We study the impact of weak identification in discrete choice models, and provide insights into the determinants of identification strength in these models. Using these insights, we propose a novel test that can consistently detect weak identification in commonly applied discrete choice models, such as probit, logit, and many of their extensions. Furthermore, we demonstrate that when the null hypothesis of weak identification is rejected, Wald-based inference can be carried out using standard formulas and critical values. A Monte Carlo study compares our proposed testing approach against commonly applied weak identification tests. The results simultaneously demonstrate the good performance of our approach and the fundamental failure of using conventional weak identification tests for linear models in the discrete choice model context. Furthermore, we compare our approach against those commonly applied in the literature in two empirical examples: married women labor force participation, and US food aid and civil conflicts.
Identification of multinomial choice models is often established by using special covariates that have full support. This paper shows how these identification results can be extended to a large class of multinomial choice models when all covariates are bounded. I also provide a new $\sqrt{n}$-consistent asymptotically normal estimator of the finite-dimensional parameters of the model.
The recent literature often cites Fang and Wang (2015) for analyzing the identification of time preferences in dynamic discrete choice under exclusion restrictions (e.g. Yao et al., 2012; Lee, 2013; Ching et al., 2013; Norets and Tang, 2014; Dubé et al., 2014; Gordon and Sun, 2015; Bajari et al., 2016; Chan, 2017; Gayle et al., 2018). Fang and Wang's Proposition 2 claims generic identification of a dynamic discrete choice model with hyperbolic discounting. This claim uses a definition of "generic" that does not preclude the possibility that a generically identified model is nowhere identified.…
We propose a framework for nonparametric identification and estimation of discrete choice models with unobserved choice sets. We recover the joint distribution of choice sets and preferences from a panel dataset on choices. We assume that either the latent choice sets are sparse or that the panel is sufficiently long. Sparsity requires the number of possible choice sets to be relatively small. It is satisfied, for instance, when the choice sets are nested, or when they form a partition. Our estimation procedure is computationally fast and uses mixed-integer optimization to recover the sparse s…
Empirical research often cites observed choice responses to variation that shifts expected discounted future utilities, but not current utilities, as an intuitive source of information on time preferences. We study the identification of dynamic discrete choice models under such economically motivated exclusion restrictions on primitive utilities. We show that each exclusion restriction leads to an easily interpretable moment condition with the discount factor as the only unknown parameter. The identified set of discount factors that solves this condition is finite, but not necessarily a single…
McFadden's random-utility model of multinomial choice has long been the workhorse of applied research. We establish shape-restrictions under which multinomial choice-probability functions can be rationalized via random-utility models with nonparametric unobserved heterogeneity and general income-effects. When combined with an additional restriction, the above conditions are equivalent to the canonical Additive Random Utility Model. The sufficiency-proof is constructive, and facilitates nonparametric identification of preference-distributions without requiring identification-at-infinity type ar…
This paper studies the semi-parametric identification and estimation of a rational inattention model with Bayesian persuasion. The identification requires the observation of a cross-section of market-level outcomes. The empirical content of the model can be characterized by three moment conditions. A two-step estimation procedure is proposed to avoid computation complexity in the structural model. In the empirical application, I study the persuasion effect of Fox News in the 2000 presidential election. Welfare analysis shows that persuasion will not influence voters with high school education …
Control variables provide an important means of controlling for endogeneity in econometric models with nonseparable and/or multidimensional heterogeneity. We allow for discrete instruments, giving identification results under a variety of restrictions on the way the endogenous variable and the control variables affect the outcome. We consider many structural objects of interest, such as average or quantile treatment effects. We illustrate our results with an empirical application to Engel curve estimation.
This paper is concerned with learning decision makers' preferences using data on observed choices from a finite set of risky alternatives. We propose a discrete choice model with unobserved heterogeneity in consideration sets and in standard risk aversion. We obtain sufficient conditions for the model's semi-nonparametric point identification, including in cases where consideration depends on preferences and on some of the exogenous variables. Our method yields an estimator that is easy to compute and is applicable in markets with large choice sets. We illustrate its properties using a dataset…
Using results from convex analysis, we investigate a novel approach to identification and estimation of discrete choice models which we call the Mass Transport Approach (MTA). We show that the conditional choice probabilities and the choice-specific payoffs in these models are related in the sense of conjugate duality, and that the identification problem is a mass transport problem. Based on this, we propose a new two-step estimator for these models; interestingly, the first step of our estimator involves solving a linear program which is identical to the classic assignment (two-sided matching…
Discrete Choice Experiments (DCE) have been widely used in health economics, environmental valuation, and other disciplines. However, there is a lack of resources disclosing the whole procedure of carrying out a DCE. This document aims to assist anyone wishing to use the power of DCEs to understand people's behavior by providing a comprehensive guide to the procedure. This guide contains all the code needed to design, implement, and analyze a DCE using only free software.
