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Discrete choice
From Wikipedia, the free encyclopedia
In economics, discrete choice problems involve choices between two or more discrete alternatives, such as entering or not
entering the labor market, or choosing between modes oftransport. Such choices contrast with standard consumptionmodels in which the quantity of each good consumed is assumed to be a continuous variable. In the continuous case,
calculus methods (e.g. first-order conditions) can be used to determine the optimum, and demand can be modeled
using regression analysis. On the other hand, discrete choice analysis examines situations in which the potential outcomes
are discrete, such that the optimum is not characterized by standard first-order conditions. Loosely,regression
analysis examines how much while discrete choice analysis examines which. However, discrete choice analysis can
be and has been used to examine the chosen quantity in particular situations, such as the number of vehicles a household
chooses to own[1]
and the number of minutes of telecommunications service a customer decides to use.[2]
Discrete choice models are statistical procedures that model choices made by people among a finite set of alternatives. The
models have been used to examine, e.g., the choice of which car to buy, [1][3] where to go to college,[4] , which mode
oftransport (car, bus, rail) to take to work[5]
among numerous other applications. Discrete choice models are also used to
examine choices by organizations, such as firms or government agencies. In the discussion below, the decision-making
unit is assumed to be a person, though the concepts are applicable more generally.Daniel McFadden won theNobel
prize in 2000 for his pioneering work in developing the theoretical basis for discrete choice.
Discrete choice models statistically relate the choice made by each person to the attributes of the person and the attributes
of the alternatives available to the person. For example, the choice of which car a person buys is statistically related to the
persons income and age as well as to price, fuel efficiency, size, and other attributes of each available car. The models
estimate the probability that a person chooses a particular alternative. The models are often used to forecast how peoples
choices will change under changes in demographics and/or attributes of the alternatives.
Contents
[hide]
1 Application Areas
2 Common Features of Discrete Choice Models
o 2.1 Choice Seto 2.2 Defining Choice Probabilitieso 2.3 Consumer Utilityo 2.4 Properties of Discrete Choice Models Implied by Utility Theory
2.4.1 Only differences matter 2.4.2 Scale must be normalized
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3 Prominent Types of Discrete Choice Modelso 3.1 Binary Choice
3.1.1 A. Logit with attributes of the person but no attributes of the alternatives
3.1.2 B. Probit with attributes of the person but no attributes of the alternatives 3.1.3 C. Logit with variables that vary over alternatives 3.1.4 D. Probit with variables that vary over alternatives
o 3.2 Multinomial Choice - No Correlation Among Alternatives 3.2.1 E. Logit with attributes of the person but no attributes of the alternatives 3.2.2 F. Logit with variables that vary over alternatives (also called conditional logit)
o 3.3 Multinomial Choice - With Correlation Among Alternatives 3.3.1 G. Nested Logit and Generalized Extreme Value (GEV) models
3.3.2 H. Multinomial Probit 3.3.3 I. Mixed Logit
o 3.4 Model Applications 3.4.1 Ranking of Alternatives
3.4.1.1 J. Exploded Logit 3.4.2 Ratings Data
3.4.2.1 K. Ordered Logit 3.4.2.2 L. Ordered Probit
4 Textbooks for further reading5 Notes
6 References
[edit]Application Areas
Marketing researchers use discrete choice models to study consumer demand and to predict competitive businessresponses, enabling choice modelers to solve a range of business problems, such aspricing,product development,
and demand estimation problems.[1]
Transportation planners use discrete choice models to predict demand for planned transportation systems, such ashighway routings andrapid transit systems.[5][6]
Energy forecasters and policymakers use discrete choice models for households and firms choice of heating system,appliance efficiency levels, and fuel efficiency level of vehicles.[7][8]
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Environmental studies utilize discrete choice models to examine the recreators choice of, e.g., fishing or skiing siteand to infer the value of amenities, such as campgrounds, fish stock, and warming huts, and to estimate the value of
water quality improvements.[9]
Labor economists use discrete choice models to examine participation in the work force, occupation choice, andchoice of college and training programs.[4]
[edit]Common Features of Discrete Choice Models
Discrete choice models take many forms, including: Binary Logit, Binary Probit, Multinomial Logit, Conditional Logit,
Multinomial Probit, Nested Logit, Generalized Extreme Value Models, Mixed Logit, and Exploded Logit. All of these
models have the features described below in common.
[edit]Choice Set
The choice set is the set of alternatives that are available to the person. For a discrete choice model, the choice set must
meet three requirements:
1. The set of alternatives must be exhaustive, meaning that the set includes all possible alternatives. Thisrequirement implies that the person necessarily does choose an alternative from the set.
2. The alternatives must be mutually exclusive, meaning that choosing one alternative means not choosing any otheralternatives. This requirement implies that the person chooses only one alternative from the set.
3. The set must contain a finite number of alternatives. This third requirement distinguishes discrete choice analysisfrom regression analysis in which the dependent variable can (theoretically) take an infinite number of values.
Example: The choice set for a person deciding which mode oftransport to take to work includes driving alone,carpooling, taking bus, etc. The choice set is complicated by the fact that a person can use multiple modes for a given
trip, such as driving a car to a train station and then taking train to work. In this case, the choice set can include each
possible combination of modes. Alternatively, the choice can be defined as the choice of primary mode, with the set
consisting of car, bus, rail, and other (e.g. walking, bicycles, etc.). Note that other alternative is used to make the
choice set exhaustive.
Different people may have different choice sets, depending on their circumstances. For instance, Toyota-owned Scion is
not sold in Canada as of 2009, so new car buyers in Canada face different choice sets from those of American consumers.
