Jörg Stoye

Axioms for minimax regret choice correspondences

This paper unifies and extends the recent axiomatic literature on minimax regret. It compares several models of minimax regret, shows how to characterize the according choice correspondences in a unified setting, extends one of them to choice from convex (through randomization) sets, and connects them by defining a behavioral notion of perceived ambiguity. Substantively, a main idea is to behaviorally identify ambiguity with failures of independence of irrelevant alternatives. Regarding proof technique, the core contribution is to uncover a dualism between choice correspondences and preferences in an environment where this dualism is not obvious. This insight can be used to generate results by importing findings from the existing literature on preference orderings.

Revealed Preferences in a Heterogeneous Population

This paper explores the empirical content of the weak axiom of revealed preference (WARP) for repeated cross-sections. In a heterogeneous population, the fraction of consumers who violate WARP is not point identified but can be bounded. These bounds, as well as some nonparametric refinements, correspond to intuitive behavioral assumptions if there are two goods. With three or more goods, such intuitions break down, and plausible assumptions can have counterintuitive implications. We also provide estimators and confidence regions. The empirical application reveals that in the British Family Expenditure Survey, upper bounds are frequently positive but lower bounds are not significantly so.

Computation of bounds on population parameters when the data are incomplete

This paper continues our research on the identification and estimation of statistical functionals when the sampling process produces incomplete data due to missing observations or interval measurement of variables. Incomplete data usually cause population parameters of interest in applications to be unidentified except under untestable and often controversial assumptions. However, it is often possible to identify sharp bounds on these parameters. The bounds are functionals of the population distribution of the available data and do not rely on untestable assumptions about the process through which data become incomplete. They contain all logically possible values of the population parameters. Moreover, every parameter value within the bounds is consistent with some model of the process that generates incomplete data. The bounds can be estimated consistently by replacing the population distribution of the data with the empirical distribution in the functionals that give the bounds. In practice, this is straightforward in some circumstances but computationally burdensome in others; in general, the bounds are the solutions to non-convex mathematical programming problems that can be difficult to solve. Horowitz and Manski (Censoring of Outcomes and Regressors Due to Survey Nonresponse: Identification and Estimation Using Weights and Imputations, Journal of Econometrics84 (1998), pp. 37–58; Nonparametric Analysis of Randomized Experiments with Missing Covariate and Outcome Data, Journal of the American Statistical Association95 (2000), pp. 77–84) studied nonparametric mean regression with missing data. In this paper, we first describe the general problem. We then present new findings on the computation of bounds on best linear predictors under square loss. We describe a genetic algorithm to compute sharp bounds and a min-imax approach yielding simple but non-sharp outer bounds. We use actual data to demonstrate the computations.

Partial identification of spread parameters

This paper analyzes partial identification of parameters that measure a distribu- tion’s spread, for example, the variance, Gini coefficient, entropy, or interquartile range. The core results are tight, two-dimensional identification regions for the ex- pectation and variance, the median and interquartile ratio, and many other com- binations of parameters. They are developed for numerous identification settings, including but not limited to cases where one can bound either the relevant cumu- lative distribution function or the relevant probability measure. Applications in- clude missing data, interval data, “short” versus “long” regressions, contaminated data, and certain forms of sensitivity analysis. The application to missing data is worked out in some detail, including closed-form worst-case bounds on some pa- rameters as well as improved bounds that rely on nonparametric restrictions on selection effects. A brief empirical application to bounds on inequality measures is provided. The bounds are very easy to compute. The ideas underlying them are explained in detail and should be readily extended to even more settings than are explicitly discussed. Keywords. Partial identification, nonparametric bounds, missing data, sensitiv- ity analysis, variance, inequality. JEL classification. C14, C24.

MINIMAX REGRET TREATMENT CHOICE WITH INCOMPLETE DATA AND MANY TREATMENTS

This note adds to the recent research project on treatment choice under ambiguity. I generalize the Manski (Journal of Econometrics, in press) analysis of minimax regret treatment choice by considering a more general setting and, more importantly, by solving for the treatment rule given finitely many (as opposed to two) treatments. The most interesting finding is that with three or more undominated treatments, the minimax regret treatment rule may assign the same treatment to all subjects; thus, the most salient feature of the two-treatment case does not generalize.I thank Chuck Manski, the co-editor, and especially an anonymous referee for helpful comments. Financial support through the Robert Eisner Memorial Fellowship, Department of Economics, Northwestern University, in addition to a Dissertation Year Fellowship, The Graduate School, Northwestern University, is gratefully acknowledged.

Bounds on Generalized Linear Predictors with Incomplete Outcome Data

This paper develops easily computed, tight bounds on Generalized Linear Predictors and instrumental variable estimators when outcome data are partially identified. A salient example is given by Best Linear Predictors under square loss, or Ordinary Least Squares regressions, with missing outcome data, in which case the setup specializes the more general but intractable problem examined by Horowitz et al. [9]. The result is illustrated by re-analyzing the data used in that paper.

Partial Identification and Robust Treatment Choice: An Application to Young Offenders

This paper applies recently developed methods to robust assessment of treatment outcomes and robust treatment choice based on nonexperimental data. The substantive question is whether young offenders should be assigned to residential or nonresidential treatment in order to prevent subsequent recidivism. A large data set on past offenders exists, but treatment assignment was by judges and not by experimenters, hence counterfactual outcomes are not identified unless one imposes strong assumptions.The analysis is carried out in two steps. First, I show how to compute identified bounds on expected outcomes under various assumptions that are too weak to restore conventional identification but may be accordingly credible. The bounds are estimated, and confidence regions that take current theoretical developments into account are computed. I then ask which treatment to assign to future offenders if the identity of the best treatment will not be learned from the data. This is a decision problem under ambiguity. I characterize and compute decision rules that are asymptotically efficient under the minimax regret criterion. The substantive conclusion is that both bounds and recommended decisions vary significantly across the assumptions. The data alone do not permit conclusions or decisions that are globally robust in the sense of holding uniformly over reasonable assumptions.

Choice theory when agents can randomize

This paper takes choice theory to risk or uncertainty. Well-known decision models are axiomatized under the premise that agents can randomize. Under a reversal of order assumption, this convexifies choice sets, and even after imposing the weak axiom of revealed preference and nonemptiness of choice correspondences, the preferences directly revealed by choice may be incomplete or cyclical.

Revealed Price Preference: Theory and Stochastic Testing

We develop a model of demand where consumers trade-off the utility of consumption against the disutility of expenditure. This model is appropriate whenever a consumer’s demand over a strict subset of all available goods is being analyzed. Data sets consistent with this model are characterized by the absence of revealed preference cycles over prices. The model is readily generalized to the random utility setting, for which we develop nonparametric statistical tests. Our application on national household consumption data provides support for the model.