑<1, 𝑑<𝑘, 𝑁₂=𝐼−𝑃₂, and 𝑃₂=𝑋₂(𝑋^′₂𝑋₂)⁻¹𝑋₂. 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 [8] and the ridge estimator proposed by Liu [10] as special cases.

(1) When 𝑘=𝑑=0, then 𝛽₁(𝑘,𝑑)=𝛽₁=𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑌₁−𝜎₁₂𝜎₂₂𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑁₂𝑌₂. (9)

(2) When 𝑑=1, then 𝛽₁(𝑘,𝑑)=𝛽₁(𝑘)=𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑌₁−𝜎₁₂𝜎₂₂𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑁₂𝑌₂. (10)

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 𝛽(𝑘,𝑑). Firstly, we give the definition of the mean square error (MSE).

Let 𝑏^∗ be an estimator of 𝛽; the MSE of 𝑏^∗ is defined as MSE𝑏^∗,𝛽=𝐸𝑏^∗−𝛽^′𝑏^∗𝑏−𝛽=trCov^∗+bias𝑏^∗′bias𝑏^∗, (11) where bias(𝑏^∗)=𝐸(𝑏^∗)−𝛽.

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 𝑉 is known,
(a) when 0<𝑑<1 fixed,
(1) if 𝜎₁₁(1/𝜆_𝑖−𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))−𝛼²_𝑖≥0, then MSE(𝛽₁)≥MSE(𝛽₁(𝑘,𝑑));
(2) if 𝜎₁₁(1/𝜆_𝑖−𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))−𝛼²_𝑖<0, when 0<𝑘<(𝜎₁₁(2𝜆_𝑖+𝑑)(1/𝜆_𝑖+𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))+𝑑𝛼²_𝑖)/(𝛼²_𝑖−𝜎₁₁(1/𝜆_𝑖−𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))), then MSE(𝛽₁)≥MSE(𝛽₁(𝑘,𝑑));
(b) when 𝑘>0 fixed,
(1) if 𝜎₁₁(2𝜆_𝑖+𝑘)(1/𝜆_𝑖+𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))≥𝑘𝛼²_𝑖, then MSE(𝛽₁)≥MSE(𝛽₁(𝑘,𝑑));
(2) if 𝜎₁₁(2𝜆_𝑖+𝑘)(1/𝜆_𝑖+𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))<𝑘𝛼²_𝑖, when (𝑘𝛼²_𝑖−𝜎₁₁(2𝜆_𝑖+𝑘)(1/𝜆_𝑖+𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖)))/(𝜎₁₁(1/𝜆_𝑖−𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))+𝛼²_𝑖)≤𝑑<1, then MSE(𝛽₁)≥MSE(𝛽₁(𝑘,𝑑)).

