Interaction-Weighted Event Studies for Staggered Adoption

Why Interact Cohort and Event Time?

When treatment begins in different periods for different groups, the same event time can correspond to different calendar periods and different cohorts. If treatment effects vary across cohorts, a conventional two-way fixed-effects (TWFE) event study can combine those effects in ways that make even a lead coefficient reflect post-treatment effects. An interaction-weighted event study first estimates effects separately for each adoption cohort and event time, then combines the cohort-specific estimates using explicit weights.

This separation matters for interpretation. A coefficient for event time \(e=2\), for example, should describe an average effect two periods after adoption among cohorts observed at that event time. It should not inadvertently reflect effects at other event times simply because those effects differ across cohorts.

Definition: Let \(G_i=g\) denote that unit \(i\) first receives treatment in period \(g\), and let \(G_i=\infty\) denote a never-treated unit. Event time for a treated unit in period \(t\) is \(e=t-g\). A cohort-by-event-time interaction identifies units in cohort \(g\) observed at event time \(e\). An interaction-weighted event-study coefficient for event time \(e\) is a weighted average of the estimated cohort-specific effects at that event time.

Define the Cohort-Specific Effects

Write \(Y_{it}(0)\) for unit \(i\)'s outcome in period \(t\) if untreated, and \(Y_{it}(1)\) for its outcome if treated. For cohort \(g\), the cohort-time average treatment effect on the treated is

ATT(g,t)=E[Y_{it}(1)-Y_{it}(0)\mid G_i=g],\qquad t\geq g.

At event time \(e\geq 0\), the corresponding cohort-specific dynamic effect is \(ATT(g,g+e)\). An event-study regression commonly uses \(e=-1\), the period immediately before adoption, as its reference period. The interaction coefficient for cohort \(g\) and event time \(e\neq -1\) is interpreted relative to that reference. For post-treatment periods, this difference corresponds to \(ATT(g,g+e)\) when the reference period is unaffected by anticipation and the identifying assumptions below hold. Lead coefficients are pre-treatment comparisons, not treatment effects.

One way to represent the interaction regression is to include unit and calendar-time fixed effects and a separate indicator for every supported cohort-event-time cell other than the omitted reference cells:

Y_{it}=\alpha_i+\lambda_t+ \sum_g\sum_{e\neq -1}\gamma_{g,e}\, \mathbf{1}\{G_i=g\}\mathbf{1}\{t-g=e\} +\varepsilon_{it}.

Here, \(\alpha_i\) is a unit fixed effect and \(\lambda_t\) is a calendar-time fixed effect. The indicator is one only for members of cohort \(g\) in calendar period \(g+e\). In this parameterization, \(\gamma_{g,e}\) describes the cohort-specific change relative to event time \(-1\), subject to the design's support and identifying assumptions. A never-treated group can provide untreated comparisons. The regression does not make unsupported cohort-event cells estimable; only cells with suitable observations and comparisons belong in the analysis.

Identification Through Untreated Comparisons

A transparent way to define a cohort-time estimate is a difference-in-differences contrast against a comparison group \(C\) that remains untreated in both periods. For a post-treatment event time \(e\), let \(t=g+e\). The contrast is

\widehat{ATT}(g,t)= (\bar{Y}_{g,t}-\bar{Y}_{g,g-1}) -(\bar{Y}_{C,t}-\bar{Y}_{C,g-1}),

where \(\bar{Y}_{g,t}\) and \(\bar{Y}_{C,t}\) are the respective groups' mean outcomes in period \(t\). The key parallel-trends condition is that, absent treatment, the cohort and comparison group would have had the same average outcome change from \(g-1\) to \(t\). No anticipation ensures that the cohort's outcome in the reference period is not already affected by future treatment. Consistency and no interference are also needed to interpret observed outcomes as the relevant potential outcomes.

