Exercise 3.9.5 Consider the model Y = X +e, E (e) = 0, Cov (e) = 2I, (1) with the additional restriction = d,

Exercise 3.9.5 Consider the model Y = Xβ +e, E

(e) = 0, Cov

(e) =σ 2I, (1)

with the additional restriction

Λβ = d, where d =Λb for some (known) vector b and Λ = PX. Model (1) with the additional restriction is equivalent to the model

(Y −Xb) = (M−MMP)γ +e. (2)
If the parameterization of model (1) is particularly appropriate, then we might be interested in estimating Xβ subject to the restriction Λβ =

d. To do this, write Xβ = E(Y) = (M−MMP)γ +Xb, and define the BLUE of λ β = ρXβ in the restricted version of (1) to be ρ(M−
MMP)γˆ+ρXb, where ρ(M−MMP)γˆ is the BLUE of ρ(M−MMP)γ in model (2).
Let ˆβ
1 be the least squares estimate of β in the unrestricted version of model (1).
Show that the BLUE of λ β in the restricted version of model (1) is λ ˆβ1−

Cov(λ ˆβ1,Λˆβ1)

Cov(Λˆβ1)
−
(Λˆβ1−d), (3)
where the covariance matrices are computed as in the unrestricted version of model (1).