KOLMOGOROV SENSITIVITY BOUNDS FOR A MEDIAN TREATMENT CONTRAST: A CLOSED-FORM SHARP INTERVAL

Abstract

Motivated by sensitivity analysis in difference-in-differences, we study a median treatment contrast when the unobserved counterfactual cumulative distribution function (CDF) is allowed to lie within Kolmogorov distance \(M\) of a reference CDF. This restriction is distinct from the usual distributional parallel-trends assumption on untreated outcome changes, but it gives a transparent local sensitivity model around any chosen reference distribution. The identified set for the difference of marginal medians is a closed-form interval whose endpoints are quantiles of the reference distribution evaluated at > \(1 / 2 \pm M\). We give the joint asymptotic distribution of the plug-in endpoints and an Imbens-Manski-type confidence interval with pointwise asymptotic validity for fixed \(M\). The construction is computationally trivial, requires no bootstrap, and reduces to the standard quantile-based confidence interval for a difference of medians when \(M=0\). A Monte Carlo study verifies the variance formula numerically and assesses finite-sample coverage.