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This paper shows that the Conditional Quantile Treatment Effect on theTreated can be identified using a combination of (i) a conditionalDistributional Difference in Differences assumption and (ii) an assumption onthe conditional dependence between the change in untreated potential outcomesand the initial level of untreated potential outcomes for the treated group.The second assumption recovers the unknown dependence from the observeddependence for the untreated group. We also consider estimation and inferencein the case where all of the covariates are discrete. We propose a uniforminference procedure based on the exchangeable bootstrap and show its validity.We conclude the paper by estimating the effect of state-level changes in theminimum wage on the distribution of earnings for subgroups defined by race,gender, and education.

Quantile Treatment Effects in Difference in Differences Models under Dependence Restrictions and with only Two Time Periods