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Synthetic Difference-in-Differences
Open accessAbstract
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
AI snapshot
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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Claims the paper makes
63 Own interpretation · 62 No basis given · 31 Own result · 20 Not checked yet · 6 Own method
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In the paper
“and likewise is competitive with (or dominates) SC in applications where SC methods have been used in the past.”
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Empirically, we find that our method is competitive with (or dominates) DID in applications where DID methods have been used in the past”
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“As with the CPS dataset, the two-way fixed effects explain most of the variation; however, the interactive component plays a larger role in determining outcomes for this dataset than for the CPS data.”
plaincorrelationaln = 111- Asserted: passes — The authors put it forward as true.
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In the paper
“As with the CPS dataset, the two-way fixed effects explain most of the variation; however, the interactive component plays a larger role in determining outcomes for this dataset than for the CPS data.”
plaincorrelationaln = 111- Asserted: passes — The authors put it forward as true.
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“In contrast, the errors from SDID are nearly centered.”
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In the paper
“In practice the α i and M i that we construct predict a sizable part of variation in D i , with R 2 varying from 15% to 30%.”
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In the paper
“When assignment is not uniformly random, the distribution of the DID errors is visibly off-center, showing the bias of the estimator.”
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In the paper
“In particular, SDID is nearly unbiased, which is important for constructing confidence intervals with accurate coverage rates.”
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“In the case with Σ = σ 2 I T × T , i.e., without any cross-observation correlations, we note that Var [ ε (˜ ω, ˜ λ ) ] = σ 2 ( N -1 tr + ‖ ˜ ω ‖ 2 2 ) ( T -1 post + ‖ ˜ λ ‖ 2 2 ) .”
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“As discussed in Abadie, Diamond, and Hainmueller [2010], the SC weights ˆ ω sc are sparse.”
What the sources say
- Not checked yetcites Abadie, Diamond, and Hainmueller [2010]
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- It may state two separate claims as one. 42
- Probably not a substantial claim. 28
- The quote only partly states this claim. 3
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“Theoretically, we establish consistency and asymptotic normality of our estimator.”
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- 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.
- The authors state this as an assumption: 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.
- The authors state this as an assumption: E_i is independent of E_i' for each pair of units i, i'.
- 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.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“| Estimate | -15.6 | -19.6 | -27.3 | -20.2 | -11.1 |”
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In the paper
“The other methods, our proposed SDID, the DIFP and the MC estimator are all smaller than the DID estimator with the SDID and DIFP estimator substantially smaller than the SC estimator.”
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“Next, and perhaps surprisingly, we see that the standard errors obtained for SDID (and also for SCIFP, and MC) are smaller than those for DID, despite our method being more flexible.”
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“Second, as demonstrated in the application and simulations based on real data, these unit fixed effects often explain much of the variation in outcomes and can improve precision.”
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In the paper
“| Estimate | -15.6 | -19.6 | -27.3 | -20.2 | -11.1 |”
- Also stated in: § Data: Y , W · p. 8 · plain
“| Standard error | (8.4) | (9.9) | (17.7) | (11.5) | (9.5) |”
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- Also stated in: § Data: Y , W · p. 8 · plain
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“where we consider various strategies for inference based on SDID and show that they attain good coverage here.”
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In the paper
“Meanwhile, when treatment assignment is uniformly random, both estimators are centered but the errors of DID are more spread out.”
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“We find that when the treatment assignment is uniformly random, all methods are essentially unbiased, but SDID is more precise.”
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“| Baseline | 0.992 | 0.100 | 0.098 | (.01,-.06) | 0.028 | 0.037 | 0.049 | 0.035 | 0.032 | 0.010 | 0.020 | 0.021 | 0.015 | 0.007 |”
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“The SDID weights ˆ ω sdid are also sparse-but less so.”
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“Finally, we note that Figure 2 shows that the error distribution of SDID is nearly unbiased and Gaussian in both designs”
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“At a high level, these results mirror the ones above: SDID again performs well in terms of both bias and root-mean squared error and across all simulation settings dominates the other estimators.”
plainn = 111- Asserted: passes — The authors put it forward as true.
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“Observe that both DID and SC have some very high influence states, that is, states with large absolute values of ˆ ω i ( ˆ δ tr -ˆ δ i ) ( e.g. , in both cases, New Hampshire). In contrast, SDID does not give any state particularly high influence”
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“In other words, we find that the jackknife is in general conservative and is exact when treated and control units are similar enough that time weights that fit the control units generalize to the treated units.”
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“while both placebo and jackknife variance estimates also deliver results that are close to the nominal 95% level.”
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In the paper
“our results require no explicit limit on the rank of L and allow for L to have to have positive singular values that are arbitrarily close to zero”
What the sources say
- Not checked yetcites Bai [2009]
- Not checked yetcites Moon and Weidner [2015, 2017]
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“Lemma 8. For any real matrix A and appropriately shaped vectors ˜ x and b , min x α 2 ‖ Ax -b ‖ 2 + β 2 ‖ x -˜ x ‖ 2 = α 2 ‖ S 1 / 2 ( A ˜ x -b ) ‖ 2 for S = I -A ( A ′ A +( β/α ) 2 I ) -1 A ′ . If β = 0 , the same holds for S = I -A ( A ′ A ) † A .”
