Lead-Lag Bias in Conventional TWFE Event Studies

Why a Lead Can Move Before Treatment

An event-study graph is often read as a timeline: coefficients before treatment are treated as evidence about pre-treatment trends, while coefficients after treatment are interpreted as dynamic treatment effects. In a staggered-adoption design, that reading can be unreliable for the conventional two-way fixed-effects (TWFE) regression. Its lead and lag coefficients are estimated jointly after removing unit and calendar-time fixed effects. As a result, a coefficient labelled as a lead can receive weight from outcomes observed after some units have been treated.

This is a feature of the regression’s algebra, not evidence that treatment literally affected outcomes before it began. When cohort-time treatment effects vary, the resulting projection can mix them into several lead and lag coefficients. Understanding this distinction is essential when interpreting a conventional event-study graph.

Write \(Y_{it}(0)\) for the outcome unit \(i\) would have in period \(t\) without treatment, and \(Y_{it}(G_i)\) for its outcome under adoption in period \(G_i\). For units in cohort \(g\), define the cohort-time average treatment effect as \(\tau_{g,t}=E[Y_{it}(g)-Y_{it}(0)\mid G_i=g]\). No anticipation means \(Y_{it}(g)=Y_{it}(0)\) for \(t<g\), so \(\tau_{g,t}=0\) before adoption. This condition says nothing by itself about whether a fitted TWFE lead coefficient must equal zero.

The Regression’s Projection Weights

The key is that conventional event-time indicators are not estimated one at a time. After removing the unit and time fixed effects, the regression estimates all included event-time coefficients together. Their joint projection determines how outcome changes in different cohorts and periods enter each coefficient.

To state that projection precisely, collect the fixed-effect regressors in a matrix \(Z\), and collect the included event-time indicators in a matrix \(Q\). The reference period is not a column of \(Q\). Let \(M_Z=I-Z(Z'Z)^{-1}Z'\) denote the residual-maker for the fixed effects, using a generalized inverse if needed. When the event-study coefficients are identified, the Frisch-Waugh-Lovell formula gives:

Here \(\hat{\beta}\) is the vector of estimated lead and lag coefficients. The matrix in this expression depends on the sample’s adoption dates, calendar periods, and event-time support. In particular, the coefficient for one event time depends on the other included indicators through \((Q'M_ZQ)^{-1}\).

For a population interpretation, suppose the untreated outcome is represented by unit and calendar-time effects, so that \(Y_{it}(0)\) is in the span of \(Z\). Let \(T\) denote the treatment-effect component of the outcome, with \(T_{it}=0\) before treatment. Then \(M_ZY=M_ZT+M_Zu\), where \(u\) collects any remaining outcome disturbance. If \(c_k'\) is row \(k\) of \((Q'M_ZQ)^{-1}Q'M_Z\), the coefficient for event time \(k\) has the form \(\hat{\beta}_k=c_k'T+c_k'u\). At the individual-observation level, the treatment component can be written as:

The entry \(c_{k,it}\) is the coefficient on observation \((i,t)\) in row \(c_k'\); these entries are determined by the design and the full set of included event-time indicators. When treatment effects vary within cohort-time cells, this individual-level sum cannot generally be reduced to cell-level weights multiplied by the unweighted cohort-time averages \(\tau_{g,t}\). The sum can include post-treatment effects even when \(k<0\). No anticipation sets actual pre-treatment effects to zero; it does not force these post-treatment weights to zero.

If untreated outcomes do not lie in the span of the unit and time fixed effects, there is an additional projected untreated-outcome component, \(c_k'Y(0)\). It can reflect departures from the additive fixed-effects representation, such as untreated trends not captured by the model. Thus, even apart from sampling error, an estimated lead can reflect both treatment-effect contamination and untreated-outcome structure. The examples below isolate those components explicitly.

Worked Examples of Lead-Lag Contamination

Why Lags Can Also Be Contaminated

The same projection argument applies to coefficients labelled as lags. A lag coefficient for event time \(k\) need not equal a simple average of the effects \(\tau_{g,g+k}\) among cohorts observed at that event time. Its row of projection weights can also place weight on other cohorts’ and periods’ effects, including effects at different event times. The resulting coefficient can therefore mix dynamic patterns as well as cohort differences.

For example, if one cohort’s treatment effect grows over time while another cohort’s effect is constant, the regression’s lag coefficients can combine both patterns according to the design-specific weights. In a TWFE decomposition, some weights may be negative. Consequently, a fitted lag can be smaller than all the effects one hoped to summarize, or even have the opposite sign. The coefficient’s event-time label identifies the regressor, not a guarantee that it estimates only the causal effect at that duration.

This is why a conventional event-study profile should not automatically be described as the sequence of dynamic average treatment effects. The earlier material on cohort-time effects distinguishes effects by adoption cohort and calendar time; the conventional regression generally compresses those effects into a single coefficient per included event time. When effects are heterogeneous, the compression is governed by the regression’s projection weights.

Derivation and Interpretation

The source of the result follows directly from the regression decomposition. Suppose the outcome vector can be expressed as \(Y=Z\gamma+T+u\), where \(Z\gamma\) is the additive unit and calendar-time component. Apply the residual-maker to both sides. Since \(M_ZZ=0\), the fixed-effect term disappears. Substitution into the Frisch-Waugh-Lovell expression yields:

The first term is a linear transformation of all treatment effects in \(T\), including effects after adoption in cohorts whose outcomes contribute to a lead coefficient. The second term is the transformed disturbance. If \(Y(0)\) is not in the span of \(Z\), include \(Y(0)\) among the components being projected as well; its contribution does not vanish automatically. This derivation explains the distinction between the treatment effect in a particular cohort-time cell and the coefficient on a conventional event-time indicator.

The interpretation is therefore conditional, not automatic. A conventional lead coefficient can be informative about pre-treatment outcomes only if its projection is not contaminated in a consequential way and the untreated-outcome component is appropriately controlled. A coefficient near zero does not prove that every cohort had parallel untreated trends: different components can cancel. A nonzero coefficient does not, by itself, establish anticipation or a violation of parallel trends.

Common Mistake

• Wrong reasoning: “The coefficient is labelled \(k=-2\), so it estimates the treatment effect two periods before adoption. A nonzero estimate proves anticipation.”
• Correction: The label names the event-time indicator in the regression, not necessarily the causal quantity being estimated. Under no anticipation, actual pre-treatment effects are zero, but the fitted lead can still receive weight from post-treatment effects or from untreated-outcome deviations. Interpret it only after considering the coefficient’s projection and the assumptions supporting the untreated-outcome model.