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It's been said that SUR is equivalent to the equation-by-equation OLS:

i. When there are no cross-equation correlations between the error terms.
ii. When each equation contains exactly the same set of regressors.

I would like to confirm whether each condition is necessary or sufficient, namely the second condition. In 4.5.1 Seemingly Unrelated Regression, the author uses an example of SUR with equations that each use the same set of regressors, which seems to violate (ii) if each condition is sufficient.

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The answer at your first link provides a link to Wikipedia as support for its statement.

If you read the relevant portion of the Wikipedia article here, it is quite explicit that the two cases are separate. It gives a justification in each case.

That is, either condition is sufficient on its own.

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    $\begingroup$ Please note: the wiki entry isn't 100% accurate. If you read the original Zellner paper from 1962 either condition is sufficient on its own for SUR to be equivalent to equation-by-equation Maximum Likelihood estimation (page 351 of: indiana.edu/~phinite/S681/Zellner.pdf). [But note that the MLE estimates will be equivalent to OLS estimates under the assumption of normal errors, and thus to the SUR estimates]. $\endgroup$ – Jordan Collins Mar 31 '16 at 17:23

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