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

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.

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.

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;...

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