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“recall that the data in Section 3.1 was generated with noise that is both Gaussian and homoskedastic across units”
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“Thus, the cross terms between ̂ ∆ i ( L ) -ˆ µ tr ( L ) and ̂ ∆ i ( ε ) -ˆ µ tr ( ε ) in (11.6) are mean-zero and concentrate out, and so”
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“By Assumption (5.3), this bound is o P (( N tr T post ) -1 ) and therefore negligible relative to V τ .”
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“In this condensed form, rows ε i · are independent gaussian vectors with mean zero and covariance matrix Σ for i ≤ N 0 and N -1 tr Σ for i = N .”
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- 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·.
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“And this characterization of ε i : ψ as a least squares projection implies that ε i : ψ -ε iT and ε i : are uncorrelated and, being jointly normal, therefore independent.”
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“We'll now show that the first term in (11.4) is o p (( N tr T post ) -1 ), so (11.4) is equivalent to a variant in which we have dropped its first term.”
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“In practice, this equality holds only approximately, in which case including the unit fixed effects in the weighted regression will remove some of the remaining bias.”
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“Like DID, our method is invariant to additive unit-level shifts, and allows for valid large-panel inference.”
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“Relative to these methods we find, both theoretically and empirically, that this 'synthetic difference in differences' estimator has desirable robustness properties”
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“The argument for including time weights in the SDID estimator is the same as the argument for including the unit weights presented earlier: The time weight can both remove bias and improve precision by eliminating the role of time periods that are very different from the post-treatment periods.”
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“but when autocorrelation is present, shinkage toward ψ reduces the variance of the SDID estimator-and enables us to gain precision over the basic DID estimator (1.2) even when the two-way fixed effects model is correctly specified.”
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“One can verify that the basic DID estimator is of the form (4.3), with constant weights ω tr = 1 /N tr , etc.”
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In the paper
“When we move to our asymptotic analysis below, we work under assumptions that make this oracle noise term dominant relative to the other error terms in (4.8).”
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In the paper
“Second, the oracle confounding bias will be small either when the pre-exposure oracle row regression fits well and generalizes to the exposed rows, i.e., ˜ ω 0 +˜ ω ⊤ co L co , pre ≈ ˜ ω ⊤ tr L tr , pre and ˜ ω 0 + ˜ ω ⊤ co L co , post ≈ ˜ ω ⊤ tr L tr , post , or when the unexposed oracle column regression fits well and generalizes to the exposed columns, ˜ λ 0 + L co , pre ˜ λ pre ≈ L co , post ˜”
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“Furthermore, these assumption are aligned with standard practice in the literature”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The error of the synthetic difference in differences estimator can now be decomposed as follows,”
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In the paper
“To this end, we consider two carefully crafted simulation studies, calibrated to datasets representative of those typically used for panel data studies.”
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Here, pre-intervention trends are obviously not parallel, so the DID estimate should be considered suspect.”
boosted- Asserted: passes — The authors put it forward as true.
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In the paper
“Allowing for interactions in L is a natural way to generalize the fixed-effects specification and discuss inference in settings where DID is misspecified [Bai, 2009, Moon and Weidner, 2015, 2017].”
What the sources say
- Not checked yetcites Bai, 2009
- Not checked yetcites Moon and Weidner, 2015, 2017
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“From this perspective, Bertrand et al. [2004] correctly argue that the error distribution of DID is centered, and that the error scale can accurately be recovered using appropriate robust estimators.”
What the sources say
- Not checked yetcites Bertrand et al. [2004]
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The estimator shares this invariance property with DID (but not SC).”
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“opening the door to a simple asymptotic characterization of the error distribution of ˆ τ sdid .”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“for example, we can assess the claim that we balance all components of Γ by examining the extent to which the method succeeds in balancing preintervention periods.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“and is in line with the empirical practice of preferring SC over DID in settings of this type.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“suggesting that after weighting, we have achieved the desired 'parallel trends' as illustrated in the top panel of Figure 1 without inducing excessive variance in the estimator by using concentrated weights.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Finally, accounting for correlation over time within observations of the same unit is widely considered to be an important ingredient to credible inference using panel data [Angrist and Pischke, 2008, Bertrand, Duflo, and Mullainathan, 2004].”
What the sources say
- Not checked yetcites Angrist and Pischke, 2008
- Not checked yetcites Bertrand, Duflo, and Mullainathan, 2004
boosted- Asserted: passes — The authors put it forward as true.
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In the paper
“Their main finding was that only methods that were robust to serial correlation of repeated observations for a given unit ( e.g., methods that clustered observations by unit) attained valid coverage.”
What the sources say
- Not checked yetcites Bertrand et al. [2004]
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“As is well known [Ashenfelter and Card, 1985], DID relies on the assumption that cigarette sales in different states would have evolved in a parallel way absent the intervention.”
