Palgrave Handbook of Econometrics: Applied Econometrics

William Greene 515

11.5 Ordered choice

In the preceding sections, the consumer is assumed to maximize utility over a pair
of alternatives. Models of ordered choice describe settings in which individuals
reveal the strength of their utility with respect to a single outcome. For example,
in a survey of voter preferences over a single issue (a new public facility or project,
a political candidate, etc.), random utility is, as before:

Ui∗=x′iβ+z′iγ+εi.

The individual reveals a censored version ofUi∗through a discrete response, e.g.,

yi=0 : strongly dislike

1 : mildly dislike

2 : indifferent

3 : mildly prefer

4 : strongly prefer.

The translation between the underlyingUi∗and the observedyi produces the
ordered choice model:

yi=0ifUi∗≤μ 0

1ifμ 0 <Ui∗≤μ 1

2ifμ 1 <Ui∗≤μ 2

···

J ifμJ− 1 <Ui∗≤μJ,

whereμ 0 ,...,μJare threshold parameters that are to be estimated with the other
model parameters subject toμj>μj− 1 for allj. Assumingβcontains a constant
term, the distribution is located by the normalizationμ 0 =0. At the upper tail,
μJ=+∞. Probabilities for the observed outcomes are derived from the laws of
probability:

Prob(yi=j|xi,zi)=Prob(μj− 1 <U∗i≤μj), whereμ− 1 =−∞.

As before, the observed data do not reveal information about the scaling ofεi,so
the variance is normalized to one. Two standard cases appear in the literature; ifεi
has a normal distribution, then the ordered probit model emerges, while if it has
the standardized logistic distribution, the ordered logit model is produced. (Other
distributions have been suggested as the model is internally consistent with any
continuous distribution over the real line. However, these two overwhelmingly
dominate the received applications.)