A Comparative Study of the Performances of the OLS and some GLS Estimators when Stochastic Regressors are both Collinear and Correlated with Error Terms

The Classical Linear Regression Model assumes that regressors are non – stochastic, independent and uncorrelated with the error terms. These assumptions are not always tenable especially where regressors are not often assumed fixed in repeated sampling. In this paper, with stochastic regressors, the performances of the Ordinary Least Square (OLS) and some Generalized Least Square (GLS) estimators are investigated and compared under various degree of non – validity of multicollinearity and correlation between regressor and error terms’ assumptions through Monte – Carlo studies at both low and high replications. The mean squared error criterion is used to examine and compare the estimators. Results show that the performances of the estimators improved with increased replication. The ML and MLGD (GLS) estimators compare favorably with the OLS estimator with low replication. However with increased replication, the OLS method is preferred among the estimators in estimating all the parameters of the model in all level of correlations.


INTRODUCTION
Regressors are assumed to be non -stochastic (fixed in repeated sampling), independent and uncorrelated with the error terms in the Classical Linear Regression Model (CLRM). These assumptions are not always satisfied especially in business, economics and social sciences. Authors like Neter and Wasserman [1] , Fomby et.al [2] , Maddala [3] have not only given situations and instances where these assumptions may be violated but have also discussed their consequences on the Ordinary Least Square (OLS) estimator when used to estimate the model parameters.Graybill [4] , Sampson [5] , Fomby et.al [2] and many others emphasized that if regressors are stochastic and independent of the error terms; the OLS estimator is unbiased and has minimum variance even though it is not Best Linear Unbiased Estimator (BLUE). They also pointed out that the traditional hypothesis testing is valid if the error terms are further assumed normal but modification would be required in the area of confidence interval calculated for each sample and the power of the test. When regressors are dependent (i.e. there exist multicollinearity), the OLS estimates are still unbiased as long as multicollinearity is not perfect [6] . However when multicollinearity is high, only imprecise estimate may be available about the individual true regression coefficients which are often statistically insignificant because of its large standard errors [7] . Neter and Wasserman [1] , Maddala [3] attributed a source of correlation between regressors and error terms to measurement errors in the regressors. They noted that if the OLS estimator is applied to the CLRM of this form, the estimates are not only bias but lack property of consistency. Maddala [3] emphasized that this does not imply that inferences about the model parameters are not possible.
With non -stochastic regressors, OLS estimator β of β given as (1) has been proved to be BLUE with variancecovariance matrix of β given as (2) [8] .
When all the assumptions of the CLRM hold except that the error terms are not homoscedastic (i.e.  [9] has shown that the GLS estimator β of β given as ( ) (3) is efficient among the class of linear unbiased estimators of β with variance -covariance matrix of β given as where Ω is assumed to be known. However, Ω is not always known, it is often estimated by Ω to have what is known as Feasible GLS estimator.
Many consistent estimates of Ω can be obtained [2] . With first order autocorrelated error terms (AR (1)), among the Feasible GLS estimators in literatures are the Cochrane and Orcutt estimator [10] , Hildreth and Lu estimator [11] , Prais -Winsten estimator [12] , Thornton estimator [13] , Durbin estimator [14] , Theil's estimator [15] , the Maximum Likelihood estimator and the Maximum Likelihood Grid estimator [16] . Some of these estimators have now been incorporated into White's SHAZAM program [17] and the new version of the time series processor (TSP) [18] . However, all of these estimators are known to be asymptotically equivalent but the question on which is to be preferred in small samples is the worry of researchers [2] . Assuming no autocorrelation of the error terms, we examine and compare the performances of some of these Feasible GLS estimators with that of the OLS estimator when stochastic regressors are both collinear and correlated with the error terms;and also identify the estimator which is prefferred in estimating all the parameters of the CLRM in all the levels of these correlations.

MATERIALS AND MEHODS
Consider the CLRM with stochastic regressors of the form x is said to have 1 ρ correlation with e , OLS estimator discussed earlier can be used to obtain estimates of the model parameters. Also, consider the GLS model with stochastic regressors and AR (1) of the form Its parameter estimation can be done using the (feasible) GLS methods. However for the purpose of comparison, model (8) is made to be equivalent with model (5) where ( ) is the value of the correlation between the two variables [19] . Monte Carlo experiments were performed for 20 = n , a small sample size representative of many time series study [20] , with four replication (R) levels (R = 10, 40, 80, 120) and nine various degree of multicollinearity and correlation between regressor and error terms (i.e 1 ρ and 2 ρ = -0.99,-0.75,-0.5,… 0.99 respectively) utilizing equation (5), (6) The values of t y in equation (5)  For each of the estimation methods, a computer program was written using TSP software to estimate the model parameters and to evaluate the criterion. The four replication levels were further grouped into low (R =10, 40) and high (R = 40, 80) and the effect of the correlations on the performances of the estimators (method) were examined via the Analysis of Variance of the criteria of each of the model parameters in the two replication groups. This was also accomplished by the LSD test of the estimated marginal means of the highest interaction effect with method that is statistically significant. At a particular set of levels of correlation, the estimated marginal means of estimators were preferred if they are not significantly different from the most preferred one. An estimator is most preferred if its estimated marginal mean is the smallest.

SIMULATION RESULTS AND DISCUSSIONS
The summary of our findings on the performances of the estimators based on the criteria for each of the model parameters in the two replication groups is given in table 1. In table 1 the Analysis of Variance table is presented. From table 1, it is observed that the error sum of square and hence the mean square error (if estimated) of all the estimated parmeters reduce with increased replications. Thus, the performances of the estimators improve with increased replication. Also, the multicollinearity effect and any interaction effect with multicollinearity are completely insignificant at the two replication groups in estimating 0 β . 1 ρ * M is significant in estimating 0 β while that of 2 ρ * M is significant in estimating 1 β and 2 β .
Hence, the performances of the estimators are affected by correlation between regressor and error terms in estimating 0 β and by multicollinearity in estimating 1 β and 2 β . Their estimated marginal means can be found in the study done by Ayinde [21] . From the study, it was observed that the estimated marginal means of 0   is not only most efficient but its performances is significantly different from others. Therefore, it can be inferred that the OLS estimator is consistently preferred in estimating all the model parameters at all the levels of correlations.

CONCLUSION
Among the four GLS estimators examined, the ML and MLGD estimators can only compete with the OLS estimator when replication is low. However, with increased replication, the OLS estimator is most preferred among the estimators in estimating all the model parameters at all the levels of correlation even though the performances of the GLS methods at times may not be bad.