Selection and parallel trends

TL;DR

This paper analyzes how selection into treatment affects the validity of difference-in-differences (DiD) estimates, providing conditions to assess parallel trends assumptions and methods to benchmark bias in empirical studies.

Contribution

It derives necessary and sufficient conditions for parallel trends under various selection mechanisms, enhancing understanding of DiD assumptions and bias decomposition.

Findings

Provides a selection-based bias decomposition for DiD

Offers strategies for benchmarking DiD bias in applications

Reanalyzes case studies demonstrating the approach

Abstract

We study the role of selection into treatment in difference-in-differences (DiD) designs. We derive necessary and sufficient conditions for parallel trends assumptions under general classes of selection mechanisms. These conditions characterize the empirical content of parallel trends and clarify the trade-offs between assumptions about selection into treatment and restrictions on the time series properties of the potential outcomes required for DiD methods. We use the necessary and sufficient conditions to provide a selection-based decomposition of the bias of DiD and provide easy-to-implement strategies for benchmarking its components. We also provide templates for justifying DiD in applications with and without covariates. Reanalyses of the causal effect of NSW training programs and the effect of the Medicaid expansion demonstrate the usefulness of our selection-based approach to benchmarking the bias of DiD.