Proof. By the definition of the covariance-improve estimator 𝛽₁, we obtain 𝐸𝛽₁=𝛽₁,Cov𝛽₁=𝜎₁₁𝑋^′₁𝑋₁⁻¹−𝜎²₁₂𝜎₂₂𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑁₂𝑋₁𝑋^′₁𝑋₁⁻¹. (12) Then by the definition of the mean square error, we obtain MSE𝛽₁=trCov𝛽₁=𝜎₁₁tr𝑋^′₁𝑋₁⁻¹−𝜎²₁₂𝜎₂₂tr𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑁₂𝑋₁𝑋^′₁𝑋₁⁻¹. (13) Since 𝑋^′₁𝑋₁>0, there exists an orthogonal matrix such that 𝑄^′𝑋^′₁𝑋₁𝑄=Λ=diag(𝜆₁,…,𝜆_𝑝1), where 𝜆₁≥⋯≥𝜆_𝑝1 is the ordered eigenvalues of 𝑋^′₁𝑋₁; thus we obtain MSE𝛽₁=𝜎₁₁^𝑝1_𝑖=11𝜆_𝑖−𝜌²₁₂𝑎_𝑖𝑖𝜆²_𝑖, (14) where 𝐴=𝑄^′𝑋^′₁𝑁₂𝑋₁𝑄=(𝑎_𝑖𝑗). Since Cov(𝛽₁)≥0 so we get that 1/𝜆_𝑖−𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖)≥0, 𝑖=1,…,𝑝1.
Now we compute the MSE of the Liu-type estimator (𝛽₁(𝑘,𝑑)) for SUR model (1). Firstly, we compute that 𝐸𝛽₁(𝑘,𝑑)=𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁+𝑑𝐼𝛽₁,Cov𝛽₁(𝑘,𝑑)=𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁𝛽+𝑑𝐼×Cov₁𝑋^′₁𝑋₁+𝑑𝐼𝑋^′₁𝑋₁+𝑘𝐼⁻¹. (15) Thus we have MSE𝛽₁(𝑘,𝑑)=trCov𝛽₁(𝛽𝑘,𝑑)+bias₁(𝑘,𝑑)^′bias𝛽₁(𝑘,𝑑)^′=𝜎₁₁^𝑝1_𝑖=1𝜆_𝑖+𝑑²𝜆_𝑖+𝑘²1𝜆_𝑖−𝜌²₁₂𝑎_𝑖𝑖𝜆²_𝑖+(𝑘−𝑑)^2𝑝1_𝑖=1𝛼²_𝑖𝜆_𝑖+𝑘², (16) where 𝛼=𝑄^′𝛽₁.
In order to compare 𝛽₁(𝑘,𝑑) and 𝛽₁, we consider the following difference: MSE𝛽₁−MSE𝛽₁(𝑘,𝑑)=𝜎₁₁^𝑝1_𝑖=11𝜆_𝑖−𝜌²₁₂𝑎_𝑖𝑖𝜆²_𝑖−𝜎₁₁^𝑝1_𝑖=1𝜆_𝑖+𝑑²𝜆_𝑖+𝑘²1𝜆_𝑖−𝜌²₁₂𝑎_𝑖𝑖𝜆²_𝑖−(𝑘−𝑑)^2𝑝1_𝑖=1𝛼²_𝑖𝜆_𝑖+𝑘²=^𝑝1_𝑖=1(𝑘−𝑑)𝜆_𝑖+𝑘²2𝜆_𝑖1+𝑘+𝑑𝜆_𝑖−𝜌²₁₂𝑎_𝑖𝑖𝜆²_𝑖−(𝑘−𝑑)𝛼²_𝑖. (17) When 0<𝑑<1 fixed, then we write (17) as follows: MSE𝛽₁−MSE𝛽₁(=𝑘,𝑑)_𝑝1𝑖=1(𝑘−𝑑)𝜆_𝑖+𝑘²𝜎₁₁2𝜆_𝑖1+𝑑𝜆_𝑖+𝜌²₁₂𝑎_𝑖𝑖𝜆²_𝑖+𝑑𝛼²_𝑖+𝑘𝜎₁₁1𝜆_𝑖−𝜌²₁₂𝑎_𝑖𝑖𝜆²_𝑖−𝛼²_𝑖. (18) So
(1) If 𝜎₁₁(1/𝜆_𝑖−𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))−𝛼²_𝑖≥0, then MSE(𝛽₁)≥MSE(𝛽₁(𝑘,𝑑));
(2) if 𝜎₁₁(1/𝜆_𝑖−𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))−𝛼²_𝑖<0, when 0<𝑘<(𝜎₁₁(2𝜆_𝑖+𝑑)(1/𝜆_𝑖+𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))+𝑑𝛼²_𝑖)/(𝛼²_𝑖−𝜎₁₁(1/𝜆_𝑖−𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))), then MSE(𝛽₁)≥MSE(𝛽₁(𝑘,𝑑)).
When 𝑘>0 fixed, then we write (17) as follows: MSE𝛽₁−MSE𝛽₁(=𝑘,𝑑)_𝑝1𝑖=1(𝑘−𝑑)𝜆_𝑖+𝑘²𝜎₁₁2𝜆_𝑖1+𝑘𝜆_𝑖+𝜌²₁₂𝑎_𝑖𝑖𝜆²_𝑖−𝑘𝛼²_𝑖+𝑑𝜎₁₁1𝜆_𝑖−𝜌²₁₂𝑎_𝑖𝑖𝜆²_𝑖+𝛼²_𝑖. (19)
So if
(1) 𝜎₁₁(2𝜆_𝑖+𝑘)(1/𝜆_𝑖+𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))≥𝑘𝛼²_𝑖, then MSE(𝛽₁)≥MSE(𝛽₁(𝑘,𝑑));
(2) 𝜎₁₁(2𝜆_𝑖+𝑘)(1/𝜆_𝑖+𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))<𝑘𝛼²_𝑖, when (𝑘𝛼²_𝑖−𝜎₁₁(2𝜆_𝑖+𝑘)(1/𝜆_𝑖+𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖)))/(𝜎₁₁(1/𝜆_𝑖−𝜌²₁₂(𝑎_𝑖𝑖/𝜆²_𝑖))+𝛼²_𝑖)≤𝑑<1, then MSE(𝛽₁)≥MSE(𝛽₁(𝑘,𝑑)).

