Group-Time ATT Estimation With Not-Yet-Treated Controls

Why Control Eligibility Depends on Time

With staggered adoption, different groups begin treatment in different periods. A group that is untreated now may adopt later, so it can serve as a comparison group for some outcome periods but not for others. This creates a precise rule for choosing controls: for an estimate at time $t$, use units that remain untreated through $t$, not units that were untreated only at an earlier baseline.

The result is a separate difference-in-differences contrast for each treated cohort $g$ and outcome period $t$. Its target is the average treatment effect for cohort $g$ at time $t$, denoted $ATT(g,t)$. The comparison group can change as $t$ advances, so each contrast must be interpreted and justified for its own timing.

Definition: Let $G_i$ be unit $i$'s first treatment period. A unit with $G_i=g$ belongs to cohort $g$; a never-treated unit has $G_i=\infty$. Treatment is absorbing: a unit remains treated after its adoption period. For a treated cohort $g$ and an outcome period $t\ge g$, the not-yet-treated control group consists of units with $G_i>t$. These units are untreated in every period through $t$, including the baseline period $g-1$.

Write $Y_{it}(0)$ for unit $i$'s outcome in period $t$ if untreated and $Y_{it}(1)$ for its outcome if treated. The observed outcome is $Y_{it}(0)$ before adoption and $Y_{it}(1)$ from adoption onward. The group-time ATT is

ATT(g,t)=E\!\left[Y_{it}(1)-Y_{it}(0)\mid G_i=g\right],\qquad t\ge g.

This is the average effect for members of cohort $g$ in period $t$, not the difference between that cohort and the controls. The controls help recover the treated cohort's unobserved untreated outcome: what its average outcome would have been at $t$ had it not adopted treatment.

The Timing-Specific Contrast

For a cohort adopting in period $g$, use period $g-1$ as the baseline. It is the last period before adoption. At outcome time $t\ge g$, compare the treated cohort's change from $g-1$ to $t$ with the change over the same periods among units with $G_i>t$.

ATT(g,t) = \left\{E[Y_{it}\mid G_i=g]-E[Y_{i,g-1}\mid G_i=g]\right\} - \left\{E[Y_{it}\mid G_i>t]-E[Y_{i,g-1}\mid G_i>t]\right\}.

The first bracket is the observed outcome change for cohort $g$. Because that cohort is treated at $t$, this change combines its treatment effect with the change it would have experienced without treatment. The second bracket estimates that untreated change using the eligible controls. The difference removes the estimated untreated trend.

Identification assumptions: For the contrast to identify $ATT(g,t)$, assume (1) consistency and no interference: observed outcomes correspond to each unit's treatment status, and one unit's treatment does not change another unit's outcome; (2) no anticipation: before adoption, cohort $g$'s outcome equals its untreated potential outcome, in particular at $g-1$; and (3) parallel trends for this comparison: $$ E[Y_{it}(0)-Y_{i,g-1}(0)\mid G_i=g] = E[Y_{it}(0)-Y_{i,g-1}(0)\mid G_i>t]. $$ There must also be eligible controls with outcomes observed at both periods. The parallel-trends assumption is specific to cohort $g$, outcome time $t$, and the set $G_i>t$.

The parallel-trends condition does not require the treated cohort and controls to have equal outcome levels at baseline. It requires their average untreated changes from $g-1$ to $t$ to be equal. Under no anticipation, the treated cohort's observed baseline outcome is untreated. The control group is untreated at both dates because $G_i>t$. Thus the second bracket is an observed change in untreated outcomes, and parallel trends lets it stand in for the treated cohort's missing untreated change. Subtracting it from the treated cohort's observed change leaves $ATT(g,t)$.

Worked Contrasts

Worked Example: Estimating an Effect for One Cohort and Period

Suppose a hypothetical program is adopted by cohort $g=3$. We want its effect in period $t=4$. Cohorts adopting in period 5 or later, together with never-treated units, are the eligible controls because all have $G_i>4$. Assume no anticipation and parallel trends between cohort 3 and this control group over periods 2 to 4.

The hypothetical mean outcomes are:

Group	Mean outcome at period 2	Mean outcome at period 4
Cohort 3	50	62
Controls with $G_i>4$	48	54

Cohort 3's observed change is $62-50=12$ outcome units. The eligible controls' change is $54-48=6$ outcome units. Therefore,

\widehat{ATT}(3,4)=(62-50)-(54-48)=12-6=6.

Under the stated assumptions, the estimated treatment effect for cohort 3 in period 4 is 6 outcome units on average. The baseline means differ, but that does not invalidate the contrast; identification requires parallel untreated changes, not equal starting levels.

Worked Example: Controls Change at a Later Outcome Time

Suppose cohorts adopt in periods 2, 4, and 6, and some units are never treated. Consider the cohort adopting in period $g=2$. At $t=3$, cohorts 4 and 6 and never-treated units are eligible controls. At $t=5$, cohort 4 is already treated and must be excluded; only cohort 6 and never-treated units remain eligible. Assume the required no-anticipation and comparison-specific parallel-trends conditions hold for each contrast.

At $t=3$, the hypothetical mean outcome for cohort 2 is 30 in baseline period 1 and 39 in period 3. The eligible controls' corresponding means are 28 and 32. Thus,

\widehat{ATT}(2,3)=(39-30)-(32-28)=9-4=5.

