Advanced Econometrics Takeshi Amemiya
Global Concavity of the Likelihood Function in the Logit and Probit Models
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Global concavity means that d2 log L/dfldf}' is a negative definite matrix for fi Є В. Because we have by a Taylor expansion
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.
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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.
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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.
