A More Credible Approach to Parallel Trends

Recent research has shown that pre-trends tests may suffer from low power.

“recent research has shown that they may suffer from low power (Freyaldenhoven et al., 2019;Bilinski and Hatfield, 2020;Kahn-Lang and Lang, 2020;Roth, 2022)”

• cites Freyaldenhoven et al., 2019 sourceSupported
• cites Bilinski and Hatfield, 2020Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Roth, 2022 sourceNot checked — no full text of the source

Pre-trends tests may be underpowered against meaningful violations of parallel trends, potentially leading to severe undercoverage of conventional confidence intervals.

“First, pre-trends tests may be underpowered against meaningful violations of parallel trends, potentially leading to severe undercoverage of conventional confidence intervals (Freyaldenhoven et al., 2019;Bilinski and Hatfield, 2020;Kahn-Lang and Lang, 2020;Roth, 2022).”

• cites Freyaldenhoven et al., 2019 sourceSupported
• cites Bilinski and Hatfield, 2020Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Roth, 2022 sourceNot checked — no full text of the source

The identification framework generalizes the partial identification framework in Manski and Pepper (2018).

“Our identification framework, which generalizes the partial identification framework in Manski and Pepper (2018), thus helps formalize the intuition motivating pre-trends testing.”

• cites Manski and Pepper (2018) sourceSupported

Parametric approaches to controlling for pre-existing trends may be sensitive to functional form assumptions.

“Third, parametric approaches to controlling for pre-existing trends may be sensitive to functional form assumptions (Wolfers, 2006;Lee and Solon, 2011).”

• cites Wolfers, 2006 sourceSupported
• cites Lee and Solon, 2011 sourceNot checked — no full text of the source

Chetty et al. (2014) estimate that a 1 standard deviation increase in teacher value-added corresponds with a 0.4 percentage point increase in adult employment.

“Chetty et al. (2014) estimate that a 1 standard deviation increase in teacher value-added (VA) corresponds with a 0.4 percentage point increase in adult employment.”

• cites Chetty et al. (2014) sourceSupported

In Monte Carlo simulations, the use of FLCIs can lead to substantial power gains over the conditional/hybrid approaches for Δ_SD when the length of the identified set is short relative to the sampling variation in the data.

“In Monte Carlo simulations, we find that the use of such FLCIs can lead to substantial power gains over the conditional/hybrid approaches for SD when the length of the identified set is short relative to the sampling variation in the data.”

Conducting inference on θ can be cast as the problem of testing a system of moment inequalities.

“This approach is based on the observation that conducting inference on θ can be cast as the problem of testing a system of moment inequalities”

The paper introduces methods that yield uniformly valid inference for the treatment effect under the imposed restrictions.

“We then introduce methods that yield uniformly valid inference for the treatment effect under the imposed restrictions.”

The paper introduces methods to conduct uniformly valid inference on a scalar parameter θ = l'τ_post under the restriction δ ∈ Δ.

“We then introduce methods to conduct uniformly valid inference on a scalar parameter of the form θ = l τ post under the restriction δ ∈ .”

• AssumptionThe authors state this as an assumption: The difference in trends δ lies in a researcher-specified set Δ.

The conditional and hybrid tests are consistent, in the sense that they have power approaching 1 against fixed alternatives outside of the identified set.

“First, the conditional and hybrid tests are consistent, in the sense that they have power approaching 1 against fixed alternatives outside of the identified set.”

These results are applicable for one of the leading examples, Δ_SD, which restricts the smoothness of the difference in trends.

“These results are applicable for one of our leading examples, SD , which restricts the smoothness of the difference in trends.”

The conditional test has optimal local asymptotic power under a linear independence constraint qualification (LICQ) assumption.

“Second, we prove that the conditional test has optimal local asymptotic power under a linear independence constraint qualification (LICQ) assumption.”

• AssumptionThe authors state this as an assumption: The linear independence constraint qualification (LICQ) holds for the optimal local asymptotic power result.

The ARP hybrid test will have near-optimal local asymptotic power under LICQ.

“Our result also implies that the ARP hybrid test will have near-optimal local asymptotic power under LICQ.”

• AssumptionThe authors state this as an assumption: The linear independence constraint qualification (LICQ) holds for the optimal local asymptotic power result.

The first general inference approach can accommodate a large class of choices for Δ.

“We first introduce a general inference approach that can accommodate a large class of choices for .”

Uniform size control for these tests follows nearly immediately from results in ARP.

“Uniform size control for these tests follows nearly immediately from results in ARP.”

• cites Andrews et al. (forthcoming, henceforth ARP)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The moments have a potentially large number of nuisance parameters that enter linearly.

“The moments have a potentially large number of nuisance parameters that enter linearly”

The causal parameter τ_post is partially identified under restrictions δ ∈ Δ for some researcher-specified set Δ.

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

• AssumptionThe authors state this as an assumption: The difference in trends δ lies in a researcher-specified set Δ.

