Developing Median Regression for SURE Models - with Application to 3-Generation Immigrants’ data in Sweden

In this paper we generalize the median regression method in order to make it applicable to systems of regression equations. Given the existence of proper systemwise medians of the errors from different equations, we apply the weighted median regression with the weights obtained from the covariance matrix of errors from different equations calculated by conventional SURE method. The Seemingly Unrelated Median Regression Equations (SUMRE) method produces results that are more robust than the usual SURE or single equations OLS estimations when the distributions of the dependent variables are not symmetric. Moreover, the estimations of the SUMRE method are also more efficient than those of the cases of single equation median regressions when the cross equations errors are correlated. More precisely, the aim of our SUMRE method is to produce a harmony of existing skewness and correlations of errors in systems of regression equations. A theorem is derived and indicates that even with the lack of statistically significant correlations between the equations, using the SMRE method instead of the SURE method will not damage the estimation of parameters. A Monte Carlo experiment was conducted to investigate the properties of the SUMRE method in situations where the number of equations in the system, number of observations, strength of the correlations of cross equations errors and the departure from the normality distribution of the errors were varied. The results show that, when the cross equations correlations are medium or high and the level of skewness of the errors of the equations are also medium or high, the SUMRE method produces estimators that are more efficient and less biased than the ordinary SURE GLS estimators. Moreover, the estimates of applying the SUMRE method are also more efficient and less biased than the estimates obtained when applying the OLS or single equation median regressions. In addition, our results from an empirical application are in accordance with what we discovered from the simulation study, with respect to the relative gain in efficiency of SUMRE estimators compared to SURE estimators, in the presence of Skewness of error terms.