Synthetic Difference-in-Differences

Empirically, SDID is competitive with or dominates SC in applications where SC methods have been used in the past.

“and likewise is competitive with (or dominates) SC in applications where SC methods have been used in the past.”

Empirically, SDID is competitive with or dominates DID in applications where DID methods have been used in the past.

“Empirically, we find that our method is competitive with (or dominates) DID in applications where DID methods have been used in the past”

The interactive component plays a larger role in determining outcomes for the Penn World Table dataset than for the CPS data.

“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.”

In the Penn World Table dataset, two-way fixed effects explain most of the variation.

“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.”

When assignment is not uniformly random, the errors from SDID are nearly centered.

“In contrast, the errors from SDID are nearly centered.”

In the CPS simulation design, the α_i and M_i constructed predict a sizable part of variation in D_i, with R^2 varying from 15% to 30%.

“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%.”

When assignment is not uniformly random, the distribution of the DID errors is visibly off-center, showing the bias of the estimator.

“When assignment is not uniformly random, the distribution of the DID errors is visibly off-center, showing the bias of the estimator.”

SDID is nearly unbiased in the Penn World Table simulations.

“In particular, SDID is nearly unbiased, which is important for constructing confidence intervals with accurate coverage rates.”

In the case without cross-observation correlations, the variance of the oracle noise term is σ^2 (N_tr^{-1} + ||ω||_2^2)(T_post^{-1} + ||λ||_2^2).

“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 ) .”

The SC weights are sparse.

“As discussed in Abadie, Diamond, and Hainmueller [2010], the SC weights ˆ ω sc are sparse.”

• cites Abadie, Diamond, and Hainmueller [2010]Not checked yet

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The noise in DID can be substantially reduced by using an estimator like SDID that can exploit predictable variation by matching on pre-exposure trends.

“Here, however, we go further and show that this noise can be substantially reduced by using an estimator like SDID that can exploit predictable variation by matching on pre-exposure trends.”

In formal results, SDID can consistently estimate τ in the latent factor design under reasonable conditions.

“In our formal results given in Section 4, we show how, despite not explicitly fitting the model (3.2), SDID can consistently estimate τ in this design under reasonable conditions.”

• AssumptionThe 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.
• AssumptionThe authors state this as an assumption: The treatment assignment cannot depend on E.
• AssumptionThe authors state this as an assumption: The treatment assignment may depend on the systematic component L; W is not randomized.
• AssumptionThe authors state this as an assumption: E_i is independent of E_i' for each pair of units i, i'.
• AssumptionThe 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.

When the interactive component M is dropped from the outcome model so that the fixed effect specification is correct, the DID estimator performs best alongside MC.

“If we drop the interactive component M from the outcome model ('No M '), so that the fixed effect specification is correct, the DID estimator performs best (alongside MC).”

When the noise component is removed so that there is only signal, the increased flexibility of the SDID estimator allows it alongside MC to outperform the SC and DID estimators dramatically.

“On the other hand, if we remove the noise component so that there is only signal, the increased flexibility of the SDID estimator allows it (alongside MC) to outperform the SC and DID estimators dramatically.”

When both parts of the systematic component are dropped and there is only noise, the superiority of the SDID estimator vanishes and all estimators are essentially equivalent.

“If we drop both parts of the systematic component, and there is only noise, the superiority of the SDID estimator vanishes and all estimators are essentially equivalent.”

DID does substantially worse in Table 3 relative to SC than in Table 2.

“The main difference between Tables 2 and 3 is that DID does substantially worse here relative to SC than before.”

When the fixed effects component is dropped but the interactive component is kept, the SC estimator does best.

“In contrast, if we drop the fixed effects component ('No F ') but keep the interactive component, the SC estimator does best.”

When treatment assignment is not uniformly random, SDID is particularly successful at mitigating bias while keeping variance in check.

“Meanwhile, when the treatment assignment is not uniformly random, SDID is particularly successful at mitigating bias while keeping variance in check.”

