Seemingly Unrelated Regressions - Equivalence To OLS

Equivalence To OLS

There are two important cases when the SUR estimates turn out to be equivalent to the equation-by-equation OLS, so that there is no gain in estimating the system jointly. These cases are:

1. When the matrix Σ is known to be diagonal, that is, there are no cross-equation correlations between the error terms. In this case the system becomes not seemingly but truly unrelated.
2. When each equation contains exactly the same set of regressors, that is X₁ = X₂ = … = X_m. That the estimators turn out to be numerically identical to OLS estimates follows from Kruskal's theorem, or can be shown via the direct calculation.