William Greene 539The multinomial logit specification implies the peculiar restriction that
∂ln Prob(choice=j)
∂xi.m=[ 1 (j=m)−Prob(choice=m)]β.Thus the impact of a change in an attribute of a particular choice on the set of choice
probabilities is the same for all (other) choices. For example, in our application,
∂lnPTRAIN
∂CostAIR
=
∂lnPBUS
∂CostAIR
=
∂lnPCAR
∂CostAIR
=(−PAIR)βCost.This striking result, termedindependence from irrelevant alternatives(IIA), follows
from the initial assumptions of independent and identical distributions forεi,j.
This is a major shortcoming of the model, and has motivated much of the research
on specification of discrete choice models. Many model extensions have been pro-
posed, including a heteroskedastic extreme value model (Bhat, 1995), the Dogit
(dodging the logit model, Gaudry and Dagenais, 1979), and a host of others. The
major extensions of the canonical multinomial logit (MNL) model have been the
multinomial probit (MNP) model, the nested logit model and the current frontier,
the mixed logit model. We consider each of these in turn.
11.7.1.1 Multinomial probit model
The MNP model (Daganzo, 1979) replaces the i.i.d. assumptions of the MNL model
with a multivariate normality assumption:
εi∼NJ[ 0 ,].This specification relaxes the independence assumption. In principle, it can also
relax the assumption of identical (marginal) distributions as well. Recall that, since
only the most preferred choice is revealed, information about utilities is obtained
in the form of differences,Ui,j–Ui,m. It follows that identification restrictions are
required, as only some, or certain combinations of, elements ofare estimable.
The simplest approach to securing identification that is used in practice is to impose
that the last row ofbe equal to (0, 0,..., 1), and one other diagonal element
also equals 1. The remaining elements ofmay be unrestricted, subject to the
requirement that the matrix be positive definite. This can be done by a Cholesky
decomposition,=CC′, whereCis a lower triangular matrix.
The MNP model relaxes the IIA assumptions. The shortcoming of the model is its
computational demands. The relevant probabilities that enter the log-likelihood
function and its derivatives must be approximated by simulation. The GHK
simulator (Manski and Lerman, 1977; Geweke, Keane and Runkle, 1994) is com-
monly used. The Gibbs sampler with non-informative priors (Allenby and Rossi,
1999) has also proved useful for estimating the model parameters. Even with the
GHK simulator, however, computation of the probabilities by simulation is time
consuming.