Contamination in Staggered Event-Study Regressions

Why Event-Time Labels Can Mislead

An event-study regression reports coefficients indexed by time relative to treatment adoption. It is tempting to read the coefficient at event time \(k\) as the average treatment effect \(k\) periods after adoption, and to read lead coefficients as evidence about whether treated units had different trends before adoption. Under staggered adoption, those interpretations can fail: the regression uses a common set of coefficients to fit observations from cohorts treated at different dates, and some treated cohorts can serve as comparisons for others.

The central problem is contamination. When treatment effects differ across cohorts or over time, a conventional two-way fixed-effects (TWFE) event-study coefficient can combine effects from multiple cohort-date cells. In particular, a coefficient labeled as a lead can receive contributions from effects after some cohorts have already adopted treatment. Thus a nonzero lead need not imply anticipation or a violation of parallel trends.

A conventional event-study specification includes unit and calendar-time fixed effects and indicators for event times, omitting one event time as the reference period, often \(k=-1\). A generic version is

The set \(\mathcal{K}\) contains the event times included in the regression. Event-time indicators are zero for never-treated units. The coefficient \(\beta_k\) is a regression coefficient relative to the omitted event time, conditional on the included fixed effects and other event-time indicators. It is not defined simply as the average of \(\operatorname{ATT}(g,g+k)\) over cohorts.

How the Regression Mixes Treatment Effects

To isolate the issue, suppose treatment begins at \(G_i\), there is no anticipation (treatment does not affect outcomes before adoption), and untreated outcomes satisfy the restrictions needed for the fixed effects to absorb their systematic differences. Parallel trends is one familiar way to motivate such restrictions. These assumptions do not require treatment effects to be identical: \(\operatorname{ATT}(g,t)\) may vary both across adoption cohorts and across dates since adoption.

The source of contamination can be stated exactly using the Frisch–Waugh–Lovell theorem. Let \(D\) be the matrix of included event-time indicators, \(Z\) the matrix of unit and time indicators, and \(M_Z=I-P_Z\) the residual-maker that removes the projection on \(Z\). Define \(A\) to contain cohort-by-calendar-time indicators for treated observations, and let \(\theta\) collect their treatment effects. After removing the fixed effects, the event-study regression estimates

Under the stated untreated-outcome restrictions, the systematic untreated-outcome component contributes zero to the population projection. Since the treatment-effect component is represented by \(A\theta\), the population coefficient vector has the form

This is a linear combination of cohort-time effects. In other words, for each included event time \(k\), there are design-dependent weights \(w_{k,g,t}\) such that

The weights depend on the adoption pattern, which event times are included, the omitted reference period, and the regression’s observation structure. They are not generally restricted to cells with \(t-g=k\); some can be negative, and they need not form a simple average. The event-time label describes the indicator being estimated, not necessarily every treatment effect contributing to its coefficient.

A Lead Coefficient Can Reflect Later Effects

No anticipation means \(\operatorname{ATT}(g,t)=0\) for \(t<g\). It does not force every estimated lead coefficient to be zero. In the linear-combination expression, a lead coefficient can place nonzero weight on cohort-time effects for which \(t\geq g\). Heterogeneous effects in those post-adoption cells can therefore make the lead coefficient nonzero, even if every cohort’s true pre-adoption effect is zero.

Contamination can also affect lag coefficients. A coefficient at \(k=1\), for example, need not be the average effect exactly one period after adoption. It can include effects from other event times and cohorts. Consequently, a conventional event-study plot may distort both the apparent timing of effects and the apparent presence of pretrends.

Reading the Projection Equations Carefully

The matrix expression also clarifies why coefficients must be solved jointly. After residualizing the event-time indicators on the fixed effects, their columns can still be correlated with one another. The normal equations therefore involve all included event-time coefficients at once. A coefficient cannot generally be read from one equation in isolation.

Heterogeneity Changes the Meaning of a Lag

A further consequence is that a lag coefficient may not equal a natural average of the effects at that lag. A natural target might average \(\operatorname{ATT}(g,g+k)\) over cohorts observed at event time \(k\), using explicit nonnegative weights that sum to one. The conventional coefficient instead follows from the regression projection. Its weights may reach other event times and may be negative, so it can differ from that target even when all relevant treatment effects are positive.

What Lead Coefficients Can and Cannot Diagnose

A pretrend diagnostic asks whether outcomes appear to differ before treatment. In a conventional TWFE event study with staggered adoption, a nonzero lead can reflect genuine anticipation or a failure of the assumptions governing untreated outcomes. But it can also arise through contamination from heterogeneous post-adoption effects. Conversely, a lead near zero does not establish that parallel trends holds: the projection weights can combine contributions in ways that cancel, and diagnostics may have limited power.

Interpret leads in light of the estimator and design, not as automatic tests that identify the cause of a nonzero coefficient. The earlier material on pretrend diagnostics and the Goodman-Bacon decomposition addresses related issues; here the key point is specific to event-study regressions: heterogeneity can make the lead coefficients themselves mixtures of effects. The event-time coefficients remain regression summaries, but their causal interpretation requires more than reading off their labels.

Common Mistake

Wrong reasoning: “The coefficient at \(k=-2\) is nonzero, so treated units must have had an effect two periods before adoption.” This treats the coefficient’s label as if it fixed the set of treatment-effect cells contributing to the estimate.

Correction: Under staggered adoption, the coefficient is a design-dependent linear combination of cohort-time effects. With heterogeneous post-adoption effects, some contributions to a lead can come from after adoption. A nonzero lead is therefore not, on its own, proof of anticipation or nonparallel untreated trends. Establishing its interpretation requires examining the regression’s comparison structure and the assumptions that identify its components.