Treatment Cohorts, Adoption Dates, and Event-Time Notation

Why Treatment Timing Needs Its Own Notation

In a simple two-period difference-in-differences design, it may be enough to label observations as before or after treatment. With staggered adoption, different units begin treatment in different periods. A unit can therefore be untreated in a period when another unit is already treated, and the same calendar period can represent different amounts of time since adoption for different units. A clear notation keeps these distinctions visible.

This tutorial defines adoption cohorts, calendar time, event time, and the treatment histories implied by an absorbing adoption process. These are descriptions of when treatment starts and which status a unit has in each period. They do not, by themselves, specify a causal effect or justify a comparison between units.

Definition: Let $i$ index units and $t$ index discrete calendar periods. For a unit that adopts treatment, $G_i$ is its adoption date: the first calendar period in which it receives treatment. An adoption cohort is the group of units sharing the same finite adoption date. The event time for an adopting unit in calendar period $t$ is $e=t-G_i$.

Calendar Time, Adoption Cohorts, and Event Time

Calendar time $t$ identifies when an observation occurs on the study's common timeline: for example, 2021, 2022, and 2023, or periods 1, 2, and 3. It is the same time index for all units. The adoption date $G_i$ uses that same calendar-time scale. If two units have different adoption dates, they belong to different cohorts even if they are observed in the same calendar period.

Event time measures a period relative to the unit's adoption date. For an adopting unit, $e=0$ is the adoption period, $e=-1$ is the immediately preceding period, and $e=2$ is the second period after adoption. Thus, event time is unit-specific even when calendar time is shared. Given $t$ and $G_i$, event time is calculated by subtraction; conversely, $t=G_i+e$.

The choice that adoption occurs in event time zero is a convention tied to defining $G_i$ as the first treated period. It is important to state this convention, since a different definition of the adoption date would shift the event-time labels. In a discrete-time setting, an event-time difference of one means one study period, not necessarily one year.

Worked Example: Assigning Cohorts and Event Times

Suppose a hypothetical policy is adopted by three regions. Region A first receives it in 2020, Region B in 2022, and Region C in 2022. Observations are indexed by calendar years.

Cohorts. Region A belongs to the 2020 cohort because $G_A=2020$. Regions B and C belong to the 2022 cohort because $G_B=G_C=2022$. The units in a cohort share an adoption date; they need not be adjacent or otherwise similar.

Event time for Region B in 2021. The calculation is $e=2021-2022=-1$, so this is one year before adoption.

Event time for Region B in 2022. The calculation is $e=2022-2022=0$, the adoption year.

Event time for Region B in 2024. The calculation is $e=2024-2022=2$, two years after adoption. In 2024, Region A is at event time $2024-2020=4$, even though both regions share the same calendar year.

Representing an Absorbing Treatment History

For staggered adoption, treatment is commonly represented as absorbing: once a unit adopts, it remains treated in every later period. Let $D_{it}$ indicate the treatment status of unit $i$ in period $t$. Its value is zero before adoption and one beginning in the adoption period. This indicator describes exposure status; it is not an outcome and does not quantify the effect of treatment.

Definition: For a unit with a finite adoption date $G_i$, an absorbing treatment history is represented by $$ D_{it}=\mathbf{1}(t\geq G_i), $$ where $\mathbf{1}(\cdot)$ equals one when its condition is true and zero otherwise. Thus, $D_{it}=0$ for $t<G_i$, and $D_{it}=1$ for $t\geq G_i$.

The absorbing condition means that, as calendar time increases, a unit's sequence of treatment indicators can switch from zero to one at most once and cannot switch back. This representation presumes that the treatment definition concerns adoption or receipt beginning at a particular date and continuing thereafter. If a policy can be withdrawn, or treatment can turn on and off, this simple adoption-date representation does not describe its full history.

Worked Example: Writing a Treatment History

Consider a hypothetical five-period panel indexed by $t=1,2,3,4,5$. Unit A adopts in period 3, Unit B adopts in period 5, and Unit C never adopts during the study.

For Unit A, $G_A=3$. Substituting each period into $D_{At}=\mathbf{1}(t\geq 3)$ gives the history $0,0,1,1,1$: periods 1 and 2 are before adoption, while periods 3 through 5 are treated.

For Unit B, $G_B=5$. The calculation $D_{Bt}=\mathbf{1}(t\geq 5)$ gives $0,0,0,0,1$. In period 4, for example, $4\geq5$ is false, so $D_{B4}=0$; in period 5, $5\geq5$ is true, so $D_{B5}=1$.

Unit C has no finite adoption date in the study. Its treatment indicator is zero in all five observed periods: $0,0,0,0,0$. One may write $G_C=\infty$ as a bookkeeping convention, so that $\mathbf{1}(t\geq\infty)=0$ for every finite observed $t$. This convention represents never-treated status; it does not assert that C could never adopt outside the observed period.

Never-Treated and Not-Yet-Treated Units

These terms describe different aspects of treatment timing. A never-treated unit has no adoption during the study's relevant horizon. A not-yet-treated unit is untreated at a particular calendar period but may adopt later. The latter label is period-specific: a unit can be not yet treated in one period and treated in a later period.

For a unit with a finite future adoption date, the not-yet-treated condition at time $t$ is $G_i>t$. A never-treated unit is also untreated at every observed period. Consequently, when describing units that are untreated at $t$, it is useful to say explicitly whether the group includes never-treated units, only future adopters, or both.

