Bounds on inequality with incomplete data

TL;DR

This paper introduces a nonparametric framework for deriving sharp bounds on inequality measures with incomplete data, enabling practical inference and optimization for various inequality indices.

Contribution

It develops a unified approach to partial identification of inequality indices with coarsened data, including computational methods and inference procedures.

Findings

Sharp bounds for Gini and quantile ratios are computed from incomplete data.

Finite support distributions attain the bounds, simplifying optimization.

Bootstrap methods provide confidence intervals for the bounds.

Abstract

We develop a unified nonparametric framework for sharp partial identification and inference on inequality indices when the data contain coarsened observations of the variable of interest. We characterize the extremal allocations for all Schur-convex inequality measures, and show that sharp bounds are attained by distributions with finite support. This reduces the computational problem to finite-dimensional optimization, and for indices admitting linear-fractional representations after suitable ordering of the data (including the Gini coefficient and quantile ratios), we express the bound problems as linear or quadratic programs. We then establish $\sqrt{n}$ inference for the upper and lower bounds using a directional delta method and bootstrap confidence intervals. In applications, we compute sharp Gini bounds from household wealth data with mixed point and interval observations and use historical U.S. grouped income tables to bound time series for the Gini and quantile ratios.