In non-staggered DiD, under no anticipation, β_s will be consistent for β_s = τ_ATT,s + δ_s, where τ_ATT,s is the ATT and δ_s is the difference in trends.

“Then, for any s, under mild regularity conditions βs will be consistent for”

• AssumptionThe authors state this as an assumption: There is no anticipation of treatment, so that Y_it(1) = Y_it(0) for all t < 1.

Under the assumption that δ ∈ Δ = {δ : δ_post = 0}, the parameter θ will typically be set-identified.

“Under the assumption that δ ∈ = {δ : δ post = 0}, the parameter θ will typically be setidentified.”

The larger the magnitude of the pre-treatment violation in parallel trends, |δ_{-1}|, the wider the range of possible post-treatment violations of parallel trends.

“The larger the magnitude of the pre-treatment violation in parallel trends, |δ -1 |, the wider the range of possible post-treatment violations of parallel trends.”

In staggered DiD, β_r will be consistent for β_r = τ_r + δ_r, where τ_r is a weighted average of ATTs and δ_r is a weighted average of differences in trends.

“Then βr will be consistent for the parameter β r = τ r + δ r , where”

• AssumptionThe authors state this as an assumption: In staggered DiD, under a no-anticipation assumption, τ_r = 0 for r < 0.

The paper obtains partial identification of θ by assuming that δ lies in a researcher-specified set of possible differences in trends Δ ⊆ R^{T+T}.

“We obtain partial identification of θ by assuming that δ lies in a researcher-specified set of possible differences in trends ⊆ R T + T .”

• AssumptionThe authors state this as an assumption: The difference in trends δ lies in a researcher-specified set Δ.

When Δ is closed and convex, the identified set for the ATT in period 1 is [β_1 - b_max, β_1 - b_min], where b_max = max_δ δ_1 s.t. (δ_{-1}, δ_1) ∈ Δ and b_min is defined analogously.

“When is closed and convex, the identified set for the ATT in period 1 is [β 1b max , β 1b min ], where b max = max δ δ 1 s.t (δ -1 , δ 1 ) ∈ is the maximum possible bias of β1 given δ -1 = β -1 and b min is defined analogously.”

In the three-period DiD model (T=1, T'=1), the ATT in period 1 is point identified if we assume that the counterfactual post-treatment difference in trends δ_1 is exactly zero (parallel trends).

“In the three-period DiD model ( T = T = 1), the ATT in period 1 is point identified if we assume that the counterfactual post-treatment difference in trends δ 1 is exactly zero (parallel trends).”

The methodology can be used in conjunction with alternative estimators to assess their sensitivity to violations of the corresponding parallel trends assumption.

“Our methodology can be used in conjunction with these alternative estimators to assess their sensitivity to violations of the corresponding parallel trends assumption; see Section 2.1 for additional details.”

For fixed a and v, the smallest value of χ that satisfies the coverage requirement is the 1-α quantile of the |N(b, vΩv)| distribution, where b is the affine estimator's worst-case bias.

“For fixed values a and v, the smallest value of χ that satisfies (10) is therefore the 1 -α quantile of the |N b, v n v | distribution, where b is the affine estimator's worst-case bias”

• AssumptionThe authors state this as an assumption: The finite-sample normal approximation (9) holds exactly with Ωn known.

FLCIs based on affine estimators offer finite-sample power guarantees (in the normal model) for certain classes of interest.

“While the conditional and hybrid confidence sets offer attractive asymptotic power guarantees under asymptotics in which sampling variation grows small relative to the length of the identified set, FLCIs offer finite-sample power guarantees (in the normal model) for certain classes of interest.”

• AssumptionThe authors state this as an assumption: The finite-sample normal approximation (9) holds exactly with Ωn known.

The paper's tests control size even when LICQ fails.

“Thus, our tests control size even when LICQ fails (and so the bounds may be non-differentiable)”

The paper's size control results do not require LICQ.

“One important distinction between our results and previous results using LICQ is that we do not require LICQ for our size control results (Proposition 3.1).”

If LICQ holds for a particular set of moments, then it also holds if one adds moments that are slack at the optimal τ*post.

“It is worth highlighting that if LICQ holds for a particular set of moments, then it also holds if one adds moments that are slack at the optimal τ * post .”

When α = 0.05, the lower bound in Proposition 4.1 evaluates to 0.72, meaning that the expected length of the shortest possible confidence set that satisfies the coverage requirement is at most 28% shorter than the length of the optimal FLCI when the conditions of the proposition hold.

“When α = 0.05, the lower bound in Proposition 4.1 evaluates to 0.72, meaning that the expected length of the shortest possible confidence set that satisfies the coverage requirement (10) is at most 28% shorter than the length of the optimal FLCI when the conditions of the proposition hold.”

• AssumptionThe authors state this as an assumption: The finite-sample normal approximation (9) holds exactly with Ωn known.
• AssumptionThe authors state this as an assumption: Assumption 8: (i) Θ is convex and centrosymmetric (i.e. δ ∈ Θ implies -δ ∈ Θ), and (ii) δ ∈ Θ is such that (δ - δ') ∈ Θ for all δ' ∈ Θ.

