Potential Outcomes Under Staggered Adoption

Why Adoption-Date Potential Outcomes Matter

With staggered adoption, units begin treatment in different calendar periods. A unit’s observed outcome after adoption reflects its treatment, while a unit not yet treated can still provide information about what outcomes might have looked like without treatment. Potential outcomes make that distinction explicit: they separate the outcome a unit would have under a particular treatment history from the one it would have if it remained untreated.

Let $G_i$ denote unit $i$’s first treatment period, using the adoption-date convention introduced for staggered adoption. Treatment is absorbing: once a unit adopts, it remains treated. Thus, for a unit with finite adoption date, $D_{it}=1$ when $t\geq G_i$ and $D_{it}=0$ when $t<G_i$. Write $G_i=\infty$ for a unit that is never treated during the study period. Event time is $t-G_i$ for a treated cohort.

Definition: $Y_{it}(g)$ is unit $i$’s potential outcome in calendar period $t$ under the treatment regime that begins in period $g$ and continues thereafter. $Y_{it}(\infty)$ is the potential outcome in period $t$ if unit $i$ remains untreated throughout. A unit’s adoption cohort is indexed by its adoption date $G_i$. These potential outcomes may vary across units, dates, and adoption regimes.

The notation indexes a treatment history, not merely a treated-versus-untreated label at one date. For example, $Y_{it}(g)$ and $Y_{it}(g+1)$ can differ in period $t$, even if both regimes have treated the unit by then: treatment may have been in place longer under adoption at $g$. Potential outcomes are not all observed for the same unit. The observed data reveal the outcome under the unit’s realized adoption regime.

Consistency and No Anticipation

Potential outcomes have a causal interpretation only when the treatment regime is sufficiently well defined. Consistency connects the observed outcome to the potential outcome under the regime actually received. In this setting, it also requires that the treatment label represent the same relevant intervention across units; if versions of treatment differ in consequential ways, the regime needs to distinguish those versions. The usual no-interference condition rules out one unit’s outcome depending on other units’ treatment assignments, unless that dependence is explicitly part of the treatment regime.

Definition: Consistency means that if unit $i$ adopts in period $g$, its observed outcome in period $t$ is $Y_{it}(g)$; if it is never treated, its observed outcome is $Y_{it}(\infty)$. No anticipation means that an adoption scheduled for period $g$ does not affect outcomes before $g$: $$ Y_{it}(g)=Y_{it}(\infty) \quad \text{for } t<g. $$

No anticipation is a restriction on potential outcomes, not a claim that units cannot know treatment is coming. It says that such knowledge has no effect on the outcome before the specified adoption date. If announcement or preparation affects outcomes earlier, the treatment onset or regime must be defined to include that anticipation, or the no-anticipation assumption is inappropriate. Consistency alone does not imply no anticipation: consistency links observed and potential outcomes, while no anticipation equates potential outcomes under two different regimes before adoption.

Worked Example: Reading Potential Outcomes for One Cohort

Suppose a hypothetical program is adopted in period 3 by a cohort of units. For one unit, the potential outcomes under adoption in period 3 and under no treatment are as follows:

Calendar period $t$	$Y_{it}(3)$	$Y_{it}(\infty)$	Observed outcome
1	10	10	10
2	12	12	12
3	16	13	16
4	18	14	18

The unit’s adoption date is $G_i=3$, so its observed outcomes follow $Y_{it}(3)$: 10, 12, 16, and 18. No anticipation holds in periods 1 and 2 because $Y_{i1}(3)=Y_{i1}(\infty)=10$ and $Y_{i2}(3)=Y_{i2}(\infty)=12$. In period 3, the contrast between the two potential outcomes is $16-13=3$ outcome units; in period 4 it is $18-14=4$ outcome units. The untreated outcomes are counterfactual for this unit after adoption, so the table does not make those values observable.

