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Cited by 1,559Advanced Causal Inference TechniquesSpatial and Panel Data AnalysisMonetary Policy and Economic Impact

The DOI matches its Crossref record; 94 of the 94 references it cites were found in a registry or an index; 18 of the 66 sources its claims cite were checked for support.

A More Credible Approach to Parallel Trends

Rambachan, Ashesh, Roth, Jonathan·2023·Review of Economic Studies
DOI 10.1093/restud/rdad018
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Abstract

Abstract This paper proposes tools for robust inference in difference-in-differences and event-study designs where the parallel trends assumption may be violated. Instead of requiring that parallel trends holds exactly, we impose restrictions on how different the post-treatment violations of parallel trends can be from the pre-treatment differences in trends (“pre-trends”). The causal parameter of interest is partially identified under these restrictions. We introduce two approaches that guarantee uniformly valid inference under the imposed restrictions, and we derive novel results showing that they have desirable power properties in our context. We illustrate how economic knowledge can inform the restrictions on the possible violations of parallel trends in two economic applications. We also highlight how our approach can be used to conduct sensitivity analyses showing what causal conclusions can be drawn under various restrictions on the possible violations of the parallel trends assumption.

Overview

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Written from the paper’s full text.

TL;DR

This paper develops partial-identification and robust inference methods for difference-in-differences and event-study designs when parallel trends may fail, by restricting how post-treatment trend violations relate to observed pre-trends. It proposes moment-inequality and fixed-length confidence interval approaches, derives uniform validity and power properties, and illustrates sensitivity analysis in two empirical applications.

Key findings

  • The causal parameter is partially identified under researcher-specified restrictions on the difference in trends, such as relative-magnitudes bounds \(RM(M)\) and smoothness/second-difference bounds \(SD(M)\).
  • The conditional and hybrid moment-inequality tests are uniformly valid and consistent; under a linear independence constraint qualification, the conditional test attains optimal local asymptotic power, and the hybrid test is near-optimal.
  • Fixed-length confidence intervals are near-optimal in finite samples for convex, centrosymmetric restrictions such as \(SD(M)\); at \(\alpha=0.05\), the shortest possible valid confidence set is at most 28% shorter than the optimal FLCI under the stated conditions.
  • Simulations calibrated to 12 event-study papers show FLCIs perform best for \(SD(M)\), especially when \(M\) is small, while conditional/hybrid methods perform well across generic restrictions; for \(RM(M)\), efficiency ratios approach about 83% when the pre-treatment pulse grows large even though LICQ fails. In applications, a VAT-cut effect for 2009 has robust interval \([0.07,0.31]\) under \(M=1\), with a null-effect breakdown around \(M=2\); for duty-to-bargain laws, the male employment effect breaks down at \(M \approx 0.01\).

Methods

  • The paper models event-study coefficients \(\hat\beta\) as consistent for \(\beta=\tau+\delta\), where \(\tau\) is the causal effect and \(\delta\) is the differential trend, then imposes restrictions \(\delta \in \Delta\) to partially identify a linear target \(\theta=l'\tau_{post}\).
  • For general polyhedral restrictions, inference is recast as testing moment inequalities with linear nuisance parameters, using the conditional and hybrid approaches of Andrews, Roth, and Pakes; for special convex restrictions, optimal fixed-length confidence intervals are constructed from affine estimators.
  • Simulations use estimated covariance matrices from 12 papers surveyed in Roth (2022), with 1,000 simulations per paper; empirical illustrations reanalyze Benzarti and Carloni’s French VAT study and Lovenheim and Willen’s duty-to-bargain study.

Limitations

  • The paper states that FLCIs are inconsistent for many restrictions, including \(RM(M)\) and sign/monotonicity restrictions, and can be the entire real line for \(RM(M)\); their finite-sample near-optimality applies only to convex, centrosymmetric classes such as \(SD(M)\).
  • The paper states that the optimal-power result for the conditional test relies on LICQ and an asymptotic regime where sampling variation is small relative to the identified set, so it may not approximate point-identified or short-identified-set cases well.
  • The paper states that inference on the breakdown frontier is left to future work, and the main simulations use a normal model; the duty-to-bargain application uses the original staggered specification, which the text notes is valid under homogeneous treatment effects. Evident from the text: conclusions depend on the researcher’s chosen restriction set and sensitivity parameter, so the approach does not remove the need for judgment about plausible violations.

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Registry

Crossref · 10.1093/restud/rdad018

Cites

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Claims the paper makes

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  • Probably not a substantial claim. 50
  • It may state two separate claims as one. 44
84 of 84

  • The authors state this as an assumption.

