Dynamic Treatment Effects by Event Time

Why Measure Effects by Event Time?

A policy may begin in different calendar periods for different groups, and its effect may change as time passes after adoption. A single post-treatment average can conceal this pattern: an effect in the first year may differ from an effect several years later. Dynamic treatment effects describe those changes using event time, the number of periods before or after a group’s treatment begins.

Event time is not calendar time. Calendar time identifies when an outcome was measured, such as 2022. Event time identifies when that measurement occurred relative to a particular group’s adoption date, such as the second year after adoption. Keeping these two clocks separate is essential when adoption dates differ across groups.

Definition: Let \(G_i\) denote the first period in which unit \(i\) receives treatment; units never treated during the study period may be assigned \(G_i=\infty\). Units sharing the same adoption period \(g\) form adoption cohort \(g\). For a unit in cohort \(g\), event time in calendar period \(t\) is \(e=t-g\). Thus \(e<0\) is before adoption, \(e=0\) is the adoption period, and \(e>0\) is after adoption.

Potential Outcomes and the Dynamic Causal Target

To define a causal effect, compare outcomes under treatment with outcomes under a specified alternative treatment history. Write \(Y_{it}(g)\) for unit \(i\)’s outcome in calendar period \(t\) if it first adopts treatment in period \(g\), and \(Y_{it}(\infty)\) for its outcome if it remains untreated throughout the relevant study period. These potential outcomes describe alternative histories for the same unit; only the outcome corresponding to the unit’s actual history is observed.

The treatment effect of interest for cohort \(g\) in calendar period \(t\) is the average treatment effect on that cohort. Expressing \(t\) as \(g+e\) gives the dynamic effect at event time \(e\):

ATT(g,g+e) = E\!\left[Y_{i,g+e}(g)-Y_{i,g+e}(\infty)\mid G_i=g\right], \qquad e\geq 0.

This quantity is cohort-specific: it averages the effect over units that adopted in period \(g\). It is also event-time-specific: it compares outcomes in the same calendar period \(g+e\) under adoption in \(g\) and under remaining untreated. For example, \(ATT(g,g)\) is the average effect in the adoption period for cohort \(g\), while \(ATT(g,g+3)\) is that cohort’s average effect three periods after adoption.

The notation makes clear that “the effect three periods after adoption” need not be the same for every cohort. Different cohorts experience different calendar conditions at their third post-adoption period, and the policy’s effect may vary across cohorts. A single dynamic effect at event time \(e\) therefore requires a defined aggregation across the cohorts for which that event time is observed.

Worked Example: Translating Between the Two Clocks

Suppose a hypothetical program is adopted by District A in calendar period 3 and by District B in calendar period 5. District A’s mean outcome is recorded in periods 2 through 6, and District B’s in periods 4 through 7. Translate several calendar periods into event time.

For District A, \(g=3\), so \(e=t-3\). In calendar period 2, event time is \(2-3=-1\); in period 3 it is \(3-3=0\); and in period 6 it is \(6-3=3\). The last observation is three periods after adoption.

For District B, \(g=5\), so \(e=t-5\). In calendar period 4, event time is \(4-5=-1\); in period 5 it is \(5-5=0\); and in period 7 it is \(7-5=2\).

The same calendar period can represent different event times: period 5 is event time 2 for District A but event time 0 for District B. Conversely, event time 0 occurs in period 3 for A and period 5 for B. The available observations also differ: this example has data through event time 3 for A but only through event time 2 for B.

Identifying a Cohort-Time Effect With Difference-in-Differences

The potential-outcome definition is a causal target, not yet a recipe for estimating it. For cohort \(g\) and a post-adoption period \(t=g+e\), a comparison group can consist of units that have not adopted by \(t\), including never-treated units. Denote this group by \(G_i>t\). The comparison units must be untreated both at the baseline period and at \(t\).

The following assumptions state the conditions for the basic cohort-time difference-in-differences contrast. Consistency means that each unit’s observed outcome equals the potential outcome under its actual treatment history. No interference means one unit’s treatment does not change another unit’s outcome. No anticipation means outcomes before adoption are not affected by the future treatment, so the cohort’s baseline outcome at \(g-1\) is untreated. Finally, parallel trends requires that, in the absence of treatment, the cohort and comparison group would have had the same average outcome change from \(g-1\) to \(t\):

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

Under these conditions, the cohort-time effect is identified by the difference between the cohort’s observed outcome change and the comparison group’s observed outcome change:

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

The justification follows by separating the treated cohort’s observed change into its untreated change and its treatment effect. By consistency and no anticipation, its observed change is the change under no treatment plus \(ATT(g,t)\). The comparison group is untreated at both periods, so its observed change is its change under no treatment. Parallel trends equates the two groups’ expected untreated changes. Subtracting the comparison-group change therefore removes the treated cohort’s counterfactual untreated change, leaving the average treatment effect.

