Suppose I have a VAR model with different regressors in different equations (this could be due to restricting some coefficients of a full VAR($p$) model to zero or having some different exogenous regressors in different equations).
Suppose also that the model is correctly specified.
I will be interested in asymptotic results (small sample properties may be more complicated, so I leave them aside for simplicity).
Such a model can be estimated by GLS or by equation-by-equation OLS.
Both GLS and OLS estimators will be unbiased.
GLS estimator will be more efficient than equation-by-equation OLS estimator. I understand this means that the variance-covariance matrix of the GLS coefficient vector will be "smaller" than that of OLS (the difference of the two matrices will be a negative semidefinite matrix).
Question: will all the diagonal elements of the variance-covariance matrix be smaller in the GLS case? In other words, will each and every coefficient have a smaller variance in GLS case?
Here is a related question comparing OLS and GLS.