If Θ = RM(M) and M > 0, then all affine estimators for τ1 have infinite worst-case bias.

“If = RM ( M) and M > 0, then all affine estimators for τ 1 have infinite worst-case bias, since δ ∈ RM ( M) can have |δ 1 | arbitrarily large if |δ -1 | is also sufficiently large.”

When M = 0, both the upper and lower bounds for SD(M) and SDPB(M) are binding, so LICQ fails.

“When M = 0, both the upper and lower bounds for SD (M) and S D P B (M) are binding, so LICQ fails.”

When Θ is a union of K polyhedra, the conditional test based on the union of confidence sets has optimal local asymptotic power when the k that determines the identified set bounds is unique and satisfies the conditions of Proposition 3.3.

“An immediate corollary, however, is that when = K k=1 k , the conditional test based on the union of confidence sets has optimal local asymptotic power when the k that determines the identified set bounds is unique and satisfies the conditions of Proposition 3.3.”

The identified set S(β, Θ) is an interval when Θ is convex, with length equal to θub(β, Θ) - θlb(β, Θ) = bmax(βpre, Θ) - bmin(βpre, Θ).

“Recall from Lemma 2.1 that the identified set S(β, ) is an interval when is convex, with length equal to θ ub (β, ) -θ lb (β, ) = b max (β pre , )b min (β pre , ).”

The LICQ condition rules out settings where θ is point identified.

“Importantly, it can be shown that the LICQ condition rules out settings where θ is point identified.”

Proposition 4.1 bounds the ratio of the expected length of the shortest possible confidence interval that controls size relative to the length of the optimal FLCI.

“The following result, which is an immediate consequence of results in Armstrong andKolesár (2018, 2020a), bounds the ratio of the expected length of the shortest possible confidence interval that controls size relative to the length of the optimal FLCI.”

• cites Armstrong andKolesár (2018, 2020a)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: The finite-sample normal approximation (9) holds exactly with Ωn known.
• AssumptionThe authors state this as an assumption: Assumption 8: (i) Θ is convex and centrosymmetric (i.e. δ ∈ Θ implies -δ ∈ Θ), and (ii) δ ∈ Θ is such that (δ - δ') ∈ Θ for all δ' ∈ Θ.

The result in Proposition 3.3 is under the asymptotic regime where sampling variation grows small relative to the length of the identified set, and thus the binding and non-binding moments are far apart relative to sampling variation.

“More generally, the result in Proposition 3.3 is under the asymptotic regime where the sampling variation grows small relative to the length of the identified set, and thus the binding and non-binding moments are”

The optimal FLCI is consistent if and only if LID(βpre, Θ) is its maximum possible value, provided that the identified set is not the entire real line.

“Our next result shows that C F LC I α,n ( βn , n ) is consistent if and only if L I D(β pre , ) is its maximum possible value, provided that the identified set is not the entire real line (in which case any procedure is trivially consistent).”

• AssumptionThe authors state this as an assumption: The finite-sample normal approximation (9) holds exactly with Ωn known.
• AssumptionThe authors state this as an assumption: Assumption 9: Identified set maximal length and finite: δ ∈ Θ such that LID(δpre, Θ) = sup_{δpre ∈ Θpre} LID(δpre, Θ) < ∞.

In the three-period DiD model, LICQ is satisfied when the bounds of the identified set are each determined by one moment.

“In the three period DiD model (where there are no nuisance parameters, since T = 1), LICQ is satisfied when the bounds of the identified set are each determined by one moment.”

The finite-sample guarantees of FLCIs do not apply for several types of restrictions of importance, including those that construct bounds using the maximum pre-treatment violation or incorporate sign and shape restrictions.

“As discussed above, these finite-sample guarantees do not apply for several types of restrictions of importance, including those that construct bounds using the maximum pre-treatment violation or incorporate sign and shape restrictions.”

Under LICQ, the conditional test's local asymptotic power converges to the power envelope for tests controlling size in the finite-sample normal model.

“Proposition 3.3 shows that under LICQ the local asymptotic power of the conditional test converges to the power envelope for tests controlling size in the finite-sample normal model.”

LICQ holds everywhere for SD(M) when M > 0.

“This holds everywhere for SD (M) when M > 0.”

In particular cases of interest, such as when Θ = SD(M), the optimal FLCIs have near-optimal expected length in the finite-sample normal model.

“In particular cases of interest, such as when = SD (M), the optimal FLCIs introduced above have near-optimal expected length in the finite-sample normal model.”

• AssumptionThe authors state this as an assumption: The finite-sample normal approximation (9) holds exactly with Ωn known.
• AssumptionThe authors state this as an assumption: Assumption 8: (i) Θ is convex and centrosymmetric (i.e. δ ∈ Θ implies -δ ∈ Θ), and (ii) δ ∈ Θ is such that (δ - δ') ∈ Θ for all δ' ∈ Θ.

