On Event Studies and Distributed-Lags in Two-Way Fixed Effects Models: Identification, Equivalence, and Generalization

Among event study papers that specify an empirical model, only 15% provide some information on the choices made at the ends of the effect window.

“among the studies that specify an empirical model, only 15% provide some information on the choices that have been made at the ends of the effect window-in many cases only verbally and not formally.”

The dynamic treatment effects β_l in the standard event study model of Equation (2) can be estimated with ordinary least squares including two-way fixed effects for units and time periods.

“The dynamic treatment effects 𝛽 𝓁 in the standard event study model of Equation ( 2) can be estimated with ordinary least squares including two-way fixed effects for units and time periods.”

Binning implies that if the reform had happened on or before t - l = 1996, D_l_i,t = D_4_i,t = 1 for all t from 2000 to 2010.

“Binning also implies that if the reform had happened on or before t = t -𝓁 = 2000 -4 = 1996 (rather than in t = 2005 as assumed in the example), D 𝓁 i,t = D 4 i,t = 1 for all t from 2000 to 2010.”

Binning of one or both endpoints allows for identification with or without the presence of never-treated units.

“Hence, binning of one or both endpoints allows for identification with or without the presence of never-treated units.”

The distributed-lag specification in Equation (8) is the first difference of the distributed-lag specification in levels Equation (9).

“The distributed-lag specification in Equation ( 8) is the first difference of the following distributed-lag specification in levels”

In the restricted model, adding a linear trend λ•l to the restricted number of dynamic treatment effects does not offset adding a linear trend to the secular time trend for observations outside the effect window.

“In this model, adding a linear trend 𝜆 • 𝓁 to the restricted number of dynamic treatment effects does not offset adding a linear trend to the secular time trend for the observations outside of the effect window”

It suffices to know all events from time period t - l + 1 = 1997 to t + |l| -1 = 2012 to estimate the model.

“It, therefore, suffices to know all events from time period t -𝓁 + 1 = 1997 to t = t + |𝓁| -1 = 2012 to estimate the model, see Remark 2.”

In the case of an always-treated unit, D_l_i,t = D_4_i,t is a constant, and its effect is absorbed in the unit fixed effect μ_i.

“In this case of an always-treated unit, D 𝓁 i,t = D 4 i,t is a constant, and its effect is absorbed in the unit fixed effect 𝜇 i .”

In the absence of both never-treated units and unit-period observations outside of the effect window, dynamic treatment effects and secular time fixed effects are not separately identified.

“In the absence of both never-treated units and unit-period observations outside of the effect window, dynamic treatment effects and secular time fixed effects are not separately identified.”

Binning of one or both endpoints generates unit-period observations outside of the effect window, which serve as control group observations.

“Binning of one or both endpoints generates unit-period observations outside of the effect window, which serve as control group observations.”

The equivalence between event study and distributed-lag models holds in the general case without restricting the effect window.

“We have proven the equivalence between event study and distributed-lag models in the general case without restricting the effect window.”

Taking first differences of the standard event study specification given in Equation (1), we can rewrite the event study specification into a distributed-lag model.

“Taking first differences of the standard event study specification given in Equation ( 1), we can rewrite the event study specification into a distributed-lag model:”

The never-treated units serve as a control group, which uniquely identifies the secular time trends θ_t if there is at least one control group observation for each time period t.

“These never-treated units will serve as a control group, which uniquely identifies the secular time trends 𝜃 t if there is at least one control group observation for each time period t.”

• AssumptionThe authors state this as an assumption: There are units that are never treated.

Restricting the effect window as in Remark 2 introduces restrictions that allow separately identifying dynamic effects β_l and secular time trends θ_t.

“In the following, we show that restricting the effect window as proposed in Remark 2 introduces restrictions that allow separately identifying dynamic effects, 𝛽 𝓁 , and secular time trends, 𝜃 t .”

To identify the dynamic treatment effects β_l, we need to observe at least one treated unit for each lag and lead j of the effect window.

