Palgrave Handbook of Econometrics: Applied Econometrics

David T. Jacho-Chávez and Pravin K. Trivedi 811

whereμi=E(yi|di,xi,li)=exp(xi$β+d$iγ+l$iλ)andψ≡ 1 /α(α>0) is the
overdispersion parameter.
As in the standard MNL model, the parameters in the MMNL are only identi-
fied up to a scale. Therefore, a normalization for the scale of the latent factors is
required. We assumeδj=1 for eachj, without loss of generality. Although the
model is identified whenzi=xi, for robust identification it would be preferable
to include some variables inzithat are not includedxi, i.e., identification via
exclusion restrictions is the preferred approach.

15.6.2 Estimation algorithm

The joint distribution of treatment and outcome variables, conditional on the
common latent factors, can be written as the product of the marginal density of
treatment and the conditional density:

Pr(yi,di|xi,zi,li) = f(yi|di,xi,li)×Pr(di|zi,li)

= f(x$iβ+d$iγ+l$iλ)

×g(z$iα 1 +δ 1 li 1 ,...,z$iαJ+δJliJ).

(15.41)

The problem in estimation arises because thelijare unknown. Assume that the
lijare i.i.d. draws from a standard normal distribution so their joint distributionh
can be integrated out of the joint density, i.e.:

Pr(yi,di|xi,zi) =

∫[

f(x$iβ+d$iγ+l$iλ)

×g(z$iα 1 +δ 1 li 1 ,...,z$iαJ+δJliJ)

]

h(li)dli.

(15.42)

The main computational problem, given suitable specifications forf,gandhj,is
that the integral (15.42) does not have, in general, a closed-form solution. As was
explained in section 15.3.3, this difficulty can be tackled using simulation-based
estimation (Gouriéroux and Monfort, 1996). Note that:

Pr(yi,di|xi,zi) = E

[

f(x$iβ+d$iγ+l$iλ)

×g(z$iα 1 +δ 1 li 1 ,...,z$iαJ+δJliJ)

]

≈

1

S

∑S

s= 1

[

f(x$iβ+d$iγ+ ̃l$isλ),

×g(z$iα 1 +δ 1 ̃li 1 s,...,z$iαJ+δJ ̃liJs)

]

,

(15.43)

where ̃lisis thesth draw (from a total ofSdraws) of a pseudo-random number from
the densityh. The simulated log-likelihood function for the data is given by:

lnl(yi,di|xi,zi) ≈

∑N

i= 1

ln

(

1

S

∑S

s= 1

[

f(x$iβ+d$iγ+ ̃l$isλ) ̃lis,

×g(z$iα 1 +δ 1 ̃li 1 s,...,z$iαJ+δJ ̃liJs)

])

.

(15.44)

Provided thatSis sufficiently large, maximization of the simulated log-likelihood is
equivalent to maximizing the log-likelihood. The covariance of the MSL estimates
may be obtained using the robust Eicker–White formula.