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17
Discrete Choicebinary outcomes and Discrete ChoiceS
Kit Baum on LPM 17.1 Introduction This is the first of three chapters that will survey models used in
microeconometrics. The analysis of individual choice that is the focus of
this field is fundamentally about modeling discrete outcomes such as
purchase decisions, for example whether or not to buy insurance, voting
behavior, choice among a set of alternative brands, travel modes or places
to live, and responses to survey questions about the strength of
preferences or about self-assessed health or well-being. In these and any
number of other cases, the "dependent variable" is not a quantitative
measure of some economic outcome, but rather an indicator of whether or not
some outcome has occurred. It follows that the regression methods we have
used up to this point are largely inappropriate. We turn, instead, to
modeling probabilities and using econometric tools to make probabilistic
statements about the occurrence of these events. We will also examine
models for counts of occurrences. These are closer to familiar regression
models, but are, once again, about discrete outcomes of behavioral choices.
As such, in this setting as well, we will be modeling probabilities of
events, rather than conditional mean functions.
The models used in this area of study are inherently (and
intrinsically) nonlinear. We have developed some of the elements of
nonlinear modeling in Chapters 7 and 14. Those elements are combined in
whole in the study of discrete choices. This chapter will focus on "binary
choices," where the model is the probability of an event. Many general
treatments of "nonlinear modeling" in econometrics in fact focus on only
this segment of the field. This is reasonable. Nearly the full set of
results used more broadly, for specification, estimation, inference and
analysis can be developed and understood in this particular application.
We will take that approach here. Several of the parts of nonlinear
modeling will be developed in detail in this chapter, then invoked or
extended in straightforward ways in the chapters to follow.
The models that are analyzed in this and the next chapter are built on a
platform of preferences of decision makers. We take a random utility view
of the choices that are observed. The decision maker is faced with a
situation or set of alternatives and reveals something about their
underlying preferences by the choice that he or shethey makes. The
choice(s) made will be affected by observable influences-this is, of
coursefor example, the ultimate objective of advertising-and by
unobservable characteristics of the chooser. The blend of these fundamental
bases for individual choice is at the core of the broad range of models
that we will examine here.[1]
This chapter and Chapter 18 will describe four broad frameworks for
analysis. The first is the simplest::
Binary Choice: The individual faces a pair oftwo choices and makes that
choice between the two that provides the greater utility. Many such
settings involve the choice between taking an action and not taking that
action, for example the decision whether or not to purchase health
insurance. In other cases, the decision might be between two distinctly
different choices, such as the decision whether to travel to and from work
via public or private transportation. In the binary choice case, the [pic]
outcome is merely a label for "no/yes"-the numerical values are a
mereathematical convenience. This chapter will present a lengthy survey of
models and methods for binary choices.
The binary choice case naturally extends to cases of more than two
outcomes. For one example, in our our travel mode case, the individual
choosing private transport might choose between private transport as driver
and private transport as passenger, or public transport by train or by bus.
Such multinomial (many named) choices are unordered. Another case is one
that is a constant staple of the online experience. Instead of being asked
"did you like our service?," a binary choice, the hapless surfer will be
asked "on a scale from 1 to 5, how much did you like our service?," an
ordered multinomial choice.
Multinomial Choice: The individual chooses among more than two choices,
once again, making the choice that provides the greatest utility. In the
previous example, private travel might involve a choice of being a driver
or passenger while public transport might involve a choice between bus and
train. At one level, this is a minor variation of the binary choice case-
the latter is, of course, a special case of the former. But, more elaborate
models of multinomial choice allow a rich specification of consumer
preferences. In the multinomial case, the observed response is simply again
a label for the selected choice; it might be a brand, the name of a place,
or the type of travel mode. Numerical assignments are not meaningful in
this setting.
Ordered Choice: The individual reveals the strength of his or her
preferences with respect to a single outcome. Familiar cases involve survey
questions about strength of feelings about a particular commodity such as a
movie, or self-assessments of social outcomes such as health in general or
self-assessed well-being. In the ordered choice setting, opinions are given
meaningful numeric values, usually [pic] for some upper limit, [pic]. For
example, opinions might be labelled [pic] to indicate the strength of
preferences, for example, for a product, a movie, a candidate or a piece of
legislation. But, in this context, the numerical values are only a ranking,
not a quantitative measure. Thus a "1" is greater than a "0" only in a
qualitative sense, but not by one unit, and the difference between a "2"
and a "1" is not the same as that between a "1" and a "0."
In these three cases, although the numerical outcomes are merely labels
of some nonquantitative outcome, the analysis will nonetheless have a
regresson-style motivation. Throughout, the models will be based on the
idea that observed "covariates" are relevant in explaining the observed
choices and in how changes in those attributes can help to explain
variation in choices. For example, in the binary outcome "did or did not
purchase health insurance," a conditioning model suggests that covariates
such as age, income, and family situation will help to explain the choice.
This cChapter 18 will describe a range of models that have been developed
around these considerations.
We will also be interested in a fourth application of discrete outcome
models:
Event Counts: The observed outcome is a count of the number of
occurrences. In many cases, this is similar to the preceding three settings
in that the "dependent variable" measures an individual choice, such as the
number of visits to the physician or the hospital, the number of derogatory
reports in one's credit history, the number of vehicles in a household's
capital stock, or the number of visits to a particular recreation site. In
other cases, the event count might be the outcome of some natural process,
such as incidencethe occurrence rate of a disease in a population or the
number of defects per unit of time in a production process. In thisese
settings, we will be doing a more familiar sort of regression modeling.
However, the models will still be constructed specifically to accommodate
the discrete (and nonnegative) nature of the observed response variable and
the modeling of probabilities of occurrences of events rather than some
measure of the events themselves..
We will consider these four cases in turn. The four broad areas have many
elements in common; however, there are also substantive differences between
the particular models and analysis techniques used in each. This chapter
will develop the first topic, models for binary choices. In each section,
we will begin with aninclude overview ofseveral applications and then
present the single basic model that is the centerpiece of the methodology,
and, finally, examine some recently developed extensions of the model. This
chapter contains a very lengthy discussion of models for binary choices.
This analysis is as long as it is because, first, the models discussed are
used throughout microeconometrics-the central model of binary choice in
this area is as ubiquitous as linear regression. Second, all the
econometric issues and features that are encountered in the other areas
will appear in the analysis of binary choice, where we can examine them in
a fairly straightforward fashion.
It will emerge that, at least in econometric terms, the models for
multinomial and ordered choice considered in Chapter 18 can be built from
the two fundamental building blocks, the model of random utility and the
translation of that model into a description of binary choices. There are
relatively few new econometric issues that arise here. Chapter 18 will be
largely devoted to suggesting different approaches to modeling choices
among multiple alternatives and models for ordered choices. Once again,
models of preference scales, such as movie or product ratings, or self-
assessments of health or well-being, can be naturally built up from the
fundamental model of random utility. Finally, Chapter 18 will develop the
well-known Poisson regression model for counts of events. We will then
extend the model to demonstra