discrete choice

8/8/2019 Discrete Choice

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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

S

t

a

n

d

a

r

d

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o

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a

l

,


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Then the

probabili

ty of

taking

the

action is

,

where () is cumulative distribution function of standard normal.

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Uni

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pe

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Th

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=

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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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