This exercise involves some details regarding Example 12.16.

(i) Show that PXf

= m

−1Jm ⊗ I2.

(ii) Show that PX = (2m)

−1Jm ⊗ J2 for X corresponding to model I and PX = m

−1Jm ⊗ (δ2δ



2) for X corresponding to model II.

(iii) Show that for model I, the right side of (12.72) is equal to τ 2, where τ 2 is the true variance of the errors.

(iv) Show that for model II, the right side of (12.74) is equal to σ2 + τ 2 +

β2 0m, where β0 is the true fixed effect under model I and σ2 and τ 2 are the true variance components.