In the baseline CPS simulation, SDID has the lowest RMSE among the compared estimators (SDID 0.028, SC 0.037, DID 0.049, MC 0.035, DIFP 0.032).

“| 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 |”

The asymptotic variance V_τ is on the order of 1/(N_tr T_post).

“Here V τ is on the order of 1 / ( N tr T post ) , i.e., N tr T post V τ is bounded and bounded away from zero.”

Under the stated assumptions, the SDID estimator is asymptotically normal.

“Our main formal result states that under these assumptions, our estimator is asymptotically normal.”

Coverage rates for DID and SC can be relatively low, especially in cases with significant bias such as the setting with the state unemployment rate as the outcome.

“In contrast, we see that coverage rates for DID and SC can be relatively low, especially in cases with significant bias such as the setting with the state unemployment rate as the outcome.”

If Assumptions 5, 6, and 8 hold with T0 ∼ N0 and log(N0), T1, and N1 are bounded away from zero, and weights are defined as in (9.3) with unit simplices and ζ ≫ m1^{1/2} log^{1/2}(m0), then with probability 1 - 2 exp(-min(T1 log(T0), N1 log(N0))) - c exp(-c N0^{1/2}), ˆτ(ˆω,ˆλ) - ˆτ( ̃λ, ̃ω) = o_p((N1 T1)^(-1/2)) if the stated bounds hold and the latter two bounds go to infinity.

“Corollary 7. Suppose Assumptions 5, 6, and 8 hold with T 0 ∼ N 0 and that log( N 0 ) , T 1 and N 1 are bounded away from zero. Let m 0 = N 0 , m 1 = √ N 1 T 1 , and ¯ m 1 = max( N 1 , T 1 ) . Consider the weights defined in (9.3) with Ω ⊆ R N 0 and Λ ⊆ R T 0 taken to be the unit simplices and ζ ≫ m 1 / 2 1 log 1 / 2 ( m 0 ) . With probability 1 -2 exp( -min( T 1 log( T 0 ) , N 1 log( N 0 ))) -c ex”

• AssumptionThe 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.
• AssumptionThe 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.
• AssumptionThe authors state this as an assumption: For the largest integer K ≤ √min(T0,N0), the properties of L stated in Assumption 8 hold.

The bootstrap appears to yield robust performance in large panels in the experiments.

“The bootstrap is simple to implement and, in our experiments, appears to yield robust performance in large panels.”

With staggered adoption, the weighted DID regression approach in SDID does not work directly.

“With staggered adoption the weighted DID regression approach in SDID does not work directly.”

Under the conditions of Theorem 1, the jackknife variance estimator described in Algorithm 3 yields conservative confidence intervals.

“Then, under the conditions of Theorem 1, the jackknife variance estimator described in Algorithm 3 yields conservative confidence intervals”

If Assumptions 5-9 hold, then the SDID estimator satisfies ˆτ(ˆλ,ˆω) - τ = ε_NT - ε_N:ψ + o_p((N1 T1)^(-1/2)).

“Theorem 3. If Assumptions 5-9 hold, then ˆ τ ( ˆ λ, ˆ ω ) -τ = ε NT -ε N : ψ + o p (( N 1 T 1 ) -1 / 2 ) .”

• AssumptionThe 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.
• AssumptionThe 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.
• AssumptionThe 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)) → ∞.
• AssumptionThe authors state this as an assumption: For the largest integer K ≤ √min(T0,N0), the properties of L stated in Assumption 8 hold.
• AssumptionThe 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.

The jackknife procedure still recovers conservative confidence intervals under considerable generality.

“however, as shown below, we still recover conservative confidence intervals under considerable generality.”

In the abstract setting, if Λ and Ω are sets with sums equal to 1, and random and deterministic weights are as specified, then on the intersection of an event of probability 1 - c exp(-u^2) and one on which the stated condition holds, the corresponding treatment effect estimators are close in the stated sense.

