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Testing Conditional Stochastic Dominance at Target Points

Federico A. Bugni, Ivan A. Canay, Deborah Kim

TL;DR

This work develops a nonparametric Kolmogorov--Smirnov-type test for conditional stochastic dominance at target points, using induced order statistics to form empirical CDFs conditional on a nearby covariate value. The test computes a data-independent critical value from a KS limit object, avoiding kernel smoothing and resampling, and relies on a tuning parameter that selects the observations closest to the target point. The authors establish asymptotic validity under two frameworks (fixed and growing local sample sizes) and show that, in the limit, the procedure reduces to an unconditional stochastic dominance testing problem, enabling a connection to permutation-based inference for continuous data. They further refine the method for discrete data with a refined critical value to boost power, demonstrate strong finite-sample performance via Monte Carlo simulations, and illustrate applicability with an empirical example on retirement effects. The approach offers a simple, kernel-free alternative for CSD testing with broad applicability in economics and policy evaluation, while leaving open questions about permutation validity when both variables are discrete.

Abstract

This paper introduces a novel test for conditional stochastic dominance (CSD) at specific values of the conditioning covariates, referred to as target points. The test is relevant for analyzing income inequality, evaluating treatment effects, and studying discrimination. We propose a Kolmogorov--Smirnov-type test statistic that utilizes induced order statistics from independent samples. Notably, the test features a data-independent critical value, eliminating the need for resampling techniques such as the bootstrap. Our approach avoids kernel smoothing and parametric assumptions, instead relying on a tuning parameter to select relevant observations. We establish the asymptotic properties of our test, showing that the induced order statistics converge to independent draws from the true conditional distributions and that the test is asymptotically of level $α$ under weak regularity conditions. While our results apply to both continuous and discrete data, in the discrete case, the critical value only provides a valid upper bound. To address this, we propose a refined critical value that significantly enhances power, requiring only knowledge of the support size of the distributions. Additionally, we analyze the test's behavior in the limit experiment, demonstrating that it reduces to a problem analogous to testing unconditional stochastic dominance in finite samples. This framework allows us to prove the validity of permutation-based tests for stochastic dominance when the random variables are continuous. Monte Carlo simulations confirm the strong finite-sample performance of our method.

Testing Conditional Stochastic Dominance at Target Points

TL;DR

This work develops a nonparametric Kolmogorov--Smirnov-type test for conditional stochastic dominance at target points, using induced order statistics to form empirical CDFs conditional on a nearby covariate value. The test computes a data-independent critical value from a KS limit object, avoiding kernel smoothing and resampling, and relies on a tuning parameter that selects the observations closest to the target point. The authors establish asymptotic validity under two frameworks (fixed and growing local sample sizes) and show that, in the limit, the procedure reduces to an unconditional stochastic dominance testing problem, enabling a connection to permutation-based inference for continuous data. They further refine the method for discrete data with a refined critical value to boost power, demonstrate strong finite-sample performance via Monte Carlo simulations, and illustrate applicability with an empirical example on retirement effects. The approach offers a simple, kernel-free alternative for CSD testing with broad applicability in economics and policy evaluation, while leaving open questions about permutation validity when both variables are discrete.

Abstract

This paper introduces a novel test for conditional stochastic dominance (CSD) at specific values of the conditioning covariates, referred to as target points. The test is relevant for analyzing income inequality, evaluating treatment effects, and studying discrimination. We propose a Kolmogorov--Smirnov-type test statistic that utilizes induced order statistics from independent samples. Notably, the test features a data-independent critical value, eliminating the need for resampling techniques such as the bootstrap. Our approach avoids kernel smoothing and parametric assumptions, instead relying on a tuning parameter to select relevant observations. We establish the asymptotic properties of our test, showing that the induced order statistics converge to independent draws from the true conditional distributions and that the test is asymptotically of level under weak regularity conditions. While our results apply to both continuous and discrete data, in the discrete case, the critical value only provides a valid upper bound. To address this, we propose a refined critical value that significantly enhances power, requiring only knowledge of the support size of the distributions. Additionally, we analyze the test's behavior in the limit experiment, demonstrating that it reduces to a problem analogous to testing unconditional stochastic dominance in finite samples. This framework allows us to prove the validity of permutation-based tests for stochastic dominance when the random variables are continuous. Monte Carlo simulations confirm the strong finite-sample performance of our method.

Paper Structure

This paper contains 25 sections, 13 theorems, 133 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Testing problem
  3. Test based on induced ordered statistics
  4. Asymptotic framework and formal results
  5. Asymptotic results for fixed q
  6. Asymptotic results for large q
  7. Discussion and extensions
  8. Data-dependent choice of tuning parameters
  9. Refined critical value for discrete data
  10. Properties of our test in the limit experiment
  11. Simulations
  12. Concluding remarks
  13. Proof of the main results
  14. Proof of Theorem \ref{['thm:SD-AsySz']}
  15. Proof of Lemma \ref{['lem:limit-exact']}
  16. ...and 10 more sections

Key Result

Theorem 1

Let Assumptions ass:Z and ass:cond-continous hold. Then, where for any $s:=(s_1,\dots,s_{q})\in \mathbf{R}^{q}$, the random vector $S$ satisfies

Figures (3)

  • Figure 1: Rejection rates under $H_0$ for Designs 1-4 in cases (a)-(c): $n=1,000$, $\alpha=10\%$, $MC=10,000$.
  • Figure 2: Rejection rates under $H_0$ for Designs 5-7 in cases (a)-(c): $n=1,000$, $\alpha=10\%$, $MC=10,000$.
  • Figure 3: Rejection rates under $H_1$ for Designs 1-7: $n=1,000$, $\alpha=10\%$, $MC=10,000$.

Theorems & Definitions (29)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Theorem 2
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Lemma 1
  • ...and 19 more