Why Aggregation Defines the Question
With staggered adoption, treatment effects can differ both by adoption cohort and by calendar time. A single cohort-time effect describes one cohort at one period after adoption; it does not, by itself, describe a common event-time profile or an overall effect. Aggregation combines these effects, but the weights determine which treated units and periods the summary represents.
This tutorial takes the cohort-time average treatment effect, \(ATT(g,t)\), as its building block. Here \(g\) denotes the period in which a cohort first adopts treatment, and \(t\) is calendar time. For a treated period \(t\geq g\), the event time is \(k=t-g\). An event-time effect therefore combines effects from different cohorts at periods that are equally far from adoption.
The formulas below describe population summaries; replacing each \(ATT(g,t)\) with an estimate gives the corresponding estimated summary. The aggregation formulas do not establish that the cohort-time effects are causally identified. Their causal interpretation depends on the validity of those effects and on a clearly specified target population. Aggregation also requires compatible outcome definitions and units across cohorts and periods.
Dynamic Effects: Align Cohorts by Event Time
A dynamic effect at event time \(k\) averages effects for cohorts observed \(k\) periods after adoption. Let \(\mathcal{G}_k\) be the set of cohorts for which the period \(g+k\) is observed and included in the target population. If each treated unit in a cohort is to receive equal weight, the cohort weight at event time \(k\) is its share of the eligible treated units:
The weights are nonnegative and sum to one over the eligible cohorts. Thus \(\theta_D(k)\) is a treated-unit-weighted average of the cohort-specific effects at event time \(k\). If instead the scientific target gives cohorts equal importance regardless of size, one could use equal cohort weights. That is a different estimand and should be stated as such.
The eligible set can change with \(k\). For example, if observations end at a fixed calendar period, early adopters may be observed for several post-adoption periods while late adopters contribute only at adoption. Consequently, the dynamic effect at one event time may describe a different mixture of cohorts from the effect at another. A changing profile can reflect both changes in effects and changes in the cohorts represented; it is not automatically a like-for-like comparison of one fixed group over time.
Worked Example: Calculating Dynamic Effects With Unequal Cohort Sizes
Suppose a hypothetical study follows treatment cohorts through calendar period 4. Cohort 2 contains 100 treated units, cohort 3 contains 60, and cohort 4 contains 40. The cohort-time effects, in outcome units, are as follows:
| Cohort \(g\) | \(N_g\) | \(ATT(g,g)\), \(k=0\) | \(ATT(g,g+1)\), \(k=1\) | \(ATT(g,g+2)\), \(k=2\) |
|---|---|---|---|---|
| 2 | 100 | 2 | 4 | 6 |
| 3 | 60 | 1 | 3 | Not observed |
| 4 | 40 | 2 | Not observed | Not observed |
At event time \(k=0\), all three cohorts contribute, so the weights are \(100/200\), \(60/200\), and \(40/200\). The dynamic effect is
At \(k=1\), only cohorts 2 and 3 have observed effects. The eligible treated population has \(100+60=160\) units, giving
At \(k=2\), only cohort 2 is observed, so its weight is 1 and \(\theta_D(2)=6\) outcome units. These values are correct for their respective eligible cohorts, but the \(k=2\) result is not an average across all three cohorts. The table and weights make this support limitation explicit.
Four Summaries, Four Weighting Questions
Dynamic effects are only one way to aggregate \(ATT(g,t)\). Three other useful summaries organize the same cohort-time effects differently. Each answers a different question, so none is universally the “overall” answer without a weighting rule.
| Summary | What is held fixed? | What is averaged? | Example unit-weighting rule |
|---|---|---|---|
| Dynamic | Event time \(k=t-g\) | Cohorts observed at that event time | \(N_g/\sum_{h\in\mathcal{G}_k}N_h\) |
| Cohort | Adoption cohort \(g\) | Post-adoption periods for that cohort | Specified period weights within cohort \(g\) |
| Calendar-time | Calendar period \(t\) | Treated cohorts active by that period | \(N_g/\sum_{h\leq t}N_h\), over included treated cohorts |
| Overall | Neither cohort nor time | Included treated cohort-period pairs | Specified weights across all included pairs |
A cohort summary describes how effects evolve over calendar time for a fixed adoption cohort, or combines that cohort’s post-adoption effects into one number. For the latter, let \(\mathcal{T}_g\) be the included post-adoption periods and choose period weights \(v_{g,t}\) that sum to one:
Equal period weights answer a different question from weights that give more emphasis to periods with more observed treated units. The choice should match the target: for instance, an average effect over observed post-adoption periods versus a summary that reflects the number of treated unit-period observations.
A calendar-time summary fixes a period \(t\) and combines the effects for cohorts already treated by then. With treated-unit weighting, a simple definition is
where only cohorts in the target population with an effect defined at \(t\) are included. It describes effects at the same calendar date, but cohorts in that average can be at different event times. In contrast, a dynamic summary holds event time fixed while combining calendar dates across cohorts.
