Event Study Designs and Dynamic Treatment Effects

Summary

An event study re-indexes a panel by time relative to treatment and estimates one coefficient per relative period: $Y_{it}={\alpha }_i+{\varphi }_t+{\sum }_{r\ne r_0}1[R_{it}=r]{\beta }_r+{ϵ}_{it}$, with one pre-period $r_0$ normalised to zero. Post-treatment coefficients trace the dynamic treatment effect path; pre-treatment coefficients are placebo estimates that should be zero under parallel trends and no anticipation. The unifying algebra (Rambachan & Roth 2023; Roth 2022) is $\beta =\tau +\delta$: coefficient = causal effect + differential trend, with ${\tau }_{pre}=0$ so that ${\beta }_{pre}={\delta }_{pre}$ is identified. With a common treatment date the dynamic TWFE regression is exactly a sequence of 2×2 DiDs against the reference period. With staggered timing and heterogeneous effects it is contaminated (Sun & Abraham 2021), and event-study coefficients should instead be built by aggregating [[Group-Time Average Treatment Effects|$ATT(g,t)$]] by event time.

Overview

The static DiD coefficient answers “what was the average effect over the post period?” Three questions force a dynamic view: Does the effect build, persist or decay? (For marketing: what is the carryover profile?) Was there anticipation? And — the usual reason — do the groups look comparable before treatment? The event-study plot answers all three in one figure, which is why Roth (2022) found 70 papers in three AEA journals (2014–mid-2018) featuring one, and why Roth et al. (2023, §4.6) “encourage researchers to continue to plot ‘event-study plots’” even while cautioning about how they are read.

Main Content

Dynamic TWFE / event-study specification ^def-event-study-spec

Let $G_i$ be unit $i$‘s first treated period ($G_i=\infty$ if never treated) and define relative time $R_{it}=t-G_i+1$, so $R_{it}=1$ in the first treated period (Roth et al. eq. 7/16):

$$Y_{it}={\alpha }_i+{\varphi }_t+\sum _{r\ne 0}1[R_{it}=r]{\beta }_r+{ϵ}_{it}$$

The omitted category ($r=0$ here, the last pre-period; equivalently "$\ell =-1$" when the first treated period is labelled $\ell =0$) is the normalisation: every ${\beta }_r$ is measured relative to that period. For a common treatment date this is $Y_{it}={\lambda }_i+{\varphi }_t+{\sum }_{s\ne 0}{\beta }_{s}1[t=s]D_i+{ϵ}_{it}$ (Rambachan & Roth eq. 2). ${\hat{\beta }}_{post}=({\hat{\beta }}_1,\dots ,{\hat{\beta }}_{\bar{T}})$ are the dynamic effects; ${\hat{\beta }}_{pre}=({\hat{\beta }}_{-\underline{T}},\dots ,{\hat{\beta }}_{-1})$ are the leads.

Non-staggered case: event-study coefficients are 2×2 DiDs ^thm-event-study-2x2

With a balanced panel and a single treatment date, the OLS coefficients are numerically

$${\hat{\beta }}_s=({\bar{Y}}_{s,1}-{\bar{Y}}_{s,0})-({\bar{Y}}_{0,1}-{\bar{Y}}_{0,0})$$

where ${\bar{Y}}_{s,d}$ is the period-$s$ mean for group $D_i=d$ (Rambachan & Roth Example 1). In error form, ${\hat{\beta }}_s={\beta }_s+\Delta {\overline{ϵ}}_s-\Delta {\overline{ϵ}}_0$ (Roth 2022, p. 316). Consequence: every coefficient shares the reference-period noise $\Delta {\overline{ϵ}}_0$, so under homoskedasticity $\mathrm{Var}[{\hat{\beta }}_k]={\sigma }^2$ and $\mathrm{Cov}({\hat{\beta }}_k,{\hat{\beta }}_j)={\sigma }^2/2$ — event-study coefficients are strongly positively correlated by construction. This correlation is what drives pre-test bias (Pre-Trend Testing and Its Pitfalls), and it means pointwise error bars overstate how “independently” each lead supports parallel trends.

