A Test for Rank Similarity and Partial Identification of the Distribution of Treatment Effects Preliminary and incomplete

We introduce a test of the rank invariance or rank similarity assumption common in treatment effects and instrumental variables models. The test probes the implication that the conditional distribution of ranks should be identical across treatment states using a regression-based test statistic. When justified by the test, natural extensions of rank similarity partially identify the distribution of treatment effects. The resulting bounds are substantially tighter than those based on the classical Frechet-Hoeffding bounds. We apply the test to school classroom interventions, and show that systematic slippages in rank can be important statistically and economically, and that when rank similarity is plausible, the implied bounds are sharp enough to rule out economically and policy-relevant hypotheses such as harm to students at the bottom of the distribution.