4. The Admissible of the New Estimator 𝛽₁(𝑘,𝑑)

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 𝛽₁(𝑘,𝑑). Firstly, we give the definition of the admissible.

Definition 3. Let 𝜃₁ and 𝜃₂ be two estimators of 𝜃, for arbitrarily risk loss function 𝑅(⋅,⋅), (1) 𝑅(𝜃₁,𝜃)≤𝑅(𝜃₂,𝜃) for arbitrarily 𝜃. (2) There at least exist a 𝜃₀ such that the inequality is satisfied, Then we see that 𝜃₁ is uniformly better than 𝜃₂ for risk loss function 𝑅(⋅,⋅). If, in some class estimators, there is not exist an estimator that is uniformly better than 𝜃, we can say that 𝜃 is admissible estimator of 𝜃 for risk loss function 𝑅(⋅,⋅) in that class estimators.

Lemma 4 (see [13]). For linear model 𝑦=𝑋𝛽+𝑒,𝐸(𝑒)=0,Cov(𝑒)=𝜎²Σ, (20) where Σ>0 is a known matrix and 𝑋 is full column rank. Let 𝑇=𝑋^′Σ⁻¹𝑋, 𝛽^∗(Σ)=𝑇⁻¹𝑋^′Σ⁻¹𝑦; then in the linear estimators of 𝑆𝛽, 𝐿𝛽^∗(Σ) is admissible estimator of 𝑆𝛽 if and only if 𝐿𝑇⁻¹𝐿^′≤𝐿𝑇⁻¹𝑆^′. (21)

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 𝑉 is know, if 𝑋^′₁𝑋₂=0, then 𝛽₁(𝑘,𝑑) is admissible estimator of 𝛽₁ in the class of linear estimators Φ.

