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466 Chapter 11 Regression with a Binary Dependent Variable

The likelihood function is the joint probability distribution, treated as a function of
the unknown coefficients. It is conventional to consider the logarithm of the likelihood.
Accordingly, the log likelihood function is

ln3fprobit(b0, c, bk; Y1, c, Yn ͉ X1i, c, Xki, i = 1, c, n)4

n

= ia= 1Yi ln 3Φ(b0 + b1X1i + g + bkXki)4

n

+ a (1 - Yi) ln31 - Φ(b0 + b1X1i + g + bkXki)4, (11.17)

i=1

where this expression incorporates the probit formula for the conditional probability,
pi = Φ(b0 + b1X1i + g + bkXki).

The MLE for the probit model maximizes the likelihood function or, equivalently, the
logarithm of the likelihood function given in Equation (11.17). Because there is no simple
formula for the MLE, the probit likelihood function must be maximized using a numerical
algorithm on the computer.

Under general conditions, maximum likelihood estimators are consistent and have a
normal sampling distribution in large samples.

MLE for the Logit Model

The likelihood for the logit model is derived in the same way as the likelihood for the
probit model. The only difference is that the conditional success probability pi for the logit
model is given by Equation (11.9). Accordingly, the log likelihood of the logit model is
given by Equation (11.17), with Φ(b0 + b1X1i + g + bkXki) replaced by
31 + e-(b0 + b1X1i + b2X2i + g+ bkXki)4-1. As with the probit model, there is no simple formula for
the MLE of the logit coefficients, so the log likelihood must be maximized numerically.

Pseudo-R2

The pseudo-R2 compares the value of the likelihood of the estimated model to the value of
the likelihood when none of the X’s are included as regressors. Specifically, the pseudo-R2
for the probit model is

pseudo@R2 = 1 - ln(fpmroabxit) , (11.18)
ln(fBmearnxoulli)

wherefpmroabxit is the value of the maximized probit likelihood (which includes the X’s) and
fBmearnxoulli is the value of the maximized Bernoulli likelihood (the probit model excluding all

the X’s).