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

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.