Название: Discrete Choice Models: Mathematical Methods, Econometrics, and Data Science
Автор: Alfred Galichon
Издательство: Princeton University Press
Год: 2026
Страниц: 377
Язык: английский
Формат: pdf (true), epub
Размер: 15.5 MB

A foundational treatment of discrete choice models, with a focus on random utility models.

Discrete choice models are essential tools for understanding decision-making when individuals must choose among alternatives. They have applications across the social sciences, notably in economics, marketing, and political science. This book offers a foundational treatment of discrete choice models, introducing the logit model and its generalizations, logistic and Poisson regressions, and generalized linear models, and demonstrates their use in analyzing important econometric models. These include international trade gravity, demand estimation, matching with and without transfers, hedonic markets, and dynamic discrete choice. Bridging theoretical clarity and practical applicability, the book is suitable for use in graduate-level coursework and will be an essential resource for researchers and practitioners.

The random utility paradigm does not allow us to predict the choice that a particular individual will make, but, as chapter 1 will show, it allows the analyst to compute the probability that a particular individual will choose one option or the other. This is closely connected to the classification problem in machine learning—in fact, many of the tools overlap. The random utility framework allows us to compute aggregate quantities, such as the aggregate welfare (the sum of the welfare of individuals in the population); the predicted market shares; and the part of the aggregate welfare that is due to systematic utilities and the part of it that is due to idiosyncratic terms—that part being defined (up to a sign) as the entropy of choice, a concept introduced in chapter 1. The analyst also needs to solve the inverse problem of recovering the systematic utilities based on the observation of the market shares, a fundamental problem called the “market share inversion problem.” As we shall see in chapter 1, convex analysis is the appropriate mathematical framework to perform these calculations without making any restrictive assumptions about the random utilities. Thanks to convex analysis, we will be able to formulate the basic calculations as a convex optimization problem, which is practically useful and will have important consequences on the analysis of the problem. For instance, it will allow one to deduce results about the existence and uniqueness of a solution to the market share inversion problem.

As we will see in chapter 2, some distributions of random utilities lead to simple formulas for the expression of the welfare, the predicted market shares, and, in some cases, the entropy of the choice and the demand inversion. The most famous case is the logit (or “logistic”) framework, which assumes that the random utilities follow an extreme value distribution, more precisely, that of independent and identically distributed (i.i.d.) Gumbel variables. The Gumbel distribution is one of the three max-stable distributions arising in extreme value theory; it is the limiting distribution (after renormalization) of the maximum of a large class of independently distributed random variables. As is probably already familiar to most readers, the logit model allows for simple formulas for the market shares and allows us to solve the market share inversion problem in closed form. It also leads to an entropy of choice that coincides (up to a sign) with the Gibbs entropy.

Yet for all its appeal, the logit paradigm is a very rigid framework that has significant shortcomings. In the transportation mode example, it would specify that the random utilities associated with taking bus, train, and plane are independent, which does not capture the fact that some travelers may dislike air travel in a manner that cannot be predicted by their observable characteristics, thereby introducing a correlation between the random utilities associated with the “bus” and “train” options. Consequently, chapter 2 moves on to exploring distributions of random utility that retain tractability with more flexibility, in particular allowing for this type of correlation. An important class of such distributions presented there is the class of multivariate extreme value distributions, discussed in section 2.2. These random variables can be obtained by an ingenious combination of i.i.d. Gumbel variables used as factors, in a way somewhat similar to how any Gaussian vector can be obtained by a linear combination of i.i.d. standard normal factors. Multivariate extreme value distributions, which, following Daniel McFadden, are often called “generalized extreme value distributions” in econometrics, lead to a closed-form expression for the welfare function, the market shares, and sometimes also the entropy of choice, as in the case of the nested logit model, one of the most important representatives of the class.

The logit framework plays a central role in structural estimation, as we begin to see in the subsequent chapter 3 on logistic regression. Consider a stochastic choice problem where the systematic utilities belong to a parametric family and the random utilities are i.i.d. Gumbel. Given a parameter vector, the model predicts the probabilities that each agent will pick the various options, which leads to the specification of a tractable parametric family of choice probability. Assuming that the observations are independently sampled, one can then form the log-likelihood of the sample.

The dynamic discrete choice models seen in chapter 6 connect with dynamic programming, more specifically reinforcement learning and Markov decision processes. And the chapter on discrete choice models with limited availability builds on the important theory of M-functions and submodular optimization. The text makes frequent appeals to tensor algebra with vectorization and Kronecker products, and to numerical optimization algorithms. A particular emphasis has been placed on coding. Finally, the Python code demos in appendix B underscore that, far from being abstract constructions, the concepts introduced in this book are practical and implementable.

• Extensive coverage of computational issues, focusing on optimization and the reformulation as generalized linear models
• Emphasis on econometric questions, including simulation, estimation, and inference methods, with estimation techniques based on both simulated and actual datasets
• A substantial set of exercises and problems at the end of each chapter
• Two appendixes, with one covering the mathematical tools needed to understand the material, and the other the Python code examples

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