Proof. If 𝑋^′₁𝑋₂=0, then 𝛽₁(𝑘,𝑑)=𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁+𝑑𝐼𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑌₁−𝜎₁₂𝜎₂₂𝑋^′₁𝑋₁+𝑘𝐼⁻¹×𝑋^′₁𝑋₁+𝑑𝐼𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑌₂. (22) For SUR model (1), let 𝑌=(𝑌^′₁,𝑌^′₂)^′, 𝑋=diag(𝑋₁,𝑋₂), 𝛽=(𝛽^′₁,𝛽^′₂), and 𝜀=(𝜀^′₁,𝜀^′₂), Then (1) can be written as 𝑌=𝑋𝛽+𝜀,𝐸(𝜀)=0,Cov(𝜀)=𝑉, (23) where 𝜎𝑉=¹¹𝐼𝜎₁₂𝐼𝜎₂₁𝐼𝜎₂₂𝐼,𝛽^∗=𝛽^′∗₁(𝑉),𝛽^′∗₂(𝑉) (24) and (24) satisfies the following equation: 𝑋^′𝑉⁻¹𝑋𝛽^∗=𝑋^′𝑉⁻¹𝑌. (25) Note that 𝑉⁻¹=𝜎₁₁𝜎₂₂−𝜎²₁₂⁻¹𝜎₂₂𝐼−𝜎₂₁𝐼−𝜎₁₂𝐼𝜎₁₁𝐼. (26) Then (25) can be written as follows: 𝜎₂₂𝑋^′₁𝑋₁−𝜎₂₁𝑋^′₁𝑋₂−𝜎₁₂𝑋^′₂𝑋₁𝜎₁₁𝑋^′₂𝑋₂𝛽^′∗₁(𝑉)𝛽^′∗₂(𝑉)=𝜎₂₂𝑋^′₁−𝜎₂₁𝑋^′₁−𝜎₁₂𝑋^′₂𝜎₁₁𝑋^′₂𝑌₁𝑌₂. (27) When 𝑋^′₁𝑋₂=0, (25) can be written as follows: 𝜎₂₂𝑋^′₁𝑋₁00𝜎₁₁𝑋^′₂𝑋₂𝛽^′∗₁(𝑉)𝛽^′∗₂(𝑉)=𝜎₂₂𝑋^′₁−𝜎₂₁𝑋^′₁−𝜎₁₂𝑋^′₂𝜎₁₁𝑋^′₂𝑌₁𝑌₂. (28) Then by (28), we may get 𝑋^′₁𝑋₁𝛽^′∗₁(𝑉)=𝑋^′₁𝑌₁−𝜎₁₂𝜎₂₂𝑋^′₁𝑌₂ (29) multiplication (𝑋^′₁𝑋₁+𝑘𝐼)⁻¹(𝑋^′₁𝑋₁+𝑑𝐼)(𝑋^′₁𝑋₁)⁻¹ in (28); we have 𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁+𝑑𝐼𝛽^′∗₁(𝑉)=𝛽₁(𝑘,𝑑). (30) Then 𝛽₁(𝑘,𝑑)=𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁+𝑑𝐼(𝐼,0)𝛽^′∗(𝑉). (31)
Let 𝐿=(𝑋^′₁𝑋₁+𝑘𝐼)⁻¹(𝑋^′₁𝑋₁+𝑑𝐼)(𝐼,0) and 𝑆=(𝐼,0); by Lemma 4, if we want to prove that 𝛽₁(𝑘,𝑑) is admissible estimator of 𝛽₁, we only need to prove 𝐿𝑇⁻¹𝐿^′≤𝐿𝑇⁻¹𝑆^′. Since 𝑋^′₁𝑋₂=0, we may conclude that 𝑇=𝑋^′𝑉⁻¹𝑋=𝜎₁₁𝜎₂₂−𝜎²₁₂⁻¹𝜎₂₂𝑋^′₁𝑋₁00𝜎₁₁𝑋^′₂𝑋₂. (32) Then 𝐿𝑇⁻¹𝐿^′=𝜎₁₁𝜎₂₂−𝜎²₁₂𝑋^′₁𝑋₁+𝑘𝐼⁻¹×𝑋^′₁𝑋₁()𝜎+𝑑𝐼𝐼,0²²𝑋^′₁𝑋₁00𝜎₁₁𝑋^′₂𝑋₂⁻¹×(𝐼,0)𝑋^′₁𝑋₁+𝑑𝐼𝑋^′₁𝑋₁+𝑘𝐼⁻¹=𝜎₁₁𝜎₂₂−𝜎²₁₂𝜎₂₂𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁×𝑋+𝑑𝐼_′1𝑋₁⁻¹𝑋^′₁𝑋₁+𝑑𝐼𝑋^′₁𝑋₁+𝑘𝐼⁻¹,𝐿𝑇⁻¹𝑆^′=𝜎₁₁𝜎₂₂−𝜎²₁₂𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁()×𝜎+𝑑𝐼𝐼,0₂₂𝑋^′₁𝑋₁00𝜎₁₁𝑋^′₂𝑋₂⁻¹()=𝜎𝐼,0₁₁𝜎₂₂−𝜎²₁₂𝜎₂₂𝑋^′₁𝑋₁+𝑘𝐼⁻¹×𝑋^′₁𝑋₁+𝑑𝐼𝑋^′₁𝑋₁⁻¹. (33) It is obvious that 𝐿^′𝑇⁻¹𝐿^′≤𝐿^′𝑇⁻¹𝑆^′. The proof is completed.

