Why can two-way fixed effects be biased when a policy rolls out at staggered dates?

## The setup A state-by-state policy rollout is the canonical case: some states adopt in 2019, some in 2021, some never. The reflexive specification is `Y = unit effects + period effects + beta * Treated_it + error` where `Treated_it` is 1 once unit `i` has adopted. This is called two-way fixed effects (TWFE), and for a decade `beta` was reported as *the* difference-in-differences estimate. With a single common adoption date it is exactly that. With staggered dates it is something else. ## The decomposition The Goodman-Bacon decomposition establishes that with staggered timing, `beta` equals a weighted average of all the two-group two-period comparisons available in the panel. Those comparisons come in three flavours: 1. **Treated cohort versus never-treated units.** Clean. This is what everyone thinks they are estimating. 2. **Earlier-adopting cohort versus later-adopting cohort, before the later one adopts.** Also clean — the later cohort is genuinely untreated over that window. 3. **Later-adopting cohort versus earlier-adopting cohort, after the earlier one has already adopted.** This one is the problem. The comparison group is already treated. The weights are functions of group sizes and of how much treatment variance each comparison contributes, and they sum to one. ## Why the third comparison misleads In a forbidden comparison, the double difference is `(late group's change) - (early group's change)` but the early group's change *includes the evolution of its own treatment effect*. If the effect is constant over time, that evolution is zero and nothing goes wrong — the already-treated group's effect is a fixed level shift that differences out. That is why TWFE is fine under homogeneous, time-constant effects. Once effects are **dynamic** — they build over periods, or decay — the early group's effect is still moving during the comparison window. Subtracting a rising treatment effect from the late group's change subtracts something that has nothing to do with the late group's treatment. The comparison contributes with an effectively negative sign. Aggregate that with positive weights on the clean comparisons and you can get an estimate that is attenuated toward zero, or, when the dynamic component is large enough and the forbidden comparisons carry enough weight, a `beta` with the **opposite sign from every unit-level effect in the data**. That is the striking result: a policy that helped every state can produce a negative TWFE coefficient. ## Heterogeneity across cohorts A second, related failure: even without dynamics, if different adoption cohorts have genuinely different effects — early adopters gained more because they were better suited to the policy — TWFE weights those cohorts by how much treatment variance they supply, not by anything you would choose. Units treated near the middle of the panel get the most weight; units treated at the very start or the very end get almost none. The resulting number is a weighted average with weights nobody would defend if they were written down. ## What to do instead **Group-time estimators.** Estimate a separate average effect for each adoption cohort `g` and each period `t`, always using a clean comparison set: never-treated units, or units not yet treated as of `t`. This gives a grid of `ATT(g, t)` estimates. Then aggregate them with weights *you* choose — by cohort, by time since adoption (which yields an event-study path), or by an overall average weighted by cohort size. The Callaway and Sant'Anna estimator is the standard reference for this approach. **Interaction-weighted event studies.** The Sun and Abraham estimator fixes the analogous contamination in a naive event-study regression, where a relative-time coefficient can absorb effects from other cohorts at other relative times, by interacting relative-time indicators with cohort and reweighting. **Stacked designs.** For each adoption cohort, build a small clean dataset of that cohort plus its not-yet-treated comparisons over a fixed window around its adoption, then stack all those datasets and estimate once. Easy to explain and audit, at the cost of using some units repeatedly. **Drop the already-treated.** The blunt version: restrict comparisons to never-treated units only. Simple and clean, but it throws away data and requires a never-treated group to exist, which it often does not in a full rollout. ## What an interviewer is listening for The mechanism, not the citation. Say: with staggered timing the coefficient is an average of many 2x2 comparisons; some use already-treated units as controls; when effects vary over time those comparisons enter with the wrong sign; therefore estimate cohort-by-period effects against clean comparisons and aggregate explicitly. Then add the honest boundary: with a single common adoption date, or genuinely constant homogeneous effects, plain TWFE is fine and this whole discussion is moot.