538 Discrete Choice Modeling
The earlier assumptions are extended as well. The axioms of choice will imply
that preferences are transitive, reflexive and complete. Thus, in any choice sit-
uation, the individual will make a choice, and that choice, ji, will be such
that:
Ui,ji>Ui,mfor allm=1,...,Jandm =ji.
Reverting back to the optimization problem, utility maximization over continuous
choices subject to a budget constraint produces the complete set of demands,di
(prices, income). Inserting the demands back into the utility function produces the
indirect utility function:
Ui∗=Ui[x(prices, income)].This formulation is convenient for discrete choice modeling, as the data typi-
cally observed on the right-hand sides of the model equations will be income,
prices, other characteristics of the individual such as age and sex, and attributes
of the choices, such as model or type. The random utility model for multinomial
unordered choices is then taken to be defined over the indirect utilities.
11.7.1 Multinomial logit and multinomial probit models
Not all stochastic specifications forεi,jare consistent with utility maximization.
McFadden (1981) showed that the i.i.d. Type 1 extreme value distribution:
F(εi,j)=exp(−exp(−εi,j)), j=1,...,J, i=1,...,n,produces a probabilistic choice model that is consistent with utility maximization.
The resulting choice probabilities are:
Prob(di,j= 1 |Xi,zi)=exp(x′i,jβ+z′iγ)
Jm= 1 exp(x′i,mβ+z′iγ),di,j=1ifUi,ji>Ui,m,,m=1,...,Jandm =j.This is themultinomial logit model. The components,xi,j, are the attributes of the
choices (prices, features, etc.), while theziare the characteristics of the individual
(income, age, sex). We noted at the outset of section 11.2 that identification of the
model parameters requires thatγvaries across the choices. Thus, the full model is:
Prob(di,j= 1 |Xi,zi)=exp(x′i,jβ+z′iγj)
mJ= 1 exp(x′i,mβ+z′iγm),γJ=0,di,j=1ifUi,ji>Ui,m,,m=1,...,Jandm =j.The log-likelihood function is:
lnL=∑n
i= 1∑J
j= 1
di,jln⎡
⎣exp(x′i,jβ+z′iγj)
mJ= 1 exp(x′i,mβ+z′iγm)⎤
⎦.