Equi-correlation. This problem is based on Baltagi (1998). Consider the regression model given in (9.1) with equi-correlated disturbances, i.e., equal variances and equal covariances: E(uu) =

σ2Ω = σ2[(1 − ρ)IT + ριT ι

T] where ιT is a vector of ones of dimension T and IT is the identity matrix. In this case, var(ut) = σ2 and cov(ut, us) = ρσ2 for t = s with t = 1, 2, . . ., T. Assume that the regression has a constant.

(a) Show that OLS on this model is equivalent to GLS. Hint: Verify Zyskind’s condition given in (9.8) using the fact that PXιT = ιT if ιT is a column of X.

(b) Show that E(s2) = σ2(1−ρ). Also, that Ω is positive semi-definite when −1/(T −1) ≤ ρ ≤ 1.

Conclude that if −1/(T −1) ≤ ρ ≤ 1, then 0 ≤ E(s2) ≤ [T/(T −1)]σ2. The lower and upper bounds are attained at ρ = 1 and ρ = −1/(T − 1), respectively, see Dufour (1986). Hint: Ω

is positive semi-definite if for every arbitrary non-zero vector a we have aΩa ≥ 0. What is this expression for a = ιT ?

(c) Show that for this equi-correlated regression model, the BLUP of yT+1 = x

T+1β + uT+1 is

yT+1 = x

T+1

βOLS as long as there is a constant in the model.