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Cited by 1,518Advanced Causal Inference TechniquesSpatial and Panel Data AnalysisStatistical Methods and Bayesian Inference

The DOI matches its Crossref record; 79 of the 97 references it cites were found in a registry or an index; none of the 29 sources its claims cite is checked for support yet.

Synthetic Difference-in-Differences

Open access
Arkhangelsky, Dmitry, Athey, Susan, Hirshberg, David A., Imbens, Guido W., Wager, Stefan·2021·American Economic Review
DOI 10.1257/aer.20190159
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Abstract

We present a new estimator for causal effects with panel data that builds on insights behind the widely used difference-in-differences and synthetic control methods. Relative to these methods we find, both theoretically and empirically, that this “synthetic difference-in-differences” estimator has desirable robustness properties, and that it performs well in settings where the conventional estimators are commonly used in practice. We study the asymptotic behavior of the estimator when the systematic part of the outcome model includes latent unit factors interacted with latent time factors, and we present conditions for consistency and asymptotic normality. (JEL C23, H25, H71, I18, L66)

Overview

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Written from the paper’s full text.

TL;DR

The paper proposes synthetic difference-in-differences (SDID), a panel-data causal estimator that combines unit and time weights with two-way fixed effects, and establishes consistency and asymptotic normality under a latent-factor model. In applications and placebo studies, SDID is competitive with or dominates DID and SC in bias and RMSE, with near-nominal inference in large panels.

Key findings

  • In the California Proposition 99 application, SDID estimates the effect on per-capita cigarette sales at −15.6 (SE 8.4), compared with SC −19.6 (9.9), DID −27.3 (17.7), MC −20.2 (11.5), and DIFP −11.1 (9.5); SDID’s standard error is smaller than DID’s despite the added flexibility.
  • In CPS placebo simulations, baseline RMSE/bias are SDID 0.028/0.010, SC 0.037/0.020, DID 0.049/0.021, MC 0.035/0.015, and DIFP 0.032/0.007; when the interactive component is dropped, DID performs best, and when fixed effects are dropped, SC performs best.
  • In Penn World Table simulations with democracy-based assignment, RMSE/bias are SDID 0.031/−0.005, SC 0.038/−0.004, DID 0.197/0.175, MC 0.058/0.043, and DIFP 0.039/−0.007; education-based assignment shows a similar pattern.
  • For nominal 95% intervals in the baseline CPS setting, bootstrap coverage is SDID 0.96, SC 0.93, DID 0.89; jackknife SDID coverage is 0.93, and placebo SDID coverage is 0.95. The paper proves asymptotic normality and shows the jackknife is conservative in general and exact under a stated condition.

Methods

  • SDID estimates treatment effects via a weighted two-way fixed-effects regression, with unit weights chosen to align pre-exposure trends and time weights chosen to balance pre- and post-exposure periods for control units; regularization is used for unit weights.
  • The California application uses 39 states from 1970–2000, with T_pre = 19, T_post = 12, N_co = 38, and N_tr = 1. Simulations use CPS data (N = 50, T = 40, 1000 replications) and Penn World Table data (N = 111, T = 48, 1000 replications) under a latent-factor outcome model.
  • Formal results derive consistency and asymptotic normality under Assumptions 1–4; inference is studied via clustered bootstrap, jackknife, and placebo variance estimators.

Limitations

  • The paper states that unequal weighting may worsen precision relative to DID if there is little systematic heterogeneity by units or time periods; identification requires no unexplained confounding, and the assumption that one oracle regression generalizes to exposed rows or columns has no clear testable implications.
  • The paper states that placebo inference relies on homoskedasticity across units, that the jackknife is not defined when N_tr = 1, and that bootstrap/jackknife methods are designed for large panels with many treated units; assumptions also require limited serial correlation and stable unit factors through treatment periods.
  • Evident from the text: the simulations are calibrated to CPS and Penn World Table settings and specific assignment models, so external validity is limited; the California application has N_tr = 1, so only placebo inference is available there.

