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A Joint Analysis of Sensitivity to Anticipation and Parallel Trends Violations

Gianna Fenaroli

Abstract

Two key identifying assumptions used to justify difference-in-differences are parallel trends and no anticipation, yet both may fail in practice. I propose a class of assumptions on anticipation and derive closed-form, sharp bounds on the average treatment effect on the treated while simultaneously relaxing parallel trends. Deviations from both assumptions are jointly disciplined using observed pre-trends. When some anticipation is imposed, the identified set under joint deviations can be shorter than under parallel trends violations alone. These bounds inform a sensitivity analysis assessing the robustness of qualitative conclusions to anticipation and parallel trends violations. I illustrate with an empirical application.

A Joint Analysis of Sensitivity to Anticipation and Parallel Trends Violations

Abstract

Two key identifying assumptions used to justify difference-in-differences are parallel trends and no anticipation, yet both may fail in practice. I propose a class of assumptions on anticipation and derive closed-form, sharp bounds on the average treatment effect on the treated while simultaneously relaxing parallel trends. Deviations from both assumptions are jointly disciplined using observed pre-trends. When some anticipation is imposed, the identified set under joint deviations can be shorter than under parallel trends violations alone. These bounds inform a sensitivity analysis assessing the robustness of qualitative conclusions to anticipation and parallel trends violations. I illustrate with an empirical application.
Paper Structure (19 sections, 6 theorems, 42 equations, 4 figures, 1 table)

This paper contains 19 sections, 6 theorems, 42 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Outline of Paper
  4. Setup
  5. Identification
  6. Parameter of Interest and Decomposition
  7. Assumptions on Anticipation and Parallel Trends Violations
  8. Results
  9. Discussion
  10. Practical Implications for Sensitivity Analysis
  11. Overview
  12. Deriving the Breakdown Frontier
  13. Estimation and Inference
  14. Empirical Illustration: Private School Supply
  15. Background
  16. ...and 4 more sections

Key Result

Lemma 1

A general expression for the $\text{ATT}_{1}$ in terms of anticipation effects $(\varphi_0)$ and post-treatment parallel trends violations $(\delta_1)$ is

Figures (4)

  • Figure 1: Estimated Breakdown Frontiers for $\text{ATT}_{2008} < 0$ with Fixed $\overline{p}$
  • Figure 2: Estimated Breakdown Frontier for $\text{ATT}_{2008} < 0$ - Singleton Cases
  • Figure 3: Estimated Identified Sets for the $\text{ATT}_{2008}$
  • Figure 4: Estimated Breakdown Frontier for $\text{ATT}_{2008} > -0.1$ and Identified Sets

Theorems & Definitions (7)

  • Lemma 1
  • Corollary 1
  • Corollary 2
  • Theorem 1
  • Corollary 3
  • Theorem 2
  • Remark 1