Cohort-Time Average Treatment Effects

Why Effects Need Both a Cohort and a Time

With staggered adoption, units begin treatment in different calendar periods. An effect for units that adopt in period $g$ may differ from the effect for units adopting later, and the effect for a given cohort may change over time. A single label such as “the treatment effect” can therefore hide which units and which period the claim concerns.

The group-time average treatment effect, written $ATT(g,t)$, makes both dimensions explicit. It is an average causal effect for the cohort whose first treatment period is $g$, evaluated at calendar time $t$. Defining this quantity is separate from estimating it: for treated units after adoption, the outcome under treatment is observed, but their outcome under no treatment is not.

Definition: Let $G_i$ denote unit $i$'s first treatment period, with $G_i=\infty$ for a unit that is never treated. Let $Y_{it}(g)$ be unit $i$'s potential outcome at time $t$ if its first treatment period were $g$, and let $Y_{it}(\infty)$ be its potential outcome at time $t$ if it were never treated. For a cohort $g$ and a post-adoption period $t\ge g$, the group-time average treatment effect is $$ ATT(g,t)=E\!\left[Y_{it}(g)-Y_{it}(\infty)\mid G_i=g\right]. $$ It is the average effect at time $t$ for units that in fact belong to adoption cohort $g$, comparing their outcomes under adoption at $g$ with their outcomes under never adopting.

The conditioning event $G_i=g$ matters: the average is taken over cohort $g$, not over all units and not over whichever comparison group might later be used to estimate the effect. The time $t$ matters too. $ATT(g,g)$ is the cohort's average effect in its first treatment period; $ATT(g,g+1)$ is its average effect one period later. These are distinct estimands and need not have the same value.

The potential outcome $Y_{it}(g)$ refers to the outcome under a treatment history that begins in period $g$, while $Y_{it}(\infty)$ refers to the outcome under a history with no treatment. The definition does not assert that these potential outcomes are equal before $g$. Equality before adoption requires an additional no-anticipation condition; without it, units may respond in advance of the formal treatment start. For $t\ge g$, the difference in the definition compares the adopted regime with never adopting, not merely with being untreated at time $t$.

What Is Observed for Each Cohort?

To connect potential outcomes to data, assume consistency: a unit's observed outcome is the potential outcome corresponding to its actual adoption history. Thus, if $G_i=g$, its observed outcome at time $t$ is $Y_{it}(g)$; if $G_i=\infty$, its observed outcome is $Y_{it}(\infty)$. This mapping presumes that adoption timing adequately describes the relevant treatment history and that one unit's treatment does not change another unit's outcome.

For a cohort $g$ at a post-adoption time $t\ge g$, the observed outcome is $Y_{it}(g)$. The alternative $Y_{it}(\infty)$ is counterfactual for those same treated units: they cannot both adopt at $g$ and never adopt. Consequently, the cohort-time ATT is not directly observed, even when every treated unit's outcome is recorded.

For a never-treated unit, the observed outcome is $Y_{it}(\infty)$. At a time before their adoption, units in a future cohort have not yet begun treatment, but their observed outcome is $Y_{it}(g)$, the potential outcome under their actual adoption schedule. If there is no anticipation, that pre-adoption outcome equals $Y_{it}(\infty)$; absent that condition, the equality cannot be assumed. A group mean among never-treated or not-yet-treated units is observable, but it is the mean for those units, not the missing never-treated potential-outcome mean for cohort $g$.

Key distinction: For a treated cohort after adoption, its treated potential-outcome mean is observable and its never-treated potential-outcome mean is not. An observed difference between the treated cohort and another group is a contrast of observed outcomes. It equals $ATT(g,t)$ only under additional assumptions that justify the comparison.

Worked Example: Defining a Cohort-Time ATT

Suppose the outcome is a hypothetical monthly symptom score, with lower values indicating fewer symptoms. Four people belong to cohort $g=2$, meaning their treatment begins in period 2. At $t=3$, suppose their potential outcomes under adoption in period 2 are 8, 10, 12, and 14, while their potential outcomes under never adopting are 6, 9, 10, and 13. These values are stipulated for illustration; in real data, both potential outcomes for a treated person are not observed.

The individual effects under the two regimes are $8-6=2$, $10-9=1$, $12-10=2$, and $14-13=1$ score points. Therefore, $$ ATT(2,3)=\frac{(8-6)+(10-9)+(12-10)+(14-13)}{4} =\frac{6}{4}=1.5\text{ score points}. $$

This is the average effect at period 3 for cohort 2, comparing adoption in period 2 with never adopting. Since lower scores are better, the positive value means the adopted regime produces a score 1.5 points higher on average for this cohort at that time. The calculation illustrates the estimand, not an empirical procedure: after adoption, only each person's outcome under adoption is observed, so the never-treated values used here would have to be supplied by a valid design or assumptions.

Observed Means Are Not Automatically Effects

Let the observed cohort-time mean be the average outcome among units in cohort $g$ at time $t$. Under consistency, for $t\ge g$ this mean estimates the cohort mean of $Y_{it}(g)$. A mean among never-treated units at the same time describes $Y_{it}(\infty)$ for the never-treated group. Subtracting these means is possible with observed data, but the groups differ in adoption status and may also differ in their outcomes even under never treatment.

The distinction can be stated directly. The ATT compares two potential-outcome means for the same target cohort. A raw treated-versus-never-treated contrast compares an observed mean for cohort $g$ with an observed mean for a different set of units. The latter does not, by definition alone, supply cohort $g$'s missing counterfactual mean.

