Global Concavity of the Likelihood Function in the Logit and Probit Models

Подпись: log L(P) = log L(j}) image552 Подпись: (9.2.18)

Global concavity means that d2 log L/dfldf}’ is a negative definite matrix for fi Є В. Because we have by a Taylor expansion

(P~P)-

where P* lies between P and p, global concavity implies that log L(P) > log L(P) for P Ф P’iiP is a solution of (9.2.8). We shall prove global concavity for logit and probit models.

Подпись: ал dx Подпись: ... , а*л .. _.. ал Л(1-Л) and ^ = <1 -2Л)—. Подпись: (9.2.19)

For the logit model we have

Inserting (9.2.19) into (9.2.12) with F— Л yields

Д2 T n

-ЩІ’———- 2А,(1-Л,)х,.х?. (9.2.20)

where = Л(х’іР). Thus the global concavity follows from Assumption 9.2.3.

Подпись: a2 log L apap' Подпись: - 2 <M>r2( 1 - Ф/Г2[(У, - ^УІФ, + Ф 1)ф( і-1 Подпись: (9.2.21)

A proof of global concavity for the probit model is a little more complicated. Putting F, = Фf, fi = фі, and/• = — х’іРФі, where ф is the density function of ЩО, 1), into (9.2.12) yields

+ {уі – Ф/)Ф,(і – ф,)х<0]х, х;. Thus we need to show the positivity of

8y(x) -(у – 2уФ + Ф2) + (у – Ф)Ф(1 – Ф)х

for y = 1 and 0. First, consider the case y = 1. Because gi(x) = (1 — Ф)2(ф + Фх), we need to show ф + Фх > 0. The inequality is clearly satisfied ifxSO, so assume x < 0. But this is equivalent to showing

ф > (1 — Ф)х for x > 0, (9.2.22)

which follows from the identity (see Feller, 1961, p. 166)

x-1 exp (—x2/2) — J (1 + y~2) exp (—y2/2) dy. (9.2.23)

Next, if у = 0, we have g0(x) = С^[ф — (1 — Ф)х], which is clearly positive if x S 0 and is positive if x > 0 because of (9.2.22). Thus we proved global concavity for the probit case.

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