An Exercise With the SURE Model

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:

                        y₁ = X₁β₁ + ε₁

                        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 ⊗ I_n) 

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.

Reference

Srivastava, V. K. and D. E. A. Giles, 1987. Seemingly Unrelated Regression Equations Models:Estimation and Inference. Marcel Dekker, New York. 

© 2014, David E. Giles