“Lemma 6. In the setting described in Section 9.1, let Λ ⊆ R T 0 and Ω ⊆ R N 0 be sets with the property that ∑ t ≤ T 0 λ t = ∑ i ≤ N 0 ω i = 1 for all λ ∈ Λ and ω ∈ Ω . Let ˆ λ 0 , ˆ λ ∈ R × Λ and ˆ ω 0 , ˆ ω ∈ R × Ω be random and ˜ λ 0 , ˜ λ ∈ R × Λ and ˜ λ 0 , ˜ λ ∈ R × Ω be deterministic. On the intersection of an event of probability 1 -c exp( -u 2 ) and one on which”

If Assumptions 5, 6, and 8 hold; T1 and N1 are bounded away from zero; N0, T0 → ∞ with N0 ≥ log^2(T0) and T0 ≥ log^2(N0); and weights are chosen as in (9.3) for unit simplices Ω ⊆ R^{N0} and Λ ⊆ R^{T0}, then the stated bounds hold on an event of probability 1 - c exp(-c min(N0^{1/2}, T0^{1/2}, N0/‖L:: ̃λ + ̃λ0 - L:T‖, T0/‖ ̃ω′L:: + ̃ω0 - LN:‖)).

“Lemma 5. If Assumptions 5, 6, and 8 hold; T 1 and N 1 are bounded away from zero; N 0 , T 0 →∞ with N 0 ≥ log 2 ( T 0 ) and T 0 ≥ log 2 ( N 0 ) ; and we choose weights as in (9.3) for unit simplices Ω ⊆ R N 0 and Λ ⊆ R T 0 , then the following bounds hold on an event of probability”

• AssumptionThe 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.
• AssumptionThe 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.
• AssumptionThe authors state this as an assumption: For the largest integer K ≤ √min(T0,N0), the properties of L stated in Assumption 8 hold.

If deterministic ̃ω, ̃λ satisfy ‖ ̃ω‖ = o(N1^(-1/2)) and ‖ ̃λ - ψ‖ = o(T1^(-1/2)) and Assumptions 5, 6, and 9(iii) hold, then ˆτ( ̃ω, ̃λ) - τ = ε_NT - ε_N:ψ + o_p((N1 T1)^(-1/2)).

“Lemma 4. If deterministic ˜ ω, ˜ λ satisfy ‖ ˜ ω ‖ = o ( N -1 / 2 1 ) and ‖ ˜ λ -ψ ‖ = o ( T -1 / 2 1 ) and Assumptions 5, 6, and 9(iii) hold, then ˆ τ (˜ ω, ˜ λ ) -τ = ε NT -ε N : ψ + o p (( N 1 T 1 ) -1 / 2 ) .”

• AssumptionThe 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.
• AssumptionThe 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.
• AssumptionThe 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.

In the SDID estimation, the bootstrap estimator performs particularly well, yielding nearly nominal 95% coverage.

“In the case of the SDID estimation, the bootstrap estimator performs particularly well, yielding nearly nominal 95% coverage”

N_tr^{-1} ̃λ′Σ ̃λ ∼_p V_τ in the proof of Lemma 4.

“It is shown that N -1 tr ˜ λ ′ Σ ˜ λ ∼ p V τ in the proof of Lemma 4, so”

The ratio between the jackknife variance estimate and the expression derived by replacing denominators with 1 converges to one.

“As each term in that expression is nonnegative, it follows that the ratio between it and the expression below, derived by replacing these denominators with 1, is in this interval and therefore converges to one.”

Under the assumptions of Theorem 1, ‖ˆω_co‖^2 ≪ (N_tr T_post log(N_co))^{-1}.

“Here we will show that ‖ ˆ ω co ‖ 2 ≪ ( N tr T post log( N co )) -1 under the assumptions of Theorem 1.”

