Exercise 1.11 For a linear model Y = X +e, E (e) = 0, Cov (e) =

Exercise 1.11 For a linear model Y = Xβ +e, E

(e) = 0, Cov

(e) = σ 2I, the residuals are

ˆ e =Y −X ˆβ = (I−M)Y, where M is the perpendicular projection operator onto C(X). Find

(a) E( ˆ e).

(b) Cov( ˆ e).

(c) Cov( ˆ e,MY).

(d) E( ˆ e ˆ e).

(e) Show that ˆ e ˆ e =YY −(YM)Y.