Building Event-Time Profiles From Group-Time Estimates

From Cohort-Time Effects to an Event-Time Profile

A group-time estimate answers a question about one adoption cohort in one calendar period. An event-time profile reorganizes those estimates by time since adoption and combines estimates from multiple cohorts. The reorganization is simple; the interpretation depends on which cohorts are included and how much weight each receives.

This tutorial concerns the construction and interpretation of that aggregate profile. It takes the group-time estimates as inputs rather than introducing a new estimator for them. A valid causal interpretation still depends on the assumptions needed to identify those inputs: in particular, an appropriate parallel-trends condition for the comparison groups used, no anticipation under the chosen treatment-onset convention, and valid untreated controls for the comparisons. The aggregation cannot repair a group-time estimate that is not identified.

Definition: Let $G=g$ denote membership in the cohort first treated in calendar period $g$. For a post-adoption period $t$, $ATT(g,t)$ is the average treatment effect for cohort $g$ in period $t$, among units in that cohort. Relative event time is $e=t-g$. Thus the estimate for cohort $g$ at event time $e$ is $\widehat{ATT}(g,g+e)$, when that group-time estimate is available and valid.

Event time $e=0$ is the adoption period under the specified onset convention; $e=1$ is one period after adoption, and so on. A given event time can correspond to different calendar periods for different cohorts. For example, event time $e=2$ means calendar period 7 for a cohort adopting in period 5, but period 10 for a cohort adopting in period 8.

The usual post-treatment profile uses $e\geq 0$, because $ATT(g,t)$ describes effects for a cohort in periods when it has been treated. A pre-adoption contrast can be useful for assessing patterns before treatment, but it is not a post-treatment $ATT(g,t)$. Keep the estimand and the event-time label clear when reporting such contrasts.

Define the Cohorts and Weights at Each Event Time

Not every cohort necessarily contributes an estimate at every event time. A cohort may not yet have reached a long event horizon in the observed data, or the group-time comparison may not be estimable using a valid untreated control. Let $E_e$ be the set of cohorts for which a group-time estimate is available and suitable for the profile at event time $e$.

Definition: An event-time aggregate at event time $e$ is $$ \widehat{\theta}(e)=\sum_{g\in E_e}q_{g,e}\widehat{ATT}(g,g+e), \qquad q_{g,e}\geq 0,\qquad \sum_{g\in E_e}q_{g,e}=1. $$ The weights $q_{g,e}$ specify the target mixture of cohorts at that event time. The corresponding target effect is $\theta(e)=\sum_{g\in E_e}q_{g,e}ATT(g,g+e)$, when the component effects are identified.

Two common choices answer different questions. Equal-cohort weights, $q_{g,e}=1/|E_e|$, describe an average across the contributing cohorts, giving a small cohort the same weight as a large cohort. Cohort-size weights give more weight to cohorts containing more treated units. If $n_g$ is the number of treated units in cohort $g$, a simple version is $$ q_{g,e}=\frac{n_g}{\sum_{h\in E_e}n_h}. $$ This targets an average across treated units in the represented cohorts, provided the cohort counts and component effects refer to the intended population. If the intended target is different, the weights should be chosen to represent it rather than adopted by habit.

Weights are normalized separately at each event time in the ordinary profile above. Consequently, a cohort can have a different weight at $e=0$ than at $e=3$, or be absent altogether at one horizon. A plotted line of aggregate estimates is therefore not automatically an average treatment trajectory for one fixed population. Each point must be read together with its cohort set and weights.

Worked Example: Assigning Estimates to Relative Event Time

Suppose three hypothetical adoption cohorts have adoption periods $g=2$, $g=4$, and $g=5$. The following group-time estimates are available:

Cohort adoption period $g$	Calendar period $t$	$\widehat{ATT}(g,t)$
2	2	0.30
2	3	0.50
4	4	0.10
4	5	0.40
5	5	0.20

The outcome is measured in hypothetical outcome units. To place each estimate on the event-time scale, calculate $e=t-g$. For the first row, $e=2-2=0$; for the second, $e=3-2=1$. Applying the same calculation gives $e=0$ for both estimates from period 4 and period 5 adoption cohorts, and $e=1$ for the estimate from the period 4 cohort.