\begin{aligned} E[Y_{it}-Y_{i,g-1}\mid G_i=g] &= ATT(g,t) + E[Y_{it}(\infty)-Y_{i,g-1}(\infty)\mid G_i=g],\\ E[Y_{it}-Y_{i,g-1}\mid G_i>t] &= E[Y_{it}(\infty)-Y_{i,g-1}(\infty)\mid G_i>t],\\ E[Y_{it}-Y_{i,g-1}\mid G_i=g] &-E[Y_{it}-Y_{i,g-1}\mid G_i>t] = ATT(g,t). \end{aligned}

Worked Example: Calculating a Cohort’s Effect at Event Time One

Consider hypothetical districts in a cohort that adopts in period \(g=3\). Their average outcome is 10 index points in baseline period 2 and 15 points in period 4. A comparison group that remains untreated through period 4 has average outcomes of 9 points in period 2 and 12 points in period 4. Calculate the cohort’s effect at event time 1.

Here \(t=4\), so \(e=t-g=4-3=1\). The treated cohort’s observed change is \(15-10=5\) index points. The comparison group’s change is \(12-9=3\) index points. The difference-in-differences contrast is

\widehat{ATT}(3,4) = (15-10)-(12-9) = 5-3 = 2\ \text{index points}.

This estimate says that, under the identifying assumptions, the program increased the cohort’s mean outcome by an estimated 2 index points in period 4, one period after adoption, relative to remaining untreated. The calculation uses the comparison group’s change to account for the counterfactual trend; it does not treat the treated cohort’s five-point increase as the treatment effect.

For this interpretation to be credible, the comparison units must be untreated through period 4, outcomes must be measured consistently, and the cohort and comparison group must satisfy parallel trends in their untreated outcomes from period 2 to period 4. If treatment affects outcomes before period 3, the baseline is not unaffected and this contrast does not identify the stated target without a different treatment-date specification.

Pre-Adoption Event Times and the Support of Effects

For \(e<0\), the group has not yet adopted. Under no anticipation, its causal effect of treatment in that period is zero because treatment has not begun. A pre-adoption comparison can still be informative as a diagnostic: it can reveal differences in changes before adoption that may call the parallel-trends assumption into question. Such a contrast is not itself a post-treatment causal effect, and a lack of a noticeable pre-period difference does not prove parallel trends.

Not every cohort contributes data at every event time. A cohort adopting near the end of the observation window may have no outcomes several periods after adoption. The set of cohorts available at event time \(e\) is therefore determined by both adoption dates and the study’s calendar-time coverage. An estimate at event time 3 may describe a narrower set of cohorts than an estimate at event time 0. This change in composition matters when describing how effects evolve over event time.

Worked Example: Choosing Weights for a Dynamic Average

Suppose two hypothetical cohorts have estimated effects at event time 0 of 2 outcome points for a cohort of 100 treated units and 5 points for a cohort of 300 treated units. An equal-cohort average gives each adoption cohort the same weight:

\frac{2+5}{2}=3.5\ \text{outcome points}.

If the target is instead the average effect for a treated unit across these 400 units, weight each cohort’s effect by its number of treated units:

\frac{100(2)+300(5)}{100+300} = \frac{200+1500}{400} = 4.25\ \text{outcome points}.

Both calculations use the same cohort-specific effects, but they answer different questions. The first describes the average across the two adoption cohorts, giving a small cohort the same influence as a large one. The second describes the average across the treated units. If a cohort lacks an observed outcome at event time 0, it cannot contribute to either average at that event time; the target population and weighting rule should be stated accordingly.

Common Pitfalls and Why the Distinction Matters

Confusing event time with calendar time. Calling period 5 “year one after treatment” for all groups is wrong when groups adopted in different periods. Compute \(e=t-g\) separately for each cohort.
Calling a raw outcome change the treatment effect. A cohort’s before-after change includes its untreated outcome trend. The difference-in-differences contrast subtracts the comparison group’s change, subject to the identifying assumptions.
Treating every event-time average as the same target. Different cohorts may contribute at different event times, and equal-cohort and treated-unit-weighted averages need not agree. State the cohorts and weights that define an aggregate.
Interpreting pre-treatment contrasts as causal treatment effects. Before adoption, the treatment has not started under the no-anticipation condition. Pre-period comparisons are diagnostics, not estimates of post-adoption effects.

Key takeaway: A dynamic treatment effect is defined for an adoption cohort and a particular time relative to its adoption. Calendar time maps that event time to an observed period; a valid comparison and stated assumptions connect the causal target to a difference-in-differences contrast. Any average across cohorts requires an explicit target and weighting rule.

Check Your Understanding

Use the definitions and assumptions above to answer the following questions.

A group adopts in period 6. What is its event time in calendar period 4, and what is its event time in period 9?
In words, what population and counterfactual outcomes does \(ATT(g,g+2)\) compare?
Why must the comparison group in the displayed cohort-time contrast remain untreated through period \(t\)?
A treated group’s outcome rises by 6 points, while an untreated comparison group’s outcome rises by 4 points over the same periods. What difference-in-differences estimate follows, and what assumption gives it a causal interpretation?
Why can an equally weighted average across adoption cohorts differ from an average weighted by the number of treated units?