Untreated Potential Outcomes and Parallel Trends

Consistency and no anticipation specify how observed outcomes relate to potential outcomes and what happens before adoption. They do not, by themselves, tell us how an adopting cohort’s untreated outcomes would have changed after adoption. A difference-in-differences interpretation therefore requires an assumption about those unobserved untreated outcomes. In its standard form, the assumption is parallel trends between a cohort and a suitable comparison group over the periods being compared.

Definition: Parallel trends in untreated outcomes between cohort $G_i=g$ and a comparison group $C$ means that, for a pre-adoption baseline period $b<g$ and a post-adoption period $t\geq g$, $$ E[Y_{it}(\infty)-Y_{ib}(\infty)\mid G_i=g] = E[Y_{it}(\infty)-Y_{ib}(\infty)\mid i\in C]. $$ The comparison group must be untreated throughout the relevant periods. For example, units with $G_i>t$, as well as never-treated units, are untreated from $b$ through $t$.

This assumption concerns changes in the untreated potential outcome, not equality of outcome levels. The cohort and comparison group may start at different levels and may have different characteristics. What must agree, on average, is the change they would have experienced without treatment. Parallel trends is not guaranteed by similar pre-treatment plots, and observed pre-treatment trends cannot directly reveal post-treatment untreated outcomes. Its plausibility must be argued from the design and subject-matter context.

The choice of comparison group and periods matters. A group already treated by period $t$ cannot serve as an untreated comparison for the full interval $b$ to $t$. Nor does the assumption say that a group’s observed change after its own adoption is its untreated change. These distinctions are essential when interpreting a staggered-adoption contrast.

Worked Example: Checking the Untreated-Trend Condition

Consider a hypothetical cohort adopting in period 3 and a comparison group that remains untreated through period 4. Suppose their mean untreated potential outcomes are:

Group	Period 1 untreated mean	Period 3 untreated mean	Change
Adoption cohort $G_i=3$	20	23	$23-20=3$
Untreated comparison group	15	18	$18-15=3$

For the period-1-to-period-3 comparison, the untreated changes are both 3 outcome units. Thus the parallel-trends condition is satisfied for these specified groups and dates. The different starting levels, 20 versus 15, do not violate the condition. This calculation is only a check of the stated potential-outcome values; in an actual study, the post-adoption untreated mean for the adopting cohort is not observed and cannot be verified directly.

How the Assumptions Support a DiD Interpretation

The role of the assumptions is clearest in an explicit derivation. Let $b<g\leq t$, and let $C$ be a comparison group untreated through $t$. Define the population difference in observed changes as the adoption cohort’s change minus the comparison group’s change. Under consistency and no anticipation, the cohort’s observed outcome at $t$ is $Y_{it}(g)$, and its observed baseline outcome is $Y_{ib}(\infty)$. The comparison group’s observed outcomes at both dates are untreated potential outcomes.

\begin{aligned} \mathrm{DiD}_{g,t,b} &= \{E[Y_{it}\mid G_i=g]-E[Y_{ib}\mid G_i=g]\}\\ &\quad-\{E[Y_{it}\mid i\in C]-E[Y_{ib}\mid i\in C]\}\\ &= \{E[Y_{it}(g)\mid G_i=g]-E[Y_{ib}(\infty)\mid G_i=g]\}\\ &\quad-\{E[Y_{it}(\infty)\mid i\in C]-E[Y_{ib}(\infty)\mid i\in C]\}\\ &= \{E[Y_{it}(g)-Y_{it}(\infty)\mid G_i=g]\}\\ &\quad+\{E[Y_{it}(\infty)-Y_{ib}(\infty)\mid G_i=g]\\ &\qquad-E[Y_{it}(\infty)-Y_{ib}(\infty)\mid i\in C]\}. \end{aligned}

The first equality after the definition of the contrast uses consistency and no anticipation for the adoption cohort, and consistency plus untreated status for the comparison group. It does not use parallel trends. The final line is an algebraic rearrangement: it adds and subtracts the cohort’s period-$t$ untreated potential outcome. Only now does parallel trends make the term in braces equal to zero. The DiD contrast then equals the cohort’s mean treatment contrast at $t$, relative to remaining untreated:

\mathrm{DiD}_{g,t,b} = E[Y_{it}(g)-Y_{it}(\infty)\mid G_i=g].