“To identify the dynamic treatment effects 𝛽 𝓁 , we need to observe at least one treated unit for each lag and lead 𝑗 of the effect window.”

Alternatively, dynamic cumulative treatment effects β_l can be estimated as a regression in first differences without unit fixed effects.

“Alternatively, dynamic cumulative treatment effects 𝛽 𝓁 can be estimated as a regression in first differences without unit fixed effects”

Event study (ES) models with binned endpoints and distributed-lag (DL) models yield identical parameter estimates.

“In this section, we show that event study (ES) models with binned endpoints and distributed-lag (DL) models yield identical parameter estimates.”

Events on or after t + |l| = 2013 imply D_l_i,t = D_-3_i,t = 1 for all t from 2000 to 2010, whose effect is absorbed in μ_i.

“By the analogous argument, events on or after t = t + |𝓁| = 2010 + 3 = 2013 imply D 𝓁 i,t = D -3 i,t = 1 for all t from 2000 to 2010 whose effect is absorbed in 𝜇 i .”

We need to estimate one fewer incremental effect γ_l than event study coefficient β_l because we always normalize one dynamic treatment effect to zero.

“Note that we need to estimate one fewer incremental effect 𝛾 𝓁 than event study coefficient 𝛽 𝓁 because we always normalize one dynamic treatment effect to zero”

Equation (12) shows how to recover the dynamic treatment effects β_l as the sums of distributed-lag parameters γ_l.

“Equation ( 12) shows how to recover the dynamic treatment effects 𝛽 𝓁 as the sums of distributed-lag parameters 𝛾 𝓁 .”

As in the event study model, we need a normalization in the distributed-lag model because parameters β_l are only identified up to a constant due to the individual fixed effect μ_i.

“As in the event study model, we need a normalization in the distributed-lag model because parameters 𝛽 𝓁 are only identified up to a constant due to the individual fixed effect 𝜇 i”

In finite samples, the estimates obtained with the fixed effect estimator differ from the ones obtained with the first difference estimator.

“In finite samples, the estimates obtained with the fixed effect estimator differ from the ones obtained with the first difference estimator.”

The event study parameters β_l can be calculated from the distributed-lag parameters γ_l by using the difference γ_l = β_l - β_{l-1}.

“The event study parameters 𝛽 𝓁 can be calculated from the distributed-lag parameters 𝛾 𝓁 by using the difference 𝛾 𝓁 = 𝛽 𝓁 -𝛽 𝓁-1 .”

If the true dynamic treatment effect continues to unfold beyond l, fixed effects and first difference estimates can differ strongly.

“If, however, the true dynamic treatment effect continues to unfold beyond 𝓁, fixed effects and first difference estimates can differ strongly.”

The dynamic incremental treatment effects γ_l of the distributed-lag model in Equation (10) can be estimated with ordinary least squares including two-way fixed effects for units and time periods.

“The dynamic incremental treatment effects 𝛾 𝓁 of the distributed-lag model in Equation ( 10) can be estimated with ordinary least squares including two-way fixed effects for units and time periods.”

If the dynamic nature of the effect is modeled correctly, that is, if the effect is truly constant l periods after the event, the two estimators will be numerically similar.

“If the dynamic nature of the effect is modeled correctly, that is, if the effect is truly constant 𝓁 periods after the event, the two estimators will be numerically similar.”

Estimation in both levels and first differences is consistent (with N→∞) as discussed in Section 2.3 when the model is correctly specified.

“Estimation in both levels and first differences is consistent (with N → ∞) as discussed in Section 2.3 when the model is correctly specified.”

• AssumptionThe authors state this as an assumption: Individual treatment effects are homogeneous.
• AssumptionThe authors state this as an assumption: There is independence across units i.
• AssumptionThe authors state this as an assumption: The treatment adoption indicator ΔT_i,t is strictly exogenous.

The event study model with a restricted effect window between l and l and binned endpoints in Equation (2) is equivalent to a distributed-lag specification with l lags and |l|-1 leads.

