Seemingly Unrelated Regressions (SUR):
Seemingly unrelated regressions (SUR) is a class of multivariate regression ( multiple regression ) models, normally belonging to the sub-class of linear regression models. A distinctive feature of SUR models is that they consist of several unrelated systems of equations “Unrelated” here means that any variable, dependent and or independent, is present in only one system or, in other words, the systems have no common variables.
Normally, one would try to solve such systems of equations independently, e.g. using the least squares method for each system separately. But in SUR models the error terms from different systems are correlated. At the same time, according to the general theory of the least square method, which takes covariances of errors into account, such systems should be solved as whole set of equations. Otherwise, the minimal variance of the errors in estimated regression parameters cannot be achieved.
SUR models arise naturally in economics, especially in micro-economics (where each firm considered gives rise to a system of equations, unrelated to other firms), and in other research areas as well. The idea and term belongs to A. Zellner, who published it in 1962 (Zellner A. “An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias.”; J. Amer. Statist. Assn.; 57:348-68, 1962).