I’m excited to share a new working paper that I’ve recently completed, which is joint work with Federico Bugni and Deborah Kim. The paper delves into an important aspect of stochastic dominance: testing conditional stochastic dominance (CSD) at specific values of a conditioning covariate. This concept is crucial in many applied fields, including evaluating treatment effects in social programs, investigating economic disparities, and exploring potential discrimination in decision-making processes.

What’s New?

In this paper, we focus on the problem of testing whether the conditional cumulative distribution function (CDF) of one variable stochastically dominates another at specific values of a conditioning variable. Formally, we test the null hypothesis:


H_0: F_Y(t | z) ≤ F_X(t | z) for all (t, z) ∈ R × Z


against the alternative hypothesis:


H_1: F_Y(t | z) > F_X(t | z) for some (t, z) ∈ R × Z


The target points (denoted by 𝒵) are a finite set of values, not the entire support of the conditioning variable Z, and the paper focuses on the case where 𝒵 consists of a small number of specific points.

Key Contributions

• A Novel Test for CSD: The primary contribution of this paper is the introduction of a novel test statistic based on induced order statistics. The test uses empirical CDFs, leveraging observations closest to the target points. Unlike traditional tests, our method does not require kernel smoothing or parametric assumptions on the conditional distributions, ensuring computational simplicity.
• Asymptotic Properties: we establish the asymptotic validity of the proposed test, showing that it controls size in large samples under mild regularity conditions. This result is significant as many existing methods assume continuous conditional distributions, but our approach is more flexible and accommodates finitely many discontinuities in the distributions.
• Connection to Permutation-Based Inference: we show that, when the random variables Y and X are both continuous, the critical value for our test coincides with that of a permutation-based test, thereby establishing a formal connection between our method and the broader literature on rank-based inference.
• Refinement for Discrete Data: For cases where Y or X is discrete, we introduce a refined critical value that enhances power, albeit with increased computational complexity.

What’s Next?

The remainder of the paper explores the theory, provides extensions, and discusses practical implementation through Monte Carlo simulations. The methods developed here have implications for fields ranging from economics to political science and public policy, offering a robust and computationally efficient approach to testing stochastic dominance. If this sounds interesting, please read the paper.