“is therefore equivalent to a distributed-lag specification with 𝓁 lags and |𝓁| -1 leads”

• Also stated in: § 3.1 Equivalence · plain

  “Remark 6 (Equivalence of event study and distributed-lag model).The event study specification with binned endpoints at 𝓁 = 𝓁 and 𝓁 = 𝓁 as specified in Equations ( 2) and ( 3) is equivalent to a distributed-lag models with 𝓁 lags and |𝓁| -1 leads as given by Equation ( 10).”

The isomorphism between ES and DL models implies that the same data requirements have to apply to both the ES and DL model.

“The isomorphism between ES and DL models is also insightful when thinking about the data requirements to estimate the model because obviously the same data requirements have to apply to both the ES and DL model.”

Remark 7: Dynamic treatment effects β_l are recovered from distributed-lag parameters γ_l as cumulative effects starting from a reference period, typically the period before treatment adoption, according to Equation (12).

“Remark 7 (Recovery of dynamic cumulative treatment effects from a distributed-lag model).Dynamic treatment effects 𝛽 𝓁 are recovered from distributed-lag parameters 𝛾 𝓁 as cumulative effects starting from a reference period, typically the period before treatment adoption, according to Equation ( 12).”

The standard event study design can be generalized to accommodate institutional set-ups where multiple treatments with known but varying treatment intensities take place.

“In this section, we show that the standard event study design can be generalized to accommodate institutional set-ups where multiple treatments with known but varying treatment intensities take place.”

• AssumptionRests on an assumption: In many applications, treatments may occur repeatedly and be of different signs and varying intensities across units and/or time.

Alternatively, the dynamic incremental treatment effects γ_l can be estimated as a regression in first differences without unit fixed effects.

“Alternatively, the dynamic incremental treatment effects 𝛾 𝓁 can be estimated as a regression in first differences without unit fixed effects”

The distributed-lag parameters γ_l are related to the event study parameters β_l by γ_l = β_l - β_{l-1}.

“The distributed-lag parameters 𝛾 𝓁 are related to the event study parameters 𝛽 𝓁 by 𝛾 𝓁 = 𝛽 𝓁 -𝛽 𝑗-1 .”

When the effect window is too short and dynamic effects have not fully materialized after l periods, the dynamic effect β_l at the endpoint often shows a pronounced spike when estimated in levels but not when estimated in first differences.

“When the effect window is too short and dynamic effects have hence not fully materialized after 𝓁 periods, the dynamic effect 𝛽 𝓁 at the endpoint often shows a pronounced spike when estimated in levels but not when estimated in first differences.”

We can construct dynamic cumulative treatment effects from the estimated incremental effects.

“Hence, we can construct these dynamic cumulative treatment effects from the estimated incremental effects.”

The distributed-lag coefficients γ_l measure incremental changes in the dynamic (cumulative) treatment effects.

“The distributed-lag coefficients 𝛾 𝓁 measure incremental changes in the dynamic (cumulative) treatment effects.”

The equivalence between ES and DL models for restricted effect windows holds if and only if endpoints of the effect window are binned as in Remark 2.

“Next, we show that the equivalence between ES and DL models for restricted effect windows holds if and only if endpoints of the effect window are binned as in Remark 2.”

The estimates of the generalized event study imply that a ten-week increase in potential benefit duration leads to a decrease in log job search activity of -0.027 (s.e.= 0.007), that is, 2.7%, after 4 months.

“The estimates of the generalized event study imply that a ten-week increase in potential benefit duration leads to a decrease in log job search activity of -0.027 (s.e.= 0.007), that is, 2.7%, after 4 months, which corresponds to the difference-in-difference estimates reported above.”

Conditional on unit fixed effects, a unit whose event takes place at t - l cannot be used to identify treatment effects.

“Conditional on unit fixed effects, this unit cannot be used to identify treatment effects.”

