When Is Parallel Trends Sensitive to Functional Form?

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.

Automated summary

This paper examines the impact of functional form on the validity of difference-in-differences. The study establishes that the parallel trends assumption holds under all strictly monotonic transformations of the outcome if a stronger parallel trends-type condition is satisfied for the cumulative distribution function of untreated potential outcomes. The condition for insensitivity of parallel trends to functional form is met when the population can be divided into a subgroup with effectively random treatment assignment and another subgroup with a stable distribution of untreated potential outcomes. The paper introduces falsification tests to assess whether parallel trends are insensitive to functional form.