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5See Zellner (1962) and Telser (1964).

1. If the equations are actually unrelated—that is, if oij = 0 for i = j—then there is obviously no payoff to GLS estimation of the full set of equations. Indeed, full GLS is equation by equation OLS.6

2. If the equations have identical explanatory variables—that is, if Xi = Xj —then OLS and GLS are identical. We will turn to this case in Section 14.2.2 and then examine an important application in Section 14.2.5.7

3. If the regressors in one block of equations are a subset of those in another, then GLS brings no efficiency gain over OLS in estimation of the smaller set of equations; thus, GLS and OLS are once again identical. We will look at an application of this result in Section 19.6.5.8

In the more general case, with unrestricted correlation of the disturbances and different regressors in the equations, the results are complicated and dependent on the data. Two propositions that apply generally are as follows:

1. The greater is the correlation of the disturbances, the greater is the efficiency gain accruing to GLS.

2. The less correlation there is between the X matrices, the greater is the gain in efficiency in using GLS.9

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