Even as the sampling variation Ωn converges to 0, the optimal FLCI will include fixed points outside of the identified set with positive probability unless certain special conditions are met.

“Specifically, we will show that even as the sampling variation n converges to 0, the optimal FLCI will include fixed points outside of the identified set with positive probability unless certain special conditions are met.”

• AssumptionThe authors state this as an assumption: The finite-sample normal approximation (9) holds exactly with Ωn known.

Proposition 3.3 requires that the asymptotic power of the test is not affected by the inclusion of slack moments.

“Proposition 3.3 thus requires that the asymptotic power of the test is not affected by the inclusion of slack moments.”

The worst-case bias of an affine estimator over SDPB(M) or SDI(M) is the same as the worst-case bias for that estimator over SD(M).

“It can be shown that the worst-case bias of an affine estimator over S D P B (M) or S DI (M) is the same as the worstcase bias for that estimator over SD (M).”

For SDRM(M) and RM(M), the choice of δ is more important than the choice of M.

“By contrast, for S D RM ( M) and RM ( M), the choice of δ is more important than the choice of M.”

If M* is the identification breakdown point for a null effect and M* is the sample breakdown point, then P(M* ≥ M*) ≥ P(0 ∈ C(βn, ˆn; (M*))).

“Note, however, that if we define M * = min M s.t.0 ∈ S(β, (M)) to be the identification breakdown point for a null effect, and M * = min M s.t.0 ∈ C( βn , ˆ n ; (M)) to be the sample breakdown point, then P( M * ≥ M * ) ≥ P(0 ∈ C( βn , ˆ n ; (M * )).”

To ensure that an FLCI satisfies the coverage requirement, its length must be at least sup_{δpre ∈ Θpre} LID(δpre, Θ).

“The intuition for the possible inconsistency of FLCIs is as follows: to ensure that an FLCI satisfies the coverage requirement (10), its length must be at least sup δpre ∈ pre L I D( δpre , ).”

FLCIs can perform quite poorly for other classes of Θ.

“However, FLCIs can perform quite poorly for other classes of .”

By contrast, the length of the conditional/hybrid confidence sets can depend on βpre and thus adapts to the length of the identified set.

“By contrast, the length of the conditional/hybrid confidence sets can depend on β pre and thus”

For M < 0.01 the robust confidence set contains only positive values.

“Indeed, for M < 0.01 the robust confidence set contains only positive values.”

The paper derives novel results on the power of these procedures.

“We provide inference procedures that are uniformly valid so long as the difference in trends satisfies these restrictions, and derive novel results on the power of these procedures.”

The point estimate for t = 15 lies above a linear extrapolation of the negative pre-trend.

“Intuitively, this is because the point estimate for t = 15 lies above a linear extrapolation of the negative pre-trend.”

For RM(M), the conditional and conditional-least favourable hybrid have similar power.

“Once again, we also find that the conditional and conditional-least favourable hybrid have similar power.”

The simulation study finds good size control for all of the procedures.

“We find good size control for all of the procedures, and therefore focus in the main text on a comparison of power to provide concrete recommendations on the best approach in practice.”

The paper provides inference procedures that are uniformly valid so long as the difference in trends satisfies the restrictions.

“We provide inference procedures that are uniformly valid so long as the difference in trends satisfies these restrictions, and derive novel results on the power of these procedures.”

When allowing for linear violations of parallel trends (M = 0), the picture changes substantially owing to the pre-existing downward trend visible in Figure 6.

“When we allow for linear violations of parallel trends (M = 0), however, the picture changes substantially owing to the pre-existing downward trend that is visible in Figure 6.”

For SD(M), FLCIs perform best for smaller values of M/σ1, including the point-identified case where M = 0, illustrating the finite-sample near-optimality results for the FLCIs when Assumption 8 holds.

“However, the FLCIs perform best for smaller values of M/σ 1 , including the point-identified case where M = 0, illustrating the finite-sample near-optimality results for the FLCIs when Assumption 8 holds.”

• AssumptionThe authors state this as an assumption: Assumption 8: (i) Θ is convex and centrosymmetric (i.e. δ ∈ Θ implies -δ ∈ Θ), and (ii) δ ∈ Θ is such that (δ - δ') ∈ Θ for all δ' ∈ Θ.

For SD(M), the conditional and hybrid confidence sets perform similarly.

“The conditional and hybrid confidence sets perform similarly.”

(-∞, M*] is a valid (1 - α)-level confidence interval for M* provided that the conditions for size control are satisfied for (M*).

“It follows that (-∞, M * ] is a valid (1 -α)-level confidence interval for M * provided that our conditions for size control are satisfied for (M * ).”

The results for women suggest a downward-sloping pre-existing trend.

“By contrast, the results for women (right panel) suggest a downward-sloping pre-existing trend.”

For RM(M), the conditions for the asymptotic (near-)optimality result for unions of convex sets do not hold in this simulation design (as the maximum pre-period violation is not unique).

“The conditions for our asymptotic (near-) optimality result for unions of convex sets do not hold in this simulation design (as the maximum pre-period violation is not unique).”

