A COMPARATIVE STUDY OF THE PERFORMANCES OF SOME ESTIMATORS OF LINEAR MODEL WITH FIXED AND STOCHASTIC REGRESSORS

In linear regression model, regressors are assumed fixed in repeated sampling. This assumption is not always satisfied especially in business, economics and social sciences. Consequently in this paper, effort is made to compare the performances of some estimators of linear model with autocorrelated error terms when normally distributed regressors are fixed (non – stochastic) with when they are stochastic. The estimators are the ordinary least square (OLS) estimator and four feasible generalized least estimators which are Cochrane Orcutt (CORC), Hidreth – Lu (HILU), Maximum Likelihood (ML), Maximum Likelihood Grid (MLGD) estimator. These estimators are compared using the finite properties of estimators’ criteria namely; sum of biases, sum of variances and sum of the mean squared error of the estimated parameter of the model at different levels of autocorrelation and sample size through Monte – Carlo studies. ) Results show that at each level of autocorrelation the estimated value of the criteria with stochastic regressor are much lesser than that of the fixed regressor for all the estimators except CORC when the sample size is small (n=20) and the level of autocorrelation is very high ( 0.9 ρ = . Mo ) re comparatively, it is observed that the same estimator(s) that is more efficient with fixed regressors is also more efficient with stochastic regressors except when the sample size is large (n = 80) and the level of autocorrelation is either low ( 0.4 ρ = or high ( 0.8) ρ = . At these instances, the CORC / HILU estimator is more efficient with fixed regressors while the ML / MLGD estimator is more efficient with stochastic regressors.


INTRODUCTION
One of the basic assumptions that are made about the regressors in linear regression model is that they are fixed in repeated sampling. This assumption is not always satisfied especially in business, economics and social sciences. This is because their regressors are often generated by stochastic process beyond their control. For instance, consider regressing daily bathing suit sales by a departmental store on the mean daily temperature. Certainly, the departmental store can not control daily temperature, so it would not be meaningful to think of repeated samples when temperature levels are the same from sample to sample (Fomby et. al, 1984). Authors like Neter and Wasserman (1974), Maddala (2002) have given situations and instances where these assumptions may be not be tenable and have also discussed their consequences on the Ordinary Least Square (OLS) estimator when used to estimate the model parameters. Graybill (1961), Sampson (1974), Fomby et.al (1984 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). When all the assumptions of the linear regression model hold except that the error terms are not homoscedastic (i.e. ) but are heteroscedastic (i.e. ), the resulting model the Generalized Least Squares (GLS) Model. Aitken (1935) has shown that the ( 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 (Fomby et. al, 1984). With first order autocorrelated error terms (AR (1)), among the Feasible GLS estimators in literature are the Cochrane andOrcutt estimator (1949), Hildreth andLu estimator (1960), Prais -Winsten estimator (1954), Thornton estimator (1982), Durbin estimator (1960), Theil's estimator (1971, the Maximum Likelihood estimator and the Maximum Likelihood Grid estimator (Beach and Mackinnon, 1978). Some of these estimators have now been incorporated into White's SHAZAM program (White, 1978) and the new version of the time series processor (TSP, 2005). Consequently, effort is made in this paper to compare the performances of some of these estimators of linear model when normally distributed regressors are fixed (nonstochastic) in repeated sampling with when they are stochastic.
linear model in the presence of autocorrelation (Johnston, 1984;Chartterjee et.al, 2000;Maddala, 2002). To compensate for this lost of efficiency, Cochrane and Orcutt (1949) suggested a transformation of the regression model via the generalized least square (GLS) estimator. Chipman (1979), Kramer (1980), Kleiber (2001 and many others did observe that the efficiency of these estimators depends on the structure of the regressors that are used. Rao and Griliches (1969) did one of the earliest Monte Carlo studies on the performances of some of these estimators with autoregressive stochastic regressor. They observed that the OLS estimator is only more efficient than any of the GLS estimators considered when 3 . 0 < ρ ; and that the performances of the GLS estimators are not far apart. Park and Mitchell (1980) observed that when regressors are trended, the estimator that uses the transformation (Paris -Winstern) is more efficient than the one that uses the transformation (Cochrane -Orcutt) and that the latter should even be avoided since it is less efficient than the OLS estimator.