• AssumptionThe authors state this as an assumption: Dynamic treatment effects are constant before and/or after the endpoints of the effect window.

In the absence of never-treated units, at least one restriction on dynamic treatment effects in addition to normalization is needed to identify dynamic treatment effects.

“In the absence of never-treated units, at least one restriction on dynamic treatment effects in addition to the normalization is needed to identify dynamic treatment effects”

Events happening after the last data year might affect pre-treatment effects.

“Likewise, events happening after t might affect pre-treatment effects.”

Restricting the effect window to a finite number of leads and/or lags requires assumptions about the nature of the effect outside of the window.

“Restricting the effect window to a finite number of leads, 𝓁, and/or lags, 𝓁, requires assumptions about the nature of the effect outside of the window.”

• AssumptionThe authors state this as an assumption: Dynamic treatment effects are constant before and/or after the endpoints of the effect window.

If treatment adoption is derived from changes in treatment status, one needs to observe treatment status from t - l to t + |l| - 1.

“If treatment adoption is derived from changes in treatment status we need to observe treatment status from t -𝓁 to t + |𝓁| -1.”

• AssumptionThe authors state this as an assumption: Dynamic treatment effects are constant before and/or after the endpoints of the effect window.

Dropping observations outside the effect window requires the presence of never-treated units to identify dynamic treatment effects.

“This approach requires the presence of never-treated units to identify dynamic treatment effects.”

For a given balanced panel of the dependent variable from [t, t] and a limited effect window [l, l], one needs to observe treatment adoption from t - l + 1 to t + |l| - 1.

“For a given balanced panel of the dependent variable from [t, t] and a limited effect window [𝓁, 𝓁],foot_5 we need to observe treatment adoption from t -𝓁 + 1 to t + |𝓁| -1.”

• AssumptionThe authors state this as an assumption: Dynamic treatment effects are constant before and/or after the endpoints of the effect window.

Details of the empirical model often imply identifying assumptions, which lead to misspecification if they are not warranted.

“We show below that these details of the empirical model often imply identifying assumptions, which lead to misspecification if they are not warranted.”

The equivalence between distributed-lag and event study models also holds in the general case with multiple treatments of different signs and varying intensities if and only if endpoints of the effect window are binned.

“We show that the equivalence result between DL and ES models also holds in the general case if and only if endpoints of the effect window are binned.”

Binning of endpoints in the event study model corresponds to limiting the number of leads and lags in the distributed-lag model.

“Binning of endpoints in the ES model thereby corresponds to limiting the number of leads and lags in the DL model.”

If observations outside the effect window are not dropped, a model that restricts the effect window without binning endpoints implicitly assumes that dynamic treatment effects drop to zero outside of the effect window.

“If observations outside of the effect window are not dropped, such a model implicitly assumes that dynamic treatment effects drop to zero outside of the effect window.”

Classic dummy variable event study designs can be generalized to models that account for multiple treatments of different signs and varying intensities.

“Third, classic dummy variable event study designs can be generalized to models that account for multiple treatments of different signs and varying intensities.”

• Also stated in: § 1 INTRODUCTION · plain

  “Third, we show that the simple event study design can be generalized to account for multiple treatments and/or treatments of different signs and varying intensities.”

An event that happens before the first data year of the dependent variable can affect the outcome like any other event happening between the first and last data years and needs to be taken into account.

“an event that happens before t, that is, the first data year of the dependent variable, can affect the outcome like any other event happening between t and t and needs to be taken into account.”

Binning, i.e. assuming constant treatment effects before and/or after some event time, is a natural restriction that identifies dynamic treatment effects in the absence of never-treated units.

“First, assuming constant treatment effects before and/or after some event time, also known as binning, is a natural restriction, which identifies dynamic treatment effects in the absence of never-treated units.”

• Also stated in: § 1 INTRODUCTION · plain

  “We show that binning imposes implicit assumptions that enable researchers to identify dynamic treatment effects in the absence of never-treated units.”