The reported confidence sets are between 40 and 80% longer than the estimated identified set.

“We find that the reported confidence sets are between 40 and 80% longer than the estimated identified set, suggesting that both sampling uncertainty and the length of the identified set play an important role in the width of the confidence sets.”

The breakdown value for a null effect is around M = 2.

“Looking further to the right, we see that the”

For SD(M), all procedures perform well as M/σ1 grows large with efficiency ratios approaching 1, illustrating the asymptotic (near-)optimality results for this design.

“All procedures perform well as M/σ 1 grows large with efficiency ratios approaching 1, illustrating our asymptotic (near-)optimality results for this design.”

For SDRM(M), both procedures perform well as δ-1/σ1 grows large with efficiency ratios approaching 1, illustrating the asymptotic (near-)optimality result for this design.

“Both procedures perform well as δ -1 /σ 1 grows large with efficiency ratios approaching 1, illustrating our asymptotic (near-) optimality result for this design.”

For RM(M), the efficiency ratios for the conditional and hybrid confidence sets approach about 83% when δ-1/σ1 grows large.

“Nonetheless, we find that the efficiency ratios for the conditional and hybrid confidence sets approach about 83% when δ -1 /σ 1 grow large.”

Under the restriction M = 1, the robust confidence set for the causal effect on restaurant profits in 2009 is [0.07, 0.31].

“The figure shows that if we impose M = 1, meaning that we restrict the post-treatment violations of parallel trends to be no larger than the maximal pretreatment violation of parallel trends, then we obtain a robust confidence set of [0.07, 0.31] for the causal effect on restaurant profits in 2009.”

The procedures perform similarly in simulations based on the empirical distribution in the original paper.

“In the supplementary material, we show that our procedures perform similarly in simulations based on the empirical distribution in the original paper.”

For SDPB(M), the efficiency ratios for the conditional and hybrid confidence sets are again (near-)optimal as M/σ1 grows large, highlighting the asymptotic (near-)optimality results for these procedures in this simulation design.

“The efficiency ratios for the conditional and hybrid confidence sets are again (near-)optimal as M/σ 1 grows large, highlighting our asymptotic (near-)optimality results for these procedures in this simulation design.”

The sample analogue to the identified set, S(β,), is a Hausdorff consistent estimator of the identified set in this example.

“Finally, in Supplementary Material, Appendix C, we compare the confidence sets reported in Figure 5 to the sample analogue to the identified set, S( β, ), which we show is a Hausdorff consistent estimator of the identified set in this example.”

The post-treatment coefficients for 2009-2011 appear to be substantially larger in magnitude than any of the pre-trends coefficients.

“Nevertheless, the post-treatment coefficients for 2009-2011 appear to be substantially larger in magnitude than any of the pre-trends coefficients.”

Restrictions on possible post-treatment differences in trends given pre-trends formalize the intuition that pre-trends are informative about counterfactual post-treatment differences in trends.

“Such restrictions formalize the intuition motivating pre-trends tests, namely that pre-trends are informative about counterfactual post-treatment differences in trends.”

Conditioning the analysis on passing pre-trends tests introduces statistical issues related to pre-testing.

“conditioning the analysis on passing pre-trends tests introduces statistical issues related to pre-testing (Roth, 2022).”

• cites Roth, 2022 sourceNot checked — no full text of the source

The pre-treatment coefficients β_pre are often imprecise estimates of δ_pre.

“As emphasized in the recent literature on pretrends testing, the pre-treatment coefficients β pre are often imprecise estimates of δ pre .”

When Δ is convex and centrosymmetric, FLCIs have near-optimal expected length in the finite-sample normal model.

“In particular, results from Armstrong andKolesár (2018, 2020b) imply that when is convex and centrosymmetric, FLCIs have near-optimal expected length in the finite-sample normal model.”

• cites Armstrong andKolesár (2018, 2020b)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Event-study estimates of the form βn = (βn,pre, βn,post) arise from non-staggered DiD as well as a variety of related estimators.

“Eventstudy estimates of this form arise from non-staggered DiD as well as a variety of related estimators, as we illustrate with several examples”

The conditional and hybrid methods proposed by ARP have (near-)optimal local asymptotic power in our setting when an LICQ condition is satisfied.

“From the perspective of power, we will show that the tests proposed by ARP have (near-)optimal local asymptotic power in our setting when an LICQ condition is satisfied.”

• cites ARPNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The testing problem (13) is a special case of the problem studied in ARP, which focuses on testing null hypotheses of the form H0: ∃τ such that E[Y(θ) - Xτ | X] ≤ 0 (a.s.).

“The testing problem (13) is a special case of the problem studied in ARP, which focuses on testing null hypotheses of the form H 0 : ∃τ s.t. E [Y (θ ) -X τ | X ] ≤ 0 (a.s.).”

• cites ARPNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The null H0 is equivalent to (13): ∃τ ∈ R^{T-1} such that E[Ỹ_n(θ) - Xτ] ≤ 0.