The identification argument follows by adding and subtracting the treated cohort's untreated outcome change:

\begin{aligned} &E[Y_{it}\mid G_i=g]-E[Y_{i,g-1}\mid G_i=g]\\ &\quad-\{E[Y_{it}\mid i\in C]-E[Y_{i,g-1}\mid i\in C]\}\\ &=E[Y_{it}(1)-Y_{it}(0)\mid G_i=g]\\ &\quad+\{E[Y_{it}(0)-Y_{i,g-1}(0)\mid G_i=g] -E[Y_{it}(0)-Y_{i,g-1}(0)\mid i\in C]\}\\ &=ATT(g,t). \end{aligned}

The final equality uses parallel trends to set the bracketed difference to zero, and no anticipation to treat the cohort's reference-period outcome as untreated. Thus the cohort-specific contrast has a causal interpretation under those assumptions. The same logic gives a zero expected lead contrast under no anticipation and parallel trends; that is a diagnostic implication, not proof that the assumptions hold.

Worked Example: Estimating Effects for Two Adoption Cohorts

Consider a hypothetical panel with two treated cohorts and a never-treated comparison group. The table gives mean outcomes in the reference and comparison periods. Assume no anticipation, no interference, and parallel untreated outcome trends between each treated cohort and the never-treated group over the periods used.

Group	2017	2018	2019
Cohort adopting in 2018	10	13	17
Cohort adopting in 2019	11	12	17
Never treated	8	9	11

For the 2018 cohort at \(e=0\), compare its change from 2017 to 2018 with the never-treated group's change over the same calendar periods:

\widehat{ATT}(2018,2018)=(13-10)-(9-8)=3-1=2.

The estimated effect at adoption for this cohort is 2 outcome units. For the 2019 cohort at \(e=0\), use 2018 as its reference period:

\widehat{ATT}(2019,2019)=(17-12)-(11-9)=5-2=3.

This cohort's estimated effect at adoption is 3 outcome units. The estimates are kept separate at this stage: neither cohort is required to have the same effect as the other.

Aggregate Only After Estimating Cohort Effects

For each event time \(e\), let \(\mathcal{G}_e\) be the set of adoption cohorts with an estimable effect at that event time. An interaction-weighted event-study coefficient is

\widehat{\theta}_e =\sum_{g\in\mathcal{G}_e}w_{g,e}\widehat{ATT}(g,g+e), \qquad w_{g,e}\geq 0,\qquad \sum_{g\in\mathcal{G}_e}w_{g,e}=1.

A common choice weights cohorts by their shares of treated units among the cohorts with support at event time \(e\). If \(N_g\) is the number of units in cohort \(g\), then

w_{g,e}=\frac{N_g}{\sum_{g'\in\mathcal{G}_e}N_{g'}}.

Other prespecified target weights are possible, but the target population should be stated. In particular, a cohort that lacks an observed comparison at a given event time cannot simply be treated as if its effect were observed. Since the set \(\mathcal{G}_e\) can change with \(e\), the aggregated profile can represent different cohort mixtures at different event times.

Worked Example: Calculating Event-Time Weights

Suppose three hypothetical cohorts have 40, 60, and 100 treated units. At \(e=0\), all three are supported, with estimated cohort-specific effects of 2, 3, and 1 outcome units, respectively. Unit-share weights are \(40/200=0.20\), \(60/200=0.30\), and \(100/200=0.50\). The aggregate is

\widehat{\theta}_0 =0.20(2)+0.30(3)+0.50(1) =0.40+0.90+0.50 =1.80

So the event-time-zero estimate is 1.80 outcome units for the supported cohort mixture. At \(e=1\), suppose only the first two cohorts have support, and their estimated effects are 4 and 5 outcome units. The weights must be renormalized over those cohorts:

w_{1,1}=\frac{40}{40+60}=0.40,\qquad w_{2,1}=\frac{60}{40+60}=0.60, \qquad \widehat{\theta}_1=0.40(4)+0.60(5)=4.60.

The \(e=1\) estimate is 4.60 outcome units for the two supported cohorts, not for all three cohorts. Its difference from the \(e=0\) estimate may reflect both changing treatment effects and changing cohort composition.