An overall summary combines a specified set \(\mathcal{S}\) of treated cohort-period pairs. If each treated unit-period receives equal weight, a cohort observed for more post-adoption periods contributes more unit-periods. For an observation window ending at \(T\), when every cohort is observed from adoption through \(T\), the formula is
This is not the same as giving each cohort equal weight, or first averaging within each cohort and then averaging those cohort means equally. The denominator and the order of averaging are part of the definition, not bookkeeping details.
Deriving the Dynamic Average From Cohort Effects
The dynamic formula follows directly from the definition of a cohort-time average effect. Let \(ATT(g,g+k)\) be the average treatment effect for treated units in cohort \(g\) at event time \(k\). If the cohort contains \(N_g\) units, the sum of their individual effects at that period is \(N_gATT(g,g+k)\), where the average is taken over those \(N_g\) units. Summing these cohort totals over eligible cohorts gives the total effect across the event-time-\(k\) target population. Dividing by the number of treated units in that population yields
This justification also shows why the cohort-size weights are appropriate for a treated-unit average. If the target population is not the observed treated units—for example, if the analysis defines a target with different cohort shares—then the weights should represent that target instead. In either case, the aggregation is a weighted mean only when the weights are nonnegative and sum to one.
Worked Example: Contrasting Dynamic and Calendar-Time Summaries
Use the hypothetical cohort effects from the preceding table. At event time \(k=1\), the dynamic effect is \(3.625\) outcome units: it combines cohort 2 at calendar period 3 with cohort 3 at calendar period 4. At calendar period \(t=4\), the calendar-time summary combines all three cohorts, which are then at event times 2, 1, and 0 respectively:
The two numbers differ because they answer different questions and average different cohort-time effects. The dynamic summary fixes time since adoption; the calendar-time summary fixes the calendar date. Neither should be described as the other.
For a cohort summary, cohort 2’s equal-period post-adoption mean over periods 2 through 4 is \((2+4+6)/3=4\) outcome units. Cohort 3’s equal-period mean over its observed post-adoption periods 3 and 4 is \((1+3)/2=2\) outcome units. Cohort 4’s mean over its only observed post-adoption period is \(2\) outcome units. These are three separate cohort summaries, not one dynamic profile.
Worked Example: Defining an Overall Effect by Unit-Period Weighting
For the same hypothetical data, define the overall effect to give every treated unit-period equal weight from adoption through period 4. Cohort 2 contributes \(100\times3=300\) unit-periods, cohort 3 contributes \(60\times2=120\), and cohort 4 contributes \(40\times1=40\). Thus there are \(460\) treated unit-periods. The weighted effect total is
Therefore,
This overall value weights each cohort-time ATT in proportion to \(N_g\), and consequently gives more total weight to cohorts observed in more post-adoption periods. For comparison, the equal-period cohort means are 4, 2, and 2; averaging those three means equally gives \((4+2+2)/3=2.667\) outcome units. That is a legitimate summary under equal cohort weighting, but it is not the unit-period-weighted overall effect just calculated.
Interpreting Estimates and Their Uncertainty
When estimates \(\widehat{ATT}(g,t)\) are available, use the same declared target set and weights to form a summary estimate. For example, the dynamic estimate is
Its point estimate is a weighted average of the cohort-time point estimates. The weights are not themselves evidence that the component effects are causally identified; the assumptions and design supporting those estimates still matter. Nor does computing a point estimate alone provide its uncertainty. Since cohort-time estimates can be statistically dependent, the variance of a weighted summary generally depends on their covariance:
Thus, averaging reported standard errors is not a valid general method for obtaining the standard error of an aggregate. Inference should account for the joint estimation and dependence structure of the component effects.
Common Mistakes
- Calling every aggregation an overall effect. A dynamic effect, a calendar-time effect, and an overall effect use different sets of cohort-time pairs. State which periods and cohorts enter.
- Using the same cohorts at every event time without checking support. Later event times may be observed only for early adopters. List the eligible cohorts and normalize weights over those cohorts.
- Confusing calendar time with event time. At a fixed calendar date, cohorts can be at different durations since adoption. At fixed event time, cohorts are observed at different calendar dates.
- Changing the weights without changing the interpretation. Equal cohort weights and treated-unit weights generally yield different estimates. Describe whose effects receive equal influence.
- Treating aggregation as identification. A weighted average of biased or otherwise unidentified component estimates does not become causally valid merely because its weights sum to one.
Check Your Understanding
For each question, identify the relevant weights or explain what the summary represents.
- At event time \(k=0\), cohorts of sizes 80 and 20 have effects of 1 and 4 outcome units. Calculate the treated-unit-weighted dynamic effect.
- Why might an estimate at event time \(k=3\) represent fewer cohorts than an estimate at \(k=0\)?
- Explain the distinction between a calendar-time summary at period \(t\) and a dynamic summary at event time \(k\).
- A researcher averages each cohort’s post-adoption effects over its observed periods, then gives each cohort mean equal weight. Is this necessarily the same as an overall effect that gives every treated unit-period equal weight? Explain.
- Why is the weighted average of cohort-time standard errors not generally the standard error of a dynamic effect estimate?