“The null H 0 is then equivalent to H 0 : ∃ τ ∈ R T -1 s.t. E Ỹn ( θ) -X τ ≤ 0, (13”

The null hypothesis H0 is equivalent to the composite null (12): ∃τ_post ∈ R^T such that l'τ_post = θ and E[Y_n - AL_post τ_post] ≤ 0.

“it is immediate that the null hypothesis H 0 is equivalent to the composite null H 0 : ∃τ post ∈ R T s.t. l τ post = θ and E βn ∼N (β, n ) Y n -AL post τ post ≤ 0. (12”

If for each k = 1, ..., K the confidence set C_{n,k} satisfies (10) with Δ = Δ_k, then the confidence set C_n = union_{k=1}^K C_{n,k} satisfies (10) with Δ = union_{k=1}^K Δ_k.

“Lemma 2.2.Suppose that for each k = 1, . . ., K , the confidence set C n,k ( βn , n ) satisfies (10) with = k .Then the confidence set C n ( βn , n ) = K k=1 C n,k ( βn , n ) satisfies (10) with = K k=1 k .”

The linear group-specific time trend approach is valid if the difference in trends is linear.

“This approach is valid if the difference in trends is linear”

In the three-period DiD model, assuming the differential trend is exactly linear is equivalent to assuming Δ = {δ : δ1 = -δ_{-1}}.

“In the three-period DiD model, assuming the differential trend is exactly linear is equivalent to assuming = {δ : δ 1 = -δ -1 }.”

Testing the null hypothesis H0: θ = θ, δ ∈ Δ is equivalent to testing a system of moment inequalities with linear nuisance parameters in the normal model.

“As a first step, we show that testing H 0 is equivalent to testing a system of moment inequalities with linear nuisance parameters in the normal model.”

All of the examples described above can be written either as polyhedral restrictions or finite unions of such restrictions.

“All of the examples described above can be written either as polyhedral restrictions or finite unions of such restrictions.”

Statistical distortions from pre-trends tests may further undermine the performance of conventional inference procedures.

“Second, statistical distortions from pre-trends tests may further undermine the performance of conventional inference procedures (Roth, 2022).”

• cites Roth, 2022 sourceNot checked — no full text of the source

The null hypothesis (13) implies γ*'μ(θ) ≤ 0.

“Moreover, the null hypothesis (13) implies γ * μ( θ) ≤ 0.”

The ARP conditional test uses the critical value max{0, c_C,α}, where c_C,α is the 1-α quantile of the truncated normal distribution under the worst-case assumption that γ*'μ(θ) = 0.

“The ARP conditional test therefore uses the critical value max{0, c C,α }, where c C,α is the 1 -α quantile of the truncated normal distribution ξ | ξ ∈ [v lo , v up ] under the worst-case assumption that γ * μ( θ) = 0.”

• cites ARPNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Suppose Assumptions 2-4 hold. Then for any ε > 0, x > 0, and α < 0.5, the local power of the conditional test converges to the power envelope for tests that control size in the finite sample normal model.

“Proposition 3.3.Suppose Assumptions 2-4 hold.Let θ ub P = sup S(β P , ).Then for any > 0, x > 0, and α < 0.5, lim n→∞ sup P∈P E P ψ C α βn , A, d, θ ub P + 1 √ n x, 1 n ˆ n -ρ * α (P, x) = 0,”

• AssumptionThe authors state this as an assumption: Uniform asymptotic normality of βn holds: the bounded-Lipschitz distance between √n(βn - βP) and a N(0, ΣP) variable converges to zero uniformly over P ∈ P, where βP = δP + L_post τP,post for δP ∈ Δ and τP,post ∈ R^T.
• AssumptionThe authors state this as an assumption: The asymptotic variance matrices ΣP have eigenvalues bounded below by λ > 0 and above by λ̄ ≥ λ, uniformly over P ∈ P.
• AssumptionThe authors state this as an assumption: There exists an estimator Σ̂n that is uniformly consistent for ΣP over P ∈ P.

In certain special cases, FLCIs may outperform the ARP tests when sampling variation is large relative to the length of the identified set.

“In certain special cases, FLCIs may thus outperform the ARP tests when sampling variation is large relative to the length of the identified set.”

Coverage in the normal model translates to uniform asymptotic coverage over a large class of data-generating processes.

“In Section 3.3 below, we show that coverage in the normal model translates to uniform asymptotic coverage over a large class of DGPs.”

Suppose Assumptions 2-5 hold. Then the conditional and LF-hybrid tests uniformly control size.

“Proposition 3.1.Suppose Assumptions 2-5 hold.Then the conditional and LF-hybrid tests uniformly control size.”

