1. Introduction

The seemingly unrelated regression (SUR) Model proposed by (zellner, 1962) which it depends on general least squares estimator (GLS) and assumes data without outliers but in some cases this cannot be achieved. The robust methods considered the one important approach to deal with outliers which allow the unequal weight for observations. (Koenker etal, 1990) introduce M-estimators method, as a robust methods, to estimate SUR model when the data within outliers, yet the asymptotic efficiency for M-estimators depend on initial estimate and have breakdown point (bp) equal 0. Therefore, (Bilodeau etal., 2000) suggested S-estimator for SUR model which interest same asymptotic properties for M-estimators and have high (bp) reach to 50% of the observations. In the same side, (Garcia etal., 2006) and (Roelandt etal., 2009) proposed for multivariate regression which have excellent robustness and high efficiency under normality of error. (Roelandt etal., 2009) extract generalized S estimates for multivariate regression which consider high breakdown estimation when analysis the independent component. The MM estimators for linear regression model was introduced by (Yohai, 1987)), which begin at the first with a highly initial robust regression estimator like S estimators which depend on a loss function and then used this initial estimator to obtain M estimator with other loss function . This estimator has two advantage, the high asymptotic efficiency for normality error and high breakdown point reach to 50%. The study of (Berrendero etal., 2007) show that, the asymptotic bias for mm estimator is lower than the τ estimators when we contaminations error lower than 0.20 and the MM estimator has best lower maximum bias curve than other robust estimators.

(Salibian etal., 2006) show that, the breakdown point of MM-estimator in finite-sample is equal to or greater than initial S-estimator. The MM estimators begin at the first with a highly initial robust regression estimator like S estimators which depend on a loss function and then used this initial estimator to obtain M estimator with other loss function . (Kudraszow, 2011) suggested MM estimator for multivariate linear model and study the consistency and asymptotic normality with elliptical distribution for error. On other hand, frequently, outliers appear with degree of multicollinearity. The ridge method solved multicollinearity problem has been discussed by (Hoerl etal., 1970). This method is a way of proceeding in solving the problem by adding specific information to remove the ill-condition. The SUR model possibly is under the influence of multicollinarity. (Srivastava etal., 1987) suggested general ridge estimator to remove the ill-condition in SUR model (Alkhamisi. etal, 2006) developed ridge estimators of the SUR model, when the data are transformed in a canonical form. The critical point in ridge estimate is how to choose the shrink parameter. So, (Firinguetti, 1997), (Kibria etal., 2003) and (Alkhamisi. etal, 2006) suggested non robust Criteria to choice the shrink parameter in ridge estimator for SUR model. These formulas assume choosing ridge parameter when data without outliers. (Jung, 2009) proposed robust cross validation in ridge regression when data are within outliers. (El-hosany etal., 2011) used robust Cross Validation Criteria to choice shrink parameter in SUR model.

In fact, frequently, outliers appear with degree of multicollinearity. Therefore, (El-hosany etal., 2011) mingle between ridge estimator and S- estimator to get robust ridge estimator. (Maronna, 2011), (Moawad etal., 2011) and (Mariam. etal., 2012) have developed MM-estimators to deal with leverage point and absorption problem of multicollinearity. This estimators builded by combined between ridge regression and mm-type to reach to mm ridge estimators.

There is no study use the MM estimator in the SUR model. So, we suggests, MM and MM ridge estimator for SUR model to deal with bad high leverage points and the multicollinearity problem associated with it. In MM ridge estimator, we develop robust Cross Validation Criteria, to choice shrink parameter, depended on MM estimator. This paper organized as follows: The SUR model and estimators defined in Section two. In section three, we study the asymptotic properties of MM and MM ridge estimators in SUR model while section four will be devoted to make the simulation study. In section five, we develop algorithm to compute MM and MM ridge estimator for SUR model.

