Exercise 3.9.5 Consider the model Y = X + e, E (e) = 0, Cov (e) =

Exercise 3.9.5 Consider the model Y = Xβ +

e, E

(e) = 0, Cov

(e) = σ2 I, (1)

with the additional restriction

Λ



β = d, where d = Λ

b 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 Λ



β =

d. To do this, write Xβ = E(Y ) = (M − MMP)γ + Xb, and define the BLUE of λ



β = ρ

Xβ in the restricted version of (1) to be ρ



(M −

MMP)γˆ + ρ

Xb, where ρ



(M − MMP)γˆ is the BLUE of ρ



(M − MMP)γ in model

(2). Let ˆ β1 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

 ˆ β1 −



Cov(λ

 ˆ β1,Λ

 ˆ β1)

 

Cov(Λ

 ˆ β1)

−

(Λ

 ˆ β1 − d), (3)

where the covariance matrices are computed as in the unrestricted version of model (1).

Hint: This exercise is actually nothing more than simplifying the terms in (3) to show that it equals ρ



(M − MMP)γˆ + ρ

Xb.