What the sources say
- Not checked yetcites Ashenfelter and Card, 1985
boosted- Asserted: passes — The authors put it forward as true.
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What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“Not only do we base the outcome model of our simulation study on real datasets, we further ensure that the treatment assignment process is realistic by seeking to emulate the distribution of real policy initiatives.”
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It may state two separate claims as one.
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In the paper
“In this setting, we show that one can still conduct valid inference, as long as N and T are large enough and the size of the treatment block is small.”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- The authors state this as an assumption: The number of treated units is small relative to the number of control units.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“homoskedasticity is used in the proof essentially only to simplify notation and allow the use of concentration inequalities which have been proven in the homoskedastic case but can be generalized.”
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As our assumptions imply the conclusions of Lemma 4 and Corollary 7, and those two results imply the conclusions of Theorem 3, this concludes our proof.”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“As argued in Bertrand, Duflo, and Mullainathan [2004], unit-level bootstrapping presents a natural approach to inference with panel data when repeated observations of the same unit may be correlated with each other.”
What the sources say
- Not checked yetcites Bertrand, Duflo, and Mullainathan [2004]
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“To address this issue we next consider an approach to inference that is more closely tailored to the SDID method and only involves running the full SDID algorithm once, thus dramatically decreasing the computational burden.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“Algorithm 4 can also be seen as an adaptation of the method of Conley and Taber [2011] for inference in DID models with few treated units and assuming homoskedasticity”
What the sources say
- Not checked yetcites Conley and Taber [2011]
plain- Asserted: passes — The authors put it forward as true.
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What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“We do not report jackknife coverage rates for SC because, as discussed in the text, the variance estimator is not well justified in this case.”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“with a linear model m ( · ), ˆ τ sdid and ˆ τ asc are very similar. In fact, had we fit ˆ ω sdid without intercept, they would be equivalent for ˆ m ( · ) fit by least squares on the controls, imposing the constraint that its coefficients are nonnegative and to sum to one”
What the sources say
- Not checked yetcites Ben-Michael, Feller, and Rothstein [2018]
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“which were assumptions that are both heavily used by the placebo estimator.”
boostedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“in our model, stationarity would imply that a simple pre-post comparison for exposed units would be an unbiased estimator of τ and, as a result, the only purpose of the unexposed units would be to help improve efficiency.”
What the sources say
- Not checked yetcites Li and Bell [2017]
- Not checked yetcites Chernozhukov, Wuthrich, and Zhu [2018]
hedgeddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- The authors state this as an assumption: In the paper's model, L is deterministic and the errors E may be non-stationary.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“it's likely that a similar result would also hold without homoskedasticity”
hedged- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Such a placebo test is closely connected to permutation tests in randomization inference”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In particular, an analogue to Algorithm 3 for SC would be severely biased upwards, and would not be exact even in the well-specified fixed effects model.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Under appropriate conditions, the estimator is asymptotically normal and zero-centered”
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“and the jackknife is not even defined when N tr = 1.”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“We do not report jackknife and bootstrap coverage rates for N tr = 1 because the estimators are not well-defined.”
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Our approach uses past data not only to check whether the trends are parallel, but also to construct the weights to make them parallel.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“However, many applications of synthetic controls have N tr = 1, e.g., the California smoking application from Section 2.”
plainSingle case- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Because the first term is nonnegative, our variance estimate is asymptotically either unbiased or upwardly biased, so our confidence intervals are conservative as claimed.”
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
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”
Claims other papers make about this one
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In the citing paper
“Recent work has begun to try to combine insights from the two literatures (e.g., Arkhangelsky, Athey, Hirshberg, Imbens and Wager, 2021; Ben-Michael, Feller and Rothstein, 2021, 2022; Doudchenko and Imbens, 2016).”
From What's trending in difference-in-differences? A synthesis of the recent econometrics literature (2023) · cites this paper as Arkhangelsky, Athey, Hirshberg, Imbens and Wager, 2021
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Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ 9.3 Theorem 1 in Condensed Form · p. 50
“And we define N 0 = N -1 and T = T 0 -1 .”
§ 9.1 Abstract Setting · p. 48
“We will use analogous notation for the parts of L and ε and let N 0 = N -1 and T 0 = T -1.”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ 5 Large-Sample Inference · p. 30
“The first three columns show coverage of confidence intervals obtained via the Placebo method. The second set of columns show coverage from the jackknife method. The last set of columns show coverage from the clustered bootstrap.”
§ 5 Large-Sample Inference · p. 30
“| | Bootstrap | Bootstrap | Bootstrap | Jackknife | Jackknife | Jackknife | Placebo | Placebo | Placebo |”
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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)
- Venue
- American Economic Review
- Publisher
- American Economic Association
- Published
- 2021-12-01
- Volume
- 111
- Issue
- 12
- Pages
- 4088-4118
- ISSN
- 0002-8282 (ISSN-L)
- 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
- green
- arxiv.org
- 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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