• AssumptionThe authors state this as an assumption: Dynamic treatment effects are constant before and/or after the endpoints of the effect window.

The distributed-lag model is a reparametrization of the event study model.

“We show that the DL model is a reparametrization of the ES model.”

The inherent imprecision in event study specifications can easily give rise to a misspecified model or a model that relies on strong implicit assumptions.

“We show below that the inherent imprecision in event study specifications can easily give rise to a misspecified model or a model that relies on strong implicit assumptions.”

Without binning it is impossible to separate dynamic policy effects from secular time effects.

“Without binning it would be impossible to separate dynamic policy effects from secular time effects.”

Event study estimates can be recovered from distributed-lag estimates by properly normalizing the distributed-lag model and cumulating the post-treatment and pre-treatment effects away from zero.

“Event study estimates can be recovered from DL estimates by properly normalizing the DL model and cumulating the post-treatment and pre-treatment effects away from zero.”

Event study designs with binned endpoints and distributed-lag models with a limited number of leads and lags are numerically identical.

“Second, event study designs with binned endpoints and distributed-lag models are numerically identical.”

• Also stated in: § 1 INTRODUCTION · plain

  “Second, we show that event study (ES) designs with binned endpoints and distributed-lag (DL) models with a limited number of leads and lags are equivalent.”

A pronounced drop between the endpoint coefficient and the preceding coefficient is an indication that dynamic effects are still unfolding.

“A pronounced drop between β𝓁 and β𝓁-1 is instead an indication that dynamic effects are still unfolding.”

• AssumptionThe authors state this as an assumption: Dynamic treatment effects are constant before and/or after the endpoints of the effect window.

The data requirements described for the standard event study design also apply to the generalized design in Section 4.

“The data requirements described for the standard event study design also apply to the generalized design in Section 4.”

If treatment adoption is staggered, the recent literature on heterogeneous treatment effects in event studies has shown that the two-way fixed effects estimates of β_l cannot be interpreted as average treatment effects.

“If, however, treatment adoption is staggered, the recent literature on heterogeneous treatment effects in event studies has shown that the two-way fixed effects estimates of 𝛽 𝓁 cannot be interpreted as average treatment effects”

• cites Borusyak et al., 2022Not checked yet

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• cites Callaway & Sant'Anna, 2021Not checked yet

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• cites de Chaisemartin & D'Haultfœuille, 2020Not checked yet

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Dropping observations outside the effect window is not straightforward when effect windows of multiple treatments overlap.

“Dropping observations outside of the effect window is not straightforward when effect windows of multiple treatments overlap.”

It is possible to restrict the effect window only pre-treatment or only post-treatment.

“As stated in Remark 2, it is possible to only restrict the effect window pre or post-treatment.”

Adding a constant to all dynamic treatment effect parameters and subtracting the same constant from the unit fixed effect for units with an event does not alter Equation (1).

“Adding a constant to 𝛽 𝓁 for all 𝓁 = -∞, … , ∞ and subtracting the same constant from the unit fixed effect 𝜇 i for units with an event between -∞ and ∞ do not alter Equation (1).”

In a model with individual fixed effects, dynamic treatment effect parameters are only identified up to a constant.

“In a model with individual fixed effect, 𝜇 i , parameters 𝛽 𝓁 are only identified up to a constant.”

Due to the leads and lags of the effect window, the treatment variable must be observed for a longer observation window than the dependent variable.

“Due to the leads and lags of the effect window, we need to observe the treatment variable for a longer observation window than the dependent variable.”

If treatment effects had fully materialized by the endpoint lag after the reform, estimates leading up to the endpoint coefficient would level off and converge to that endpoint coefficient.

“If treatment effects had fully materialized by 𝓁 after the reform, we would expect estimates leading up to β𝓁 to level off and converge to β𝓁 .”

• AssumptionThe authors state this as an assumption: Dynamic treatment effects are constant before and/or after the endpoints of the effect window.

