Event-Time Support and Changing Cohort Composition

Why Event-Time Support Matters

An event-study profile is often read as a sequence describing how an intervention’s effect evolves after adoption. But the estimate at each event time may draw on a different set of adoption cohorts. At short horizons, many cohorts may contribute; at distant horizons, only early adopters may have enough observed follow-up. If those cohorts differ in their treatment effects, an aggregate can change even when each cohort’s effect is constant over time.

This is a support and interpretation problem, not merely a question of whether a coefficient is statistically significant. A distant-horizon estimate can be based on few cohorts, few units, or a distinctive subset of adopters. The event-time label alone does not tell the reader who contributes to the estimate or which population its average describes.

Defining Cohort Support

Let \(G_i=g\) denote that unit \(i\) first adopts treatment in calendar period \(g\). Event time \(k\) for that cohort occurs in calendar period \(t=g+k\). For a cohort and event time, the group-time average treatment effect can be written as

where \(Y_t(1)\) and \(Y_t(0)\) are the potential outcomes at time \(t\) under treatment and no treatment. This definition describes an effect for the units in adoption cohort \(g\), at event time \(k\). It does not, by itself, assert that the effect is identified from observed data.

For example, if the data end in period \(T\), a cohort can have calendar support at event time \(k\) only if \(g+k\leq T\). It may still lack usable support if no suitable untreated comparison units are available, if relevant outcomes are missing, or if the estimator’s requirements are not met. A support table should therefore distinguish the cohorts present in the data from the cohorts actually used to estimate each effect.

The causal interpretation of a cohort-specific estimate still depends on the design. For difference-in-differences comparisons, the chosen untreated comparison group must support the relevant parallel-trends argument; assumptions such as no anticipation and adequate overlap may also be needed for the estimator being used. Sparse support does not establish or repair these assumptions. It instead makes clear which cohort-time effects the data and design can speak to.

Available-Support Aggregates and Changing Populations

A common way to summarize cohort-specific effects at event time \(k\) is a weighted average over the cohorts supported at that horizon. Write \(S_k\) for that support set and \(w_{g,k}\) for nonnegative weights that sum to one over \(S_k\). Then the event-time aggregate is

The weights may give each cohort equal influence or may reflect cohort sizes, depending on the target being reported. If the support set or weights vary with \(k\), then \(\theta_k\) can describe a different weighted population at each event time. The expression is a valid summary for each horizon, but treating the sequence as the trajectory of one fixed population requires additional care.

The source of the interpretive problem can be seen by separating within-cohort change from weight change on a set of cohorts observed at both horizons. For two event times \(k\) and \(\ell\), let \(C\) be a common set of cohorts and use weights \(w_{g,k}\) and \(w_{g,\ell}\) that each sum to one over \(C\). The difference between the two averages has the identity

To verify it, add and subtract \(\sum_{g\in C}w_{g,k}\tau_{g,\ell}\) on the left. The first resulting term is the weighted change in cohort effects; the second is the change in weights applied to the effects at \(\ell\). Thus even on common support, an aggregate difference can combine effect evolution with a change in weighting. When support sets differ, cohorts outside their intersection add another complication: a comparison of the available-support averages is not automatically a comparison for the same population.

Endpoint Bias and How to Diagnose It

At the ends of an event-time profile, support often narrows. Early event times can lack cohorts whose adoption occurred before the observation window; late event times can lack later adopters whose follow-up extends beyond the study. These endpoints may therefore represent different cohort mixtures from the middle of the profile. If cohort effects differ, a changing mixture can make an endpoint aggregate differ from an interior aggregate even without within-cohort change.

This is sometimes described as endpoint or composition bias. The term should be used precisely: the estimate may be correctly calculated for its available-support weighted average, while still being a poor estimate of a different, fixed-population dynamic effect that a reader assumes it represents. The issue is not necessarily bias for the horizon-specific estimand actually computed. It is a mismatch between that estimand and an unstated interpretation as a stable-population trajectory.

• Tabulate support by horizon. Report the number and identity of cohorts contributing to each event time, not just the plotted coefficients.
• Show the weights. Cohort counts alone do not reveal whether one cohort dominates an aggregate. State whether weights are equal across cohorts, proportional to cohort size, or defined another way.
• Inspect units and precision. A horizon with few cohorts or few contributing units may be imprecise even if its point estimate is large. Cohort count is not a substitute for uncertainty estimates.
• Check comparator availability. A cohort observed at a horizon may not be usable for the estimator if the required untreated comparison group is absent or unsuitable.
• Compare the support pattern with the study window. Identify which calendar-time boundary or missing-data feature removes cohorts at each endpoint.

Reporting Choices for an Interpretable Profile

There is no single aggregation rule that is best for every research question. The important requirement is to define the target clearly and show how it relates to observed support. An available-support profile can be useful for describing the effects estimable at each horizon, provided readers can see that its cohort composition may change. A common-support profile is more directly comparable over the selected horizons, but it changes the target population to the cohorts observed throughout that window.

A responsible report should state the event-time range, the support rule, the aggregation weights, and the number or identity of contributing cohorts at each horizon. If the analysis restricts the profile to common support, name the retained cohorts or describe the restriction and explain that the resulting target is limited to them. If the profile uses available support, flag horizons at which cohorts drop out and avoid presenting the endpoint as a continuation for an unchanged population.

It is also useful to show cohort-specific effects or a support table alongside the aggregate when feasible. This can reveal whether an endpoint is driven by a single cohort and makes the distinction between within-cohort dynamics and cohort composition visible. Truncating a plot to a range with adequate support may improve readability, but it should not conceal which horizons were omitted or why. Conversely, extending the plotted range does not create information where support is absent.

Common Mistake

A stable support set does not by itself guarantee causal identification, and a changing support set does not automatically invalidate every horizon-specific estimate. Support tells us which comparisons and populations are represented; the design assumptions determine whether those comparisons identify causal effects. Keeping these questions distinct makes event-time results easier to interpret and harder to overstate.