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Abstract： The Seemingly Unrelated Regressions （SUR） model proposed in 1962 by Arnold Zellner has gained a wide acceptability and its practical use is enormous. In this research， two methods of estimation techniques were examined in the presence of varying degrees offirst order Autoregressive [AR（1）] coefficients in the error terms of the model. Data was simulated using bootstrapping approach for sample sizes of 20， 50， 100， 500 and 1000. Performances of Ordinary Least Squares （OLS） and Generalized Least Squares （GLS） estimators were examined under a definite form of the variance-covariance matrix used for estimation in all the sample sizes considered. The results revealed that the GLS estimator was efficient both in small and large sample sizes. Comparative performances of the estimators were studied with 0.3 and 0.5 as assumed coefficients of AR（1） in thefirst and second regressions and these coefficients were further interchanged for each regression equation， it was deduced that standard errors of the parameters decreased with increase in the coefficients of AR（1） for both estimators with the SUR estimator performing better as sample size increased. Examining the performances of the SUR estimator with varying degrees of AR（1） using Mean Square Error （MSE）， the SUR estimator performed better with autocorrelation coefficient of 0.3 than that of 0.5 in both regression equations with best MSE obtained to be 0.8185 usingρ= 0.3 in the second regression equation for sample size of 50.
Key words： Autocorrelation； Bootstrapping； Generalized least squares； Ordinary least squares； Seemingly unrelated regressions
1. INTRODUCTION Seemingly Unrelated Regression （SUR） is a system of regression equations which consists of a set of M regression equations， each of which contains different explanatory variables and satisfies the assumptions of the Classical Linear Regression Model （CLRM）. The SUR estimation technique which allows for an efficient joint estimation of all the regression parameters wasfirst reported by Zellner [21] which involves the application of Aitken’s Generalised Least Squares （AGLS） [2] to the whole system of equations. Several scholars have also developed other estimators for diverse SUR models to address different situations being examined. Dwivedi and Srivastava [6]， Zellner [21] cited in William [18] have shown that the estimation procedure of SUR model was based on Generalized Least Squares （GLS） approach. In answering how much efficiency is gained by using GLS instead of OLS， Zellner [21] has shown in his two-stage approach the gain in efficiency of SUR model over separate equation by equation， that efficiency would be attained when contemporaneous correlation between the disturbances is high and explanatory variables in different equations are uncorrelated. Youssef [19，20] studied the properties of seemingly unrelated regression equation estimators. In an additional paper， he considered a general distribution function for the coefficients of seemingly unrelated regression equations （SURE） model when we unrestricted regression （SUUR） equations.Viraswami [17] presented a working paper on some efficiency results on SURE model. In his work， he considered a two equation seemingly unrelated regressions model in which the equations have some common independent variables and obtained the asymptotic efficiency of the OLS estimator of a parameter of interest relative to its FGLS estimator. He also provided the small-sample relative efficiency of the ordinary least squares estimator and the seemingly unrelated residuals estimator. Alaba et al. [3] recently examined the efficiency gain of the GLS estimator over the Ordinary Least Squares （OLS） estimator. This paper thus examines the performances of OLS and GLS estimators when the disturbances are both autoregressively and contemporaneously correlated.
The remainder of the paper is organized as follows. In section 2， the parametric SUR framework is presented while the simulation studies carried out in the work is discussed in Section 3. Results and detailed discussions are presented in Section 4 while Section 5 gives some concluding remarks.
The diagonal structure of the R matrix implies that each equation or crosssection unit exhibits its own serial correlation coefficient， and the innovations v（t） are contemporaneously correlated with covariance matrixΣ.
The most general model that is usually considered involves the diagonal R matrix， with M parameters， specifying the serial correlation together with a full， symmetricΣmatrix， with M（M + 1）/2 parameters， specifying the contemporaneous covariance. This implies that in （4），？=Σ？I， and？？1=Σ？1？l.
If？is known and denoting the ijth element ofΣ？1byσij， the generalized least squares estimator for the coefficients in this model is：
The summary of the results when the model is estimated by interchanging the coefficients of autocorrelated errors are presented below.
4.2. Discussion of Results
Table 1 gives the comparative performance of the estimators with the coefficient of AR（1） for thefirst regression equation to be 0.3 （i.e.，ρ1= 0.3） and that of the second regression equation to be 0.5 （i.e.，ρ2= 0.5）.