• AssumptionThe authors state this as an assumption: Uniform asymptotic normality of βn holds: the bounded-Lipschitz distance between √n(βn - βP) and a N(0, ΣP) variable converges to zero uniformly over P ∈ P, where βP = δP + L_post τP,post for δP ∈ Δ and τP,post ∈ R^T.
• AssumptionThe authors state this as an assumption: The asymptotic variance matrices ΣP have eigenvalues bounded below by λ > 0 and above by λ̄ ≥ λ, uniformly over P ∈ P.
• AssumptionThe authors state this as an assumption: There exists an estimator Σ̂n that is uniformly consistent for ΣP over P ∈ P.
• AssumptionThe authors state this as an assumption: The matrix A satisfies at least one of two regularity conditions: (A) A can be written as TQ with Q full row-rank and T a specified block matrix; or (B) for the vertices γ1, ..., γK of V(I), for all k either γk'A = 0 or inf_{a≥0} inf_{j≠k} (γk - aγj)'A > 0.

The conditional test has optimal local asymptotic power for SDRM(M) when there is a unique maximum non-linearity in the pre-treatment period.

“Likewise, the conditional test has optimal local asymptotic power for S D RM ( M) when there is a unique maximum non-linearity in the pre-treatment period.”

The optimal FLCI constructed using SDPB(M) or SDI(M) is the same as the one constructed using SD(M).

“Since the construction of the optimal FLCI depends only on the worst-case bias and variance of the affine estimator, it follows that the optimal FLCI constructed using S D P B (M) or S DI (M) is the same as the one constructed using SD (M).”

For the case where Δ is the finite union of polyhedra, a valid confidence set can be constructed by taking the union of the confidence sets for each of its components.

“For the case where is the finite union of polyhedra, a valid confidence set can be constructed by taking the union of the confidence sets for each of its components.”

The assumptions stated above are sufficient for the conditions in Proposition 2 in ARP, which establishes uniform size control for the conditional and hybrid tests.

“The assumptions stated above are sufficient for the conditions in Proposition 2 in ARP, which establishes uniform size control for the conditional and hybrid tests.”

• cites ARPNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: Uniform asymptotic normality of βn holds: the bounded-Lipschitz distance between √n(βn - βP) and a N(0, ΣP) variable converges to zero uniformly over P ∈ P, where βP = δP + L_post τP,post for δP ∈ Δ and τP,post ∈ R^T.
• AssumptionThe authors state this as an assumption: The asymptotic variance matrices ΣP have eigenvalues bounded below by λ > 0 and above by λ̄ ≥ λ, uniformly over P ∈ P.
• AssumptionThe authors state this as an assumption: There exists an estimator Σ̂n that is uniformly consistent for ΣP over P ∈ P.
• AssumptionThe authors state this as an assumption: The matrix A satisfies at least one of two regularity conditions: (A) A can be written as TQ with Q full row-rank and T a specified block matrix; or (B) for the vertices γ1, ..., γK of V(I), for all k either γk'A = 0 or inf_{a≥0} inf_{j≠k} (γk - aγj)'A > 0.

Thus, the only valid FLCI for RM(M) with M > 0 is the entire real line.

“Thus, the only valid FLCI is the entire real line.”

The optimal FLCI does not adapt to additional sign or monotonicity restrictions.

“Therefore, the optimal FLCI does not adapt to additional sign or monotonicity restrictions.”

The asymptotics considered in Proposition 3.3 may not provide a good approximation to the finite-sample performance of the conditional test in settings where θ is point-identified, or when the length of the identified set is small relative to sampling variation.

“Thus, the asymptotics considered in Proposition 3.3 may not provide a good approximation to the finite-sample performance of the conditional test in settings where θ is point-identified, or when the length of the identified set is”

Since the LF-hybrid test rejects whenever the conditional test with size (α-κ)/(1-κ) rejects, its local asymptotic power is at least as good as the power of the optimal size-((α-κ)/(1-κ)) test.

“Since the LF-hybrid test rejects whenever the conditional test with size (α -κ)/(1 -κ) rejects, it is immediate that the local asymptotic power of the LF-hybrid test is at least as good as the power of the optimal size-((α -κ)/(1 -κ)) test.”

Under the conditions of Proposition 3.3, the local asymptotic power of the LF-hybrid test is at least as good as the power of the optimal size-((α-κ)/(1-κ)) test.

“Corollary 3.1. Under the conditions of Proposition 3.3, lim inf n→∞ inf P∈P E P ψ C-L F κ,α βn , A, d, θ ub P + 1 √ n x, 1 n ˆ n -ρ * (α-κ)/(1-κ) (P, x) ≥ 0.”

The hybrid confidence set satisfies coverage criterion (10) in the finite-sample normal model.

“As with the conditional test, by construction the hybrid confidence set satisfies that coverage criterion (10) in the finite-sample normal model.”

ARP show that all quantiles of the conditional distribution of η are increasing in γ*'μ(θ).

“ARP show that all quantiles of the conditional distribution of η in the previous display are increasing in γ * μ( θ).”

• cites ARPNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

If Assumption 9 fails, then the optimal FLCI is inconsistent in the strong sense that it includes fixed points outside of the identified set with non-vanishing probability.

