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

When Is Parallel Trends Sensitive to Functional Form?