At $e=0$, the available cohort effects are 0.30, 0.10, and 0.20 outcome units. With equal-cohort weights, the profile point is $$ \widehat{\theta}(0)=\frac{0.30+0.10+0.20}{3}=0.20. $$ At $e=1$, only the cohorts adopting in periods 2 and 4 contribute, so the equal-cohort aggregate is $$ \widehat{\theta}(1)=\frac{0.50+0.40}{2}=0.45. $$ The second aggregate is 0.25 outcome units higher than the first. But the two values average over different sets of cohorts, so this difference alone does not establish that the same cohorts experienced a 0.25-unit increase.

Worked Aggregation With Explicit Cohort-Size Weights

The following hypothetical estimates show how a profile can change both because cohort effects evolve and because the cohorts represented at each horizon differ. Suppose the outcome is measured in points and the group-time estimates and cohort sizes are as follows:

Cohort $g$	Treated units $n_g$	Estimate at $e=0$	Estimate at $e=1$	Estimate at $e=2$
2	100	0.80	1.00	1.20
3	200	0.40	0.60	Not available
4	100	-0.20	Not available	Not available

Worked Example: Calculating a Profile With Treated-Unit Weights

At event time $e=0$, all three cohorts contribute. The normalized cohort-size weights are $100/400=0.25$, $200/400=0.50$, and $100/400=0.25$. Therefore, $$ \widehat{\theta}(0) =\frac{100(0.80)+200(0.40)+100(-0.20)}{100+200+100} =\frac{80+80-20}{400} =0.350. $$ The estimate is an average effect of 0.350 points for the treated units in these three represented cohorts, under the stated weighting target.

At $e=1$, only cohorts 2 and 3 contribute. Their weights are $100/300=1/3$ and $200/300=2/3$, not 0.25 and 0.50, because the weights must sum to one over the cohorts included at this event time. Thus $$ \widehat{\theta}(1) =\frac{100(1.00)+200(0.60)}{100+200} =\frac{100+120}{300} =0.7333. $$ At $e=2$, only cohort 2 contributes. Its weight is 1, so $$ \widehat{\theta}(2)=1.20\text{ points}. $$

The profile rises from 0.350 to 0.7333 to 1.20 points. Those points have different cohort compositions: all three cohorts contribute at $e=0$, two at $e=1$, and one at $e=2$. In particular, the final point describes cohort 2 alone. It should not be presented as if it were a weighted average over the same population of cohorts as the first point.

Why the Composition of a Profile Matters

A change in an aggregate profile across event times may reflect changes in the cohort-specific effects, changes in the weights, or both. This distinction can be made precise when the same cohorts are observed at two adjacent event times. Write $\tau_{g,e}=ATT(g,g+e)$, and suppose the same cohort set $E$ is available at $e$ and $e+1$. Then:

\begin{aligned} \theta(e+1)-\theta(e) &=\sum_{g\in E}q_{g,e+1}\tau_{g,e+1} -\sum_{g\in E}q_{g,e}\tau_{g,e}\\ &=\sum_{g\in E}q_{g,e+1}\bigl(\tau_{g,e+1}-\tau_{g,e}\bigr) +\sum_{g\in E}\bigl(q_{g,e+1}-q_{g,e}\bigr)\tau_{g,e}. \end{aligned}

The equality follows by adding and subtracting $\sum_{g\in E}q_{g,e+1}\tau_{g,e}$ and collecting terms. The first sum reflects within-cohort changes, using the later weights; the second reflects the effect of changing weights applied to the earlier cohort effects. If cohort sets differ, this same decomposition cannot be applied without additional definitions for the missing cohort-time effects. Entry and exit of cohorts are then an additional reason the aggregate points may not be directly comparable.

This identity is a reminder that even when all estimates are causally identified, a changing aggregate need not represent a change in treatment effect for a fixed population. Cohort effects can genuinely differ because adoption cohorts differ, and the event-time profile can change its mixture of those cohorts as the horizon lengthens.

Worked Example: Comparing a Changing Mix With a Fixed Mix

Return to the hypothetical table above. To compare event times 0 and 1 using only cohorts observed at both horizons, retain cohorts 2 and 3. Their treated-unit weights are $1/3$ and $2/3$ at both event times. The fixed-composition estimates are $$ \widehat{\theta}_{\{2,3\}}(0) =\frac{100(0.80)+200(0.40)}{300} =\frac{160}{300} =0.5333, $$ and $$ \widehat{\theta}_{\{2,3\}}(1) =\frac{100(1.00)+200(0.60)}{300} =\frac{220}{300} =0.7333. $$ For these two cohorts, the fixed-composition profile increases by $0.7333-0.5333=0.2000$ points.