Table 2 gives the comparative performance of the estimators with the coefficient of AR（1） for thefirst regression equation to be 0.5 （i.e.，ρ1= 0.5） and that of the second regression equation to be 0.3 （i.e.，ρ2= 0.3）. From Table 1 and 2， we found that the standard errors of the parameter estimates decreased as sample size increased with varying degrees of coefficients of AR（1） with the SUR estimator performing better than the OLS estimator in both cases（that is， the standard errors of the parameter estimates obtained using the SUR estimator were consistently lower than that of the OLS estimator in both cases as the sample size increased）. Significantly， we found that higher coefficient offirst order autoregressive scheme accounts for better efficiencies of the estimators.
From Table 3， we found on the basis of the Mean Square Error （MSE） for the SUR estimator that the MSE of the regression equations increased with increase in the coefficients of AR（1） with the second model for sample size of 50 performing best with coefficients of AR（1） [0.3，0.5] and MSE [0.8185，0.8852] respectively. Thus， the best MSE obtained for the SUR estimator is 0.8185 usingρ= 0.3 for the second regression for sample size of 50.
The results obtained showed that the standard errors of the SUR estimator is consistently lower than that of the OLS estimator when the model is estimated with varied coefficients of AR（1）. It is revealed that higher coefficient of AR（1） accounts for better efficiencies of the estimators.
REFERENCES
[1] Adebayo， S. B. （2003）. Semi parametric Bayesian regression for multivariate responses. Munich： Hieronymus.
[2] Aitken， A. C. （1934-35）. On least-squares and linear combination of observations. Proceedings of The Royal Society of Edinburgh， 55， 42-48.
[3] Alaba， O. O.， Olubusoye， E. O.， & Ojo， S. O. （2010）. Efficiency of seemingly unrelated regression estimator over the ordinary least squares. European Journal of Scientific Research， 39（1）， 153-160.
[4] Bartels， R.， & Fiebig， D. G. （1991）. A simple characterization of seemingly unrelated regressions models in which OLS is BLUE. The American Statistician， 45（2）， 137-140.
[5] Binkley， J. K.， & Nelson， C. H. （1988）. A note on the efficiency of seemingly unrelated regression. The American Statistician， 42（2）， 137-139.
[6] Dwivedi， T. D.， & Srivastava， V. K. （1978）. Optimality of Least squares in seemingly unrelated regression equation model. Journal of Econometrics， 7， 391-395.
[7] Judge， G. G.， Griffiths， W. E.， Hill， R. C.， & L¨utkepohl， H. （1985）. Disturbancerelated sets of regression equations. The theory and practice of econometrics（2nd ed.） （pp. 465-514）. New York： Wiley. [8] Judge， G. G.， Hill， R. C.， Griffiths， W. E.， & L¨utkepohl， H. （1988）. Introduction to the theory and practice of econometrics （2nd ed.） （pp. 444-468）. New York： Wiley.
[9] Kmenta， J.， & Gilbert， R. F. （1968）. Small sample properties of alternative estimators of seemingly unrelated regression. Journal of American Statistical Association， 63， 1180-1200.
[10] Parks， R. W. （1967）. Efficient estimation of a system of regression equations when disturbances are both serially and contemporaneously correlated. Journal of the American Statistical Association， 62（318）， 500-509.
[11] Percy， D. F. （1992）. Prediction for seemingly unrelated regressions. Journal of the Royal Statistical Society Series， 54（1）， 243-252.
[12] Powell， A. A. （1965）. Aitken estimators as a tool in allocating predetermined aggregates. Journal of the American Statistical Association， 64， 913-922.
[13] Rao， P.， & Griliches， Z. （1969）. Small-sample properties of several two-stage regression methods in the context of auto-correlated errors. Journal of the American Statistical Association， 64（325）， 253-272.
[14] Revankar， N. S. （1974）. Somefinite sample results in the context of two seemingly unrelated regression equations. Journal of American Statistical Association， 69， 187-190.
[15] Rocke， D. M. （1989）. Bootstrap Bartlett adjustment in seemingly unrelated regression. Journal of the American Statistical Association， 84（406）， 598-601.
