One Panel, Three Summaries
In staggered adoption, different groups begin treatment in different periods. A single panel can therefore support several summaries: a conventional two-way fixed-effects (TWFE) coefficient, cohort-and-time-specific treatment effects, and averages organized by time since adoption. These quantities may differ even when the untreated outcome trends are parallel. The reason is that they combine treatment effects across cohorts and periods in different ways.
We will use a small hypothetical panel with three equally sized groups and four periods, labeled 0 through 3. Group A adopts in period 2, group B adopts in period 3, and group N is never treated. The entries below are group-period mean outcomes. Assume each group contains the same number of units, and each unit is observed in all four periods, so the displayed cell means represent an equally weighted balanced panel.
| Group | Adoption period | Period 0 | Period 1 | Period 2 | Period 3 |
|---|---|---|---|---|---|
| A | 2 | 10 | 11 | 14 | 17 |
| B | 3 | 12 | 13 | 14 | 25 |
| N | Never | 8 | 9 | 10 | 11 |
The untreated outcomes in this example would be 10, 11, 12, and 13 for A; 12, 13, 14, and 15 for B; and 8, 9, 10, and 11 for N. Thus, absent treatment, each group’s outcome rises by one unit per period. The observed outcomes imply effects of 2 and 4 units for A in periods 2 and 3, respectively, and an effect of 10 units for B in period 3. Treatment effects are therefore heterogeneous across cohorts and exposure durations.
Worked Example: The TWFE Coefficient
Worked Example: The TWFE Coefficient
Regress the outcome on group fixed effects, period fixed effects, and the binary treatment indicator \(D_{gt}\), which equals 1 in periods at or after a group’s adoption. Calculate the treatment coefficient from this balanced panel.
With two-way fixed effects, the Frisch–Waugh–Lovell result says the treatment coefficient is the inner product of treatment residuals and outcome residuals divided by the sum of squared treatment residuals. Equivalently, because the residualized treatment is orthogonal to group and period indicators, the numerator can be calculated by multiplying those residuals by outcomes.
Here the group treatment means are \(1/2\), \(1/4\), and 0 for A, B, and N. The period treatment means are 0, 0, \(1/3\), and \(2/3\), and the overall treatment mean is \(1/4\). Substituting these values gives the residualized treatment indicators:
For the numerator, group and period components of the outcome contribute zero because the treatment residuals sum to zero within each group and period. Only the treatment effects contribute: \( (5/12)(2)+(1/12)(4)+(1/3)(10)=4.5\). The denominator is the sum of the squared entries in the residual table:
The fitted TWFE coefficient is 5.4 outcome units. In this particular panel, it can also be written as a weighted average of the three observed cohort-period effects: \(0.5(2)+0.1(4)+0.4(10)=5.4\). Those weights come from the residualized treatment, not from a decision to average cohorts equally. This numerical coefficient is not automatically a common treatment effect; its causal interpretation depends on the design and on how the regression combines effects.
Group-Time Effects Using the Never-Treated Cohort
A group-time estimate compares the outcome change for a treated cohort with the change over the same calendar periods for an untreated comparison group. With N as the comparison group and the period just before adoption as the baseline, the difference-in-differences calculation is:
For a causal interpretation, this comparison requires parallel trends in untreated outcomes between the treated cohort and N over the periods used. It also requires no anticipation: outcomes before adoption are not affected by future treatment. In the hypothetical panel, untreated outcomes rise by one unit per period for all groups, and the specified effects begin only at adoption. The assumptions therefore hold by construction.
Worked Example: Cohort A’s Effects
Estimate the effects for A in periods 2 and 3. The pre-adoption period is 1. In period 2, A’s outcome rises from 11 to 14, while N’s rises from 9 to 10:
In period 3, use the same period-1 baseline for A and N. A’s outcome change is \(17-11=6\), and N’s is \(11-9=2\):
Under the stated assumptions, these estimates identify A’s effects at event times 0 and 1: 2 and 4 outcome units, respectively. Subtracting N’s change removes the common untreated increase; comparing A’s change alone with zero would incorrectly include that increase.