• AssumptionThe 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.
• AssumptionThe 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.
• AssumptionThe 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)) → ∞.
• AssumptionThe authors state this as an assumption: For the largest integer K ≤ √min(T0,N0), the properties of L stated in Assumption 8 hold.
• AssumptionThe 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.

When treatment effects are constant across units, they do not contribute to ˆV_jack_τ.

“Now, focusing further on (11.5) we note that, when treatment effects are constant across units, we can verify that they do not contribute to ̂ V jack τ and so”

• AssumptionThe authors state this as an assumption: Throughout the proof of Theorem 2, treatment effects are constant: τ_ij = τ.

The variance of ε_i:ψ - ε_iT is on the order of 1/T_post.

“Furthermore, it implies the variance of ε i : ψ -ε iT is on the order of 1 /T post .”

The first term in (11.4) is negligible relative to the second.

“We will simplify this further by showing that the first term is negligible relative to the second.”

As σ_min(Σ) is bounded away from zero, the mean of the second term in (11.4) is on the order of (N_tr T_post)^{-1} or larger.

“As σ min (Σ) is bounded away from zero, it follows that the mean of the second term in (11.4) is on the order of ( N tr T post ) -1 or larger.”

It suffices to show that max_{i ≤ N_co} ˆΔ_i^2 ≪ log(N_co).

“And it suffices to show that max i ≤ N co ̂ ∆ 2 i ≪ log( N co ), as ( ̂ ∆ i -ˆ µ co ) 2 ≤ 2 ̂ ∆ 2 i + 2ˆ µ 2 co and ˆ µ co is a convex combination of ̂ ∆ 1 . . . ̂ ∆ co .”

The denominators 1 - ˆω_i lie in the interval [1 - o_p(1), 1].

“This bound implies that the denominators 1 -ˆ ω i appearing in the expression above all lie in the interval [1 -max( ‖ ˆ ω co ‖ , N -1 tr ) , 1] = [1 -o p (1) , 1].”

The second term in (11.7) is equivalent to a variant in which ̃λ replaces ˆλ.

“We will now show that the second term is equivalent to a variant in which ˜ λ replaces ˆ λ .”

Because N_co^{-1/2} ≪ T_post^{-1/2} log^{-1/2}(N_co) under Assumption 2, the first quantity is negligible relative to the second.

“Thus, because N -1 / 2 co ≪ T -1 / 2 post log -1 / 2 ( N co ) under Assumption 2, the first quantity is negligible relative to the second.”

SDID combines attractive features of both DID and SC.

“We then propose a new method, Synthetic Difference in Differences (SDID), that combines attractive features of both.”

SC provides a reduced and generally considered more credible estimate of -19.6.

“SC provides a reduced (and generally considered more credible) estimate of -19.6.”

Localization can improve the SDID estimator's precision by implicitly removing systematic (predictable) parts of the outcome.

“Perhaps less intuitively, the use of the weights can also improve the estimator's precision by implicitly removing systematic (predictable) parts of the outcome.”

The use of weights in SDID makes the two-way fixed effect regression local, emphasizing units that are similar in their past to the treated units and periods that are similar to the treated periods.

“The use of weights in the SDID estimator effectively makes the two-way fixed effect regression 'local,' in that it emphasizes (puts more weight on) units that on average are similar in terms of their past to the target (treated) units, and it emphasizes periods that are on average similar to the target (treated) periods.”

Each of the three estimators (DID, SC, SDID) can be rewritten as a weighted average difference in adjusted outcomes with appropriate sample weights.

“To facilitate direct comparisons, we observe that each of the three estimators can be rewritten as a weighted average difference in adjusted outcomes ˆ δ i for appropriate sample weights ˆ ω i :”

Under some conditions, SC weighting can account for the unit fixed effects on its own.

“Under some conditions, SC weighting can account for the unit fixed effects on its own.”

The DID estimate of -27.3 likely overstates the effect of the policy change on smoking.

“As argued in Abadie et al. [2010], the assumptions underlying the DID estimator are suspect here, and the -27.3 point estimate likely overstates the effect of the policy change on smoking.”

