Why does SUR improve efficiency of parameter estimation over OLS? If residuals for the same observation from two 'seemingly unrelated' equations are correlated, then it is often stated that there are efficiency gains from estimating the parameters of the two equations jointly using Zellner’s (1962) over separate OLS.
Gains on 'efficiency' grounds is the reasoning presented in this IMF paper for example examples, for using SUR over OLS on each estimating equation.
However, the intuition behind the so-called improved efficiency of SUR over OLS, when cross-equation contemporaneous residuals are correlated, is not clear to me.
Something I thought is that because SUR system, by construction, combines information on different equations it utilizes more observations at once than OLS does and more observations are generally welcomed. Still, the idea that SUR improves OLS on efficiency grounds is not clear to me. Is it that parameters estimated from SUR will have have smaller standard errors than those from OLS? Will the SUR equations have smaller RMSE than the OLS equations? 
Any insights on how to think about this would be helpful.
 A: Your fallacy is thinking that "efficiency" merely boils down to number of observations. SUR does not "utilize more observations" than OLS. The relative efficiency of two estimators is the ratio of their squared standard errors. You must also consider the bias of the estimators and distinguish between attributes of regressors ($\beta$) and the resulting predictions. The RMSE is generally discussed as a measure of predictive accuracy, e.g. the variance of the residuals.
When the linear model is correctly specified, any arbitrarily weighted linear regression model, including the "unweighted" OLS, is unbiased. This holds even when the "independence of errors" assumption is violated. However, the Gauss-Markov theorem tells us the only BLUE (Best Linear Unbiased Estimator) is the inverse-variance weighted linear regression.
SUR (seemingly unrelated regressions) proposes to use the (inverse of the) first-step estimate of the covariance structure as a plug-in weight for the second-step regression. Provided the appropriate assumptions hold, this will do us better than pretending there is no between-regression variance as would be the case by fitting a bunch of OLS models and saying, "done". However, both of these are still less efficient than simply knowing the between regression variance and fixing the weights accordingly. This is the only model that would, in finite samples (and asymptotically?), achieve the Cramer-Rao lower bound.