    The estimator βn is consistent for the reduced-form parameter β and asymptotically normally distributed.

    “We assume that β is consistent for the reduced-form parameter β, which can be decomposed as”
    “Under mild regularity conditions, all of the estimators described above will be asymptotically normally distributed, satisfying √ n( βn -β) → N (0, * ) for some parameter vector β.”

  • The authors state this as an assumption.

    The parameter vector β can be decomposed as β = τ + δ with τ_pre = 0 (Assumption 1).

    “Assumption 1.The parameter vector β can be decomposed as”
    “with τ pre = 0.”

  • The authors state this as an assumption.

    The treatment has no causal effect prior to its implementation (τ_pre = 0).

    “We assume the treatment has no causal effect prior to its implementation, so τ pre = 0.”

  • The authors state this as an assumption.

    There is no anticipation of treatment, so that Y_it(1) = Y_it(0) for all t < 1.

    “Assume further that there is no anticipation of treatment, so that Y it (1) = Y it (0) for all t < 1.”

  • The authors state this as an assumption.

    In staggered DiD, under a no-anticipation assumption, τ_r = 0 for r < 0.

    “Under a no-anticipation assumption, τ r = 0 for r < 0, which again yields the decomposition (3).”

  • The authors state this as an assumption.

    The difference in trends δ lies in a researcher-specified set Δ.

    “we assume that δ ∈ for some researcher-specified set , and show that the causal parameter τ post is partially identified under such restrictions.”

  • The authors state this as an assumption.

    The linear independence constraint qualification (LICQ) holds for the optimal local asymptotic power result.

    “under a linear independence constraint qualification (LICQ) assumption.”

  • The authors state this as an assumption.

    Uniform asymptotic normality of βn holds: the bounded-Lipschitz distance between √n(βn - βP) and a N(0, ΣP) variable converges to zero uniformly over P ∈ P, where βP = δP + L_post τP,post for δP ∈ Δ and τP,post ∈ R^T.

    “Assumption 2. Let B L 1 denote the set of Lipschitz functions which are bounded by 1 in absolute value and have Lipschitz constant bounded by 1.We assume lim n→∞ sup P∈P sup f ∈B L 1 E P f ( √ n( βn -β P )) -E f (ξ P ) = 0, where ξ P ∼ N (0, P ), and β P = δ P + L post τ P, post for δ P ∈ and τ P, post ∈ R T .”

  • The authors state this as an assumption.

    The asymptotic variance matrices ΣP have eigenvalues bounded below by λ > 0 and above by λ̄ ≥ λ, uniformly over P ∈ P.

    “Assumption 3. Let S denote the set of matrices with eigenvalues bounded below by λ > 0 and above by λ ≥ λ.For all P ∈ P, P ∈ S.”

  • The authors state this as an assumption.

    There exists an estimator Σ̂n that is uniformly consistent for ΣP over P ∈ P.

    “Assumption 4. We have an estimator ˆ n that is uniformly consistent for P , lim n→∞ sup P∈P P P ˆ n -P > = 0, for all > 0.”

  • The authors state this as an assumption.

    The matrix A satisfies at least one of two regularity conditions: (A) A can be written as TQ with Q full row-rank and T a specified block matrix; or (B) for the vertices γ1, ..., γK of V(I), for all k either γk'A = 0 or inf_{a≥0} inf_{j≠k} (γk - aγj)'A > 0.

    “Assumption 5. At least one of the following holds. (A) For k 1 + k 2 = dim(δ), the matrix A can be written as T Q, where Q has full row-rank and T = I k 1 0 -I k 1 0 0 I k 2 . (We allow for the case where one of k 1 or k 2 is 0, in which case the zero-dimensional blocks can be ignored).(B) Let γ1 , . . ., γK be the elements of V (I ).Then for all k, either γ k A = 0 or inf a≥0 inf j =k ( γka γ j )”

  • The authors state this as an assumption.

    The vector Wn = ((β̂n - βP)', (vec(Σ̂n) - vec(ΣP))') has a joint normal asymptotic distribution uniformly over P ∈ P, with covariance matrix VP.

    “Assumption 6.Let W n = (( βn -β P ) , (vec( ˆ n )vec( P )) ) , where vec( ) is the vector of the elements of the matrix .We assume lim n→∞ sup P∈P sup f ∈B L 1 E P f ( √ nW n ) -E f (ξ + P ) = 0, where ξ + P ∼ N (0, V P ), V P = P V P,β V P, β V P, .”

  • The authors state this as an assumption.