“Thus, if Assumption 9 fails, then C F LC I α,n ( βn , n ) is inconsistent in the strong sense that it includes fixed points outside of the identified set with non-vanishing probability.”

• AssumptionThe authors state this as an assumption: Assumption 9: Identified set maximal length and finite: δ ∈ Θ such that LID(δpre, Θ) = sup_{δpre ∈ Θpre} LID(δpre, Θ) < ∞.

Suppose Assumptions 4-7 hold. Then for any x > 0 and α < 0.5, the conditional and LF-hybrid tests are uniformly asymptotically consistent against fixed alternatives outside the identified set.

“Proposition 3.2.Suppose Assumptions 4-7 hold.Then for any x > 0 and α < 0.5, lim n→∞ inf P∈P E P ψ C α βn , A, d, θ ub P + x, 1 n ˆ n = 1 lim n→∞ inf P∈P E P ψ C-L F κ,α βn , A, d, θ ub P + x, 1 n ˆ n = 1,”

• AssumptionThe authors state this as an assumption: There exists an estimator Σ̂n that is uniformly consistent for ΣP over P ∈ P.
• AssumptionThe authors state this as an assumption: The matrix A satisfies at least one of two regularity conditions: (A) A can be written as TQ with Q full row-rank and T a specified block matrix; or (B) for the vertices γ1, ..., γK of V(I), for all k either γk'A = 0 or inf_{a≥0} inf_{j≠k} (γk - aγj)'A > 0.
• AssumptionThe authors state this as an assumption: The vector Wn = ((β̂n - βP)', (vec(Σ̂n) - vec(ΣP))') has a joint normal asymptotic distribution uniformly over P ∈ P, with covariance matrix VP.
• AssumptionThe authors state this as an assumption: For all P ∈ P, ΣP ∈ S and VP lies in a compact set V; additionally, (ΣP - VP,β VP† VP,β') has eigenvalues bounded below by λ > 0, where † denotes the Moore-Penrose inverse.

The restrictiveness of Assumption 9 depends greatly on Θ.

“The restrictiveness of Assumption 9 thus depends greatly on .”

• AssumptionThe authors state this as an assumption: Assumption 9: Identified set maximal length and finite: δ ∈ Θ such that LID(δpre, Θ) = sup_{δpre ∈ Θpre} LID(δpre, Θ) < ∞.

There will be some δ ∈ Θ such that the FLCI is inconsistent under δ unless the identified set is always the same length.

“It follows that there will be some δ ∈ such that the FLCI is inconsistent under δ unless the identified set is always the same length.”

The inconsistency result shows that FLCIs may perform poorly for other choices of Θ that may be of interest in empirical applications, such as those that construct bounds using a pre-treatment maximum or incorporate sign and monotonicity restrictions.

“Our inconsistency result shows, however, that FLCIs may perform poorly for other choices of that may be of interest in empirical applications, such as those that construct bounds using a pre-treatment maximum or incorporate sign and monotonicity restrictions.”

If one bounds the magnitude of economic shocks by the maximum pre-treatment violation, one will typically obtain wider confidence sets for parameters involving later periods.

“If we are willing to bound the magnitude of economic shocks by the max in the pre-treatment period, we will thus typically obtain wider confidence sets for parameters involving later periods.”

A value of M = 0.01 would correspond with allowing the slope of the differential trend to change by the equivalent of a one-fourtieth of a standard deviation of teacher VA across consecutive periods.

“Hence, a value of M = 0.01 would correspond with allowing the slope of the differential trend to change by the equivalent of a one-fourtieth of a standard deviation of teacher VA across consecutive periods.”

The strength with which one can rule out a null effect depends on one's assessment of the economic plausibility of such non-linearities.

“Since the robust confidence sets for both men and women begin to include zero around this value of M, the strength with which we can rule out a null effect depends on our assessment of the economic plausibility of such non-linearities.”

• AssumptionRests on an assumption: Long-run trends are likely to evolve smoothly over time.

The identified set will be larger for later periods because the treatment and control groups have more time to diverge.

“Thus, the identified set will be larger for later periods, since the treatment and control groups have more time to diverge (e.g. the identified set for the second period will be twice as larger as for the first period).”

The procedures can have excess length within 17% of the optimum even in cases where LICQ fails.

“We find it somewhat encouraging that these procedures can have excess length within 17% of the optimum even in cases where LICQ fails.”

Large normalized values of M or δ-1 correspond with the case where the identified set is large relative to sampling variation, mimicking the asymptotic power results in which sampling variation grows small relative to the identified set.

“Large normalized values of M or δ -1 correspond with the case where the identified set is large relative to sampling variation, mimicking our asymptotic power results in which sampling variation grows small relative to the identified set.”

The conclusion of a significant effect on restaurant profits depends on whether one is willing to restrict post-treatment violations of parallel trends to be no more than twice as large as the maximal pre-treatment violation.

“Thus, our conclusion of a significant effect on restaurant profits depends on whether we are willing to restrict that the post-treatment violations of parallel trends can be no more than twice as large as the maximal pre-treatment violation.”

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