Exercise 3.9.7 Consider a linear model Y = X + e, E (e) = 0 with the (not necessarily estimable) linear constraint

Exercise 3.9.7 Consider a linear model Y = Xβ +

e, E

(e) = 0 with the (not necessarily estimable) linear constraint Λ



β =

d. Consider two solutions to the constraint, b1 and b2, so that Λ

bk =

d, k = 1, 2. Define appropriate least squares fitted values Yˆk from the model Y = X0γ + Xbk + e where X0 = XU with C(U) =

C(Λ)

⊥. Show that Yˆ1 = Yˆ2. Hint: After finding Yˆk , show that (I − M0)X(b1 −

b2) = 0.