We provide a tractable characterization of the sharp identi…cation region of the parameters in a broad class of incomplete econometric models.Models in this class have set valued predictions that yield a convex set of conditional or unconditional moments for the observable model variables.In short, we call these models with convex moment predictions.Examples include static, simultaneous move …nite games of complete and incomplete information in the presence of multiple equilibria; best linear predictors with interval outcome and covariate data; and random utility models of multinomial choice in the presence of interval regressors data.Given a candidate value for ; we establish that the convex set of moments yielded by the model predictions can be represented as the Aumann expectation of a properly de…ned random set.The sharp identi…cation region of ; denoted I ; can then be obtained as the set of minimizers of the distance from a properly speci…ed vector of moments of random variables to this Aumann expectation.Algorithms in convex programming can be exploited to e¢ ciently verify whether a candidate is in I : We use examples analyzed in the literature to illustrate the gains in identi…cation and computational tractability a¤orded by our method.

We propose inference procedures for partially identified population features for which the population identification region can be written as a transformation of the Aumann expectation of a properly defined set valued random variable (SVRV). An SVRV is a mapping that associates a set (rather than a real number) with each element of the sample space. Examples of population features in this class include interval-identified scalar parameters, best linear predictors with interval outcome data, and parameters of semiparametric binary models with interval regressor data. We extend the analogy principle to SVRVs and show that the sample analog estimator of the population identification region is given by a transformation of a Minkowski average of SVRVs. Using the results of the mathematics literature on SVRVs, we show that this estimator converges in probability to the population identification region with respect to the Hausdorff distance. We then show that the Hausdorff distance and the directed Hausdorff distance between the population identification region and the estimator, when properly normalized by , converge in distribution to functions of a Gaussian process whose covariance kernel depends on parameters of the population identification region. We provide consistent bootstrap procedures to approximate these limiting distributions. Using similar arguments as those applied for vector valued random variables, we develop a methodology to test assumptions about the true identification region and its subsets. We show that these results can be used to construct a confidence collection and a directed confidence collection. Those are (respectively) collection of sets that, when specified as a null hypothesis for the true value (a subset of values) of the population identification region, cannot be rejected by our tests.

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 on property insurance purchases. (JEL D81, D83, D91, G22, G52)

We propose a bootstrap‐based calibrated projection procedure to build confidence intervals for single components and for smooth functions of a partially identified parameter vector in moment (in)equality models. The method controls asymptotic coverage uniformly over a large class of data generating processes. The extreme points of the calibrated projection confidence interval are obtained by extremizing the value of the function of interest subject to a proper relaxation of studentized sample analogs of the moment (in)equality conditions. The degree of relaxation, or critical level, is calibrated so that the function of θ , not θ itself, is uniformly asymptotically covered with prespecified probability. This calibration is based on repeatedly checking feasibility of linear programming problems, rendering it computationally attractive. Nonetheless, the program defining an extreme point of the confidence interval is generally nonlinear and potentially intricate. We provide an algorithm, based on the response surface method for global optimization, that approximates the solution rapidly and accurately, and we establish its rate of convergence. The algorithm is of independent interest for optimization problems with simple objectives and complicated constraints. An empirical application estimating an entry game illustrates the usefulness of the method. Monte Carlo simulations confirm the accuracy of the solution algorithm, the good statistical as well as computational performance of calibrated projection (including in comparison to other methods), and the algorithm's potential to greatly accelerate computation of other confidence intervals.

We survey the literature on estimating risk preferences using field data. We concentrate our attention on studies in which risk preferences are the focal object and estimating their structure is the core enterprise. We review a number of models of risk preferences—including both expected utility (EU) theory and non-EU models—that have been estimated using field data, and we highlight issues related to identification and estimation of such models using field data. We then survey the literature, giving separate treatment to research that uses individual-level data (e.g., property-insurance data) and research that uses aggregate data (e.g., betting-market data). We conclude by discussing directions for future research. ( JEL C51, D11, D81, D82, D83, G22, I13)

We use data on insurance deductible choices to estimate a structural model of risky choice that incorporates “standard” risk aversion (diminishing marginal utility for wealth) and probability distortions. We find that probability distortions—characterized by substantial overweighting of small probabilities and only mild insensitivity to probability changes—play an important role in explaining the aversion to risk manifested in deductible choices. This finding is robust to allowing for observed and unobserved heterogeneity in preferences. We demonstrate that neither Kőszegi-Rabin loss aversion alone nor Gul disappointment aversion alone can explain our estimated probability distortions, signifying a key role for probability weighting. (JEL D14, D81, G22)

We outline a strategy for distinguishing rank-dependent probability weighting from systematic risk misperceptions in field data. Our strategy relies on singling out a field environment with two key properties: (i) the objects of choice are money lotteries with more than two outcomes; and (ii) the ranking of outcomes differs across lotteries. We first present an abstract model of risky choice that elucidates the identification problem and our strategy. The model has numerous applications, including insurance choices and gambling. We then consider the application of insurance deductible choices and illustrate our strategy using simulated data.

We propose a robust method of discrete choice analysis when agents' choice sets are unobserved. Our core model assumes nothing about agents' choice sets apart from their minimum size. Importantly, it leaves unrestricted the dependence, conditional on observables, between choice sets and preferences. We first characterize the sharp identification region of the model's parameters by a finite set of conditional moment inequalities. We then apply our theoretical findings to learn about households' risk preferences and choice sets from data on their deductible choices in auto collision insurance. We find that the data can be explained by expected utility theory with low levels of risk aversion and heterogeneous non‐singleton choice sets, and that more than three in four households require limited choice sets to explain their deductible choices. We also provide simulation evidence on the computational tractability of our method in applications with larger feasible sets or higher‐dimensional unobserved heterogeneity.