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Registry

Crossref · 10.1257/aer.20190159

Cites

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Statistics

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Verification

Claims the paper makes

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Show all 116 claims

  • It may state two separate claims as one. 42
  • Probably not a substantial claim. 28
  • The quote only partly states this claim. 3
66 of 66

  • The authors state this as an assumption.

    The conditional expectation of the error matrix E given the assignment matrix W and the systematic component L is zero.

    “We assume that the conditional expectation of the error matrix E given the assignment matrix W and the systematic component L is zero.”

  • The authors state this as an assumption.

    The treatment assignment cannot depend on E.

    “That is, the treatment assignment cannot depend on E .”

  • The authors state this as an assumption.

    The treatment assignment may depend on the systematic component L; W is not randomized.

    “However, the treatment assignment may in general depend on the systematic component L ( i.e. , we do not take W to be randomized).”

  • The authors state this as an assumption.

    E_i is independent of E_i' for each pair of units i, i'.

    “We assume that E i is independent of E i ′ for each pair of units i, i ′ , but we allow for correlation across time periods within a unit.”

  • The authors state this as an assumption.

    D_i is independent of E_i, i.e., the assignment is strictly exogenous.

    “In particular, the distribution of D i may depend on α i and M i ; however, D i is independent of E i , i.e., the assignment is strictly exogenous.”

  • The authors state this as an assumption.

    The systematic component L follows a latent factor model with latent unit factors γ_i and latent time factors υ_t.

    “Here, we work with the following latent factor model (also referred to as an 'interactive fixed-effects model', Xu [2017], see also Athey et al. [2017]),”
    “where γ i is a vector of latent unit factors of dimension R , and υ t is a vector of latent time factors of dimension R .”

  • The authors state this as an assumption.

    The data generating process follows a generalization of the latent factor model (3.2).

    “For the remainder of the paper, we assume that the data generating process follows a generalization of the latent factor model (3.2),”

  • The authors state this as an assumption.

    Treatment assignment is block assignment: W_it = 1({i > N_co, t > T_pre}).

    “As above, we assume block assignment W it = 1( { i > N co , t > T pre } ),”

  • The authors state this as an assumption.

    The errors E_i. are homoskedastic across units (but not across time), i.e., Var[E_i.] = Σ for all units i.

    “Throughout our analysis, we will assume that the errors E i. are homoskedastic across units (but not across time), i.e. , that Var [ E i. ] = Σ ∈ R T × T for all units i = 1 , . . . , n .”

  • The authors state this as an assumption.

    Assumption 1: The rows E_i of the noise matrix are independent and identically distributed Gaussian vectors and the eigenvalues of its covariance matrix Σ are bounded and bounded away from zero.

    “Assumption 1. (Properties of Errors) The rows E i of the noise matrix are independent and identically distributed Gaussian vectors and the eigenvalues of its covariance matrix Σ are bounded and bounded away from zero.”

  • The authors state this as an assumption.

    Assumption 3: L cannot have too many large singular values, and the √min(T_pre, N_co)th singular value of L_co,pre is sufficiently small.

    “Assumption 3. (Properties of L ) Letting σ 1 ( Γ ) , σ 2 ( Γ ) , . . . denote the singular values of the matrix Γ in decreasing order and R the largest integer less than √ min( T pre , N co ) ,”

  • The authors state this as an assumption.

    Assumption 4: The oracle unit weights, oracle time weights, and their joint behavior satisfy specified properties that allow them to cancel out L via weighted double-differencing.

    “Assumption 4. (Properties of Weights and L ) The oracle unit weights ˜ ω satisfy”

  • The authors state this as an assumption.

    One of the oracle regressions generalizes to exposed rows or columns; this is an identification assumption without clear testable implications.

    “the assumption that one of these regressions generalizes to exposed rows or columns is an identification assumption without clear testable implications.”

  • The authors state this as an assumption.

    There is no unexplained confounding: any exceptional behavior of the exposed observations, whether due to exposure or not, can be ascribed to the treatment.

    “It is essentially an assumption of no unexplained confounding: any exceptional behavior of the exposed observations, whether due to exposure or not, can be ascribed to it.”

