Aggregating Group-Time Effects

Why Aggregate Group-Time Effects?

In a staggered-adoption design, treatment effects can differ across both adoption cohorts and calendar periods. A group-time effect preserves that variation: it describes the effect for a particular cohort at a particular time. But an analysis may also need a concise summary, such as an average over all included post-treatment observations or a profile of effects by time since adoption. Those summaries are not uniquely determined by the group-time effects alone. Their weights decide which cohorts, periods, and treated units the summary represents.

The aim is therefore not simply to average a table of estimates. First specify the target summary; then choose weights that correspond to it. A summary weighted equally over group-time cells answers a different question from one weighted by the number of treated units observed in each cell.

Group-Time Effects and the Aggregation Target

Let $G=g$ denote membership in a cohort first treated in calendar period $g$, and let $t$ denote calendar time. For a post-treatment cell, $t\geq g$. Write $ATT(g,t)$ for the average treatment effect for cohort $g$ in period $t$, using the group-time effect convention established in group-time average treatment effect estimation. An estimated effect for that cell is written $\widehat{ATT}(g,t)$.

The group-time effects must be identified and estimated under the design's relevant assumptions, including the applicable parallel-trends and comparison-group conditions. Aggregation does not supply those conditions or make an unidentified effect causal. It combines effects that have already been estimated for the chosen cells.

Definition: An aggregation is a weighted average of selected group-time effects. For a set of included cells $\mathcal C$, with nonnegative weights $w_{gt}$ summing to one, the aggregate estimate is $\sum_{(g,t)\in\mathcal C} w_{gt}\widehat{ATT}(g,t)$. The included cells and the weights jointly define the summary's target.

To describe weights based on observed treated units, let $N_{gt}$ be the number of treated units from cohort $g$ represented in the data in period $t$. This count can vary over time because of attrition, unbalanced panels, or other changes in observation. It need not equal the cohort's original size. When all cohort members are observed in every included period, $N_{gt}=N_g$, where $N_g$ is the cohort size; otherwise, the period-specific count is the relevant one for an average across the treated units represented in that cell.

A count-weighted summary describes observed treated unit-periods, not necessarily unique treated individuals. A person observed in multiple periods contributes multiple unit-periods. Also, if an estimate is intended to represent a broader population than the units observed in a cell, the data and identification assumptions must justify that interpretation; the count $N_{gt}$ alone does not correct for selective observation.

Four Common Aggregations

The same group-time effects can be summarized in several useful ways. In each case, define the included cells first. A cell that is unavailable or intentionally excluded receives no weight.

Overall treated-unit-period average: For an included set of post-treatment cells $\mathcal C$, weight each cell in proportion to its observed treated count: $$ \widehat{\theta}_{\mathrm{overall}} = \sum_{(g,t)\in\mathcal C} \frac{N_{gt}}{\sum_{(h,s)\in\mathcal C}N_{hs}} \widehat{ATT}(g,t). $$ This targets an average over the observed treated unit-periods in those cells, when each cell estimate is an average for its represented treated units.

An equal-cell average instead assigns $w_{gt}=1/|\mathcal C|$ to every included cell. It represents an average over cells, not an average over treated unit-periods. The two summaries coincide only in special cases, such as equal cell counts or equal effects.

A cohort-specific summary combines the post-treatment effects for one cohort over a selected set of periods $\mathcal T_g$. For an average across that cohort's observed treated unit-periods, use weights proportional to $N_{gt}$:

\widehat{\theta}_g = \sum_{t\in\mathcal T_g} \frac{N_{gt}}{\sum_{s\in\mathcal T_g}N_{gs}} \widehat{ATT}(g,t).

Equal weights over periods are another valid choice if the intended target is an average period for that cohort. Calendar-time summaries fix $t$ and combine all eligible treated cohorts $\mathcal G_t$. To average over represented treated units in period $t$, the weights are:

\widehat{\theta}_t = \sum_{g\in\mathcal G_t} \frac{N_{gt}}{\sum_{h\in\mathcal G_t}N_{ht}} \widehat{ATT}(g,t).

If all members of every included cohort are represented in period $t$, then $N_{gt}=N_g$ and these calendar-time weights reduce to weights proportional to cohort size. With changing observation counts, using original cohort sizes instead would generally describe a different population from the treated units actually represented in that period.

Finally, define event time as $e=t-g$, the number of periods since cohort $g$ first received treatment. An event-time summary combines the effects from the cohorts observed at that relative time. Let $\mathcal G_e$ be the included cohorts with an available effect at $t=g+e$. To average across represented treated units at event time $e$, weight by their counts at the corresponding calendar times:

\widehat{\theta}_e = \sum_{g\in\mathcal G_e} \frac{N_{g,g+e}}{\sum_{h\in\mathcal G_e}N_{h,h+e}} \widehat{ATT}(g,g+e).

The set of contributing cohorts may change with $e$, particularly at long event times. Consequently, differences between two event-time summaries can reflect both changes in treatment effects and changes in the mix of cohorts represented. Reporting the contributing cohorts and the weights helps make that limitation visible.

Worked Aggregations

Worked Example: Overall Average With Unequal Cell Counts

Suppose a hypothetical analysis estimates effects in five post-treatment cells. Effects are measured in outcome points; $N_{gt}$ counts the treated units observed in each cell.

