Palgrave Handbook of Econometrics: Applied Econometrics

William Greene 483

proper probability models for the implied regressions. The logit and probit mod-
els described in the next section are the overwhelming choices in the received
literature.

11.3.2 Estimation and inference in parametric binary choice models

A parametric model is completed by specifying a distribution forεi. Many candi-
dates have been proposed, though there is little in the way of observable evidence
that one can use to choose among the candidates.^5 For convenience, we will assume
a symmetric distribution, such as the normal or logistic which are used in the
overwhelming majority of studies. For a symmetric distribution:

1 −Prob[εi≤−(x′iβ+z′iγ)]=Prob(εi≤x′iβ+z′iγ)

=F(x′iβ+z′iγ).

Once again relying on the symmetry of the distribution, the probabilities associated
with the two outcomes are:

Prob(di= 1 |xi,zi)=F(x′iβ+z′iγ),

and:

Prob(di= 0 |xi,zi)=F[−(x′iβ+z′iγ)].

For the two outcomesdi=j,j=0, 1, these may be combined in the form suggested
earlier:

F(j,x′iβ+z′iγ)=F[( 2 j− 1 )(x′iβ+z′iγ)],

where:

F(t)=%(t)=

exp(t)

1 +exp(t)

for the logistic distribution,

and:

F(t)=#(t)=

∫t

−∞

1

√

2 π

exp(−

1

2

z^2 )dzfor the normal distribution.

The assumption of the logistic distribution gives rise to the logit model, while the
normal distribution produces the probit model.

11.3.2.1 Parameter estimation

The model is now fully parameterized, so the analysis can proceed based either
on the likelihood function or the posterior density. We consider the maximum
likelihood estimator (MLE) first, and the Bayesian estimator in section 11.3.3.