  • The authors state this as an assumption.

    Γ is stable over time, particularly through the treatment periods.

    “furthermore, Γ should be stable over time, particularly through the treatment periods.”

  • The authors state this as an assumption.

    There were no other shocks disproportionately affecting the treatment units at the time of the treatment.

    “Historical context may be needed to justify the assumption that that there were no other shocks disproportionately affecting the treatment units at the time of the treatment.”

  • The authors state this as an assumption.

    Homoskedasticity across units is effectively a necessary assumption for inference to be possible in the placebo approach.

    “and so homoskedasticity across units is effectively a necessary assumption in order for inference to be possible here.”

  • The authors state this as an assumption.

    The data in Section 3.1 was generated with noise that is both Gaussian and homoskedastic across units, assumptions heavily used by the placebo estimator.

    “recall that the data in Section 3.1 was generated with noise that is both Gaussian and homoskedastic across units-which were assumptions that are both heavily used by the placebo estimator.”

  • The authors state this as an assumption.

    In Theorem 1, the errors are homoskedastic.

    “In Theorem 1, we also assumed homoskedasticity.”

  • The authors state this as an assumption.

    The number of treated units is small relative to the number of control units.

    “the assumption that the number of treated units is small relative to the number of control units”

  • The authors state this as an assumption.

    In the paper's model, L is deterministic and the errors E may be non-stationary.

    “L is taken as deterministic and the errors E may be non-stationary.”

  • The authors state this as an assumption.

    We observe Y_it = L_it + 1(i=N,t=T)τ + ε_it for deterministic τ ∈ R and L ∈ R^{N×T} and random ε ∈ R^{N×T}, and define N0 = N - 1 and T = T0 - 1.

    “Assumption 5 (Model) . We observe Y it = L it +1( i = N,t = T ) τ + ε it for deterministic τ ∈ R and L ∈ R N × T and random ε ∈ R N × T . And we define N 0 = N -1 and T = T 0 -1 .”

  • The authors state this as an assumption.

    The rows ε_i· of the noise matrix are independent gaussian vectors with mean zero and covariance matrix Σ for i ≤ N0 and N1^{-1}Σ for i = N, where the eigenvalues of Σ:: are bounded and bounded away from zero; N1 > 0 can be arbitrary and T1 = 1/Var[ε_i:ψ - ε_iT] and ψ = arg min_v E(ε_i:v - ε_iT)^2.

    “Assumption 6 (Properties of Errors) . The rows ε i · of the noise matrix are independent gaussian vectors with mean zero and covariance matrix Σ for i ≤ N 0 and N -1 1 Σ for i = N where the eigenvalues of Σ :: are bounded and bounded away from zero. Here N 1 > 0 can be arbitrary and we define T 1 = 1 / Var[ ε i : ψ -ε iT ] and ψ = arg min v ∈ R T 0 E ( ε i : v -ε iT ) 2 .”

  • The authors state this as an assumption.

    We consider a sequence of problems where T0/N0 is bounded and bounded away from zero, T1 and N1 are bounded away from zero, and N0/(N1 T1 max(N1,T1) log^2(N0)) → ∞.

    “Assumption 7 (Sample Sizes) . We consider a sequence of problems where T 0 /N 0 is bounded and bounded away from zero, T 1 and N 1 are bounded away from zero, and N 0 / ( N 1 T 1 max( N 1 , T 1 ) log 2 ( N 0 )) → ∞ .”

  • The authors state this as an assumption.

    For the largest integer K ≤ √min(T0,N0), the properties of L stated in Assumption 8 hold.

    “Assumption 8 (Properties of L ) . For the largest integer K ≤ √ min( T 0 , N 0 ) ,”

  • The authors state this as an assumption.

    We use weights as in (9.3) for ζ ≫ (N1 T1)^{1/4} log^{1/2}(N0) and the oracle weights satisfy the stated conditions.

    “Assumption 9 (Properties of Oracle Weights) . We use weights as in (9.3) for ζ ≫ ( N 1 T 1 ) 1 / 4 log 1 / 2 ( N 0 ) and the oracle weights satisfy”

  • The authors state this as an assumption.

