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.