(This continues exercises 9 and 10: hard.) Show that Ln() is a concave function of , and...

(This continues exercises 9 and 10: hard.) Show that Ln(β) is a concave function of β, and strictly concave if X has full rank. Hints. Let the parameter vector β be p × 1. Let c be a p × 1 vector with c > 0. You need to show c

L

n(β)c ≤ 0, with strict inequality if X has full rank. Let Xi be the ith row of X,a1×p vector. Confirm that L

n(β) =

i X

iXi

ϕ(Xi

β), where ϕ was defined in exercise 9. Check that c

X

iXi c ≥ 0 and ϕ(Xiβ) ≤ m < 0 for all i = 1,...,n, where m is a real number that depends on β.

On the probit model