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