Anticipation and the Choice of Treatment Onset

Why Recorded Adoption May Not Be the Relevant Onset

In a staggered-adoption design, a policy or program is often assigned a recorded adoption date for each cohort. That date is not necessarily the first time outcomes could respond. A firm may change hiring when a regulation is announced but before it takes legal effect; patients may change behavior when a treatment is scheduled but before their first dose. If responses begin before the recorded date, labeling those periods “untreated” can distort both the comparison period and the causal contrast.

The key decision is therefore not simply which calendar date appears in an administrative record. It is which date marks the start of possible causal effects for the outcome and population being studied. The choice determines which periods can serve as untreated baselines or controls, and what treatment effect the resulting comparison describes.

When \(L=0\), this condition rules out effects before the recorded adoption period. When \(L>0\), the period immediately before adoption is not automatically untreated. An anticipation window is a substantive restriction on when effects could begin, not an estimate of the effect size in each pre-adoption period. It can be cohort-specific if the timing of announcements, preparation, or behavioral response differs across cohorts.

How an Affected Baseline Changes the Contrast

Consider a treated cohort \(g\), a post-adoption period \(t\geq g\), and a baseline period \(b<g\). Let the cohort-time average treatment effect be \(\operatorname{ATT}(g,t)=E[Y_{it}(g)-Y_{it}(\infty)\mid G_i=g]\). A difference-in-differences contrast compares the treated cohort’s outcome change from \(b\) to \(t\) with the change for a comparison group that remains untreated in both periods. Its causal interpretation requires parallel trends in untreated potential outcomes for the treated cohort and the comparison group over those periods. It also requires that the comparison group is not itself affected by anticipation or treatment during the comparison.

If the treated cohort is already affected in the baseline, the contrast subtracts that baseline effect. Define the anticipation effect in period \(s<g\) as \(a_g(s)=E[Y_{is}(g)-Y_{is}(\infty)\mid G_i=g]\). Under the stated parallel-trends condition and an unaffected comparison group, the expected contrast is:

The last equality follows because the treated cohort’s observed outcome at \(t\) includes its treatment effect, while its outcome at \(b\) includes the anticipation effect \(a_g(b)\). The comparison group’s change removes the change in untreated outcomes that would otherwise confound the contrast. Thus, an affected baseline generally produces a difference between the target post-treatment effect and the baseline anticipation effect, not the target effect alone.

Choosing and Interpreting the Onset

There are two related ways to handle possible pre-adoption responses. First, keep the formal adoption date \(g\) as the reference date, explicitly mark the anticipation window, and exclude potentially affected periods from the untreated baseline and from any untreated comparison. Second, define a substantive onset that occurs before the formal date and re-index treatment timing to that onset. The second approach can make event time align more closely with when exposure begins, but it changes the meaning of event time and the treatment contrast.

If the anticipation window has length \(L\), one may set the effective onset to \(g^*=g-L\). This is a conservative boundary for possible response: it does not assert that a response definitely starts at \(g^*\), only that periods from \(g^*\) onward might be affected. If evidence supports a specific earlier behavioral onset, that date can instead be used, with the assumptions stated explicitly. In either case, an outcome difference after the redefined onset is an effect relative to the relevant no-exposure potential outcome, not automatically an effect of the formal policy alone.

A baseline must be chosen relative to the onset assumption. For an adoption cohort \(g\), a baseline \(b<g-L\) is outside the stated anticipation window. A baseline \(b\geq g-L\) is potentially affected and cannot be treated as untreated merely because it precedes formal adoption. Similarly, a comparison cohort is useful for a given period contrast only if it remains untreated and outside its own possible anticipation window at both dates. These requirements are period-specific: a cohort that is a valid comparison early in the calendar timeline may cease to be valid later.

A Practical Decision Sequence

The onset and comparison choices should be made before interpreting dynamic contrasts. A useful sequence is to identify the earliest plausible response period, state the no-anticipation boundary, and then determine which observations remain valid as untreated baselines and controls.

Why the Choice Matters

Changing the onset or the baseline changes more than the labels on a graph. A baseline before the possible response window compares post-onset outcomes with a plausibly unaffected reference. A baseline inside that window subtracts some response from the contrast. Shifting the onset can make a behavioral interpretation more coherent, but it also changes which periods count as event time zero and which effects are being summarized.

The no-anticipation condition is not verified just by observing flat pre-treatment estimates. Estimates can be imprecise, and a response may be too small to detect. Conversely, if substantive knowledge indicates that behavior could respond at announcement, treating that period as unaffected solely because a test is not significant is not a sound timing argument. The assumption and the uncertainty in the data should be discussed separately.

• Do not equate formal adoption with first exposure. Record dates may lag the first event capable of changing outcomes.
• Do not call every pre-adoption period untreated. Periods inside the anticipation window are potentially affected.
• Do not interpret a contaminated-baseline contrast as the post-treatment effect. Under the stated conditions, it subtracts the baseline anticipation effect.
• Do not treat re-indexing as a remedy for all identification problems. A new onset does not establish parallel trends, eliminate confounding, or guarantee unaffected controls.