P Q
More recently, Nwabueze (2005) examined the performance of some of these estimators with exponential independent variable. His result, among other things, show that the OLS estimator compares favorably with the Maximum Likelihood (ML) and Maximum Likelihood Grid (MLGD) estimators for small value of ρ but it appears to be superior to Cochrane -Orcutt (CORC) and the Hidreth and Lu (HILU) especially when ρ is large. Some other recent works that are done with different specification of regressors include that of Iyaniwura and Nwabuwze (2004a), Iyaniwura and Nwabuwze (2004b) and Olaomi and Iyaniwura (2006). Consequently, this paper compares the performances of some of these estimators when normally distributed regressors are fixed in repeated sampling with when they are stochastic.

METHODOLOGY
Consider the GLS model with AR (1) of the form Furthermore, and were generated.
Hence, the values of in equation (1) were also calculated by setting the true regression coefficients as This process continued until all replications in this scenario were obtained. Another scenario then started until all the scenarios were completed. The only difference in these procedures with stochastic regressors is that at each replication the and were newly generated.

K. AYINDE and J. O. IYANIWURA
Evaluation and comparison of estimators were examined using the finite sampling properties of estimators which include bias (B), and variance (Var) and the mean squared error (MSE) criteria. Mathematically, for any estimato for i = 0, 1, 2 and j= 1,2,…,120.

SBIAS of
For each of the estimation methods, a computer program was written using TSP software to estimate all the model parameters and to evaluate the criteria. Often times, preference of estimators are based on bias (closest to zero), minimum variance and minimum (root) mean squared error. In this study, we utilized the criteria of sum of bias (SBIAS), sum of variance (SVAR), and the root mean squared error (SRMSE) of the estimated model parameters to compare the performances of the estimators. This approach has also been used by Iyaniwura and Nwabueze (2004a), Iyaniwura and Nwabueze (2004b), Nwab (2006) and some others.

RMSEB RMS
β their SRMSE, they are simply said to be more The efficiency of the estimators was further examined using the sum of SRMSE. An estimator with the smallest SRMSE is most efficient whereas if two estimators are nearly equal in terms of efficient.

S AND DISCUSSION
The summary of the performances of the estimators on the basis of the sum of the criteria is given in table 1, 2 and ppendix while for all other sample ssor espec 3 in the appendix. Also, the summary of the most/ more efficient estimator(s) is shown in table 4 in the appendix. However, the estimated criteria of the model parameters for n = 20 for both fixed and stochastic regressors are given in table 5, 6,7,8,9 and 10 in the a sizes, see Ayinde (2006).
From the tables, it is observed that at each level of autocorrelation the estimated value of the criteria with stochastic regressor are much lesser than that of the fixed regressor for all the estimators except CORC when the sample size is small (n=20) where the CORC estimator has estimated value of the criteria greater with stochastic regressor than the fixed regre ially when the level of autocorrelation is very high ( 0 ) .9 ρ = . Also, with both fixed and stochastic regressor, as ρ increases the estimated criteria of the estimators in all the sample sizes increase. Asymptotically, it is also observed that the estimated value of the criteria reduce at ach lev

CONCLUSION
The performances of estimators of linear model in the presence of autocorrelated error terms with stochastic regressors are often much lesser than that of the fixed regressors on the basis of finite properties of estimator criteria. However, the estimators' performances in terms of their efficiency are much alike except when the sample and the level of a          . 367750 .192151 .192541 .184822 .184908 .400525 .199529 .199462 .194689 .195110 6.389459 5.859107 5.301591 5.932051 5.940578