    Rows of ε are independent and subgaussian, and for i ≤ N0 they are identically distributed with linear post-on-pretreatment autoregression function E[ε_iT | ε_i:] = ε_i:ψ and covariance Σ = E ε'_i· ε_i·, with Σ_N the covariance matrix of ε_N·.

    “We assume that rows of ε are independent and subgaussian and that for i ≤ N 0 they are identically distributed with linear post-on-pretreatment autoregression function E[ ε iT | ε i : ] = ε i : ψ and covariance Σ = E ε ′ i · ε i · and let Σ N be the covariance matrix of ε N · .”

  • The authors state this as an assumption.

    Throughout the proof of Theorem 2, treatment effects are constant: τ_ij = τ.

    “Throughout this proof, we will assume constant treatment effects τ ij = τ .”

  • The paper relies on this without support.

    There is no cross-unit correlation.

    “given that there is no cross-unit correlation”
29 of 29

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  • Assumption 5 reverses the definition of T0

    Minor

    Section 9.1 defines T0 as T minus one. Assumption 5 instead states that T equals T0 minus one; both cannot hold.

  • Table 4 caption reverses the bootstrap and placebo columns

    Minor

    The caption says the first three columns are placebo results and the last three are bootstrap results, but the table headers label them Bootstrap and Placebo, respectively.

2 of 2

Citations

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Show all 97 references

18 of the 97 references couldn’t be matched to a known paper.

Research works printed but not linked: 8

  1. Amjad Muhammad, 2018, Journal of Machine Learning Research, 19, 1
  2. Arkhangelsky, Dmitry, Susan Athey, David A. Hirshberg, Guido W. Imbens, and Stefan Wager. 2021. "Replication Data for: Synthetic Difference-in-Differences." American Economic Association [publisher], Inter-university Consortium for Social and Political Research [distributor]. https://doi. org/10.3886/E146381V1. · DOI
  3. Athey Susan, 2021, Journal of the American Statistical Association 116. https://doi.org/10., 1080, 2021
  4. Athey Susan, 2021, Journal of Econometrics. https://doi.org/10.1016/j. jeconom., 2020, 10 · DOI
  5. Callaway Brantly, 2020, Journal of Econometrics. https://doi.org/10.1016/j.jeconom., 2020, 12 · DOI
  6. Kallus Nathan, 2020, Journal of Machine Learning Research, 21, 1
  7. Miller Rupert G, 1974, Biometrika, 61, 1
  8. Orzechowski, & Walker. 2005. The Tax Burden on Tobacco. Historical Compilation, Vol. 40, Arlington, VA: Orzechowski & Walker.

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Metadata

Authors
Arkhangelsky, D., Athey, S., Hirshberg, D. A., Imbens, G. W., & Wager, S.
Affiliations
  • Centro de Estudios Monetarios y Financieros (ES)
  • Stanford Institute for Economic Policy Research (US)
  • Emory University (US)
  • Stanford University (US)
ORCID
Venue
American Economic Review
Publisher
American Economic Association
Published
2021-12-01
Volume
111
Issue
12
Pages
4088-4118
ISSN
0002-8282 (ISSN-L)
DOI
10.1257/aer.20190159
OpenAlex
W2952814763
Cited by
1,518
References
97
Citations by year
2026: 456 · 2025: 453 · 2024: 306 · 2023: 172 · 2022: 87 · 2021: 17
FWCI
108.51
Type
journal article
Language
en
Open access
Locations
  • American Economic Review (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • RePEc: Research Papers in Economics (submittedVersion)
  • SSRN Electronic Journal (submittedVersion)
  • arXiv (Cornell University)
Funders
  • Alfred P. Sloan Foundation
  • Office of Naval Research (N00014-17-1-2131), N00014-17-1, 17-1-2131, N00014, N00014-17)
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
  • Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
Keywords
synthetic difference-in-differences, causal effects, panel data, synthetic control method, difference-in-differences, estimator robustness, asymptotic normality, consistency

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