Here's an exercise that I sometimes set for students if we're studying the Seemingly Unrelated Regression equations (SURE) model. In fact, I used it as part of a question in the final examination that my grad. students sat last week.
Suppose that we have a 2-equation SURE model:
y2 = X2β2 + ε2 ,
where the sample is "balanced" (i.e,. we have n observations on all of the variables in both equations), and the errors satisfy the usual assumptions for a SURE model:
E[ε] = 0 ; V(ε) = (Σ ⊗ In)
where ε' = [ε1' , ε2']' .
Exercise: Prove that the SURE estimators of β1 and β2 are identical to the OLS estimators of β1 and β2 if the condition, X1 (X1'X1)-1 X1' = X2 (X2'X2)-1 X2' , is satisfied.
Viren Srivastava and I gave this as Exercise 2.14 in our 1987 book on the SURE model. However, we didn't give the solution there - so don't think you can cheat in that way!
You can see that the above condition is satisfied if X1 = X2, and the latter condition is one that is mentioned in most econometrics textbooks. However, it's much more stringent than is needed to get the result.
Also, the above condition is necessary, as well as sufficient, for the OLS and SURE estimators to coincide. However, that's another matter.
I'll post the "solution" to the exercise in a few days' time.
Srivastava, V. K. and D. E. A. Giles, 1987. Seemingly Unrelated Regression Equations Models:Estimation and Inference. Marcel Dekker, New York.