Cohort $g$	Period $t$	$N_{gt}$	$\widehat{ATT}(g,t)$, points
2	2	100	2
2	3	90	3
3	3	200	1
3	4	180	2
4	4	50	4

The total number of observed treated unit-periods is $100+90+200+180+50=620$. The count-weighted overall effect is:

\widehat{\theta}_{\mathrm{overall}} = \frac{100(2)+90(3)+200(1)+180(2)+50(4)}{620} = \frac{1230}{620} \approx 1.984 \text{ points}.

Thus, across these five cells, the estimated average effect is about 1.984 outcome points per observed treated unit-period. For comparison, the equal-cell mean is $(2+3+1+2+4)/5=2.400$ points. That larger value gives the small cell with 50 observations the same weight as the cell with 200; it answers an equal-cell question rather than the observed-unit-period question.

Worked Example: Cohort and Calendar-Time Summaries

Use the same hypothetical cell estimates and counts. For cohort 2, suppose the target is its average effect over periods 2 and 3, weighted by observed treated unit-periods. The calculation is:

\widehat{\theta}_2 = \frac{100(2)+90(3)}{100+90} = \frac{470}{190} \approx 2.474 \text{ points}.

This summary gives more weight to period 2 because 100 cohort-2 units are represented there, compared with 90 in period 3. It is not the equal-period average, which would be $(2+3)/2=2.500$ points.

Now fix calendar period 3. Cohort 2 contributes 90 observed treated units with an estimated effect of 3 points, while cohort 3 contributes 200 with an estimated effect of 1 point. The represented-unit average is:

\widehat{\theta}_3 = \frac{90(3)+200(1)}{90+200} = \frac{470}{290} \approx 1.621 \text{ points}.

This is an average for the treated units represented in period 3. Using original cohort sizes in place of these period-specific counts would be inappropriate if cohort 2's 90 observed units differ from its original size. The weights must match the chosen target and the data actually represented.

Worked Example: Event-Time Effects and Changing Cohort Composition

At event time $e=0$, the included cells are the first treatment periods for cohorts 2, 3, and 4. Their observed treated counts are 100, 200, and 50, and their estimated effects are 2, 1, and 4 points. Therefore:

\widehat{\theta}_0 = \frac{100(2)+200(1)+50(4)}{100+200+50} = \frac{600}{350} \approx 1.714 \text{ points}.

At event time $e=1$, suppose the available cells are cohort 2 in period 3 and cohort 3 in period 4. Cohort 4 has no included observation at this event time. The summary is:

\widehat{\theta}_1 = \frac{90(3)+180(2)}{90+180} = \frac{630}{270} \approx 2.333 \text{ points}.

The estimated average rises from about 1.714 to 2.333 points. This comparison is descriptive of these event-time-specific represented populations. It is not automatically the average change for a fixed cohort mix: cohort 4 contributes at $e=0$, but not at $e=1$, and the relative weights also differ. A fixed-cohort comparison would require restricting both summaries to the same cohorts and choosing a consistent weighting target.

Why the Weighting Choice Matters

Aggregation is part of defining the estimand, not a cosmetic step after estimation. Different defensible choices answer different questions. Equal weighting by cell treats each group-time combination as equally important. Weighting by $N_{gt}$ gives more influence to cells with more observed treated unit-periods. Weighting cohorts equally gives each adoption cohort the same influence, regardless of its size. A period average restricted to a particular set of cohorts answers a narrower question than one that includes every eligible cohort.

A clear report should state the included cohorts and periods, the weighting rule, and the population represented. For an event-time profile, it is especially useful to show how many cohorts contribute at each event time. If observation counts vary, also describe whether the weights use the observed cell counts or another prespecified target distribution.

Key takeaway: An aggregate is interpretable only together with its included cells and weights. Count weights based on $N_{gt}$ describe observed treated unit-periods in those cells; they do not, without further assumptions, represent unobserved cohort members or a fixed population across periods.

Uncertainty and a Common Pitfall

If the vector of group-time estimates has estimated covariance matrix $\widehat{\Sigma}$, and the aggregation weights are treated as fixed, the variance of a linear aggregate is obtained by applying those weights to the full covariance matrix:

\widehat{\operatorname{Var}}(\widehat{\theta}) = \mathbf{w}^{\mathsf T}\widehat{\Sigma}\mathbf{w}.

The covariance terms matter because estimates from different cells may share units, comparison groups, or time periods. Averaging the cell standard errors is not a valid general method for finding the aggregate standard error. Inference must also respect the design's dependence structure.

Common pitfall: using fixed cohort sizes when observed counts change. Suppose cohorts A and B originally contain 100 and 300 treated units. If, in a particular calendar period, only 50 units from A and 300 from B are observed, weights based on original cohort sizes are $100/400=0.25$ and $300/400=0.75$. But the average across treated units represented in that period uses weights $50/350\approx0.143$ and $300/350\approx0.857$. The first calculation is not the observed-period treated-unit average. Use $N_{gt}/\sum_h N_{ht}$ for that target, and state a different weighting target explicitly if one is intended.

Check Your Understanding

For each question, identify the aggregation target before calculating or interpreting a weight.

What population of observations does an overall summary weighted by $N_{gt}$ represent?
For a calendar-time average, cohort A has 40 observed treated units and cohort B has 160. What are the observed-unit weights?
Why can an equal-cell average differ from an average across observed treated unit-periods?
At event time $e=2$, one cohort has no available group-time effect. Should it receive a zero effect or be excluded from the included cohort set? Explain.
Why might comparing event-time averages at $e=0$ and $e=1$ not describe changes for a fixed cohort population?

What You'll Learn