Key distinction: At calendar time $t$, an eventual adopter is not yet treated when $G_i>t$, and is treated when $G_i\leq t$. A never-treated unit has $D_{it}=0$ throughout the observed horizon and has no finite adoption cohort within that horizon.

Worked Example: Classifying Units at a Calendar Period

Use the hypothetical five-period panel above: A adopts in period 3, B in period 5, and C does not adopt during the study. Classify each unit at calendar period $t=4$.

For A, $G_A=3\leq4$, so $D_{A4}=\mathbf{1}(4\geq3)=1$. A has already adopted and is treated in period 4.

For B, $G_B=5>4$, so $D_{B4}=\mathbf{1}(4\geq5)=0$. B is an eventual adopter who is not yet treated in period 4.

For C, there is no finite adoption date in the study, and $D_{C4}=0$. C is never-treated over the observed horizon. Thus, all three units are not treated in period 4 only if the phrase is being used to mean $D_{i4}=0$; among these untreated units, B is not-yet-treated and C is never-treated. A is treated.

How the Indicators Follow From Event Time

For an adopter, calendar time and event time are related by $e=t-G_i$. This relation also shows how event-time labels correspond to the treatment indicator. The derivation is direct: subtracting $G_i$ from both sides of the comparison $t\geq G_i$ gives $t-G_i\geq0$. By the definition of event time, that is $e\geq0$. Therefore, under the stated adoption convention, a finite adopter is treated exactly at event time zero and thereafter.

D_{it}=\mathbf{1}(t\geq G_i) =\mathbf{1}(t-G_i\geq0) =\mathbf{1}(e\geq0), \qquad G_i<\infty

For the same reason, a finite adopter is not yet treated exactly when $t<G_i$, which is equivalent to $e<0$. This equivalence applies only to units with a finite adoption date, since event time $e=t-G_i$ is not defined in the usual way for a never-treated unit. For never-treated units, treatment status can still be recorded as zero in each observed period.

These identities are useful checks on a treatment-history table. If an eventual adopter is labelled with a negative event time but a treatment indicator of one, the entries conflict with the convention that $G_i$ is the first treated period. Likewise, an event time of zero must correspond to the adoption period, not the period before it.

A Compact Timeline

A table with both calendar time and event time can prevent a common source of confusion: comparing the same event-time label across units does not mean comparing the same calendar period.

Unit	Adoption date $G_i$	Calendar period $t$	Event time $e=t-G_i$	Treatment status $D_{it}$
A	3	2	$2-3=-1$	0
A	3	3	$3-3=0$	1
B	5	4	$4-5=-1$	0
B	5	5	$5-5=0$	1

Both units are at event time $-1$ in the table, but A is observed in calendar period 2 and B in calendar period 4. Both are at event time zero at adoption, but their calendar adoption periods differ. The timeline records treatment timing; it does not imply that the two units' outcomes or circumstances are otherwise comparable.

Common Mistakes and Why the Notation Matters

Confusing calendar time with event time. Saying “period 2” is ambiguous without specifying the scale. Calendar period 2 is a shared date; event time 2 means two periods after a unit's own adoption.
Putting event time zero before adoption. With $G_i$ defined as the first treated period, $e=0$ is treated. The preceding period is $e=-1$.
Treating not-yet-treated and never-treated as synonyms. An eventual adopter may be untreated now but treated later. A never-treated unit has no adoption in the study horizon.
Assuming treatment can reverse while using an absorbing indicator. The formula $D_{it}=\mathbf{1}(t\geq G_i)$ forces all later indicators to remain one. If the actual treatment history includes withdrawal, this encoding is inadequate.
Reading a status indicator as an effect. $D_{it}=1$ records treatment status, not what treatment caused. Causal interpretation requires additional definitions and assumptions beyond this timing notation.

The notation matters because staggered adoption creates several distinct clocks: the calendar period shared across units, the adoption date specific to each adopter, and event time measured relative to that date. Keeping these clocks separate makes it possible to describe who has adopted, when, and for how long without prematurely treating a timing label as a causal comparison.

Key takeaway: Record each adopter's first treated period as $G_i$, define event time as $e=t-G_i$, and represent absorbing treatment by $D_{it}=\mathbf{1}(t\geq G_i)$. Identify never-treated units separately from eventual adopters who are not yet treated at a particular time.

Check Your Understanding

Use the definitions in this tutorial to answer the following questions.

A unit adopts in calendar period 7. What is its event time in period 5, in period 7, and in period 9?
Two units adopt in periods 3 and 6. In which calendar period is each at event time 1?
A unit will adopt in period 8. Is it treated, not-yet-treated, or never-treated in period 6?
Explain why an absorbing treatment history cannot be represented correctly if the treatment is withdrawn after adoption.
Can a never-treated unit be assigned an event time of $-1$ using $e=t-G_i$? Explain why or why not.

Unit	Adoption date \(G_i\)	Calendar period \(t\)	Event time \(e=t-G_i\)	Treatment status \(D_{it}\)
A	3	2	\(2-3=-1\)	0
A	3	3	\(3-3=0\)	1
B	5	4	\(4-5=-1\)	0
B	5	5	\(5-5=0\)	1