Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounds on inequality with incomplete data

James Banks, Thomas Glinnan, Tatiana Komarova

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.

Bounds on inequality with incomplete data

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 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.

Paper Structure

This paper contains 17 sections, 20 theorems, 41 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Data Scenarios and Inequality Indices
  3. Computation in Scenario 1
  4. Charnes-Cooper transformation for optimizing linear-fractional inequality indices
  5. Solution form for linear-fractional inequality indices
  6. General results for Schur-convex indices
  7. Computation in Scenario 2
  8. Asymptotics of sharp bounds
  9. Example Applications
  10. Household level wealth data
  11. Income inequality in the U.S. from historical data
  12. Conclusion
  13. Appendix
  14. Proofs of results in Section \ref{['sec:GeneralTheory']}
  15. Proofs of results in Section \ref{['sec:asymptotics']}
  16. ...and 2 more sections

Key Result

Proposition 1

With only constraints ordering and LF3, an optimal solution $\mathbf{y}^*$ to LF1 lies at the interval boundaries, i.e. $y^*_i \in \{\underline{a}_{d(i)}, \overline{a}_{d(i)}\}$, where $d(i)$ is the group index of $\mathcal{G}_{d(i)}=[\underline{a}_{d(i)}, \overline{a}_{d(i)}]$ containing $y_i$. Ad

Figures (5)

  • Figure 1: Bootstrap distributions for the Gini inequality index (narrow savings definition)
  • Figure 2: Sharp bounds on Gini inequality for U.S. historical data.
  • Figure 3: Sharp bounds on 90/50 quantile ratio for U.S. historical data.
  • Figure A.1: Example for Scenario 1A: 1945 income distribution tables from Census1948P60-02.
  • Figure A.2: Example for Scenario 1B: income tables from the U.S. Department of Commerce, Office of Business Economics, U.S. Income and Output (1958) OBE1958USIncomeOutput.

Theorems & Definitions (22)

  • Definition 1: Scenario 1
  • Definition 2: Scenario 2
  • Proposition 1: Scenario 1A
  • Proposition 2: Scenario 1B
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 3
  • Lemma 1
  • Proposition 4
  • ...and 12 more