# discrete choice

• F. Logit with variables that vary over alternatives (also called conditional logit) The utility for each alternative depends on attributes of that alternative, interacted
perhaps with attributes of the person: where J is the total number of alternatives.

• Thus, instead of examining “how much” as in problems with continuous choice variables, discrete choice analysis examines “which one”.

• Properties of discrete choice models implied by utility theory 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: As the last term indicates, the choice probability depends only on the difference in utilities between alternatives, not on the absolute level
of utilities.

• * 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 In addition,
specific forms of the models are available for examining rankings of alternatives (i.e., first choice, second choice, third choice, etc.)

• D. Probit with variables that vary over alternatives The description of the model is the same as model C, except the difference of the two unobserved terms are distributed

• Therefore, discrete choice models rely on stochastic assumptions and specifications to account for unobserved factors related to a) choice alternatives, b) taste variation
over people (interpersonal heterogeneity) and over time (intra-individual choice dynamics), and c) heterogeneous choice sets.

• Multinomial choice without correlation among alternatives E. Logit with attributes of the person but no attributes of the alternatives Further information: Multinomial
logit The utility for all alternatives depends on the same variables, sn, but the coefficients are different for different alternatives: • Since only differences in utility matter, it is necessary to normalize for one alternative.

• Prominent types of discrete choice models Discrete choice models can first be classified according to the number of available alternatives.

• Discrete choice models theoretically or empirically model choices made by people among a finite set of alternatives.

• For instance, in the example of the helping people facing foreclosure, the person chooses for some real numbers a, b, c, d. Defining Logistic, then the probability of each
possible response is: The parameters of the model are the coefficients β and the cut-off points a − d, one of which must be normalized for identification.

• Discrete choice models specify the probability that an individual chooses an option among a set of alternatives.

• Binary choice Further information: binary regression A. Logit with attributes of the person but no attributes of the alternatives Further information: Logistic
regression Un is the utility (or net benefit) that person n obtains from taking an action (as opposed to not taking the action).

• 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.

• However, discrete choice analysis can also be used to examine the chosen quantity when only a few distinct quantities must be chosen from, such as the number of vehicles a
household chooses to own [1] and the number of minutes of telecommunications service a customer decides to purchase.

• In the mode of transport 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.

• This third requirement distinguishes discrete choice analysis from forms of regression analysis in which the dependent variable can (theoretically) take an infinite number
of values.

• first-order conditions) can be used to determine the optimum amount chosen, and demand can be modeled empirically using regression analysis.

• The choice probability is then Given β, the choice probability is the probability that the random terms, εnj − εni (which are random from the researcher’s perspective, since
the researcher does not observe them) are below the respective quantities Different choice models (i.e.

• In economics, discrete choice models, or qualitative choice models, describe, explain, and predict choices between two or more discrete alternatives, such as entering or not
entering the labor market, or choosing between modes of transport.

• The scale of utility is often defined by the variance of the error term in discrete choice models.

• The models estimate the probability that a person chooses a particular alternative.

• [27][28] Estimation of such models is usually done via parametric, semi-parametric and non-parametric maximum likelihood methods,[29] but can also be done with the Partial
least squares path modeling approach.

• [16] A number of models have been proposed to allow correlation over alternatives and more general substitution patterns: • Nested Logit Model – Captures correlations between
alternatives by partitioning the choice set into ‘nests’ o Cross-nested Logit model[17] (CNL) – Alternatives may belong to more than one nest o C-logit Model[18] – Captures correlations between alternatives using ‘commonality factor’ o Paired
Combinatorial Logit Model[19] – Suitable for route choice problems.

• The exploded logit model is the product of standard logit models with the choice set decreasing as each alternative is ranked and leaves the set of available choices in the
subsequent choice.

• Multinomial choice with correlation among alternatives A standard logit model is not always suitable, since it assumes that there is no correlation in unobserved factors
over alternatives.

• However, these models are derived under the concept that the respondent obtains some utility for each possible answer and gives the answer that provides the greatest utility.

• [30] Estimation from rankings In many situations, a person’s ranking of alternatives is observed, rather than just their chosen alternative.

• Such considerations are taken into account in the formulation of discrete choice models.

• The specification is written succinctly as: B. Probit with attributes of the person but no attributes of the alternatives Further information: Probit model The description
of the model is the same as model A, except the unobserved terms are distributed standard normal instead of logistic.

• • Generalized Extreme Value Model[20] – General class of model, derived from the random utility model[16] to which multinomial logit and nested logit belong • Conditional
probit[21][22] – Allows full covariance among alternatives using a joint normal distribution.

• In this case, the choice set can include each possible combination of modes.

• In particular, Pn1 can also be expressed as Note that if two error terms are iid extreme value,[nb 1] their difference is distributed logistic, which is the basis for the
equivalence of the two specifications.

• For a discrete choice model, the choice set must meet three requirements: 1.

• In addition, McFadden and Train have shown that any true choice model can be approximated, to any degree of accuracy by a mixed logit with appropriate specification of explanatory
variables and distribution of coefficients.

• The models are often used to forecast how people’s choices will change under changes in demographics and/or attributes of the alternatives.

• In the discussion below, the decision-making unit is assumed to be a person, though the concepts are applicable more generally.

• In its general form, the probability that person n chooses alternative i is expressed as: 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, is a vector of characteristics of person n, and is a set of parameters giving the effects of variables on probabilities, which are estimated statistically.

• Different people may have different choice sets, depending on their circumstances.

• [26] Estimation from choices Discrete choice models are often estimated using maximum likelihood estimation.

• First, the model allows to be random in addition to .

• [5][7] The first applications of discrete choice models were in transportation planning, and much of the most advanced research in discrete choice models is conducted by transportation
researchers.

• 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.

• 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.

• Rather, it is the lack of information that leads us to describe choice in a probabilistic fashion.

• 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.

• Different models (i.e., models using a different function G) have different properties.

• This lack of correlation translates into a particular pattern of substitution among alternatives that might not always be realistic in a given situation.

• [6] Applications • Marketing researchers use discrete choice models to study consumer demand and to predict competitive business responses, enabling choice modelers to solve
a range of business problems, such as pricing, product development, and demand estimation problems.

• G. Nested Logit and Generalized Extreme Value (GEV) models The model is the same as model F except that the unobserved component of utility is correlated over alternatives
rather than being independent over alternatives.

• Then the probability of taking the action is where Φ is the cumulative distribution function of standard normal.

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Photo credit: https://www.flickr.com/photos/chitrasudar/2788276815/’]