Exercise 3.4 Show that XMMPX = 0 if and only if = 0. Earlier, we

Exercise 3.4 Show that β XMMPXβ = 0 if and only if Λβ = 0.

Earlier, we found the reduced model matrix X0 = XU directly and then, for

Λ = PX, we showed that C(MP) = C(X0)⊥

C(X), which led to the numerator sum of squares. An alternative derivation of the test arrives at C(M−M0) =C(X0)⊥

C(X) =

C(MP) more directly for estimable constraints. The reduced model is Y = Xβ +e and PXβ = 0, or Y = Xβ +e and PMXβ = 0,

or E(Y) ∈ C(X) and E(Y) ⊥C(MP), or E(Y) ∈ C(X)∩ C(MP)⊥
.
The reduced model matrix X0 must satisfy C(X0) =C(X)∩ C(MP)⊥ ≡C(MP)⊥
C(X).
It follows immediately that C(X0)⊥
C(X) =C(MP). Moreover, it is easily seen that X0 can be taken as X0 = (I−MMP)X.