Worked Example: A Post-Adoption Contrast Is Observable but Not the ATT

Consider hypothetical outcome means in period 4. Cohort $g=3$ has adopted treatment, and its observed mean is 27 units. Never-treated units have an observed mean of 19 units. The observed difference is $$ 27-19=8\text{ outcome units}. $$

The value 8 is an observable treated-cohort-versus-never-treated contrast. But $ATT(3,4)$ is $E[Y_{i4}(3)-Y_{i4}(\infty)\mid G_i=3]$. The value 19 is the mean for never-treated units, not necessarily the mean of $Y_{i4}(\infty)$ among cohort 3. For example, if cohort 3's missing never-treated mean were 23, the cohort-time effect would be $27-23=4$ units, not 8. The number 23 is hypothetical and unobserved; it demonstrates why the observed contrast alone does not determine the ATT.

The difference could equal the ATT under suitable conditions linking the comparison group's outcomes to the treated cohort's missing counterfactual outcomes. The arithmetic does not establish those conditions. The contrast and the causal estimand should therefore be named separately.

Calendar Time and Adoption Time Must Both Be Tracked

A cohort can be observed in several periods, and each period has a different relationship to its adoption date. Before $g$, the cohort has not formally adopted; at $g$, it starts treatment; after $g$, it has been exposed for a longer duration. Keeping $g$ fixed while changing $t$ changes the time-specific effect being asked about. Keeping $t$ fixed while changing $g$ changes the target population.

Worked Example: Classifying Cohort-Time Cells

Suppose four hypothetical groups are observed in periods 1 through 4. Group A first adopts in period 2, group B first adopts in period 4, and group N is never treated. Consider what is observed for cohort B and group N at periods 3 and 4.

At period 3, B has not yet adopted. Its observed outcome is the outcome under its actual adoption schedule, $Y_{i3}(4)$. If there is no anticipation, this equals $Y_{i3}(\infty)$; without that condition, the equivalence is not guaranteed. Group N's observed outcome is $Y_{i3}(\infty)$.

At period 4, B has adopted. Its observed outcome is $Y_{i4}(4)$, while its $Y_{i4}(\infty)$ is counterfactual. Group N's observed $Y_{i4}(\infty)$ is available, but it remains the outcome for group N, not for B. The target $ATT(4,4)$ is the average difference between $Y_{i4}(4)$ and $Y_{i4}(\infty)$ among B units. A comparison of B's period-4 mean with N's period-4 mean is observable, but the cell labels alone do not make it that ATT.

Group A is already treated in both periods 3 and 4. Its outcomes are outcomes under adoption in period 2 at both times. They are relevant to $ATT(2,3)$ and $ATT(2,4)$, respectively, but are not untreated outcomes for group B. Treatment status is determined by adoption timing, so an already-treated group's outcome should not be mistaken for an untreated comparison outcome.

A Short Justification for the Observed-Outcome Mapping

The relationship between the estimand and the observed data follows from consistency. For any unit with $G_i=g$, consistency gives $Y_{it}^{\mathrm{obs}}=Y_{it}(g)$. Restricting to that cohort and averaging at a post-adoption time yields $$ E[Y_{it}^{\mathrm{obs}}\mid G_i=g] =E[Y_{it}(g)\mid G_i=g]. $$

But the ATT also subtracts $E[Y_{it}(\infty)\mid G_i=g]$. Consistency does not reveal that second mean, because the cohort did not follow the never-treated regime. Thus the observed treated-cohort mean supplies one term in the ATT and leaves the other term counterfactual. An identification argument must justify how to learn or represent the missing term; simply observing outcomes for another group does not complete the equality.

Common Mistakes and Why the Distinction Matters

Calling a raw mean difference the ATT. A comparison between the treated cohort and never-treated units is observable, but the comparison group's mean need not equal the treated cohort's mean outcome under never treatment. State the observed contrast and the causal estimand separately.
Ignoring the cohort in the target. $ATT(g,t)$ averages effects among units with $G_i=g$. It is not automatically an effect for all treated units, all units, or the comparison group.
Ignoring calendar time. $ATT(g,t)$ refers to one cohort at one calendar time. An effect at adoption need not equal an effect in later periods.
Assuming pre-adoption outcomes equal never-treated outcomes without qualification. They coincide under no anticipation, but not by notation alone. Be explicit about whether that condition is being used.
Treating an already-treated cohort as untreated. A cohort that adopted earlier has outcomes under its treatment history. Its outcome is not an untreated potential outcome for a later-adopting cohort.

These distinctions matter because staggered designs offer many possible group-and-time comparisons, but not every observed comparison targets the same causal quantity. Naming $g$, $t$, the target cohort, and the counterfactual regime keeps the scientific question clear before any estimation method is chosen.

Check Your Understanding

Answer each question by identifying the target cohort, time, or potential outcome precisely.

In words, what population and time does $ATT(5,7)$ refer to?
For a unit in cohort $g=3$ observed at $t=4$, which potential outcome is observed under consistency, and which term in $ATT(3,4)$ remains counterfactual?
A period-4 mean for cohort 2 is compared with a period-4 mean for never-treated units. Why is the difference not automatically $ATT(2,4)$?
When can a future-adopting cohort's pre-adoption outcome be treated as its never-treated potential outcome?
What does $ATT(g,g+2)$ describe, and why should it not be assumed equal to $ATT(g,g)$?