Exercise 10.5 Prove observations (a), (b), and (c).

(a) In practice, when a covariance matrix other than σ 2I is appropriate, most of the time the condition C(X) ⊂C(V) will be satisfied. When C(X) ⊂C(V), then Y ∈ C(X,V) =C(V) a.s., so the condition 1Xβ ∈ Y +C(V) a.s. merely indicates that 1Xβ ∈C(V) a.s. In fact, any estimate of Xβ satisfying condition (i) of consistency also satisfies condition (ii). (Note: The last claim is not restricted to linear estimates or even unbiased estimates.)

(b) An estimate AY satisfies condition (i) of consistency if and only if A = XB for some matrix B.

(c) An estimate AY satisfies condition (ii) of consistency if and only if (I−A)Y ∈
C(V) for any Y ∈ C(X,V).