• cites Abadie et al. [2010]Not checked yet

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If there is little systematic heterogeneity in outcomes by either units or time periods, the unequal weighting of units and time periods may worsen the precision of the estimators relative to DID.

“However, the latter is not guaranteed: If there is little systematic heterogeneity in outcomes by either units or time periods, the unequal weighting of units and time periods may worsen the precision of the estimators relative to the DID estimator.”

The difference in point estimates implies that the use of time weights and unit fixed effects in SDID materially affects conclusions.

“At the very least, this difference in point estimates implies that the use of time weights and unit fixed effects in (1.1) materially affects conclusions”

The fundamental assumptions that justify DID and SC methods are closely related.

“In this paper, we argue that although the empirical settings where DID and SC methods are typically used differ, the fundamental assumptions that justify both methods are closely related.”

When the SC and SDID estimates differ, the SDID estimate is often more credible.

“and, throughout this paper, we will argue that when ˆ τ sc and ˆ τ sdid differ, the latter is often more credible.”

Like SC, SDID re-weights and matches pre-exposure trends to weaken the reliance on parallel trend type assumptions.

“Like SC, our method re-weights and matches pre-exposure trends to weaken the reliance on parallel trend type assumptions.”

Together, the unit and time weights make the DID strategy more plausible.

“Together, these weights make the DID strategy more plausible.”

The SDID estimator performs well in settings where conventional estimators are commonly used in practice.

“and that it performs well in settings where the conventional estimators are commonly used in practice”

The smaller standard errors for SDID are a result of the local fit of SDID (and SC) being improved by the weighting.

“This is a result of the local fit of SDID (and SC) being improved by the weighting.”

Localization can make the SDID estimator more robust.

“Intuitively, using only similar units and similar periods makes the estimator more robust.”

SDID addresses pretesting concerns recently expressed in Roth [2018].

“From this point of view, SDID addresses pretesting concerns recently expressed in Roth [2018].”

• cites Roth [2018]Not checked yet

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SC weighting accounts for unit fixed effects when the weighted average of the outcomes for the control units in the pre-treatment periods is exactly equal to the average of outcomes for the treated units during those pre-treatment periods.

“In particular, this happens when the weighted average of the outcomes for the control units in the pre-treatment periods is exactly equal to the average of outcomes for the treated units during those pre-treatment periods.”

The benefits of including unit fixed effects in the SC regression can also be obtained by applying the synthetic control method after centering the data by subtracting each unit's pre-treatment mean.

“The benefits of including unit fixed effects in the SC regression (1.3) can also be obtained by applying the synthetic control method after centering the data by subtracting, from each unit's trajectory, its pre-treatment mean.”

• cites Doudchenko and Imbens [2016]Not checked yet

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• cites Ferman and Pinto [2019]Not checked yet

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Including unit fixed effects strengthens the model's robustness properties by making the model more flexible.

“First, by making the model more flexible, we strengthen its robustness properties.”

When Σ is not diagonal, the effective regularization term does not shrink the pre-treatment time weights toward zero, but rather toward an autoregression vector.

“When Σ is not diagonal, the effective regularization term in (4.7) does not shrink ˜ λ pre towards zero, but rather toward an autoregression vector”

The use of two-way fixed effects and intercept terms makes the SDID estimator invariant to additive shocks to any row or column.

“Second, the use of two-way fixed effects in (1.1) and intercept terms in (2.1) and (2.3) makes the SDID estimator invariant to additive shocks to any row or column, i.e., if we modify L it ← L it + α i + β t for any choices α i and β t the estimator ˆ τ sdid remains unchanged.”

In model (4.1), the oracle weights admit a simplified characterization.

“In the case of our model (4.1) these weights admit a simplified characterization”

Even if neither model generalizes sufficiently well on its own, it suffices for one model to predict the generalization error of the other.

