When OLS and SUR Disagree

Under Zellner’s classical SUR assumptions—the same regressors in each equation and a common sample of observations—OLS and SUR produce the same coefficient estimates, but SUR yields smaller standard errors. Differences in coefficients appear in practice for good and bad reasons:

Efficient, “good” differences arise from intentional GLS re‑weighting. If the equations remain correctly specified but the set of regressors is not identical across them—say, the demand equation includes $x_3$ while the supply equation does not—and contemporaneous errors are correlated, OLS is still unbiased for each equation in isolation, but it is no longer the minimum‑variance (BLUE) estimator once cross‑equation information is admissible. SUR treats the stacked system as a generalized least‑squares (GLS) problem, uses the estimated error‑covariance matrix to re‑weight the equations, and produces new slope estimates that remain unbiased yet have lower variance. Those shifts are desirable; they merely reflect the fact that SUR attains the BLUE property for the whole system while equation‑by‑equation OLS does not.

Undesirable differences stem from data or model problems that SUR cannot fix. The most common culprit is sample mismatch. If one equation keeps rows that another drops—for instance, because its dependent variable is missing—OLS and SUR are fitted to different datasets, and the resulting coefficients can diverge mechanically. Balanced list‑wise deletion under missing completely at random (MCAR) restores equality: drop the same rows from every equation, and SUR again equals OLS, albeit with larger standard errors. Imbalance must be solved by aligning samples or by principled imputation. The second culprit is misspecification. Omitting a relevant variable, mis‑transforming a regressor, or mishandling missing data under MAR or MNAR biases the affected equation; SUR not only fails to repair that bias but can transmit it to the other equations through GLS weights. Efficiency gains never compensate for misspecification, so the model—not the estimator—must be corrected.