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Partial identification of spread parameters


  • Jörg Stoye

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    • I am greatly indebted to Chuck Manski for his guidance and feedback on this project. For useful comments, I thank Elie Tamer and three extremely helpful referees as well as Adeline Delavande, Joel Horowitz, Peter Klibanoff, Francesca Molinari, seminar participants at Cal Tech, Chicago, Mannheim, NYU, Northwestern, Wisconsin–Madison, and conference audiences at Carlos III Madrid, Columbia, and Nürnberg-Erlangen universities. Steven Laufer provided able research assistance. Of course, any and all errors are mine. I gratefully acknowledge financial support through the Robert Eisner Memorial Fellowship, a Dissertation Year Fellowship, and a Graduate Research Travel Grant, all from Northwestern University, as well as through a University Research Challenge Grant from New York University.


This paper analyzes partial identification of parameters that measure a distribution's spread, for example, the variance, Gini coefficient, entropy, or interquartile range. The core results are tight, two-dimensional identification regions for the expectation and variance, the median and interquartile ratio, and many other combinations of parameters. They are developed for numerous identification settings, including but not limited to cases where one can bound either the relevant cumulative distribution function or the relevant probability measure. Applications include 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 parameters 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.