[edit]Defining Choice Probabilities
A discrete choice model specifies the probability that a person chooses a particular alternative, with the probability
expressed as a function of observed variables that relate to the alternatives and the person. In its general form, the
probability that person n chooses alternative i is expressed as:
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where
is a vector of attributes of alternative i faced by person n,
is a vector of attributes of the other alternatives (other than i) faced by person n,
sn is a vector of characteristics of person n, and
is a set of parameters that relate variables to probabilities, which are estimated statistically.
In the mode oftransport example above, the attributes of modes (xni), such as travel time
and cost, and the characteristics of consumer (sn), such as annual income, age, and gender,
can be used to calculate choice probabilities. The attributes of the alternatives can differ
over people; e.g., cost and time for travel to work by car, bus, and rail are different for each
person depending on the location of home and work of that person.
Properties:
Pni is between 0 and 1 where J is the total number of alternatives. Expected share choosing i where N is the number of
people making the choice.
Different models (i.e. different function G) have different properties. Prominent models are
introduced below.
[edit]Consumer Utility
Discrete choice models can be derived from utility theory. This derivation is useful for
three reasons:
1. It gives a precise meaning to the probabilities Pni2. It motivates and distinguishes alternative model specifications, e.g., Gs.3. It provides the theoretical basis for calculation of changes in consumer surplus
(compensating variation) from changes in the attributes of the alternatives.
Uni is the utility (or net benefit or well-being) that person n obtains from choosingalternative i. The behavior of the person is utility-maximizing: person n chooses the
alternative that provides the highest utility. The choice of the person is designated by
dummy variables,yni, for each alternative:
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Consider now the researcher who is examining the choice. The persons choice
depends on many factors, some of which the researcher observes and some of which
the researcher does not. The utility that the person obtains from choosing an
alternative is decomposed into a part that depends on variables that the researcher
observes and a part that depends on variables that the researcher does not observe. In
a linear form, this decomposition is expressed as
where
is a vector of observed variables relating to alternative i for person n that depends on attributes of the
alternative,xni, interacted perhaps with attributes of the person,sn, such that it can be expressed as
for some numerical functionz,
is a corresponding vector of coefficients of the observed variables, and
captures the impact of all unobserved factors that affect the persons choice.
The choice probability is then
Given, the choice probability is the probability that
the random terms, nj ni (which are random fromthe researchers perspective, since the researcher
does not observe them) are below the respective
quantities . Different
choice models (i.e. different specifications of G)
arise from different distributions ofni for all i and
different treatments of.
[edit]Properties ofDiscrete Choice
Models Implied by Utility Theory
[edit]Only differences matter
The probability that a person chooses a particular
alternative is determined by comparing the utility of
choosing that alternative to the utility of choosing
other alternatives:
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As the last term indicates, the choiceprobability depends only on the difference in
utilities between alternatives, not on the
absolute level of utilities. Equivalently, adding
a constant to the utilities of all the alternatives
does not change the choice probabilities.
[edit]Scale must be normalized
Since utility has no units, it is necessary to
normalize the scale of utilities. The scale of
utility is often defined by the variance of the
error term in discrete choice models. This
variance may differ depending on the
characteristics of the dataset, such as when or
where the data are collected. Normalization of
the variance therefore affects the interpretation
of parameters estimated across diverse
datasets.
[edit]Prominent Types of
Discrete Choice Models
Discrete choice models can first be classified
according to the number of available
alternatives.
* Binomial choice models (dichotomous): 2 available alternatives
* Multinomial choice models (polytomous): 3 or more available alternatives
Multinomial choice models can
further be classified according to
the model specification:
* Models, such as standard logit, that assume no correlation in unobserved factors over alternatives
* Models that allow correlation in unobserved factors among alternatives
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In addition, specific
forms of the models
are available for
examining rankings
of alternatives (i.e.,
first choice, second
choice, third choice,
etc.) and for ratings
data.
Details for each
model are provided in
the following
sections.
[edit]Binary
Choice
[edit]A. Logit with
attributes of the
person but no
attributes of the
alternatives
Main article:Logistic
regression
Un is the utility (or
net benefit) that
person n obtains from
taking an action (as
opposed to not taking
the action). The
utility the person
obtains from taking
the action depends on
the characteristics of
the person, some of
which are observed
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by the researcher and
some are not:
The persontakes the
action,yn = 1,
ifUn > 0. The
unobserved
term, n , is
assumed to
have a logistic
distribution.
The
specification is
written
succinctly as:
Un =
sn +
n
L
ogisti
c,
Then the
probability of
taking the
action is
[edit]B.
Probit
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Un
=
s
n
+
n
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t
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d
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o
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l
,
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Then the
probabili
ty of
taking
the
action is
,
where () is cumulative distribution function of standard normal.
[e
dit
]C
.
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Uni
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de
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pe
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1
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=
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e
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r
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o
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s
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where is the cumulative distribution function of standard normal.
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where J is the total number of alternatives.
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where J is the total number of alternatives.
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where is the joint normal density with mean zero and covariance .
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where
is logit probability evaluated at
Jis the total number of alternatives.
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Example: Please give your rating of how well the President is doing.
1: Very badly
2: Badly
3: Fine
4: Good
5: Very good
Example: On a 1-5 scale where 1 means disagree completely and 5 means agree completely, how much do you
agree with the following statement. "The Federal government should do more to help people facing foreclosure
on their homes."
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Prob(choosing1) = (a zn), Prob(choosing2) = (b zn) (a zn),
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