This paper establishes a general equivalence between discrete choice and rational inattention models. Matejka and McKay (2015, AER) showed that when information costs are modelled using the Shannon entropy function, the resulting choice probabilities in the rational inattention model take the multinomial logit form. By exploiting convex-analytic properties of the discrete choice model, we show that when information costs are modelled using a class of generalized entropy functions, the choice probabilities in any rational inattention model are observationally equivalent to some additive random …
Standard methods in preference learning involve estimating the parameters of discrete choice models from data of selections (choices) made by individuals from a discrete set of alternatives (the choice set). While there are many models for individual preferences, existing learning methods overlook how choice set assignment affects the data. Often, the choice set itself is influenced by an individual's preferences; for instance, a consumer choosing a product from an online retailer is often presented with options from a recommender system that depend on information about the consumer's preferen…
AI Lab, CyberAgent, Inc.（CyberAgent公司AI实验室）；School of Electrical Engineering and Computer Science, KTH（KTH皇家理工学院电气工程与计算机科学学院）；Stern School of Business, New York University（纽约大学斯特恩商学院）；Fox School of Business, Temple University（天普大学福克斯商学院）；Columbia Business School, Columbia University（哥伦比亚大学商学院）
We consider the "policy choice" problem -- otherwise known as best arm identification in the bandit literature -- proposed by Kasy and Sautmann (2021) for adaptive experimental design. Theorem 1 of Kasy and Sautmann (2021) provides three asymptotic results that give theoretical guarantees for exploration sampling developed for this setting. We first show that the proof of Theorem 1 (1) has technical issues, and the proof and statement of Theorem 1 (2) are incorrect. We then show, through a counterexample, that Theorem 1 (3) is false. For the former two, we correct the statements and provide ri…
In a nonparametric instrumental regression model, we strengthen the conventional moment independence assumption towards full statistical independence between instrument and error term. This allows us to prove identification results and develop estimators for a structural function of interest when the instrument is discrete, and in particular binary. When the regressor of interest is also discrete with more mass points than the instrument, we state straightforward conditions under which the structural function is partially identified, and give modified assumptions which imply point identificati…
We propose an estimation procedure for discrete choice models of differentiated products with possibly high-dimensional product attributes. In our model, high-dimensional attributes can be determinants of both mean and variance of the indirect utility of a product. The key restriction in our model is that the high-dimensional attributes affect the variance of indirect utilities only through finitely many indices. In a framework of the random-coefficients logit model, we show a bound on the error rate of a $l_1$-regularized minimum distance estimator and prove the asymptotic linearity of the de…
This paper studies inference in models of discrete choice with social interactions when the data consists of a single large network. We provide theoretical justification for the use of spatial and network HAC variance estimators in applied work, the latter constructed by using network path distance in place of spatial distance. Toward this end, we prove new central limit theorems for network moments in a large class of social interactions models. The results are applicable to discrete games on networks and dynamic models where social interactions enter through lagged dependent variables. We il…
This paper studies the joint estimation problem of a discrete choice model and the arrival rate of potential customers when unobserved stock-out events occur. In this paper, we generalize [Anupindi et al., 1998] and [Conlon and Mortimer, 2013] in the sense that (1) we work with generic choice models, (2) we allow arbitrary numbers of products and stock-out events, and (3) we consider the existence of the null alternative, and estimates the overall arrival rate of potential customers. In addition, we point out that the modeling in [Conlon and Mortimer, 2013] is problematic, and present the corr…
A family of models of individual discrete choice are constructed by means of statistical averaging of choices made by a subject in a reinforcement learning process, where the subject has short, k-term memory span. The choice probabilities in these models combine in a non-trivial, non-linear way the initial learning bias and the experience gained through learning. The properties of such models are discussed and, in particular, it is shown that probabilities deviate from Luce's Choice Axiom, even if the initial bias adheres to it. Moreover, we shown that the latter property is recovered as the m…
We present a mixed multinomial logit (MNL) model, which leverages the truncated stick-breaking process representation of the Dirichlet process as a flexible nonparametric mixing distribution. The proposed model is a Dirichlet process mixture model and accommodates discrete representations of heterogeneity, like a latent class MNL model. Yet, unlike a latent class MNL model, the proposed discrete choice model does not require the analyst to fix the number of mixture components prior to estimation, as the complexity of the discrete mixing distribution is inferred from the evidence. For posterior…
Estimation and inference in dynamic discrete choice models often relies on approximation to lower the computational burden of dynamic programming. Unfortunately, the use of approximation can impart substantial bias in estimation and results in invalid confidence sets. We present a method for set estimation and inference that explicitly accounts for the use of approximation and is thus valid regardless of the approximation error. We show how one can account for the error from approximation at low computational cost. Our methodology allows researchers to assess the estimation error due to the us…
Dynamic Discrete Choice Models (DDCMs) are important in the structural estimation literature. Since the structural errors are practically always continuous and unbounded in nature, researchers often use the expected value function. The idea to solve for the expected value function made solution more practical and estimation feasible. However, as we show in this paper, the expected value function is impractical compared to an alternative: the integrated (ex ante) value function. We provide brief descriptions of the inefficacy of the former, and benchmarks on actual problems with varying cardina…
Dynamic discrete choice models often discretize the state vector and restrict its dimension in order to achieve valid inference. I propose a novel two-stage estimator for the set-identified structural parameter that incorporates a high-dimensional state space into the dynamic model of imperfect competition. In the first stage, I estimate the state variable's law of motion and the equilibrium policy function using machine learning tools. In the second stage, I plug the first-stage estimates into a moment inequality and solve for the structural parameter. The moment function is presented as the …
We study the use of Temporal-Difference learning for estimating the structural parameters in dynamic discrete choice models. Our algorithms are based on the conditional choice probability approach but use functional approximations to estimate various terms in the pseudo-likelihood function. We suggest two approaches: The first - linear semi-gradient - provides approximations to the recursive terms using basis functions. The second - Approximate Value Iteration - builds a sequence of approximations to the recursive terms by solving non-parametric estimation problems. Our approaches are fast and…