2. GLS, S, MM and Weight MM Estimator for SUR Model

Conceder the SUR model

(1)

where is a nq × 1 vector of dependant variable in q equation, is a nq×kq matrix of independent variable and is a nq × 1 vector of random error in q equation. Suppose for all i=1,2,…q and for all i,j=1,2,…q. We can right the SUR model in the multivariate form

(2)

where is a matrix of n q observations with , is a matrix, is a kq×q matrix of coefficients. and is a nq matrix of residual with vector. Let where is a matrix and is vector contains one at position i and zero elsewhere. (Srivastava and Giles, 1987) suggested the GLS estimator for SUR model

(3)

where is q×q variance covariance matrix for error between equations. The GLS estimator calculate by use as consistent estimator for (Bilodeau etal, 2000) introduce S estimator to deal with outliers for SUR model. This estimator used the Huber function at the form

(4)

where is symmetric, continuous, differentiable and for it is strictly increasing on [0; c], constant on [c,∞] and ρ(0)= 0. The percentage of breakdown point depend on C. (Ruppert, 1992) show that, when choice C=1.5476 the of breakdown point reach to 50%. The S estimator for SUR model minimize

where is the standard normal distribution. The S estimator satisfy the following equation

(5)

(6)

Where

Lemma (1):

Let the and have symmetric, continuous, differentiable, for it is strictly increasing on [0; c], constant on [c,∞] and ρ(0)= 0. In adding, the function ρ_o and ρ₁ achieve ρ₁ (µ) ≤ ρo (µ) and Sup ρ₁ (µ) = Sup ρ₀ (µ). We use Huber function at the form (4). In the SUR model in (1). We can extracted the MM-estimator for SUR model (new) by

(7)

Where, estimated by

(8)

Where and are initial estimators.

Then the MM-estimator for SUR model satisfy the following equations

(9)

(10)

Where

Proof of lemma (1)

If we differentiate (7) with respect to β and equalize the result to zero then

Where

Then

Where

and

Then complete the proof.

Lemma (2)

If and satisfy the condition in lemma (1), then we extract MM ridge estimator for SUR model (new) by

(11)

Where, estimated by (8) and is ridge parameter. The MM ridge estimator for SUR model (new) in (1) can be written as

(12)

Proof of lemma (2)

If we differentiate (11) with respect to β and equalize the result to zero then:

Then complete the proof.

3. Asymptotic Properties for MM and MM Ridge Estimators for SUR Model

In this section, we study the asymptotic properties MM and MM ridge estimators.

Lemma (3)

Consider the SUR model with observe, which independent random vectors distributed as pq-variate normal distribution with mean μ= Zβ and variance ∑s, where is distribution of X and is distribution of µ.

Then

Where

Proof of lemma (3):

We use the proof for the Theorem (4-1) in (Yohai, 1987) and the asymptotical properties for S-estimator for SUR model in (Bilodeau etal, 2000) to access the prove. The MM-estimator of in SUR model, can be defined as a solution of M-type equation, then the anther form for equations (8) and (9) are

(14)

Where:

By using Mean Value Theorem (MVT),

For (14)

Then

For uniform laws of large numbers (ULLN)

By central limit theorem (CLT), we get

And then

Under the assumption u elliptical. We study the asymptotic variance for

Hence

Let:

while is spherical then

and then

Extension for Lemma (5-1) (Lopuhaa, 1989). We get

Where

Then

Then

Then completes the proof.

Lemma (4):

Consider the same condition in proposition (3-1) and then

(15)

Where

and then Following by the proof for lemma (3) then completes the proof.

Where

and

Proof of lemma (4):

The MM ridge estimator of β,∑, in SUR model, can be defined as a solution of M-type equation, then the anther form for equations (9) and (12) are

(16)

Where

By using (MVT, (ULLN) and (CLT)) theorem in proofs for proportion (3-2). In adding, consider the condition then

Under the assumption u elliptical. We study the asymptotic variance for

Hence

Let:

While is spherical then

Where

and then

Following by the proof for lemma (3) then completes the proof.

4. The Simulation Study

In this section, we provide a simulation study to illustrate the performance of four estimators, the (GLS), S, MM and MM ridge estimator for SUR model. This simulation process is executed to generate data for the following equation Where In this simulation, we set the initial value for β= [1,2,3] for k=3. The explanatory variables are generated by multivariate normal distribution MNNk=3 (0,∑x) where diag(∑x)=1, off-diag(∑x)= ρX= 0.15 for low interdependency and ρx= 0.70 for high interdependency. Where ρx is correlation between explanatory variables. We chose two sample size 25 for small sample and 100 for large sample. The specific error in equations μi, i=1,2,…..,n, we generated by MVNk=3 (0, ∑ε), ∑ε the variance covariance matrix of errors, diag(∑ε)= 1, off-diag(∑ε)= ρε= 0.15. To investigate the robustness of the estimators against outliers, we chosen different percentages of outliers ( 20%, 45%). We choose shrink parameter in (12) by minimize the new robust Cross Validation (CV_MM) criterion which avoided outliers. This criterion depend on MM estimators and takes the form Where is a residual depend on MM estimators and

