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I've been reading some about Generalized Least Squares (GLS) and trying to tie it back to my basic econometric background. I recall in grad school using Seemingly Unrelated Regression (SUR) which seems somewhat similar to GLS. One paper I stumbled on even referred to SUR as "special case" of GLS. But I still can't wrap my brain around the similarities and differences.

so the question:

What are the similarities and differences between GLS and SUR? What are the hallmarks of a problem which should use one method over the other?

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In a narrow sense, GLS (and in particular Feasible GLS or FGLS) is an estimation method applied to SUR models.

SUR implies a system of m equations that are assumed to have correlated errors, and (F)GLS helps to recover from this -- see Wikipedia on Seemingly Unrelated Regressions.

GLS, on the other hand, is a method of incorporating information from the covariance structure of your model. See Wikipedia on GLS.

To recap, you can use the latter (GLS) to estimate the former (SUR).

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