This paper discusses testable implications of rank invariance or rank similarity, assumptions that are common in program evaluation and in the quantile treatment effect (QTE) literature. We nonparametrically identify, estimate, and test the counterfactual distribution of potential ranks, or features of the distribution. The proposed tests allow treatment to be endogenous, with exogenous treatment following as a special case. The tests essentially do not require any additional assumptions other than those to identify and estimate QTEs. We apply the proposed tests to investigate whether the Job Training Partnership Act training causes trainees to systematically change their ranks in the earnings distribution.

Regression discontinuity models are commonly used to nonparametrically identify and estimate a local average treatment effect (LATE).We show that the derivative of the treatment effect with respect to the running variable at the cutoff, referred to as the treatment effect derivative (TED), is nonparametrically identified, easily estimated, and has implications for testing external validity and extrapolating the estimated LATE away from the cutoff. Given a local policy invariance assumption, we further show this TED equals the change in the treatment effect that would result from a marginal change in the threshold, which we call the marginal threshold treatment effect (MTTE). We apply these results to Goodman (2008), who estimates the effect of a scholarship program on college choice. MTTE in this case identifies how this treatment effect would change if the test score threshold to qualify for a scholarship were changed, even though no such change in threshold is actually observed.