Probit Maximum Likelihood Estimator
The Tobit likelihood function (10.2.5) can be trivially rewritten as
т = П [І – Ф(х’,0/о) П Ф(x’,0/a) (10.4.1)
П Ф(х’іР/о)~1а~1Ф[(Уі ~ *іР)/о].
Then the first two products of the right-hand side of (10.4.1) constitute the likelihood function of a probit model, and the last product is the likelihood function of the truncated Tobit model as given in (10.2.6). The probit ML estimator ofa = fifa, denoted a, is obtained by maximizing the logarithm of the first two products.
Note that we can only estimate the ratio fi/a by this method and not flora separately. Because the estimator ignores a part of the likelihood function that involves fi and a, it is not fully efficient. This loss of efficiency is not surprising when we realize that the estimator uses only the sign of yf, ignoring its numerical value even when it is observed.
From the results of Section 9.2 we see that the probit MLE is consistent and follows
a – a = (X’D. Xr’X’D. D^fw – £w), (10.4.2)
where D0 is the n X n diagonal matrix the rth element of which is ф(х'(а Di is the nXn diagonal matrix the fth element of which is Ф(х<а)_1[1— Ф(х’О!)]-1ф(х’а02, and w is the л-vector the rth element wt of which is defined by
w( = 1 if yf > 0 (10.4.3)
= 0 if 0.
Note that the rth element of £w is equal to Ф(х|а). The symbol = means that both sides have the same asymptotic distribution.1 As shown in Section 9.2.2, a is asymptotically normal with mean a and asymptotic variance-covariance matrix given by
Vol = (X’D, X)_1. (10.4.4)