Distributional Change in Ordinal Data with Missing Observations: Minimal Mobility and Partial Identification

TL;DR

This paper introduces a framework for analyzing distributional changes in ordinal data with missing observations, using optimal transport and partial identification to measure discrepancy and assess sensitivity.

Contribution

It develops a novel approach combining optimal transport and partial identification to analyze distributional change in ordinal data with missing observations.

Findings

Provides sharp bounds on marginal distributions under missing data.

Introduces minimal-mobility configurations for distributional change.

Supports inference with standard resampling methods.

Abstract

Empirical analyses of ordinal outcomes using repeated cross-sectional data rely on marginal distributions, leaving the joint distribution unobserved and the sources of distributional change unidentified. This paper develops a framework to measure and interpret such changes under limited information. The $L_1$ distance between cumulative distribution functions admits an optimal transport representation as the minimal reallocation of probability mass across ordered categories, which provides a foundation for the analysis. This yields both a scalar measure of discrepancy and a structured characterization of how distributional change must occur, which I term minimal-mobility configurations. To address missing data, I adopt a partial identification approach that delivers sharp bounds on the marginal distributions and, in turn, on both the discrepancy measure and its associated configurations. The resulting framework supports inference using standard resampling methods and provides a transparent basis for assessing sensitivity to nonresponse. An application to Arab Barometer data illustrates the approach.