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. Such choices contrast with standard consumption models 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 amount chosen, and demand can be modeled empirically 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. Thus, instead of examining “how much” as in problems with continuous choice variables, discrete choice analysis examines “which one.” 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  and the number of minutes of telecommunications service a customer decides to purchase. Techniques such as logistic regression and probit regression can be used for empirical analysis of discrete choice.
Discrete choice models theoretically or empirically 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, where to go to college, which mode of transport (car, bus, rail) to take to work 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 the Nobel 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 person’s 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 people’s choices will change under changes in demographics and/or attributes of the alternatives.
Discrete choice models specify the probability that an individual chooses an option among a set of alternatives. The probabilistic description of discrete choice behavior is used not to reflect individual behavior that is viewed as intrinsically probabilistic. Rather, it is the lack of information that leads us to describe choice in a probabilistic fashion. In practice, we cannot know all factors affecting individual choice decisions as their determinants are partially observed or imperfectly measured. 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. The different formulations have been summarized and classified into groups of models.
- 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. In market research, this is commonly called conjoint analysis.
- Transportation planners use discrete choice models to predict demand for planned transportation systems, such as which route a driver will take and whether someone will take rapid transit systems.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.
- 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.
- Environmental studies utilize discrete choice models to examine the recreators’ choice of, e.g., fishing or skiing site and to infer the value of amenities, such as campgrounds, fish stock, and warming huts, and to estimate the value of water quality improvements.
- Labor economists use discrete choice models to examine participation in the work force, occupation choice, and choice of college and training programs.
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.
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:
- The set of alternatives must be collectively exhaustive, meaning that the set includes all possible alternatives. This requirement implies that the person necessarily does choose an alternative from the set.
- The alternatives must be mutually exclusive, meaning that choosing one alternative means not choosing any other alternatives. This requirement implies that the person chooses only one alternative from the set.
- The set must contain a finitenumber of alternatives. 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.
As an example, the choice set for a person deciding which mode of transport 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 the alternative “other” is included in order to make the choice set exhaustive.
Different people may have different choice sets, depending on their circumstances. For instance, the Scion automobile was not sold in Canada as of 2009, so new car buyers in Canada faced different choice sets from those of American consumers. Such considerations are taken into account in the formulation of discrete choice models.
- ^ Jump up to:ab c Train, K. (1986). Qualitative Choice Analysis: Theory, Econometrics, and an Application to Automobile Demand. MIT Press. Chapter 8.
- ^Train, K.; McFadden, D.; Ben-Akiva, M. (1987). “The Demand for Local Telephone Service: A Fully Discrete Model of Residential Call Patterns and Service Choice”. RAND Journal of Economics. 18 (1): 109–123. doi:10.2307/2555538. JSTOR 2555538.
- ^Train, K.; Winston, C. (2007). “Vehicle Choice Behavior and the Declining Market Share of US Automakers”. International Economic Review. 48 (4): 1469–1496. doi:10.1111/j.1468-2354.2007.00471.x.
- ^ Jump up to:ab Fuller, W. C.; Manski, C.; Wise, D. (1982). “New Evidence on the Economic Determinants of Post-secondary Schooling Choices”. Journal of Human Resources. 17 (4): 477–498. doi:10.2307/145612. JSTOR 145612.
- ^ Jump up to:ab Train, K. (1978). “A Validation Test of a Disaggregate Mode Choice Model” (PDF). Transportation Research. 12 (3): 167–174. doi:10.1016/0041-1647(78)90120-x.
- ^Baltas, George; Doyle, Peter (2001). “Random utility models in marketing research: a survey”. Journal of Business Research. 51 (2): 115–125. doi:10.1016/S0148-2963(99)00058-2.
- ^Ramming, M. S. (2001). “Network Knowledge and Route Choice”. Unpublished Ph.D. Thesis, Massachusetts Institute of Technology. MIT catalogue. hdl:1721.1/49797.
- ^Goett, Andrew; Hudson, Kathleen; Train, Kenneth E. (2002). “Customer Choice Among Retail Energy Suppliers”. Energy Journal. 21 (4): 1–28.
- ^ Jump up to:ab Revelt, David; Train, Kenneth E. (1998). “Mixed Logit with Repeated Choices: Households’ Choices of Appliance Efficiency Level”. Review of Economics and Statistics. 80(4): 647–657. doi:10.1162/003465398557735. JSTOR 2646846.
- ^ Jump up to:ab Train, Kenneth E. (1998). “Recreation Demand Models with Taste Variation”. Land Economics. 74 (2): 230–239. CiteSeerX 188.8.131.5279. doi:10.2307/3147053. JSTOR 3147053.