We measure the goodness of fit of estimated model for β by use the median average squared error [Median(ASE_j)] where ASE_j is defined by

where j=1,2,……N. Where is MM estimator for β and N is a number of iteration. We run N=1,2,…, 500 replications by using software MATHCAD.

5. Algorithm

In this paper, we use two algorithm to compute the MM and MM ridge estimators. and is initial estimate for and respectively, then iterative solve for (9) is

(16)

And the then iterative solve for (12) is

(17)

Where N is a number of iteration,

The equation (17) is a weight version for ridge estimator. This equation is rewrite as normal equations

(18)

Where N is a number of iteration,

and

The equation (14) is a weight version for ridge estimator. This equation is rewrite as normal equations

(15)

Where and

Algorithm (1)

In this algorithm, we compute MM estimators for SUR model by developed the algorithms for (Tharmaratnam et al., 2008) and (Al-hosany et al., 2011) We drawing the algorithm (1) by the following steps:

Step (1): Let is initial candidates estimate for β.

Step (2): Design the variable For each

a. Compute

b. Compute S estimators as in (8) for some ρ-function ρ0 by set m=0, and get the following steps:

i. Let

ii. If either m equal the maximum number of iterations

or where is affixed small

iii. constant, then go to step (2)

iv. Else Compute , and put m←m+1

Step (3): Select which active

Step (4): Compute MM estimator as in (7) for some ρ-function ρ1: Let g=0 and compute and let

Step (5): If either m equal the maximum number of iterations

or where is affixed small constant, then

break

Else Compute and put g←g+1

Step (6): Select which active

The J random subsample for initial candidates ware chosen as set for OLS estimators. To avoid ill condition, we take J by large size.

Algorithm (2)

In this algorithm, we compute MM ridge estimators for SUR model by develop (Maronna R, 2011), (Tharmaratnam et al., 2008) and (Al-hosany et al., 2011) algorithms.

We drawing the algorithm (2) by the following steps:

Step (1): We use the steps (1,2,3) in algorithm (1).

Step (2): Compute MM ridge estimator as in (11) for some ρ-function ρ₁. Let g = 0 and compute and let

Step (3): If either m equal the maximum number of iterations

Or where is affixed small constant, then

break

Else Compute and put g←g+1

Step (4): Select active

6. Result for Simulation

We summarize the simulation results in tables (1, 2). These tables shows the median ASE for GLS,S, MM and MM ridge estimators for SUR model under the study factors. When analysis the result of simulation study, we can say that, the GLS estimators, in all case, works better than anther estimators when the percentages of outliers close to zero percentage. Although S estimator has clear advantage when percentages of outliers reach to 20% and 45%, but the MM estimators have more efficiency. When the number of observations were increasing, we note improvement in the work of all the estimators. On other hand, when the number of equations increase, all estimators going to be less efficient. works good when the percentage of outliers and correlation between explanatory variables increases. The MM estimators and MM ridge estimators works good when the percentage of outliers and correlation between explanatory variables increases. Nevertheless, MM ridge estimators get more efficient than MM estimators.

Table (1). Median ASE for GLS, S, MM and MM ridge estimators for SUR model, (percentages of outliers(ν) = 20%

Table (2). Median ASE for (GLS), S, MM and MM ridge estimator for SUR model, (percentages of outliers(ν) = 45%

7. Conclusions

To reach the best estimator for the SUR model which have advantages of the robust parameter estimation, we compare between the robust and nonrobust estimators. To attain the robust high level break down point, we used S, MM and MM ridge estimator for SUR model. The MM ridge estimators avoids the multicollinearity problem, which accompany the outliers. The median average squared error [Median (ASE)] was used for trade-off between estimators. The result of simulation show that, when the percentage of outlier and correlation between explanatory are increasing, the MM and MM ridge estimators are the best estimators between the robust and nonrobust estimators under the all factors of study.