When Is Parallel Trends Sensitive to Functional Form?

This paper assesses when the validity of difference-in-differences and related estimators 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. We introduce falsification tests for the insensitivity of parallel trends to functional form. We also show that it is impossible to construct any estimator that is consistent for the average treatment effect on the treated (ATT) without either imposing functional form restrictions or imposing assumptions that identify the full distribution of untreated potential outcomes. Our results suggest that researchers who wish to point-identify the ATT should either (i) argue treatment is as-if randomly assigned, (ii) provide a method for inferring the full counterfactual distribution for the treated group, or (iii) justify the validity of the specific chosen functional form.