Is the effectiveness of seemingly unrelated regression an example of Stein's paradox? The existence of Stein's Example prima facie appears similar to seemingly unrelated regression (SUR) insofar as simultaneously estimating multiple parameters seems more effective than training them separately.
In the Wikipedia articles linked above there seems to be a somewhat different emphasis. The article on Stein's example is focused on a minimum example to show the seemingly paradoxical decrease in mean squared error. The article on SUR emphasizes the effectiveness of training multiple regression equations together that could have been trained separately. But neither article refers to the other.
Is there a precise relationship between these notions? For instance, might SUR be an example of Stein's Example/Paradox?

No, it's not related to Stein's Paradox; instead, it's related to Generalized Least Squares (GLS.)
In SUR, there's an assumption that the regression errors of the individual equations are correlated across equations. This leads to a situation where you can, conceptually, construct a single, large, regression out of the component regressions, one which will have a non-diagonal error correlation matrix.  This naturally leads to a feasible generalized least squares approach to estimating the parameters of the joint system of equations; if you knew the cross-equation correlations a priori,  you would use GLS instead.