Worked Example: Cohort B’s Effect at Adoption
B adopts in period 3, so period 2 is its pre-adoption baseline. Compare B’s change from period 2 to period 3 with N’s change over those same calendar periods:
This is B’s estimated effect at event time 0. It is not directly comparable to A’s period-3 outcome as though both groups had the same exposure duration: B is just treated in period 3, while A has been treated for two periods. The estimate uses a different baseline period as well, but the common-trends assumption supports both comparisons.
Aggregating by Event Time
An event-time profile collects cohort-time effects with the same duration since adoption. To make the aggregation rule explicit, use equal weights across cohorts with an effect observed at that event time:
where \(\mathcal{G}_e\) is the set of cohorts with observed outcomes at event time \(e\). This weighting rule is a choice; a unit-weighted or otherwise weighted profile would answer a different aggregation question.
Worked Example: Aggregating the Event-Time Effects
At event time 0, both A and B have an estimated effect: A’s is 2 and B’s is 10. With equal cohort weights, the event-time-0 average is:
At event time 1, A has an observed effect of 4 in period 3, but B would require an outcome in period 4, which is outside this panel. Therefore only A contributes:
The event-time-0 estimate is an average across two cohorts; the event-time-1 estimate describes A alone. Thus the profile’s cohort composition changes with event time. The two values describe averages for the available cohorts at each duration, not necessarily the trajectory of one fixed population of cohorts.
Why the Three Estimates Differ
The calculations give a TWFE coefficient of 5.4, an event-time-0 average of 6, and an event-time-1 average of 4. None is an arithmetic error. The TWFE coefficient uses residualized-treatment weights over treated cohort-period observations. The event-time profile instead groups effects by time since adoption and averages available cohorts according to its specified rule. In this panel, the TWFE weights on A’s effects at event times 0 and 1 and B’s event-time-0 effect are 0.5, 0.1, and 0.4. The event-time-0 average gives A and B equal weight and excludes A’s later effect; the event-time-1 average includes only A.
This distinction matters for interpretation. A coefficient from a conventional TWFE regression is not automatically the average effect for all treated units, nor is it automatically an effect at a particular event time. When effects vary, its implicit weights can yield a summary that differs from a deliberately defined cohort-time or event-time estimand. In other staggered designs, some TWFE comparisons may also involve already-treated groups as controls; the numerical contrast then requires particular care. Here, the TWFE coefficient is calculated to show its weighting, while the group-time calculations use the never-treated cohort as the comparison group.
Decide whether the question concerns a regression coefficient, a specific \(ATT(g,t)\), or an average at a given event time.
For a group-time difference-in-differences estimate, identify the untreated comparison group, baseline period, no-anticipation condition, and parallel-trends assumption.
For an event-time average, specify which cohorts contribute at each event time and how they are weighted.
Common Pitfall
A tempting but incorrect conclusion is: “The TWFE coefficient is 5.4, so treatment raises the outcome by 5.4 units for every treated group in every post-adoption period.” That conclusion treats a regression summary as a constant effect despite the panel’s heterogeneous effects. The correction is to describe 5.4 as the fitted TWFE coefficient for this design, explain its implicit weighting, and use group-time estimates or a clearly specified event-time aggregation when those are the target quantities. A numerical estimate alone does not establish the causal assumptions; those must be justified by the design and substantive knowledge.
Check Your Understanding
Use the hypothetical panel and calculations above to answer these questions.
- Why is group N’s outcome change subtracted when estimating \(ATT(2,2)\)?
- What makes the event-time-1 estimate a different cohort summary from the event-time-0 estimate?
- Which three cohort-period effects receive positive weights in the TWFE calculation, and what are those weights?
- Would an event-time-0 average weighted by cohort size differ here if A and B had equal cohort sizes? Explain.
- State one assumption needed to interpret the group-time difference-in-differences calculations causally.