18.2 Essays and Longer Questions 1) Write an essay on the difference between the OLS estimator and the GLS estimator. Answer: Answers will vary by student, but some of the following points should be made. The multiple regression model is Yi = β0 + β1 X1i + β0 X2i + ... + βkXki + ui, i = 1, …, n which, in matrix form, can be written as Y = Xβ + U. The OLS estimator is derived by minimizing the ^ squared prediction mistakes and results in the following formula: β = (X′X)-1 X′Y. There are two GLS ^
estimators. The infeasible GLS estimator is β GLS = (X′Ω-1 X)-1 (X′Ω-1 Y). Since Ω is typically unknown, the estimator cannot be calculated, and hence its name. However, a feasible GLS estimator can be calculated if Ω is a known function of a number of parameters which can be estimated. Once ^
these parameters have been estimated, they can then be used to calculate Ω, the estimator of Ω. The ^ ^ ^ feasible GLS estimator is defined as β GLS= (X′Ω -1 )-1 (X′Ω -1 Y). There are extended least squares assumptions. · ·
E(ui Xi) = 0 (ui has conditional mean zero); (Xi,Yi), i = 1, …, n are independently and identically distributed (i.i.d.) draws from their
joint
distribution; Xi and ui have nonzero finite fourth moments; · ·
X has full column rank (there is no perfect multicollinearity);
·
2 var(ui Xi) = σ u (homoskedasticity);
·
the conditional distribution of ui given Xi is normal (normal errors),
2 These assumptions imply E(U X) = 0 n and E(UU′ X) = σ u In, the Gauss-Markov conditions for multiple regression. If these hold, then OLS is BLUE. If assumptions 5 and 6 do not hold, but assumptions 1 to 4 still hold, then OLS is consistent and asymptotically normally distributed. Small sample statistics can be derived for the case where the errors are i.i.d. and normally distributed, conditional on X. The GLS assumptions are 1.
E(U X) = 0 n;
2. 3.
E(UU′ X) = Ω(X), where Ω(X) is n×n matrix-valued that can depend on X; Xi and ui have nonzero finite fourth moments;
4.
X has full column rank (there is no perfect multicollinearity).
The major differences between the two sets of assumptions relevant to the estimators themselves are that (i) GLS allows for homoskedastic errors to be serially correlated (dropping assumption 2 of OLS list), and (ii) there is the possibility that the errors are heteroskedastic (adding assumption 2 to GLS list). For 2 the case of independent sampling, replacing E(UU′ X) =Ω(X) with E(UU′ X) = σ u In turns the GLS estimator into the OLS estimator. In the case of the infeasible GLS estimator, the model can be transformed in such a way that the Gauss-Markov assumptions apply to the transformed model, if the four GLS assumptions hold. In that case, GLS is BLUE and therefore more efficient than the OLS estimator. This is of little practical value Stock/Watson 2e -- CVC2 8/23/06 -- Page 421