Theorem 6. For the SUR model (1), when 𝑉 is known, if 𝑃₁𝑃₂=𝑃₂𝑃₁, then 𝛽₁(𝑘,𝑑) is admissible estimator of 𝛽₁ in the class of linear estimators Φ.

Proof. For SUR model (1), when 𝑃₁𝑃₂=𝑃₂𝑃₁, by Lin [14], we obtain 𝛽₁=𝛽^∗₁(𝑉); thus we obtain 𝛽₁(𝑘,𝑑)=𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁+𝑑𝐼𝛽₁=𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁+𝑑𝐼𝛽^∗₁(𝑉)=𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁+𝑑𝐼(𝐼,0)𝛽^∗(𝑉)=𝐿𝛽^∗(𝑉). (34) Similar to Theorem 5, we only need to prove 𝐿𝑇⁻¹𝐿^′≤𝐿𝑇⁻¹𝑆^′; 𝐿 and 𝑆 are defined as Theorem 5: 𝑇=𝑋^′𝑉⁻¹𝑋=𝜎₁₁𝜎₂₂−𝜎²₁₂⁻¹𝜎₂₂𝑋^′₁𝑋₁−𝜎₂₁𝑋^′₁𝑋₂−𝜎₁₂𝑋^′₂𝑋₁𝜎₁₁𝑋^′₂𝑋₂. (35) Denote 𝑇⁻¹=^𝑇11𝑇12_𝑇21𝑇22; by Rao et al. [15], we have 𝑇¹¹=𝜎₁₁𝜎₂₂−𝜎²₁₂𝜎₂₂𝐼−𝜌²₁₂𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑃₂𝑋₁⁻¹×𝑋^′₁𝑋₁⁻¹. (36) Then we have 𝐿𝑇⁻¹𝐿^′=𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁+𝑑𝐼𝑇¹¹×𝑋^′₁𝑋₁+𝑑𝐼𝑋^′₁𝑋₁+𝑘𝐼⁻¹=𝜎₁₁𝜎₂₂−𝜎²₁₂𝜎₂₂𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁×+𝑑𝐼𝐼−𝜌²₁₂𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑃₂𝑋₁⁻¹𝑋^′₁𝑋₁⁻¹×𝑋^′₁𝑋₁+𝑑𝐼𝑋^′₁𝑋₁+𝑘𝐼⁻¹,𝐿𝑇⁻¹𝑆^′=𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁+𝑑𝐼𝑇¹¹=𝜎₁₁𝜎₂₂−𝜎²₁₂𝜎₂₂𝑋^′₁𝑋₁+𝑘𝐼⁻¹𝑋^′₁𝑋₁×+𝑑𝐼𝐼−𝜌²₁₂𝑋^′₁𝑋₁⁻¹𝑋^′₁𝑃₂𝑋₁⁻¹𝑋^′₁𝑋₁⁻¹. (37) Obviously 𝐿^′𝑇⁻¹𝐿^′≤𝐿^′𝑇⁻¹𝑆^′. 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.

Acknowledgments

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).