Even if individual effects were homogeneous, the two-way fixed effect estimates of binned endpoints cannot be interpreted as ATTs when the true dynamic treatment effects are not constant beyond the endpoints.

“Even if individual effects were homogeneous (as we assume throughout this paper), the two-way fixed effect estimates of binned endpoints cannot be interpreted as ATTs when the true dynamic treatment effects are not constant beyond the endpoints.”

• AssumptionThe authors state this as an assumption: Individual treatment effects are homogeneous.

A model that restricts all effects before the event and allows the treatment effect to continue indefinitely does not allow estimation and visualization of pre-trends.

“But clearly, such a model does not allow us to estimate and visualize pre-trends.”

Relying on a never-treated control group is not a fully convincing strategy in many applications as it raises the question of whether units that are never treated are comparable with units that receive treatments.

“Hence, relying on a never-treated control group is not a fully convincing strategy in many applications as it raises the question of whether units that are never treated are comparable with units that receive treatments.”

• AssumptionRests on an assumption: All units can be manipulated, at least conceptually, so that treatment happens.

These biased secular time period effects pin down the counterfactual for other treated units whose lags/leads of outcomes are still within the effect window.

“These biased secular time period effects, in turn, pin down the counterfactual for other treated units whose lags/leads of outcomes are still within the effect window.”

Estimation in first differences does not solve this problem.

“We discuss in Section 3.2 that estimation in first differences does not solve this problem.”

This problem is more severe in the absence of never-treated units, where the secular time period effects are only identified by the restrictions imposed by binning.

“This problem is more severe in the absence of never-treated units, where the secular time period effects are only identified by the restrictions imposed by binning”

The interpretation also holds for the standard difference-in-differences (DiD) model, which is the most extreme form of binning.

“It therefore also holds for the standard difference-in-differences (DiD) model, which is the most extreme form of binning.”

When treatment effects are allowed to have effects into the infinite future (or affect the infinite past), no unit is known to be never-treated with certainty.

“When treatment effects are allowed to have effects into the infinite future (or affect the infinite past), no unit is known to be never-treated with certainty.”

Consequently, all estimated dynamic treatment effects are biased because two-way fixed effects estimation fails to correctly separate dynamic causal effects from time trends when treatment is staggered.

“Consequently, all estimated dynamic treatment effects are biased because two-way fixed effects estimation fails to correctly separate dynamic causal effects from time trends when treatment is staggered.”

For a cohort that experiences treatment at the same point in time and in the presence of never-treated units, two-way fixed effects estimates for this cohort can be interpreted as average treatment effects (ATT) when individual treatment effects are heterogeneous.

“For a cohort that experiences treatment at the same point in time and in the presence of never-treated units, two-way fixed effects estimates for this cohort can be interpreted as average treatment effects (ATT) when individual treatment effects are heterogeneous.”

• cites Sun and Abraham (2021)Not checked yet

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As an alternative to binning, identification is also achieved by dropping observations outside of the effect window and setting another dynamic treatment effect β_l to zero, for example, β_-2 = 0 in addition to β_-1 = 0.

“As an alternative to binning, identification is also achieved by dropping observations outside of the effect window and setting another dynamic treatment effect 𝛽 𝓁 to zero, for example, 𝛽 -2 = 0 in addition to 𝛽 -1 = 0 (as proposed in earlier versions of Borusyak et al., 2022).”

• cites Borusyak et al., 2022Not checked yet

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If a unit cannot possibly be treated for its special characteristics, the very same special characteristics are potential confounders.

“But if a unit cannot possibly be treated for its special characteristics, the very same special characteristics are potential confounders.”

Borusyak et al. (2022) show that with an infinite effect window, [l,l] = [-∞, ∞], dynamic effects are only identified up to a linear trend.

“Borusyak et al. (2022) show that with an infinite effect window, [𝓁, 𝓁] = [-∞, ∞], dynamic effects are only identified up to a linear trend.”

• cites Borusyak et al. (2022)Not checked yet

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The non-identification of the absolute size of dynamic effects can be solved by setting a reference period to which the effects are compared.