“Moreover, even if neither model generalizes sufficiently well on its own, it suffices for one model to predict the generalization error of the other:”

The SDID estimator incorporates time weights that seek to balance out Υ, in addition to unit weights that balance out Γ.

“First, instead of only making use of unit weights ˆ ω that can be used to balance out Γ , the estimator also incorporates time weights ˆ λ that seek to balance out Υ .”

In the absence of autocorrelation, ψ is zero.

“In the absence of autocorrelation, ψ is zero”

The SDID estimator will be close to the oracle when the oracle unit and time weights look promising on their respective training sets.

“Third, our core theoretical claim, formalized in our asymptotic analysis, is that the SDID estimator will be close to the oracle when the oracle unit and time weights look promising on their respective training sets”

The assumptions required for the SDID formal analysis are considerably weaker than what is required in results of Bai [2009] or Moon and Weidner [2015, 2017] for methods that require explicitly estimating L.

“However, as discussed further in Section 6, they are considerably weaker than what is required in results of Bai [2009] or Moon and Weidner [2015, 2017] for methods that require explicitly estimating L in (4.1).”

• cites Bai [2009]Not checked yet

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• cites Moon and Weidner [2015, 2017]Not checked yet

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The double robustness property of SDID is similar to that of the augmented inverse probability weighting estimator.

“This double robustness property is similar to that of the augmented inverse probability weighting estimator”

• cites Ben-Michael, Feller, and Rothstein, 2018Not checked yet

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• cites Scharfstein, Rotnitzky, and Robins, 1999Not checked yet

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If the SC weights succeed in balancing out the factors in Γ, the SC estimator can be approximated as a nearly unbiased estimate of τ.

“Then, if ˆ ω sc in fact succeeds in balancing out the factors in Γ , the SC estimator can be approximated as ˆ τ sc ≈ τ + ∑ N i =1 (2 W i -1)ˆ ω sc i ¯ ε i with ¯ ε i = T -1 post ∑ T t = T pre +1 ε it ; in words, SC weighting has succeeded in removing the bias associated with the systematic component L and in delivering a nearly unbiased estimate of τ .”

Under completely random assignment, a number of methods, including DID, perform well because the presence of L in (3.2) introduces zero bias.

“Under completely random assignment, a number of methods, including DID, perform well because the presence of L in (3.2) introduces zero bias.”

Even if one of the sets of weights fails to remove the bias from the presence of L, the combination of the unit and time weights can compensate for such failures.

“The upshot is even if one of the sets of weights fails to remove the bias from the presence of L , the combination of weights ˜ ω and ˜ λ can compensate for such failures.”

The lesser sparsity of SDID weights is due to regularization and the use of the intercept, which allows greater flexibility in solving the weight optimization and enables more balanced weighting.

“This is due to regularization and the use of the intercept ω 0 , which allows greater flexibility in solving (2.1), enabling more balanced weighting.”

Under appropriate conditions, the difference between the oracle and feasible estimators is negligible relative to the error of the oracle estimator.

“Under appropriate conditions, we find the latter term negligible relative to the error of the oracle estimator”

With a non-uniform random assignment (treatment assignment correlated with systematic effects), methods that do not account for the presence of L will be biased.

“In contrast, with a non-uniform random assignment ( i.e. , treatment assignment is correlated with systematic effects), methods that do not account for the presence of L will be biased.”

The proposed SDID estimator can be written as (4.3) with weights solving (2.1) and (2.3).

“The proposed SDID estimator (1.1) can also be written as (4.3), but now with weights ˆ ω sdid and ˆ λ sdid solving (2.1) and (2.3) respectively.”

When the matrix L takes on an additive form, the DID regression will consistently recover τ.

“When the matrix L takes on an additive form, i.e., L it = α i + β t , then the DID regression will consistently recover τ .”

• AssumptionThe 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.

Synthetic control methods address confounding bias without explicitly estimating L.

“Synthetic control methods address confounding bias without explicitly estimatin L in (4.1).”

