Group-Time Average Treatment Effect Estimation

Why Estimate Effects by Cohort and Period?

When units begin treatment in different periods, a single before-and-after comparison can combine effects from different treatment cohorts and different durations of exposure. A more transparent target is the treatment effect for one adoption cohort at one calendar period. This group-time average treatment effect keeps the treated group and evaluation period explicit, and it allows the comparison group to be chosen so that its members are untreated during the relevant comparison.

Let $G_i$ denote unit $i$'s first treatment period, with $G_i=\infty$ for a unit that is never treated. Treatment is absorbing: after adoption, the unit remains treated. Write $Y_{it}(g)$ for unit $i$'s potential outcome in period $t$ if treatment first begins in period $g$, and $Y_{it}(0)$ for its potential outcome if it remains untreated. The observed outcome is the potential outcome corresponding to the unit's actual treatment history.

Definition: For cohort $g$ and period $t\ge g$, the group-time average treatment effect is $$ ATT(g,t)=E\left[Y_{it}(g)-Y_{it}(0)\mid G_i=g\right]. $$ It is the average effect in period $t$ for units first treated in period $g$, relative to those same units remaining untreated. It is specific to both the adoption cohort and the calendar period.

This is an average for the cohort that adopted in $g$, not for all treated units and not necessarily for the population as a whole. The outcome's units carry through to the effect: if $Y$ is measured in dollars, $ATT(g,t)$ is measured in dollars. The target is causal only under assumptions that allow the untreated outcomes of the treated cohort to be recovered from comparison units.

Identification With an Untreated Comparison Group

For a particular $(g,t)$, choose a comparison group $C_{g,t}$ whose members are untreated both at the baseline period $g-1$ and at the evaluation period $t$. Two common choices are never-treated units and not-yet-treated units. The latter are units with $G_i>t$; because adoption is absorbing, they have not yet adopted by $t$. A unit treated in period $g$ is not an eligible control for an effect at $t\ge g$.

Assume no anticipation: before its adoption period, a unit's outcome is the same as it would have been if it remained untreated. In particular, the treated cohort's outcome at $g-1$ is untreated, and not-yet-treated comparison units are untreated through $t$. Define the change in the untreated potential outcome from baseline to evaluation as $Y_{it}(0)-Y_{i,g-1}(0)$.

Conditional parallel trends: For the chosen comparison group and covariates $X_i$ measured before treatment, $$ E\left[Y_{it}(0)-Y_{i,g-1}(0)\mid G_i=g,X_i=x\right] = E\left[Y_{it}(0)-Y_{i,g-1}(0)\mid i\in C_{g,t},X_i=x\right] $$ for covariate values represented in the treated cohort. Conditional on $X_i$, the treated cohort and comparison group would have had the same average untreated outcome change.

The assumption concerns changes in untreated potential outcomes, not equality of outcome levels. The groups may start at different levels. It also does not say that observed changes must match after treatment: a genuine treatment effect can create that difference.

For covariate-adjusted identification, there must also be overlap: at covariate values occurring among cohort $g$, there are comparison units from which to estimate the untreated change. Equivalently, the conditional probability of belonging to cohort $g$, rather than the chosen comparison group, is strictly between zero and one on the treated cohort's covariate support. Without overlap, the comparison group's conditional change at those values cannot be learned from the data without extrapolation.

Identification result: Under no anticipation, conditional parallel trends, overlap, and consistency of observed outcomes with the assigned treatment history, $$ ATT(g,t) = E\left[ E\left[Y_{it}-Y_{i,g-1}\mid G_i=g,X_i\right] - E\left[Y_{it}-Y_{i,g-1}\mid i\in C_{g,t},X_i\right] \;\middle|\;G_i=g \right]. $$ If there are no covariates, or the groups are already comparable without adjustment, this reduces to the difference between their average outcome changes.

Why the Identification Formula Works

The key step is to separate the treated cohort's observed change into its untreated change and its treatment effect. No anticipation makes its baseline outcome untreated. At $t\ge g$,

Y_{it}-Y_{i,g-1} = \left[Y_{it}(0)-Y_{i,g-1}(0)\right] + \left[Y_{it}(g)-Y_{it}(0)\right], \qquad G_i=g.

Taking conditional expectations given $G_i=g,X_i=x$, the second bracket has conditional mean treatment effect for that cohort and covariate value. Conditional parallel trends says the first bracket has the same conditional mean as the untreated change among comparison units. Comparison units are untreated at both periods, so their observed change is their untreated potential-outcome change. Subtracting that comparison change therefore removes the treated cohort's expected untreated change, leaving the treatment effect. Averaging the result over the treated cohort's covariate distribution gives $ATT(g,t)$.

Worked Estimation

Worked Example: Never-Treated Controls Without Covariates

Suppose a hypothetical program begins in period 3 for a cohort of 40 districts. A separate set of never-treated districts supplies the comparison. The outcome is annual spending per resident in dollars. The cohort's mean is \$50 before adoption, in period 2, and \$58 in period 4. The never-treated comparison group's mean is \$45 in period 2 and \$50 in period 4. Estimate $ATT(3,4)$.

The estimated change for the treated cohort is $$ \widehat{\Delta Y}_{G=3}=58-50=8\text{ dollars per resident}. $$ The comparison group's change is $$ \widehat{\Delta Y}_{C}=50-45=5\text{ dollars per resident}. $$ Thus the difference-in-differences estimate is $$ \widehat{ATT}(3,4)=(58-50)-(50-45)=8-5=3\text{ dollars per resident}. $$

Under the identification assumptions, the estimate says that the program increased mean spending per resident in period 4 by an estimated \$3 for districts that adopted in period 3, relative to their untreated outcomes. The four means alone cannot establish parallel trends; that assumption must be justified using the design and subject-matter knowledge. Never-treated controls also need to be comparable in untreated trends to this cohort.

