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Difference-in-Discontinuities: Estimation, Inference and Validity Tests

Pedro Picchetti, Cristine C. X. Pinto, Stephanie T. Shinoki

TL;DR

The Difference-in-Discontinuities design ($DiDC$) addresses limitations of both RD and DiD by exploiting temporal and threshold-based variation. The authors develop a nonparametric local polynomial estimator on the time-differenced outcomes, derive identification under continuity and sharp assignment, and provide robust bias-corrected inference along with validity tests. They further offer partial identification and sensitivity analysis under bounded variation and modularity assumptions, supported by Monte Carlo simulations and an empirical illustration (Grembi et al. 2016) that demonstrates gains in bias reduction and improved coverage under favorable conditions, while highlighting the importance of testing time-invariance of confounding effects. Overall, the work extends the causal-inference toolkit for settings with threshold rules and time-varying confounding, offering practical guidance on estimation, inference, and robustness.

Abstract

This paper provides a formal econometric framework behind the newly developed difference-in-discontinuities design (DiDC). Despite its increasing use in applied research, there are currently limited studies of its properties. We formalize the theory behind the difference-in-discontinuity approach by stating the identification assumptions, proposing a nonparametric estimator, and deriving its asymptotic properties. We also provide comprehensive tests for one of the identification assumption of the DiDC and sensitivity analysis methods that allow researchers to evaluate the robustness of DiDC estimates under violations of the identifying assumptions. Monte Carlo simulation studies show that the estimators have desirable finite-sample properties. Finally, we revisit Grembi et al. (2016), which studies the effects of relaxing fiscal rules on public finance outcomes. Our results show that most of the qualitative takeaways of the original work are robust to time-varying confounding effects.

Difference-in-Discontinuities: Estimation, Inference and Validity Tests

TL;DR

The Difference-in-Discontinuities design () addresses limitations of both RD and DiD by exploiting temporal and threshold-based variation. The authors develop a nonparametric local polynomial estimator on the time-differenced outcomes, derive identification under continuity and sharp assignment, and provide robust bias-corrected inference along with validity tests. They further offer partial identification and sensitivity analysis under bounded variation and modularity assumptions, supported by Monte Carlo simulations and an empirical illustration (Grembi et al. 2016) that demonstrates gains in bias reduction and improved coverage under favorable conditions, while highlighting the importance of testing time-invariance of confounding effects. Overall, the work extends the causal-inference toolkit for settings with threshold rules and time-varying confounding, offering practical guidance on estimation, inference, and robustness.

Abstract

This paper provides a formal econometric framework behind the newly developed difference-in-discontinuities design (DiDC). Despite its increasing use in applied research, there are currently limited studies of its properties. We formalize the theory behind the difference-in-discontinuity approach by stating the identification assumptions, proposing a nonparametric estimator, and deriving its asymptotic properties. We also provide comprehensive tests for one of the identification assumption of the DiDC and sensitivity analysis methods that allow researchers to evaluate the robustness of DiDC estimates under violations of the identifying assumptions. Monte Carlo simulation studies show that the estimators have desirable finite-sample properties. Finally, we revisit Grembi et al. (2016), which studies the effects of relaxing fiscal rules on public finance outcomes. Our results show that most of the qualitative takeaways of the original work are robust to time-varying confounding effects.
Paper Structure (42 sections, 17 theorems, 122 equations, 26 tables)

This paper contains 42 sections, 17 theorems, 122 equations, 26 tables.

Table of Contents

  1. Introduction
  2. Identification
  3. Framework
  4. The DiDC Estimand
  5. Relaxing the No-Interaction Assumption
  6. Estimation and Inference for the Sharp DiDC
  7. Local Polynomial Estimator
  8. Bandwidth choice
  9. Inference
  10. Bias Correction
  11. Asymptotic Properties and Robust Confidence Interval
  12. Validity Tests
  13. Testing if Confounding Effect is Time-Invariant
  14. Testing the Time-Invariance of Potential Outcomes Functional Form
  15. Partial Identification and Sensitivity Analysis
  16. ...and 27 more sections

Key Result

Lemma 1

Under Assumptions assump:continuity, assump:random and assump:disc_prob_treat, the difference of RDs, and the RD of the differences are equivalent:

Theorems & Definitions (35)

  • Lemma 1
  • proof
  • Lemma 2: grembi2016fiscal
  • Corollary 1
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 25 more