[16] Telser， L. G. （1964）. Iterative estimation of a set of linear regression equations. Journal of American Statistical Association， 59， 845-862.
[17] Viraswami， K. （1998）. Some efficiency results on seemingly unrelated regression equations.
[18] William， H. G. （2003）. Econometric analysis （5th ed.）. Upper Saddle River， New Jersey： Prentice Hall.
[20] Youssef， A. H. （1997）. The statistical curvature of seemingly unrelated unrestricted regression equations. The Egyptian Statistical Journal， 41， 43-50.
[21] Zellner， A. （1962）. An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. Journal of the American Statistical Association 57（298）， 348-368.
Key words： Autocorrelation； Bootstrapping； Generalized least squares； Ordinary least squares； Seemingly unrelated regressions
1. INTRODUCTION Seemingly Unrelated Regression （SUR） is a system of regression equations which consists of a set of M regression equations， each of which contains different explanatory variables and satisfies the assumptions of the Classical Linear Regression Model （CLRM）. The SUR estimation technique which allows for an efficient joint estimation of all the regression parameters wasfirst reported by Zellner [21] which involves the application of Aitken’s Generalised Least Squares （AGLS） [2] to the whole system of equations. Several scholars have also developed other estimators for diverse SUR models to address different situations being examined. Dwivedi and Srivastava [6]， Zellner [21] cited in William [18] have shown that the estimation procedure of SUR model was based on Generalized Least Squares （GLS） approach. In answering how much efficiency is gained by using GLS instead of OLS， Zellner [21] has shown in his two-stage approach the gain in efficiency of SUR model over separate equation by equation， that efficiency would be attained when contemporaneous correlation between the disturbances is high and explanatory variables in different equations are uncorrelated. Youssef [19，20] studied the properties of seemingly unrelated regression equation estimators. In an additional paper， he considered a general distribution function for the coefficients of seemingly unrelated regression equations （SURE） model when we unrestricted regression （SUUR） equations.Viraswami [17] presented a working paper on some efficiency results on SURE model. In his work， he considered a two equation seemingly unrelated regressions model in which the equations have some common independent variables and obtained the asymptotic efficiency of the OLS estimator of a parameter of interest relative to its FGLS estimator. He also provided the small-sample relative efficiency of the ordinary least squares estimator and the seemingly unrelated residuals estimator. Alaba et al. [3] recently examined the efficiency gain of the GLS estimator over the Ordinary Least Squares （OLS） estimator. This paper thus examines the performances of OLS and GLS estimators when the disturbances are both autoregressively and contemporaneously correlated.
The remainder of the paper is organized as follows. In section 2， the parametric SUR framework is presented while the simulation studies carried out in the work is discussed in Section 3. Results and detailed discussions are presented in Section 4 while Section 5 gives some concluding remarks.
The diagonal structure of the R matrix implies that each equation or crosssection unit exhibits its own serial correlation coefficient， and the innovations v（t） are contemporaneously correlated with covariance matrixΣ.
The most general model that is usually considered involves the diagonal R matrix， with M parameters， specifying the serial correlation together with a full， symmetricΣmatrix， with M（M + 1）/2 parameters， specifying the contemporaneous covariance. This implies that in （4），？=Σ？I， and？？1=Σ？1？l.
If？is known and denoting the ijth element ofΣ？1byσij， the generalized least squares estimator for the coefficients in this model is：
The summary of the results when the model is estimated by interchanging the coefficients of autocorrelated errors are presented below.
4.2. Discussion of Results
Table 1 gives the comparative performance of the estimators with the coefficient of AR（1） for thefirst regression equation to be 0.3 （i.e.，ρ1= 0.3） and that of the second regression equation to be 0.5 （i.e.，ρ2= 0.5）.
Table 2 gives the comparative performance of the estimators with the coefficient of AR（1） for thefirst regression equation to be 0.5 （i.e.，ρ1= 0.5） and that of the second regression equation to be 0.3 （i.e.，ρ2= 0.3）. From Table 1 and 2， we found that the standard errors of the parameter estimates decreased as sample size increased with varying degrees of coefficients of AR（1） with the SUR estimator performing better than the OLS estimator in both cases（that is， the standard errors of the parameter estimates obtained using the SUR estimator were consistently lower than that of the OLS estimator in both cases as the sample size increased）. Significantly， we found that higher coefficient offirst order autoregressive scheme accounts for better efficiencies of the estimators.
