478 Discrete Choice Modeling
choices. The termbinary choiceis often used interchangeably with the former. A
bivariate choiceormultivariate choiceis the set of two or more choices made in a
single choice situation. In one of our applications, an individual chooses not to
visit a physician or to visit at least once; this is a binomial choice. This coupled with
a second decision, whether to visit the hospital, constitutes a bivariate choice. In a
different application, the choice of which of four modes to use for travel constitutes
a multinomial choice.
11.2.2 Estimation and inference
“Estimation” in this setting is less clearly defined than in the familiar linear regres-
sion model. If the model is fully parametric, then the way that the parameters
interact with the variables in the model, and the particular function that applies
to the problem, are all fully specified. The model is then:
Pit,j=Fit(
j,Xit,zit,β,γ,uit)
j=1,...,Ju.We will consider models that accommodate unobserved individual heterogene-
ity,uit, in sections 11.6 and 11.7. For the present, to avoid an inconvenience in
the formulation, we consider a model involving only the observed data. Various
approaches to estimation of parameters and derivative quantities in this model
have been proposed, but the likelihood based estimator is by far the method of
choice in the received literature. The log-likelihood for the model is:
lnL=∑ni= 1∑Tit= 1∑Jitj= 1dit,jlnFit(j,Xit,zit,β,γ), i=1,...,nt=1,...,Ti.The maximum likelihood estimator is that function of the data that maximizes
lnL.^4 (See, e.g., Greene, 2008a, Ch. 14, for discussion of maximum likelihood
estimation.) The Bayesian estimator will be the mean of the posterior density:
p(β,γ|D,X,Z)=
∫ L×g(β,γ)
β,γL×g(β,γ)dβdγ.whereg(β,γ)is the prior density for the model parameters and (D,X,Z) is the full
sample of data on all variables in the model. (General discussions of Bayesian
methods may be found in Koop, 2003; Lancaster, 2004; Geweke, 2005.) Semipara-
metric methods, generally in the index form, but without a specific distributional
assumption, are common in the received literature, particularly in the analysis of
binary choices and panel data. These will be considered briefly in sections 11.3.4
and 11.7.2. Nonparametric analysis of discrete choice data is on the frontier of the
theory, and does not play much of a role in the empirical literature. We will note
this segment of the development briefly in section 11.3.4.
Estimation and inference about model parameters is discussed in the sections to
follow. Though the model is commonly formulated as an “index function” model,
i.e.:
Pit,j=Fit
(
j,X′itβ,z′itγ)
j=1,...,Jit,