Palgrave Handbook of Econometrics: Applied Econometrics

484 Discrete Choice Modeling

The log-likelihood function for the observed data is:

lnL=

∑n

i= 1

lnProb(di|xi,zi)

=

∑

di= 1

ln Prob(di= 1 |xi,zi)+

∑

di= 0

ln Prob(di= 0 |xi,zi)

=

∑n

i= 1 lnF[(^2 di−^1 )(x

′

iβ+z

′

iγ)].

Estimation by maximizing the log-likelihood is straightforward for this model. The
gradient of the log-likelihood is:

∂lnL

∂

(

β

γ

)=

∑n

i= 1

( 2 di− 1 )

F′[( 2 di− 1 )(x′iβ+z′iγ)]

F[( 2 di− 1 )(x′iβ+z′iγ)]

(

xi

zi

)

=

∑n

i= 1

gi=g.

The maximum likelihood estimators of the parameters are found by equatinggto
zero, an optimization problem that requires an iterative solution.^6 For convenience
in what follows, we will define:

qi=( 2 di− 1 ),wi=

(

xi

zi

)

,θ=

(

β

γ

)

,ti=qiw′iθ,Fi=F(ti),F′i=dFi/dti=fi.

(Thus,Fiis the cumulative density function (c.d.f.) andfiis the density for the
assumed distribution.) It follows that:

gi=qiFi′(ti)wi=qifiwi.

Statistical inference about the parameters is made using one of the three con-
ventional estimators of the asymptotic covariance matrix: the Berndt, Hall, Hall
and Hausman (BHHH) (1974) estimator, based on the outer products of the first
derivatives:

VBHHH=

[∑n

i= 1 gig

′

i

]− 1

,

the actual Hessian:

VH=

[

−

∑n

i= 1

∂^2 lnL

∂θ∂θ′

]− 1

=

[

−

∑n

i= 1

FiF′′i−(Fi′)^2

Fi^2

,wiw′i

]− 1

,

or the expected Hessian, which can be shown to equal:

VEH=

[

−

∑n

i= 1 Edi

(

∂^2 lnL

∂θ∂θ′

)]− 1

=

[

−

∑n

i= 1

f(w′iθ)f(−w′iθ)

Fi( 1 −Fi)

wiw′i

]− 1

.

It has become common, evende rigueur, to compute a “robust” covariance matrix

for the MLE usingVH×V−BHHH^1 ×VH, under the assumption that the MLE is robust
to failures of the specification of the model. In fact, there is no obvious failure of