  This paper assesses when the validity of difference-in-differences depends on
functional form. We provide a novel characterization: the parallel trends
assumption holds under all strictly monotonic transformations of the outcome if
and only if a stronger ``parallel trends''-type condition holds for the
cumulative distribution function of untreated potential outcomes. This
condition for parallel trends to be insensitive to functional form is satisfied
if and essentially only if the population can be partitioned into a subgroup
for which treatment is effectively randomly assigned and a remaining subgroup
for which the distribution of untreated potential outcomes is stable over time.
These conditions have testable implications, and we introduce falsification
tests for the null that parallel trends is insensitive to functional form.
