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 on the entire distribution of untreated potential outcomes. This condition is satisfied if (i) treatment is as-if randomly assigned, (ii) the distribution of potential outcomes is stationary, or (iii) treatment is as-if randomly assigned among a subset of the population and the remainder of the population has stationary potential outcomes. We show further that it is impossible to construct any estimator that is consistent (or unbiased) 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 justify one of the following: (i) why treatment is as-if randomly assigned, (ii) why the chosen functional form is correct at the exclusion of others, or (iii) a method for inferring the entire counterfactual distribution of untreated potential outcomes.