# 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?

• Do you understand why generalized least squares (GLS) is more efficient than OLS when the errors have nonzero correlations? SUR is just a Feasible GLS with a special structure imposed on the error covariance matrix. – jbowman Feb 10 at 17:44

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.