The use of both unit and time weights provides a type of double robustness property, whereby if one of the balancing approaches is effective, the dependence on L is approximately removed.

“This provides a type of double robustness property, whereby if one of the balancing approaches is effective, the dependence on L is approximately removed.”

The oracle noise term tends to be small when the weights are not too concentrated and there are a sufficient number of exposed units and time periods.

“First, the oracle noise term tends to be small when the weights are not too concentrated, i.e., when ‖ ˜ ω ‖ 2 and ‖ ˜ λ ‖ 2 are small, and we have a sufficient number of exposed units and time periods.”

SDID being nearly unbiased is important for constructing confidence intervals with accurate coverage rates.

“In particular, SDID is nearly unbiased, which is important for constructing confidence intervals with accurate coverage rates.”

For any choice of weights satisfying the conditions, the error of a weighted double-differencing estimator can be decomposed as the sum of a bias and a noise component.

“Thus, we can decompose the error of any weighted double-differencing estimator with weights satisfying these conditions as the sum of a bias and a noise component:”

The worse performance of DID relative to SC in Table 3 appears to be due to the presence of a stronger interactive component in the Penn World Table dataset.

“This appears to be due to the presence of a stronger interactive component in the Penn World Table dataset”

Using a non-uniformly random assignment is important because it allows differentiation between various estimators in ways that completely random assignment would not.

“Using a non-uniformly random assignment is important because it allows us to differentiate between various estimators in ways that completely random assignment would not.”

It should be possible to use the SDID estimator as the basis for valid inference.

“thus suggesting that it should be possible to use ˆ τ sdid as the basis for valid inference.”

The SDID estimator only needs the combination of double differencing, unit weights, and time weights to reduce the bias to a sufficiently small level.

“Rather, we only need the combination of the three bias-reducing components in the SDID estimator, ( i ) double differencing, ( ii ) the unit weights, and ( iii ) the time weights, to reduce the bias to a sufficiently small level.”

The SDID assumptions do not require that unit comparisons alone are sufficient to remove the biases in comparisons between treated and control units as the SC assumptions do.

“Furthermore, we do not require that unit comparisons alone are sufficient to remove the biases in comparisons between treated and control units as the SC assumptions do.”

The SDID method may be more robust than the least squares method in cases where the analyst wishes to be as agnostic as possible regarding properties of L.

“thus suggesting that the SDID method may be more robust than the least squares method in cases where the analyst wishes to be as agnostic as possible regarding properties of L .”

• cites Bai [2009]Not checked yet

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• cites Moon and Weidner [2015, 2017]Not checked yet

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The validity of the placebo approach relies fundamentally on homoskedasticity across units.

“Validity of the placebo approach relies fundamentally on homoskedasticity across units”

If the conditions hold and a consistent estimator for the asymptotic variance V_τ is available, conventional confidence intervals can be used.

“thus, if these conditions hold and we have a consistent estimator for its asymptotic variance V τ , we can use conventional confidence intervals”

Homoskedasticity across units is effectively a necessary assumption for inference to be possible in this setting.

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

Non-parametric variance estimation for treatment effect estimators is in general impossible if there is only one treated unit.

“We also note that non-parametric variance estimation for treatment effect estimators is in general impossible if we only have one treated unit”

When the exposed unit was not chosen at random, placebo tests do not have the formal properties of randomization tests.

“in which case placebo tests do not have the formal properties of randomization tests”

• cites Firpo and Possebom, 2018Not checked yet

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• cites Hahn and Shi, 2016Not checked yet

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Applying the SDID estimator repeatedly, once for every adoption date, is a simple modification to estimate the average treatment effect for the treated under staggered adoption.

“Here we discuss a simple modification to estimate the average treatment effect for the treated in that setting by applying the SDID estimator repeatedly, once for every adoption date.”

Accounting for balancing unobserved factors forces a different formal approach than existing studies using balancing methods.

“accounting for this forces us to follow a different formal approach than existing studies using balancing methods.”

