568 Inferences About Normal Linear Modelsmultivariate normal distribution with meanβand variance–covariance matrix
σ^2 (X′X)−^1.
(c)Show that(Y−Xβ)′(Y−Xβ)=(βˆ−β)′(X′X)(βˆ−β)+(Y−Xβˆ)′(Y−Xβˆ),
For the remainder of the exercise, letQdenote the quadratic form on the left
side of this expression andQ 1 andQ 2 denote the respective quadratic forms
on the right side. Hence,Q=Q 1 +Q 2.(d)Show thatQ 1 /σ^2 isχ^2 (p).
(e)Show thatQ 1 andQ 2 are independent.(f)Argue thatQ 2 /σ^2 isχ^2 (n−p).
(g)Findcso thatcQ 1 /Q 2 has anF-distribution.(h)Thefactthatavaluedcan be found so thatP(cQ 1 /Q 2 ≤d)=1−αcould
be used to find a 100(1−α)% confidence ellipsoid forβ. Explain.
9.9.11.Say that G.P.A. (Y) is thought to be a linear function of a “coded” high
school rank (x 2 ) and a “coded” American College Testing score (x 3 ), namely,β 1 +
β 2 x 2 +β 3 x 3 .Notethatallx 1 values equal 1. We observe the following five points:
x 1 x 2 x 3 Y
1123
1436
1224
1424
1324(a)ComputeX′Xandβˆ=(X′X)−^1 X′Y.(b)Compute a 95% confidence ellipsoid forβ′ =(β 1 ,β 2 ,β 3 ). See part (h) of
Exercise 9.9.10.
9.9.12.Assume thatXis ann×pmatrix. Then the kernel ofXis defined to be
the space ker(X)={b:Xb= 0 }.
(a)Show that ker(X) is a subspace ofRp.(b)The dimension of ker(X) is called thenullityofXand is denoted byν(X).
Letρ(X) denote the rank ofX. A fundamental theorem of linear algebra says
thatρ(X)+ν(X)=p. Use this to show that ifXhas full column rank, then
ker(X)={ 0 }.9.9.13.SupposeXis ann×pmatrix with rankp.
(a)Show that ker(X′X)=ker(X).
(b)Use part (a) and the last exercise to show that ifXhas full column rank, then
X′Xis nonsingular.