  This article extends the widely-used synthetic controls estimator for
evaluating causal effects of policy changes to quantile functions. The proposed
method provides a geometrically faithful estimate of the entire counterfactual
quantile function of the treated unit. Its appeal stems from an efficient
implementation via a constrained quantile-on-quantile regression. This
constitutes a novel concept of independent interest. The method provides a
unique counterfactual quantile function in any scenario: for continuous,
discrete or mixed distributions. It operates in both repeated cross-sections
and panel data with as little as a single pre-treatment period. The article
also provides abstract identification results by showing that any synthetic
controls method, classical or our generalization, provides the correct
counterfactual for causal models that preserve distances between the outcome
distributions. Working with whole quantile functions instead of aggregate
values allows for tests of equality and stochastic dominance of the
counterfactual- and the observed distribution. It can provide causal inference
on standard outcomes like average- or quantile treatment effects, but also more
general concepts such as counterfactual Lorenz curves or interquartile ranges.
