Inferring Inequality: Testing for Median-Preserving Spreads in Ordinal Data

The median-preserving spread (MPS) ordering for ordinal variables (Allison and
Foster, 2004) has become ubiquitous in the inequality literature. However, the literature lacks an explicit frequentist method for inferring whether an ordered multinomial distribution G is more unequal than F according to the MPS criterion. We devise formal statistical tests of the hypothesis that G is not an MPS of F. Rejection of this hypothesis enables the conclusion that G is robustly more unequal than F. Using Monte Carlo simulations and novel graphical techniques, we nd that the choice between Z and Likelihood Ratio test statistics does not have a large impact on the properties of the tests, but that the method of inference does: bootstrap inference has generally better size and power properties than asymptotic inference. We illustrate the usefulness of our tests with three applications: (i) happiness inequality in the United States, (ii) self-assessed health in Europe and (iii) sanitation ladders in Pakistan.