Worked Example: Conditional Adjustment Across Two Covariate Strata

Consider a hypothetical treatment cohort and never-treated controls, with a pre-treatment covariate $X$ dividing units into low- and high-need strata. The outcome is a test score in points, and the table gives mean changes from $g-1$ to $t$. Suppose the treated cohort is 75% low-need and 25% high-need.

Pre-treatment stratum	Treated-cohort mean change	Never-treated mean change	Treated-cohort share
Low need	8 points	3 points	0.75
High need	10 points	7 points	0.25

Within the low-need stratum, the estimated difference in changes is $8-3=5$ points. Within the high-need stratum, it is $10-7=3$ points. To estimate the cohort effect, average these differences using the treated cohort's covariate shares: $$ \widehat{ATT}(g,t) =0.75(8-3)+0.25(10-7) =0.75(5)+0.25(3) =3.75+0.75 =4.50\text{ points}. $$

The estimate is 4.50 points for the treated cohort, standardized to its own distribution of need. Both strata have treated and comparison units, so overlap holds in this example. The calculation relies on conditional parallel trends within each stratum; different overall covariate distributions are allowed, but stratum-specific untreated trends must be comparable.

Worked Example: Not-Yet-Treated Controls at a Later Period

In a hypothetical staggered rollout, a cohort first adopts in period 2, and its effect is evaluated in period 4. Use period 1 as the baseline. A comparison group consists of units whose adoption occurs after period 4, together with never-treated units. The outcome is employment per 100 working-age residents. Suppose the treated cohort's mean is 61.0 in period 1 and 67.2 in period 4; the eligible comparison group's means are 59.5 and 61.6.

The treated cohort's change is $67.2-61.0=6.2$ employed residents per 100. The comparison group's change is $61.6-59.5=2.1$ per 100. Therefore, $$ \widehat{ATT}(2,4)=(67.2-61.0)-(61.6-59.5) =6.2-2.1 =4.1 $$ employed residents per 100 working-age residents.

This comparison is eligible because its units remain untreated through period 4. If any comparison units had adopted by period 4, their observed outcomes could include treatment effects and their changes would not represent untreated trends. The estimate still requires no anticipation and parallel untreated trends between this cohort and the selected not-yet-treated group. It is a cohort- and period-specific estimate, not an average over all adoption cohorts or periods.

From the Formula to a Sample Estimator

In practice, conditional expectations are estimated from data. Let $\Delta Y_i=Y_{it}-Y_{i,g-1}$, and let $\widehat m_C(x)$ estimate the mean change among eligible comparison units with covariates $x$. A direct standardization estimator is

\widehat{ATT}(g,t) = \frac{1}{n_g} \sum_{i:G_i=g} \left[\Delta Y_i-\widehat m_C(X_i)\right],

where $n_g$ is the number of observed units in cohort $g$. The comparison-change model can be estimated using suitable strata, regression, or other methods, provided the procedure estimates the relevant conditional mean and does not obscure the required overlap. The average is taken over treated units, so the result targets their covariate distribution. Inference must reflect how treatment is assigned and how outcomes are dependent; for example, when policy assignment occurs at a higher level than the unit, standard errors generally need to account for that assignment level.

Never-treated and not-yet-treated controls are alternative designs for this calculation, not interchangeable labels. Never-treated units can provide a stable comparison set but may differ substantially from adopting units. Not-yet-treated units may be more similar, but the eligible set can change with $t$, and its members must remain untreated through that evaluation period. The choice should be made for each group-time comparison and defended through the design's assumptions.

Common Mistakes and Practical Checks

Using already-treated units as controls. A unit treated before or during $t$ may have a treatment-affected outcome. Restrict the comparison group to units untreated at both $g-1$ and $t$.
Assuming parallel trends means equal outcome levels. It means equal expected untreated changes, conditionally if covariates are used. Different baseline levels do not by themselves violate the assumption.
Adjusting for post-treatment covariates. A covariate affected by treatment can be part of the causal pathway. Use covariates measured before treatment for the conditional adjustment described here.
Ignoring overlap. If the treated cohort has covariate values absent from the controls, a fitted regression may still return a number, but that number relies on extrapolation rather than an observed conditional comparison.
Calling a difference in observed changes automatically causal. The arithmetic is a difference-in-differences estimate; its causal interpretation requires no anticipation, an untreated comparison group, and the relevant parallel-trends and overlap conditions.

Group-time estimation matters because it makes explicit which cohort's effect is being estimated, when it is measured, and which units provide its counterfactual trend. That specificity is especially important when adoption is staggered: it avoids treating a single pooled coefficient as though it necessarily represented every cohort and exposure duration. The individual group-time effects can later be summarized for a broader question, but that is a distinct step.

Check Your Understanding

Try these questions using the definitions and assumptions above.

In your own words, what population does $ATT(g,t)$ describe, and why must both $g$ and $t$ be specified?
A proposed not-yet-treated comparison unit adopts in period $t$. Is it eligible as a control for an effect measured in period $t$? Explain.
Why can conditional parallel trends hold even if the treated and comparison groups have different average outcomes at $g-1$?
In the stratified example, what would the estimate be if the treated cohort were 50% low-need and 50% high-need, with the same within-stratum differences?
What does overlap require when estimating the effect for a treated cohort with covariate adjustment?