Why Conventional TWFE Leads Can Be Contaminated

A conventional event-study regression uses one lead or lag indicator for each event time, pooling units across adoption cohorts. This imposes a common coefficient for that event time even when the underlying cohort-specific effects differ. After accounting for unit and time fixed effects, the pooled indicators need not be orthogonal to the cohort-specific treatment-effect pattern. Consequently, a fitted coefficient for a lead can load on post-treatment effects.

The projection argument makes the issue explicit. Let \(D\) denote the matrix of pooled event-time indicators, let \(F\) denote the unit and time fixed-effect regressors, and let \(M_F\) residualize variables with respect to \(F\). Let \(Z\) contain the cohort-by-event-time indicators, and let \(\tau\) be their true cohort-specific effects. In a linear projection, the contribution of those effects to the conventional event-study coefficients is

\widetilde{\beta} =(D'M_FD)^{-1}D'M_FZ\tau.

The matrix multiplying \(\tau\) need not map a conventional lead coefficient only to true lead effects. Its entries can also reflect post-treatment cohort-event cells. This is the source of lead-lag contamination under heterogeneous effects; it does not require anticipation. Interacting event time with cohort avoids imposing a single event-time coefficient before estimation. The cohort-specific estimates can then be aggregated deliberately, with nonnegative weights that define the target average.

Worked Example: Reading a Contaminated Lead Projection

Consider a hypothetical design in which the residualized projection for one conventional lead coefficient has the following mapping to two relevant cohort-specific effects:

\widetilde{\beta}_{-2} =0.50\,\tau_{A,-2}+0.50\,\tau_{B,0}.

Here \(\tau_{A,-2}\) is the true lead effect for cohort A and \(\tau_{B,0}\) is the true effect at adoption for cohort B. Suppose there is no anticipation, so \(\tau_{A,-2}=0\), while cohort B's adoption effect is 6 outcome units. Then

\widetilde{\beta}_{-2}=0.50(0)+0.50(6)=3.

The conventional lead coefficient is 3 outcome units even though the true lead effect in this example is zero. The displayed projection weights are specific to this hypothetical design; they are not universal weights for conventional event studies. An interaction-based analysis estimates the cohort-specific lead and adoption effects separately, so the lead estimate is not defined by this pooled projection.

Practical Interpretation and Common Pitfalls

Interaction-weighted estimates clarify which cohorts contribute to each point of a dynamic profile. Report the reference period, comparison group, supported cohorts, aggregation weights, and outcome units. A lead coefficient should be described as a pre-treatment comparison relative to the reference period. A post-treatment coefficient should be described as an average causal effect only when the identifying assumptions and comparison design support that interpretation.

Pooling before estimating. Treating one conventional event-time coefficient as if it were an average of same-event-time effects can conceal cross-event-time contamination. Estimate cohort-specific effects first, then aggregate.
Ignoring support. Applying weights to cohorts without a valid comparison or observed event-time outcome does not recover their effects. Define \(\mathcal{G}_e\) using actual support.
Comparing changing mixtures as if fixed. If \(\mathcal{G}_e\) changes over event time, the profile may change partly because the contributing cohorts change. State this when interpreting dynamics.
Reading a zero reference coefficient as evidence. The coefficient at \(e=-1\) is omitted and normalized to zero. That normalization is not an empirical finding that outcomes or effects are zero in that period.
Calling a nonsignificant lead proof of parallel trends. A lead estimate is only a diagnostic. Failure to detect a pre-treatment difference does not establish parallel trends.

The central distinction is between estimation and aggregation: interaction terms preserve cohort-specific information, while the weights determine which average is reported. This makes heterogeneity visible and prevents the conventional pooled lead-lag parameterization from silently defining the estimand.

Check Your Understanding

Answer each question by identifying the cohort, event time, comparison, or target average involved.

For a cohort adopting in period 2020, which calendar period corresponds to event time \(e=2\)?
What assumptions allow a difference-in-differences contrast against never-treated units to identify a cohort-time ATT?
Three cohorts have supported effects at \(e=0\). How should unit-share weights be normalized if one cohort is not supported at \(e=1\)?
Why can a conventional TWFE lead coefficient be nonzero even when there is no anticipation?
What does the omitted \(e=-1\) coefficient mean, and what does it not establish?