  We propose a nonlinear difference-in-differences method to estimate
multivariate counterfactual distributions in classical treatment and control
study designs with observational data. Our approach sheds a new light on
existing approaches like the changes-in-changes and the classical
semiparametric difference-in-differences estimator and generalizes them to
settings with multivariate heterogeneity in the outcomes. The main benefit of
this extension is that it allows for arbitrary dependence and heterogeneity in
the joint outcomes. We demonstrate its utility both on synthetic and real data.
In particular, we revisit the classical Card \& Krueger dataset, examining the
effect of a minimum wage increase on employment in fast food restaurants; a
reanalysis with our method reveals that restaurants tend to substitute
full-time with part-time labor after a minimum wage increase at a faster pace.
A previous version of this work was entitled "An optimal transport approach to
causal inference.
