Abstract

This study has investigated the effect of multicollinearity and the sensitivity of three different estimation methods of a Simultaneous equation econometric model. A Monte Carlo approach was employed for a three equation just identified model. The anlaysis for sample size n = 100 replication, R = 200 revealed the preference order of 3SLS, 2SLS and OLS using both mean and bias estimation criteria.

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INTRODUCTION

Multicillinearity is one of the known associated problems with the best linear regression model. When the measured explanatory variables are too highly inter- corelated to allow precise analysis of their individual effects, we have multicollinearity. This problem which is peculiar in single equation estimation is addressed here by the introduction of simultaneous equation model. However, this problem may still exist in individual equations of the Simultaneous equation model.

To this end, we have employed a Monte Carlo approach to under study the comparative performance of three estimation methods under various degrees of correlation among the three specified exogenous variables.

The use of Monte Carlo approach has been supported by Wagnar (1958), Parker (1972), Hendry (1976), Johnston (1984) and Kutsoyiannis (2001), the approach has found extensive use in the fields of operational research, nuclear physics and a host of others where there are varieties of problem beyond the available tools of theoretical mathematics.

Studies on estimation under multicollinearity effects of simultaneous models revealed that a high degree of multicollinearity among the explanatory variables has a disastrous effect on estimation of the coefficients, β by the OLS (Mishra, 2004). This method was considered by Pleli and Tankovic (2005) as naive approach because the estimators are biased and inconsistent (2SLS, ILS) and full-information approach (3SLS, FIML). Adenomon and Fesojaiye (2008) merely compared the Seemingly Unrelated Regression (SUR) with the OLS technique and confirmed the superiority of the SUR estimator to the OLS estimators. In the opinion of Ayinde (2007), where he compared OLS with some GLS estimators, he observed that with increasing replications OLS estimator is preferred in estimating all the model parameters at all levels of correlation. However, this opinion negates that of Pleli and Tankovic (2005) in which they advised an econometrician to avoid the use of naive approach (OLS) in estimating the parameters of a system of simultaneous equations.

MATERIALS AND METHODS

Presentation of the model design: The following model with three structural equations are designed and presented thus:

(1)

Where, y_1t-y_3t are the endogenous variables and x_1t-x_3t are the exogenous or predetermined variables. The u_1t-u_3t are the random disturbance terms which are assumed to be independently and identically normally distributed with zero means and finite matrix, i.e., U NID (0, ∑).

Procedure for data generation: In order to ensure that the data conform to all the specifications of the econometric model, the data series are generated as follows:

•

The vector [(x1t-x3t), t = 1, 2, …, N)] of independent or exogenous variables are randomly and uniformly generated using Kmenta and Joseph (1963) and Kmenta (1971) to ensure they conform to the following scenario:

•

When the exogenous variables (x1t-x3t) selected are relatively highly negatively correlated (ρxixj<-0.05). This is considered as the incidence of high negative multicollinearity and so referred to as category I

•

When the exogenous variables (x1t-x3t) selected are feebly negatively or positively correlated (-0.05<ρxixj≤+0.05). These are considered as the incidence of feebly negative or positive multicollinearity and henceforth referred to as category II

•

When the exogenous variables (x1t-x3t) selected are relatively highly positively correlated (ρxixj>+0.05). This is considered as the incidence of high positive multicollinearity and thus referred to as category III

•	Three corresponding mutually independent N(0, 1) sequences say {(ξ1t-ξ3t): t = 1, 2, …, T} are generated and are manipulated as follows to ensure that the disturbance terms are distributed as N(0, ∑) and inter temporarily independent. Let:

(2)

Then by construction:

(3)

Consider now the random vectors:

By construction, the vectors {u_t: 1, 2, …, T} have the properties:

Where, is the kronecker product or direct product. In this fashion, the desire error terms having the prescribed variance covariance matrix is obtained.

•

According to this approach, specific values have to be assigned to the structural parameters:

(4)

Say:

(5)

putting;

(6)

and;

The reduced form of the system can be written as:

(7)

Since, all quantities on the right hand side of Eq. 7 are known, vectors of the determined variables {(y_1t-y_3t): t = 1, 2, …, T} are then generated.

At the end of this 3rd step, the data set {( y_1t-y_3t, x_1t-x_3t): t = 1, 2, …, T} have been generated by a model describe in Eq. 1 with fixed parameter vector δ⁰ and error terms that are inter temporarily independent and distributed as N (0, ∑), the matrix ∑ being known.

The next step at this stage is to determine the known parameters as if they were not known using different methods and comparing the performances or sensitivity of the estimators.

RESULTS AND DISCUSSION

In this study, the samples were restricted to size n = 100, replicated in 200 times (R = 200) for an upper triangular matrix P to determine the effects of multicollinearity and the sensitivity of the estimation methods in a simultaneous equation model.

Meanwhile, we have employed Ordinary Least Square (OLS), 2 Stage Least Square (2SLS) and 3 Stage Least Square (3SLS) as the estimation methods. We have equally chosen the mean of estimates and bias of estimates as criteria for evaluating the performance of the estimators (Table 1-10). Table 1-10 have essentially shown the true parameters values, average estimated values, standard derivation (sample), standard deviation (population) and bias of estimates for the estimation methods employed at the three different levels of correlation desired.

 

Table 1:	Mean estimates and bias estimates of OLS for category I

 

 

Table 2:	Mean estimates and bias estimates of OLS for category II

 

 

Table 3:	Mean estimates and bias estimates of OLS for category III

 

 

Table 4:	Mean estimates and bias estimates of 2SLS for category I

 

 

Table 5:	Mean estimates and bias estimates of 2SLS for category II

 

 

Table 6:	Mean estimates and bias estimates of 2SLS for category III

 

 

Table 7:	Mean estimates and bias estimates of 3SLS for category I

 

 

Table 8:	Mean estimates and bias estimates of 3SLS for category II

 

Table 9:	Mean estimates and bias estimates of 3SLS for category III

 

Table 10:	Average of parameter estimates for data set (N = 100, R = 200)

 

However, Table 10 has revealed a true state of the comparative performance of those estimates. On the overall, the following order of preference of estimators can be deduced: 3SLS>2SLS>OLS.

CONCLUSION

This study has investigated the sensitivity of three different estimators to the problem of multicollinearity among the exogenous variables in a Simultaneous equation model. A three equation just identified model was used for the Monte Carlo analysis. We simulated sample of n = 100 for a replication of R = 200 for an upper triangular matrix for three exogenous variables which were later used to generate corresponding three endogenous variables. Three levels of correlation on three different estimation methods were considered. The results obtained were analyzed and estimated using the mean of estimates and its bias. Although, the study did not produce identical estimates as the assumed parameter but some estimates are quite close, especially in 2 and 3SLS. The presence of multicollinearity may accounts for the situation. The bias estimate also confirmed the preference of this order of estimation methods 3SLS>2SLS>OLS.

How to cite this article:

D.A. Agunbiade. Effect of Multicollinearity and the Sensitivity of the Estimation Methods in Simultaneous Equation Model.
DOI: https://doi.org/10.36478/jmmstat.2011.9.12
URL: https://www.makhillpublications.co/view-article/1994-5388/jmmstat.2011.9.12