New M-Estimator Objective Function in Simultaneous Equations Model ( A Comparative Study )

In this paper, a suggested M-estimator (S-M) objective function will be introduced. In addition, A comparative study among different two-stage Mestimation functions including S-M and two stage least squares (2SLS) has been introduced using Monte Carlo Simulations to investigate the properties of these functions, when the errors term of the reduced form follows heavy tail distributions.


2-M-Estimator
M-estimator is considered as the next step in the direction of the robust estimation, where Huber (1964) has introduced it as a new approach towards a theory of robust estimation.In (1973), he extended the idea of M-estimator for regression by minimizing a symmetric smooth function of the residuals over the parameter estimates, which can presented as follows: ( )

New M-estimator objective function 1009
Where, σ the estimated scale of residuals, ρ is a symmetric, positive definite function with a unique minimum at zero, and is chosen to be less increasing than square.Let, Where, ) ( i r ψ is called Psi function.Also, it is called the influence function, which measures the influence of a datum i on the value of the parameter estimate.
The M-estimator for β , based on the function ρ and the data is the value β ˆ which minimizes ), ( Equation ( 4) can be combined into the following single matrix equation (5) Equation ( 5) can be solved as weighed least squares (WLS).Bhar (2010).
It is noted that weights depend upon the residuals, the residuals depend upon the estimated coefficients, and at the same time the estimated coefficients depend upon the weights, so an iterative solution, which is called iteratively reweighed least-squares ( IRLS) is required in this case as: 1. Select initial estimates 0 β such as the least-squares estimates.
2. At each iteration t, calculate residuals ( 1)   t i i r − and associated weights ⎦ from the previous iteration.
3. Solve WLS estimates as: where X is the matrix of the model, and ( 1) is the current weighted matrix.

3-The 2SLS Estimation
Consider the structural equation model as ( , ) K K K = + , and jt V is the error term of the reduced form.
The 2SLS estimator is obtained basing on LS regression of t y on t Y and 1 x .Thus, the 2SLS name arises from the two regressions in the procedure:

4-Two-Stage Huber Estimation
The first stage of M-estimation that yields ∧ Π as estimate of j Π in (7) would be obtained as the solution of 1 ( ) The second stage of Huber estimation that yields α as estimate of 0 ( , ) where H ρ a smooth and symmetric function, c is a tuning value, we use c =2

b-Huber Weight Function
The Huber weight function of the residuals for equation (10) according to Flavin (1999) is defined as where i r is a general concept for residuals whether in the first stage or in the second stage in 2SLS.

5-Tukey M-Estimator
In addition to Huber M-estimator, Tukey M-estimator is considered as one of the most famous M-estimator.Tukey objective function is defined as where k is a tuning value, k = 4.685, and the Tukey weight function ( ) w r is defined as: where ( ) w r denotes Tukey residuals weight function.Fox (2002)

6-Suggested M -Estimator (S-M)
In this section, suggested M-objective function (S-M) has been introduced, which combined between LS objective function and another logistic function that approaches the idea of Cauchy function that has the form, Where c is a tuning value, the 95% asymptotic efficiency on the standard normal distribution is obtained with the tuning constant c=2.3849.Bhar (2010).
According to LS, and Cauchy function in ( 14), the suggested function can be defined as where k is a tuning value and -( ) i S M r ρ satisfies the same properties of ρ above.
The first derivative of (15), which is called influence function, w − is defined as ( ) We graph the ρ-function, Ψ-function and the weight function in Fig. 1.

Simulation Study for comparing Different M-estimators
The Simulation Study was designed to investigate the properties of some M-estimators compared with 2SLS, when there are outliers in exogenous variables (vertical outliers).Outliers in exogenous variables were reflected as a heavy tailed errors distribution.

a. Simulations Model
The Simulation Study was executed considering two simultaneous equations: x is the second element of x t , and t u is the first element of U t and the second equation is,  18) and ( 19) can be represented as: By considering the model (20), the reduced form of equations ( 18) and (19) was conducted as: Where, [ ]

b. Simulation Frame Work
Simulations for two simultaneous equations model as Kim and Muller (2007) have been executed by considering the equations ( 18) to (21) as follows: in equation ( 21) were drown from the standard normal N (0, 1), the Lognormal with log-mean = 0 and log-standard deviation = 1, denoted by LN (0, 1), the Student-t with 4 degrees of freedom, denoted by t(4), and Chisq (2).2-The second to the fourth columns in X are drawn from the normal distribution with mean (0.5, 1,−0.1),variances equal to 1. X is fixed at each replication.

3-[
] y Y were generated using the reduced form parameters basing on the generated exogenous variables X, and the generated errors as equation ( 21).4-1000 replications were used for different sample sizes 15, 30, 50, and 100.For each replication, the value of the parameters (

1-Normal Errors:
In table (3), when n =15, it is noted that 2SH (2) and S-M(3) gave the best results respectively while at n = 30, 2SLS results are better than the other methods of estimation.In table (4), we can conclude generally, that 2SLS is the most efficient estimator at large samples sizes, and its bias decreases as sample size increases, where it has the least MES at large sample sizes.The estimated values of robust methods converges each other.

2-Log Normal Errors
In table (3), when n= 15, 2SLS and 2SH (2) gave the best results compared with the other two methods.When n= 30, it can be seen generally that 2ST (4.685) has a better results among the other methods of estimation.In Table (4), we can see that the behavior of S-M(3) and 2ST(4.685) is the best compared with the other two methods.

3-Student-t Errors
In table (5), when n = 15 or 30, it can be seen generally that 2SH (2) has a better results among the other method of estimation.In table (6), when n = 50, it can be noted that 2SH (2) and S-M(3) gave the best results respectively while at n=100, 2SLS and 2SH (2) gave the best results respectively compared with the other two methods.

4-Chi-Square Errors
In table (5), when n = 15, 2SLS is the most efficient method compared to other estimators while at n = 30, 2SLS and 2SH (2) gave the best results respectively compared with the other two methods.In table (6), when n = 50, 2SH (2) and S-M(3) gave the best results respectively while at n = 100, it is noted that 2SH (2) gave the best results compared with the other methods.

1 (
of regression model parameters including the intercept term, The M-estimator of β based on the function ) ( i r ρ is the vector β which is the solution of the following p equations; determined by solving the set of p
) has been estimated.5-Thefollowing methods have been compared: a-Two Stage Least Square (2SLS) b-Two stage Huber M-estimator with a tuning value =2, which is denoted as 2SH(2).c-Two stage Tukey M-estimators with a tuning value =4.685, 2ST(4.685).d-Suggested M-estimator with tuning value =3, S-M(3).6-The comparison among these M-estimators based on Bias, MSE, and relative efficiency (RE) that is defined as Var of 2SLS Var of the robust M -estimator 7-Estimating the parameter using M-estimators based on iterative reweighted least square (IRLS), which depends on weight functions as in equations (11), (13), and (17).

Table ( 2): Estimated parameters, Bias and MSE: (n = 50,100)
and x 3t are, respectively, the third and the fourth elements of x t , and t ε is the second element of U t .The compact form for the model of equations (