- ^Ben-Akiva, M.; Lerman, S. (1985). Discrete Choice Analysis: Theory and Application to Travel Demand. Transportation Studies. Massachusetts: MIT Press.
- ^ Jump up to:ab c Ben-Akiva, M.; Bierlaire, M. (1999). “Discrete Choice Methods and Their Applications to Short Term Travel Decisions” (PDF). In Hall, R. W. (ed.). Handbook of Transportation Science.
- ^Vovsha, P. (1997). “Application of Cross-Nested Logit Model to Mode Choice in Tel Aviv, Israel, Metropolitan Area”. Transportation Research Record. 1607: 6–15. doi:10.3141/1607-02. Archived from the original on 2013-01-29.
- ^Cascetta, E.; Nuzzolo, A.; Russo, F.; Vitetta, A. (1996). “A Modified Logit Route Choice Model Overcoming Path Overlapping Problems: Specification and Some Calibration Results for Interurban Networks” (PDF). In Lesort, J. B. (ed.). Transportation and Traffic Theory. Proceedings from the Thirteenth International Symposium on Transportation and Traffic Theory. Lyon, France: Pergamon. pp. 697–711.
- ^Chu, C. (1989). “A Paired Combinatorial Logit Model for Travel Demand Analysis”. Proceedings of the 5th World Conference on Transportation Research. 4. Ventura, CA. pp. 295–309.
- ^McFadden, D. (1978). “Modeling the Choice of Residential Location” (PDF). In Karlqvist, A.; et al. (eds.). Spatial Interaction Theory and Residential Location. Amsterdam: North Holland. pp. 75–96.
- ^Hausman, J.; Wise, D. (1978). “A Conditional Probit Model for Qualitative Choice: Discrete Decisions Recognizing Interdependence and Heterogenous Preferences”. Econometrica. 48(2): 403–426. doi:10.2307/1913909. JSTOR 1913909.
- ^ Jump up to:ab Train, K. (2003). Discrete Choice Methods with Simulation. Massachusetts: Cambridge University Press.
- ^ Jump up to:ab McFadden, D.; Train, K. (2000). “Mixed MNL Models for Discrete Response” (PDF). Journal of Applied Econometrics. 15 (5): 447–470. CiteSeerX 184.108.40.20671. doi:10.1002/1099-1255(200009/10)15:5<447::AID-JAE570>3.0.CO;2-1.
- ^Ben-Akiva, M.; Bolduc, D. (1996). “Multinomial Probit with a Logit Kernel and a General Parametric Specification of the Covariance Structure” (PDF). Working Paper.
- ^Bekhor, S.; Ben-Akiva, M.; Ramming, M. S. (2002). “Adaptation of Logit Kernel to Route Choice Situation”. Transportation Research Record. 1805: 78–85. doi:10.3141/1805-10. Archived from the original on 2012-07-17.
- ^. Also see Mixed logit for further details.
- ^Manski, Charles F. (1975). “Maximum score estimation of the stochastic utility model of choice”. Journal of Econometrics. Elsevier BV. 3 (3): 205–228. doi:10.1016/0304-4076(75)90032-9. ISSN 0304-4076.
- ^Horowitz, Joel L. (1992). “A Smoothed Maximum Score Estimator for the Binary Response Model”. Econometrica. JSTOR. 60 (3): 505–531. doi:10.2307/2951582. ISSN 0012-9682. JSTOR 2951582.
- ^Park, Byeong U.; Simar, Léopold; Zelenyuk, Valentin (2017). “Nonparametric estimation of dynamic discrete choice models for time series data” (PDF). Computational Statistics & Data Analysis. 108: 97–120. doi:10.1016/j.csda.2016.10.024.
- ^Beggs, S.; Cardell, S.; Hausman, J. (1981). “Assessing the Potential Demand for Electric Cars”. Journal of Econometrics. 17 (1): 1–19. doi:10.1016/0304-4076(81)90056-7.
- ^ Jump up to:ab c Combes, Pierre-Philippe; Linnemer, Laurent; Visser, Michael (2008). “Publish or Peer-Rich? The Role of Skills and Networks in Hiring Economics Professors”. Labour Economics. 15 (3): 423–441. doi:10.1016/j.labeco.2007.04.003.
- ^Plackett, R. L. (1975). “The Analysis of Permutations”. Journal of the Royal Statistical Society, Series C. 24 (2): 193–202. JSTOR 2346567.
- ^Luce, R. D. (1959). Individual Choice Behavior: A Theoretical Analysis. Wiley.
Ofer Abarbanel is a 25 year securities lending broker and expert who has advised many Israeli regulators, among them the Israel Tax Authority, with respect to stock loans, repurchase agreements and credit derivatives. Founder of TBIL.co STATX Fund.