    For all P ∈ P, ΣP ∈ S and VP lies in a compact set V; additionally, (ΣP - VP,β VP† VP,β') has eigenvalues bounded below by λ > 0, where † denotes the Moore-Penrose inverse.

    “Assumption 7.For all P ∈ P, P ∈ S and the matrix V P defined in Assumption 6 lies in a compact set V. Additionally, ( P -V P,β V † P, V P, β ) has eigenvalues bounded below by λ > 0, where † denotes the Moore-Penrose inverse.”

  • The authors state this as an assumption.

    The finite-sample normal approximation (9) holds exactly with Ωn known.

    “For brevity of exposition, we focus on the properties of FLCIs in the case where the finite-sample normal approximation (9) holds exactly with n known; Armstrong and Kolesár (2020b) provide uniform asymptotic results for FLCIs under conditions similar to those in Section 3.3.”

  • The authors state this as an assumption.

    Assumption 8: (i) Θ is convex and centrosymmetric (i.e. δ ∈ Θ implies -δ ∈ Θ), and (ii) δ ∈ Θ is such that (δ - δ') ∈ Θ for all δ' ∈ Θ.

    “Assumption 8. Assume (i) is convex and centrosymmetric (i.e.δ ∈ implies -δ ∈ ), and ii) δ ∈ is such that ( δ -δ) ∈ for all δ ∈ .”

  • The authors state this as an assumption.

    Assumption 9: Identified set maximal length and finite: δ ∈ Θ such that LID(δpre, Θ) = sup_{δpre ∈ Θpre} LID(δpre, Θ) < ∞.

    “Assumption 9 (Identified set maximal length and finite).Suppose δ ∈ is such that L I D(δ pre , ) = sup δpre ∈ pre L I D( δpre , ) < ∞, where pre = {δ pre ∈ R T : ∃δ post s.t.(δ pre , δ post ) ∈ } is the set of possible values for δ pre .”

  • The paper relies on this without support.

    The first year after the treatment coincided with a large recession in France (2009), and it may be plausible that the differential factors affecting restaurants were larger in that year than in the pre-treatment period.

    “Given that the first year after the treatment coincided with a large recession in France (2009), it may be plausible that the differential factors affecting restaurants were larger in that year than in the pre-treatment period.”

  • The paper relies on this without support.

    Long-run trends are likely to evolve smoothly over time.

    “Given that the concern is long-run trends that are likely to evolve smoothly over time, smoothness restrictions of the form SD seem natural in this context.”

  • The authors state this as an assumption.

    Treatment effects are homogeneous across adoption cohorts.

    “For simplicity, we focus on the robustness of the results to violations of parallel trends using the original specification in LW, which is valid under the assumption of homogeneous treatment effects.”

  • The authors state this as an assumption.

    There is no causal effect in the pre-period (τ pre = 0).

    “Since our robust confidence sets assume that there is no causal effect in the pre-period (τ pre = 0), we instead treat event-time -2 as the reference period in our analysis.”
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  • Hybrid test truncates at its lower bound

    Minor

    The hybrid test sets its upper truncation point to the minimum of the lower truncation point and the first-stage cutoff. This makes the stated conditional interval empty or a single point, rather than a truncated interval usable for the second-stage test.

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Metadata

Authors
Rambachan, A., & Roth, J.
Affiliations
  • Microsoft Research New England (United States) (US)
  • Brown University (US)
Corresponding author
Roth, Jonathan
Venue
Review of Economic Studies
Publisher
Oxford University Press (OUP)
Published
2023-02-15
Volume
90
Issue
5
Pages
2555-2591
ISSN
  • 0034-6527 (ISSN-L)
  • 1467-937X
DOI
10.1093/restud/rdad018
OpenAlex
W4320855218
Cited by
1,559
References
73
Citations by year
2026: 664 · 2025: 466 · 2024: 243 · 2023: 125
FWCI
304.14
Type
journal article
Language
en
Open access
closed
Licence
academic.oup.com/pages/standard-publication-reuse-rights (vor)
Locations
  • The Review of Economic Studies (publishedVersion)
  • RePEc: Research Papers in Economics (submittedVersion)
Funders
  • National Science Foundation (1745303, 1144152)
  • Yale University
  • Dartmouth College
  • Booth School of Business, University of Chicago
Topics
  • Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
  • Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Spatial and Panel Data Analysis
  • Social Sciences › Economics, Econometrics and Finance › General Economics, Econometrics and Finance › Monetary Policy and Economic Impact
Keywords
parallel trends assumption, difference-in-differences, event study design, partial identification, robust inference, sensitivity analysis, causal inference, power properties

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