That increase can also be checked cohort by cohort. Cohort 2 changes by $1.00-0.80=0.20$ points and cohort 3 changes by $0.60-0.40=0.20$ points. Applying the fixed weights gives $$ \frac{1}{3}(0.20)+\frac{2}{3}(0.20)=0.20\text{ points}. $$ The comparison is now for the same treated-unit mixture at both horizons. It does not include cohort 4, whose estimate at $e=1$ is unavailable. The fixed-composition result therefore has a narrower target than the original $e=0$ aggregate of 0.350 points.

Equal weighting would answer a different question for cohorts 2 and 3: their means would be $(0.80+0.40)/2=0.600$ points at $e=0$ and $(1.00+0.60)/2=0.800$ points at $e=1$. The difference remains 0.200 points here because the two cohorts have equal changes, but the profile levels differ from the treated-unit-weighted levels. In other data, the choice of weighting scheme can affect both levels and changes.

A Practical Construction and Reporting Sequence

A transparent event-time profile can be built by keeping the mapping, eligibility, and weighting decisions visible rather than treating aggregation as an automatic final step.

Fix the event-time convention.
Record the adoption date $g$ and calculate $e=t-g$ for each group-time estimate. Make clear which calendar period counts as onset.

Determine eligibility separately by event time.
For each $e$, list the cohorts with an available group-time estimate and a valid comparison. Call this set $E_e$; do not fill unavailable cells with zero.

Choose the target mixture.
State whether cohorts receive equal weights, treated-unit weights, or other prespecified target weights. Normalize the weights over the eligible cohort set at each event time.

Calculate and label each aggregate.
Apply the weighted sum to the cohort-time estimates. Report its outcome units, the cohorts included, and the weights or enough information to reconstruct them.

Qualify comparisons across event times.
State when the contributing cohorts or weights change. If a common-cohort comparison is useful and supported, calculate and label that profile separately rather than silently substituting its target.

Common Mistakes and Interpretation

Confusing calendar time with event time. A period-8 estimate is not necessarily an $e=8$ estimate. Calculate $e=t-g$ using that cohort’s adoption period.
Averaging available estimates without defining the target. An unweighted average gives each cohort equal influence, regardless of size. That can be appropriate, but it is not the same as averaging across treated units.
Assuming the profile always represents one population. If $E_e$ or the weights change, different points may describe different mixtures of cohorts. Report that change rather than attributing all movement to effects evolving within the same group.
Treating an unavailable estimate as zero. Missing support is not evidence of no effect. Include only estimable components and state which cohorts are omitted.
Calling a weighted average causal by construction. Aggregation preserves a causal interpretation only if the component group-time effects are identified for their stated targets and the weights define the intended aggregate. Weighting does not establish parallel trends or make an invalid comparison valid.

Uncertainty also belongs to the profile, but it is not obtained by simply averaging the component standard errors. The aggregate’s uncertainty depends on the uncertainty of the group-time estimates and their covariance. Report inference using an appropriate procedure for the estimated components; the weighted point-estimate formula alone does not provide a standard error or confidence interval.

Key takeaway: An event-time profile is a sequence of weighted averages of cohort-time effects. Its interpretation requires three items at every event time: the relative-time mapping, the eligible cohort set, and the weights. When any of these change, disclose how the target mixture changes.

Check Your Understanding

Use the definitions and calculations above to answer the following.

A cohort adopts in period 6 and has an estimate for calendar period 9. What is its relative event time, and how would you label the estimate?
At one event time, two cohorts have effects of 0.5 and 1.1 outcome units, with 50 and 150 treated units. Calculate the equal-cohort and treated-unit-weighted aggregates. What target does each weighting scheme emphasize?
Why can the difference between two adjacent points on a dynamic profile fail to describe the effect change for any fixed cohort mixture?
If a cohort’s group-time estimate is unavailable at $e=2$, why is it incorrect to enter zero for that cohort in the $e=2$ average?
What information should accompany a reported event-time point so a reader can interpret its target population?