This exercise explains a stepwise procedure for computing in exercise 14. There are hints, but there

This exercise explains a stepwise procedure for computing βˆ in exercise 14. There are hints, but there is also some work to do. Let M be the first p − 1 columns of X, so M is n×(p − 1). Let N be the last column of X, so N is n×1.

(i) Let γˆ1 = (M

M)−1M

Y and f = Y − Mγˆ1 .

(ii) Let γˆ2 = (M

M)−1M

N and g = N − Mγˆ2 .

(iii) Let γˆ3 = f · g/g2 and e = f − gγˆ3 .

Show that e ⊥ X. (Hint: begin by checking f ⊥ M and g ⊥ M.)

Finally, show that

βˆ =

γˆ1 − ˆγ2γˆ3

γˆ3



Note. The procedure amounts to (i) regressing Y on M, (ii) regressing N on M, then (iii) regressing the first set of residuals on the second.