“This non-identification of the absolute size of the dynamic effects can be solved by setting a reference period to which the effects are compared.”

Binning endpoints goes along with an important but often unstated assumption about the dynamic treatment effects.

“However, it goes along with an important but often unstated assumption about the dynamic treatment effects, which is summarized in the following Remark 2.”

• AssumptionThe authors state this as an assumption: Dynamic treatment effects are constant before and/or after the endpoints of the effect window.

Restrictions on the effect window determine the requirements on data availability for the dependent variable, treatment status, and event indicator.

“Restrictions on the effect window determine the requirements on data availability for dependent variable Y i,t , treatment status T i,t , and event indicator ΔT i,t .”

The generalization works only with the definition of the binned treatment adoption measure D l i,t in Equation (3) and not with the more common definition in Equation (4).

“Importantly, the generalization does only work with the definition of the binned treatment adoption measure D 𝓁 i,t in Equation ( 3) and not with the more common definition in Equation ( 4).”

The dichotomization of the treatment variable eliminates valuable information, which could otherwise be used to identify the magnitude of the effect.

“Moreover, the dichotomization of the treatment variable eliminates valuable information, which could otherwise be used to identify the magnitude of the effect.”

Estimating dynamic treatment effects using the generalized event study only produces consistent estimates under a linearity and additivity assumption.

“Estimating dynamic treatment effects using the generalized event study only produces consistent estimates under a linearity and additivity assumption, which is summarized in the following remark.”

In a specification that dichotomizes treatment variables, estimated dynamic treatment effects can be larger, smaller, or identical to a specification that considers all reforms with their intensity.

“In such a specification, estimated dynamic treatment effects can be larger, smaller, or identical to a specification that considers all reforms with their intensity.”

The parameter estimates of dichotomizing treatment variables may be difficult to interpret both in magnitude and direction.

“However, the parameter estimates of such a dichotomization may be difficult to interpret both in magnitude and direction.”

If one regards the distributed-lag model with a limited number of leads and lags as a sensible econometric model, binning of the endpoints in event study designs must be equally sensible.

“An implication of Remark 6 is that if one regards the distributed-lag model with a limited number of leads and lags as a sensible econometric model, binning of the endpoints in event study designs must be equally sensible.”

Under the assumption that the treatment effect is proportional to the observed treatment intensity, the generalized event study described by Equations (2) and (3) delivers consistent estimates of the dynamic treatment effects.

“Assuming that the treatment effect is proportional to the observed treatment intensity, the generalized event study described by Equations ( 2) and (3) delivers consistent estimates of the dynamic treatment effects.”

• AssumptionThe authors state this as an assumption: The treatment effect is proportional to the observed treatment intensity.

Numerically identical treatment effects can also be obtained from a distributed-lag model as specified in Equation (10).

“Numerically identical treatment effects can also be obtained from a distributed-lag model as specified in Equation ( 10).”

These biased secular time fixed effects will lead to biases when pinning down the counterfactual for all incremental effects.

“As above, these biased secular time fixed effects will lead to biases when pinning down the counterfactual for all incremental effects.”

As in levels, the problem is more severe in the absence of never-treated units, where the secular time period effects are only identified by the restrictions imposed by binning.

“As in levels, the problem is more severe in the absence of never-treated units, where the secular time period effects are only identified by the restrictions imposed by binning”

Event study estimates can be used to infer long-term effects on an intuitive scale.

“This way, event study estimates can be used to infer long-term effects on an intuitive scale.”

This feature also leads to the often observed difference between fixed effects and first difference estimates in standard difference-in-difference models if the treatment effect builds up over time.

“This feature also leads to the often observed difference between fixed effects and first difference estimates in standard difference-in-difference models if the treatment effect builds up over time.”

BF2017's dynamic event studies only weakly support this finding.

“However, their dynamic event studies only weakly support this finding.”