At $t=5$, cohort 2's mean is 30 in period 1 and 44 in period 5. For the newly eligible control set—cohort 6 and never-treated units—suppose the means are 29 and 35. The separate contrast is

\widehat{ATT}(2,5)=(44-30)-(35-29)=14-6=8.

These are estimates for the same treated cohort at different outcome times, but they use different control groups. In particular, cohort 4's outcomes can help estimate the period-3 contrast while it is untreated, but cannot serve as a control at period 5 after adopting. The two calculations each require parallel trends for the control group actually used at that time.

Worked Example: Pooling Eligible Controls by Units

Suppose that for cohort $g=2$ and outcome time $t=3$, the eligible controls consist of 20 units in cohort 4 and 80 never-treated units. Both groups are untreated through period 3. Their average changes from period 1 to period 3 are, respectively, 2 and 8 outcome units. Assume cohort 2's average observed change is 12 outcome units and that the relevant assumptions hold.

If the control contrast uses the average change across eligible control units, its change is the count-weighted mean:

\frac{20(2)+80(8)}{20+80} = \frac{40+640}{100} = 6.8.

The resulting difference-in-differences estimate is

\widehat{ATT}(2,3)=12-6.8=5.2.

The estimate is 5.2 outcome units. Giving the two control cohorts equal weight would instead produce $(2+8)/2=5$, a different calculation that does not equal the average change among the 100 eligible control units. The intended weighting must match the estimator and target population; a cohort average is not automatically the same as a unit average.

What Changes as the Calendar Advances

The eligible control set is defined by $G_i>t$, so it can shrink as $t$ increases. Each later-adopting cohort leaves the control pool once its adoption period is reached. Never-treated units, if present, remain eligible in every period, but not-yet-treated controls need not include any never-treated units. For each $(g,t)$, identify the eligible set anew rather than treating the controls at one date as a fixed comparison group for all dates.

Choose the effect to estimate.
Specify a treated cohort $g$ and an outcome period $t\ge g$. The target is $ATT(g,t)$.

Determine eligibility at time $t$.
Include units with $G_i>t$, including never-treated units when available. Exclude every unit already treated by $t$.

Form the baseline-to-outcome changes.
For both groups, use periods $g-1$ and $t$. Check that outcomes are observed and that the comparison group is untreated at both dates.

State the identifying comparison.
Justify no anticipation and parallel untreated trends between cohort $g$ and the eligible controls for these dates.

Changing controls has two implications. First, later estimates rely on a potentially different parallel-trends comparison. Evidence or substantive reasoning that supports the assumption for one date does not automatically establish it for another. Second, the comparison group's baseline level and trend can change as cohorts enter or leave the eligible pool. That affects the observed control change; it is not, by itself, evidence that the treatment effect changed.

If no units with $G_i>t$ have usable outcomes, this not-yet-treated contrast is unavailable for that cohort and time. The problem is not repaired by keeping already-treated cohorts in the control group: their outcomes may reflect treatment. A different identification strategy or additional eligible comparison units would be needed.

Worked Example: Recognizing a Period with No Eligible Controls

Suppose all units in a hypothetical study have adopted treatment by period 6: the latest cohort has $G_i=6$, and there are no never-treated units. For an effect at $t=6$, a cohort adopting in period 3 would require controls with $G_i>6$. No such units exist.

The contrast would require the control change \[ E[Y_{i6}\mid G_i>6]-E[Y_{i2}\mid G_i>6], \] but the conditioning group is empty, so these means cannot be calculated. Consequently, the not-yet-treated difference-in-differences contrast does not identify $ATT(3,6)$ in this sample. Units from the period-6 cohort are not valid controls at $t=6$, because they are treated in that period.

Common Mistakes and Practical Interpretation

Using units that are untreated only at baseline. A cohort that adopts between $g-1$ and $t$ is treated by the outcome period. The eligible condition is $G_i>t$, not merely $G_i>g-1$.
Assuming that one parallel-trends statement covers every date. The relevant comparison is cohort $g$ against the controls with $G_i>t$, over periods $g-1$ and $t$. A different $t$ can mean a different control group and a different assumption.
Interpreting the contrast as a raw outcome difference. The estimate subtracts the controls' change from the treated cohort's change. A treated-control level gap at $t$ alone is not the group-time ATT contrast.
Treating a changing estimate as proof of changing effects. Contrasts at different times can use different control populations. Their difference may reflect treatment dynamics, changes in the comparison group, or violations of the assumptions; the arithmetic alone does not separate these possibilities.

Key takeaway: For $ATT(g,t)$, compare cohort $g$'s change from $g-1$ to $t$ with the same-period change among units with $G_i>t$. Interpret the result as a cohort-specific effect only under no anticipation, appropriate parallel trends for that timing-specific comparison, and sufficient eligible controls.

Check Your Understanding

Answer each question by identifying the cohort, outcome time, and eligible comparison group where relevant.

A treated cohort adopts in period 4, and the target outcome period is 7. Which units are eligible not-yet-treated controls?
Why is period $g-1$ a suitable baseline for a cohort adopting in period $g$, under the no-anticipation assumption?
State the parallel-trends condition needed to identify $ATT(g,t)$ with controls satisfying $G_i>t$.
A cohort is untreated at baseline but adopts before $t$. Explain why it cannot remain in the not-yet-treated control group for estimating an effect at $t$.
What does it mean if no units satisfy $G_i>t$, and why is using already-treated units not a valid automatic substitute?