From Table 3， we found on the basis of the Mean Square Error （MSE） for the SUR estimator that the MSE of the regression equations increased with increase in the coefficients of AR（1） with the second model for sample size of 50 performing best with coefficients of AR（1） [0.3，0.5] and MSE [0.8185，0.8852] respectively. Thus， the best MSE obtained for the SUR estimator is 0.8185 usingρ= 0.3 for the second regression for sample size of 50.
The results obtained showed that the standard errors of the SUR estimator is consistently lower than that of the OLS estimator when the model is estimated with varied coefficients of AR（1）. It is revealed that higher coefficient of AR（1） accounts for better efficiencies of the estimators.
REFERENCES
[1] Adebayo， S. B. （2003）. Semi parametric Bayesian regression for multivariate responses. Munich： Hieronymus.
[2] Aitken， A. C. （1934-35）. On least-squares and linear combination of observations. Proceedings of The Royal Society of Edinburgh， 55， 42-48.
[3] Alaba， O. O.， Olubusoye， E. O.， & Ojo， S. O. （2010）. Efficiency of seemingly unrelated regression estimator over the ordinary least squares. European Journal of Scientific Research， 39（1）， 153-160.
[4] Bartels， R.， & Fiebig， D. G. （1991）. A simple characterization of seemingly unrelated regressions models in which OLS is BLUE. The American Statistician， 45（2）， 137-140.
[5] Binkley， J. K.， & Nelson， C. H. （1988）. A note on the efficiency of seemingly unrelated regression. The American Statistician， 42（2）， 137-139.
[6] Dwivedi， T. D.， & Srivastava， V. K. （1978）. Optimality of Least squares in seemingly unrelated regression equation model. Journal of Econometrics， 7， 391-395.
[7] Judge， G. G.， Griffiths， W. E.， Hill， R. C.， & L¨utkepohl， H. （1985）. Disturbancerelated sets of regression equations. The theory and practice of econometrics（2nd ed.） （pp. 465-514）. New York： Wiley. [8] Judge， G. G.， Hill， R. C.， Griffiths， W. E.， & L¨utkepohl， H. （1988）. Introduction to the theory and practice of econometrics （2nd ed.） （pp. 444-468）. New York： Wiley.
[9] Kmenta， J.， & Gilbert， R. F. （1968）. Small sample properties of alternative estimators of seemingly unrelated regression. Journal of American Statistical Association， 63， 1180-1200.
[10] Parks， R. W. （1967）. Efficient estimation of a system of regression equations when disturbances are both serially and contemporaneously correlated. Journal of the American Statistical Association， 62（318）， 500-509.
[11] Percy， D. F. （1992）. Prediction for seemingly unrelated regressions. Journal of the Royal Statistical Society Series， 54（1）， 243-252.
[12] Powell， A. A. （1965）. Aitken estimators as a tool in allocating predetermined aggregates. Journal of the American Statistical Association， 64， 913-922.
[13] Rao， P.， & Griliches， Z. （1969）. Small-sample properties of several two-stage regression methods in the context of auto-correlated errors. Journal of the American Statistical Association， 64（325）， 253-272.
[14] Revankar， N. S. （1974）. Somefinite sample results in the context of two seemingly unrelated regression equations. Journal of American Statistical Association， 69， 187-190.
[15] Rocke， D. M. （1989）. Bootstrap Bartlett adjustment in seemingly unrelated regression. Journal of the American Statistical Association， 84（406）， 598-601.
[16] Telser， L. G. （1964）. Iterative estimation of a set of linear regression equations. Journal of American Statistical Association， 59， 845-862.
[17] Viraswami， K. （1998）. Some efficiency results on seemingly unrelated regression equations.
[18] William， H. G. （2003）. Econometric analysis （5th ed.）. Upper Saddle River， New Jersey： Prentice Hall.
[20] Youssef， A. H. （1997）. The statistical curvature of seemingly unrelated unrestricted regression equations. The Egyptian Statistical Journal， 41， 43-50.
[21] Zellner， A. （1962）. An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. Journal of the American Statistical Association 57（298）， 348-368.