• cites Baker & Fradkin, 2017Not checked yet

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The equivalence between event study and distributed-lag models holds in the generalized design exactly as in the standard event study specification.

“The equivalence between event study and distributed-lag models, therefore, holds in the generalized design exactly as in the standard events study specification.”

• Also stated in: § 6 CONCLUSION · boosted

  “We show that the event study methodology is perfectly applicable to such environments and that the equivalence between event study and distributed-lag models also holds in the general case.”

Considering only the first of potentially many events in a standard event study framework leads to inconsistent estimates unless the second and subsequent events are known to have no additional effect.

“Sometimes, only the first of potentially many events is considered in a standard event study framework as developed in Section 2. This approach leads to inconsistent estimates unless the second and subsequent events are known to have no additional effect.”

Excluding units with small events from the sample leads to the model being estimated on a different and possibly selected sample.

“A possible fix for this ambiguity is to exclude units with small events from the sample, in which case, the model is, however, estimated on a different and possibly selected sample.”

The equivalence also holds when the effect window is restricted through binning of the endpoints in the generalized design.

“As in the standard event study design, the equivalence also holds when the effect window is restricted through binning of the endpoints.”

Changes in the outcome variable beyond the endpoint are fully attributed to the secular time period effects.

“This is because changes in the outcome variable beyond the endpoint are fully attributed to the secular time period effects.”

As the generalized event study specification incorporates the intensity of treatment, estimated effects can be interpreted as the effect of a one-unit increase akin to the interpretation in a generalized differences-in-differences model.

“As the generalized event study specification incorporates the intensity of treatment, estimated effects can be interpreted as the effect of a one-unit increase akin to the interpretation in a generalized differences-in-differences model.”

Treatments with opposite signs may have asymmetric effects on the outcome.

“For example, treatments with opposite signs may have asymmetric effects on the outcome (see, e.g., Benzarti et al., 2020;Fuest et al., 2018).”

• cites Benzarti et al., 2020Not checked yet

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• cites Fuest et al., 2018Not checked yet

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The choice of effect window is far from being innocuous.

“Although this choice is a practical necessity due to limited data availability, it is far from being innocuous.”

Event study estimates can be recovered from distributed-lag models by cumulating the post-treatment and pre-treatment effects away from zero.

“Event study estimates can be recovered from distributed-lag models by cumulating the post-treatment and pre-treatment effects away from zero.”

The event study methodology is perfectly applicable to environments with multiple treatments and/or treatments of different signs and varying intensities.

“We show that the event study methodology is perfectly applicable to such environments and that the equivalence between event study and distributed-lag models also holds in the general case.”

Dropping either dimension of fixed effects typically leads to omitted variable bias unless treatment is randomized.

“However, dropping either dimension of fixed effects typically leads to omitted variable bias unless treatment is randomized.”

Coefficients from the fixed effects and the first difference specification will deviate if the effect has not fully materialized within the given effect window.

“As a result, coefficients from the fixed effects and the first difference specification will deviate if the effect has not fully materialized within the given effect window.”

The distributed-lag model is a reparametrization of an event study design with binned endpoints.

“To be precise, the distributed-lag model is a reparametrization of an event study design with binned endpoints.”

Binning determines which unit-period observations are assigned to the treatment or control group and thus directly affects the identifying assumption.

“We show that binning determines which unit-period observations are assigned to the treatment or control group and thus directly affects the identifying assumption.”

The event study results reinforce BF2017's important main result that longer potential benefit duration in unemployment insurance reduces search effort.

“Hence, the event study results reinforce BF2017's important main result that longer potential benefit duration in unemployment insurance reduces search effort.”

Binning introduces important parameter restrictions, which help to identify the model econometrically.

“At the same time, binning introduces important parameter restrictions, which help to identify the model econometrically.”

Covariates can cause problems in the case of treatment heterogeneity even if there is no staggered adoption.

“Covariates can cause problems in the case of treatment heterogeneity even if there is no staggered adoption(Powell, 2021).”

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