The SDID assumptions do not require that double differencing alone removes the individual and time effects as the DID assumptions do.

“We do not require that double differencing alone removes the individual and time effects as the DID assumptions do.”

The bootstrap and jackknife-based methods may be less reliable when the number of treated units N_tr is small.

“These methods may be less reliable when the number of treated units N tr is small”

The bootstrap may be computationally costly because it involves running the full SDID algorithm for each bootstrap replication.

“The main downside of the bootstrap is that it may be computationally costly as it involves running the full SDID algorithm for each bootstrap replication, and for large datasets this can be prohibitively expensive.”

The validity of the jackknife procedure is not implied directly by asymptotic linearity.

“The validity of this procedure is not implied directly by asymptotic linearity as in (4.14)”

In many synthetic controls applications, the exposed unit was not chosen at random.

“however, in many synthetic controls applications, the exposed unit was not chosen at random”

If the point estimates from DID and SC are dominated by bias, then confidence intervals that only focus on variance should not be expected to achieve coverage.

“If the point estimates ˆ τ from DID and SC are dominated by bias, then we should not expect confidence intervals that only focus on variance to achieve coverage.”

Assumptions 1-4 are substantially weaker than those used to establish asymptotic normality of comparable methods.

“Assumptions 1-4 are substantially weaker than those used to establish asymptotic normality of comparable methods.”

The asymptotic variance of the SDID estimator is optimal, coinciding with the variance that would be obtained if L and Σ were known a-priori.

“Furthermore, its asymptotic variance is optimal, coinciding with the variance we would get if we knew L and Σ a-priori”

To achieve consistency, the paper's method needs to balance unobserved factors Γ and Υ in (3.2).

“rather than balancing observed covariates, we here need to balance unobserved factors Γ and Υ in (3.2) to achieve consistency”

The assumptions are satisfied in the well-specified two-way fixed effect setting model.

“In particular, note that our assumptions are satisfied in the well-specified two-way fixed effect setting model.”

The jackknife result depends on the specific structure of the SDID estimator and does not hold for related methods such as the SC estimator.

“This result depends on specific structure of the SDID estimator, and does not hold for related methods such as the SC estimator.”

The behavior of the oracle time weights under autocorrelation explains some of the behavior noted in the simulations.

“This explains some of the behavior noted in the simulations.”

The SDID assumptions do not require a low rank factor model to be correctly specified, as is often assumed in the analysis of methods that estimate L explicitly.

“Finally, we do not require a low rank factor model to be correctly specified, as is often assumed in the analysis of methods that estimate L explicitly”

• cites Bai, 2009Not checked yet

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• cites Moon and Weidner, 2015, 2017Not checked yet

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In the paper's analysis, using unexposed units for double-differencing is crucial for identification.

“In contrast, in our analysis, using unexposed units for double-differencing is crucial for identification.”

If the exposed and unexposed units have different noise distributions, there is no way to learn V_τ from unexposed units alone.

“because if the exposed and unexposed units have different noise distributions then there is no way we can learn V τ from unexposed units alone.”

That the eigenvalues of non-condensed-form Σ are bounded and bounded away from zero implies that the eigenvalues of the submatrix Σ:: = Σ_pre,pre are bounded and bounded away from zero.

“That the eigenvalues of non-condensed-form Σ are bounded and bounded away from zero implies that the eigenvalues of the submatrix Σ :: = Σ pre , pre are bounded and bounded away from zero.”

When treatment effects are not constant, the jackknife variance estimate will include an additional nonnegative term that depends on the amount of treatment heterogeneity, making the inference conservative.

“When treatment effects are not constant, the jackknife variance estimate will include an additional nonnegative term that depends on the amount of treatment heterogeneity, making the inference conservative.”

Under the conditions of Theorem 1, all the noise in ˆτ comes from the exposed units.

“This simplification is as we would hope given that, under the conditions of Theorem 1, we found that all the noise in ˆ τ comes from the exposed units.”