When Is Parallel Trends Sensitive to Functional Form?

TL;DR

This paper explores when the difference-in-differences method's validity depends on the functional form of the outcome, providing a new characterization and testable conditions for insensitivity to functional form.

Contribution

It introduces a novel characterization of when parallel trends are invariant to functional form transformations and proposes falsification tests for this condition.

Findings

Parallel trends hold under all monotonic transformations if a specific distributional condition is met.

The condition is satisfied if the population can be divided into a randomly treated subgroup and a stable subgroup.

Falsification tests are developed to assess the insensitivity of parallel trends to functional form.

Abstract

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.