This derivation pinpoints what each assumption contributes. Without parallel trends, the DiD contrast equals the cohort’s mean treatment contrast plus the difference in untreated changes. Consistency and no anticipation do not remove that difference. The equation also clarifies that the comparison group need not have the same baseline mean as the adoption cohort.

Worked Example: A Trend Difference Changes the DiD Contrast

Suppose hypothetical population means for an adoption cohort and an untreated comparison group are:

Group	Baseline observed mean	Follow-up observed mean	Follow-up untreated mean
Adoption cohort	10	15	13
Comparison group, untreated throughout	8	10	10

For the adoption cohort, no anticipation and consistency imply that the baseline observed mean 10 is its baseline untreated mean. At follow-up, the observed mean is 15, while the hypothetical untreated mean is 13. Its mean treatment contrast is therefore $15-13=2$ outcome units. The cohort’s untreated change is $13-10=3$, and the comparison group’s untreated change is $10-8=2$.

The observed difference in changes is $$ (15-10)-(10-8)=5-2=3. $$ The decomposition gives the same answer: $$ 2+\{3-2\}=3. $$ The DiD contrast is 3 outcome units, not the cohort’s treatment contrast of 2, because the cohort’s untreated trend exceeds the comparison group’s by 1. If parallel trends held, that trend difference would be zero and the contrast would equal 2. This example shows why the first substitution in the derivation cannot replace the comparison group’s untreated change with the cohort’s untreated change before invoking parallel trends.

Common Mistakes and Practical Meaning

Treating consistency as a trend assumption. Consistency identifies the potential outcome corresponding to the realized treatment regime. It does not say how untreated outcomes change across groups.
Assuming no anticipation means no effects at any time. It restricts effects only before the specified adoption date. After adoption, $Y_{it}(g)$ can differ from $Y_{it}(\infty)$.
Calling parallel trends equality of levels. The assumption equates untreated changes, not untreated means at each date.
Claiming the DiD contrast is automatically a treatment effect. Without the untreated-trend condition, the contrast includes the difference in untreated changes, as the worked calculation shows.
Using treated units as if they were untreated controls. A comparison group must remain untreated over the periods used for the contrast; otherwise its observed change need not represent an untreated change.

Potential outcomes make the causal question precise, but they do not make assumptions empirically certain. In staggered adoption, the adoption date indexes a treatment history, no anticipation links pre-adoption outcomes across regimes, consistency links observed outcomes to the realized regime, and parallel trends supplies the missing counterfactual change. Each plays a distinct role; keeping those roles separate is necessary for a defensible causal interpretation.

Check Your Understanding

Use the notation and assumptions in this tutorial to answer the following.

What does $Y_{it}(g)$ represent, and how does it differ from $Y_{it}(\infty)$?
A unit adopts in period 4. Which equality does no anticipation require for its potential outcomes in period 2?
Does parallel trends require the adoption cohort and comparison group to have equal untreated means at baseline? Explain.
In the DiD derivation, which step uses consistency and no anticipation, and which step uses parallel trends?
If the adoption cohort’s untreated change is 4 and the comparison group’s untreated change is 1, by how much does the DiD contrast differ from the cohort’s mean treatment contrast?
Why can a group already treated by follow-up period $t$ not automatically be treated as an untreated comparison group?

Group	Period 1 untreated mean	Period 3 untreated mean	Change
Adoption cohort \(G_i=3\)	20	23	\(23-20=3\)
Untreated comparison group	15	18	\(18-15=3\)