Causal decomposition $\beta =\tau +\delta$

Under no anticipation ($Y_{it}(1)=Y_{it}(0)$ for all pre-treatment $t$),

$${\beta }_s=\underset{{\tau }_{ATT,s}}{\underbrace{\mathbb{E}[Y_{is}(1)-Y_{is}(0)\mid D_i=1]}}+\underset{{\delta }_s\text{(differential trend)}}{\underbrace{\mathbb{E}[Y_{is}(0)-Y_{i0}(0)\mid D_i=1]-\mathbb{E}[Y_{is}(0)-Y_{i0}(0)\mid D_i=0]}}$$

and stacking,

$$\beta =\left(\begin{array}{c}{\tau }_{pre}\\ {\tau }_{post}\end{array}\right)+\left(\begin{array}{c}{\delta }_{pre}\\ {\delta }_{post}\end{array}\right),{\tau }_{pre}=0$$

(Rambachan & Roth Assumption 1; Roth 2022 eq. 2). Parallel trends is ${\delta }_{post}=0$. The pre-period coefficients identify ${\delta }_{pre}$ — not ${\delta }_{post}$. Everything in the pre-trend debate is about what ${\delta }_{pre}$ licenses one to believe about ${\delta }_{post}$.

Reading the plot: the conventions

• Normalisation. One pre-period is dropped; its coefficient is zero with no error bar. Papers that do not normalise to a pre-treatment period (3 of Roth’s 70) cannot be read as DiD event studies.
• Anticipation. If units respond before formal treatment (announced campaigns, pre-launch buzz), leads near zero are not placebos. The standard fix is to move the reference period earlier than the anticipation window; the Callaway–Sant’Anna framework encodes this as an anticipation horizon (see Identifying Assumptions for Staggered DiD).
• Which baseline? Event studies implicitly put all baseline weight on the single reference period, DiD weights all pre-periods equally, and SDID chooses the weights from data (Arkhangelsky et al., p. 9) — see SDID Estimator - Unit and Time Weights. Relatedly, Callaway–Sant’Anna use period $g-1$ as the baseline, while imputation estimators (Borusyak, Jaravel & Spiess) use the average of all pre-periods: more efficient under full parallel trends, less robust if only recent trends are parallel (Roth et al. §3.3).
• Endpoint binning. Distant leads/lags are often pooled (“5+ years since treatment”). Roth et al. (fn. 8) flag that the clean interpretation of dynamic TWFE assumes all relative-time indicators are included and that “problems may arise if one ‘bins’ endpoints,” citing Sun & Abraham, Baker–Larcker–Wang, and Schmidheiny & Siegloch. Binning imposes that effects are constant beyond the bin edge; if they are not, the misspecification leaks into other coefficients. (Schmidheiny & Siegloch also show the formal equivalence between binned event-study and distributed-lag specifications — compare Carryover Effects and Distributed Lags; that paper is not yet in the vault.)
• Bands. Use simultaneous rather than pointwise confidence bands for the coefficient path (Roth et al. §4.6, citing Freyaldenhoven et al. 2021; Olea & Plagborg-Møller 2019; Callaway & Sant’Anna 2021) — implemented in Simultaneous Inference via Multiplier Bootstrap.

Staggered adoption: dynamic TWFE is contaminated (Sun & Abraham 2021, as summarised by Roth et al. §3.2) ^thm-event-study-contamination

Under staggered parallel trends (Assumption 4) and no anticipation (Assumption 5):

1. If effects depend only on time since treatment, ${\tau }_{i,t}(g)={\sum }_{s\ge 0}{\tau }_{s}1[t-g=s]$, then ${\beta }_s={\tau }_s$: dynamic TWFE is fine (unlike static TWFE, which already fails here).
2. If dynamic effects are heterogeneous across adoption cohorts, then (a) ${\beta }_r$ may put negative weight on some units’ lag-$r$ effects, and (b) ${\beta }_r$ puts non-zero weight on effects at other lags $r^{\prime }\ne r$ — cross-lag contamination.
3. Hence “the ‘treatment lead’ coefficients … are not guaranteed to be zero even if parallel trends is satisfied in all periods (and vice versa), and thus evaluation of pre-trends based on these coefficients can be very misleading.”

The root cause is the same as for static TWFE: OLS uses “forbidden comparisons” in which already-treated units serve as controls.

Heterogeneity-robust event study ^def-event-study-robust

Estimate each cohort-period effect by a clean 2×2 DiD against not-yet- or never-treated units (Roth et al. eq. 8),

$$\widehat{ATT}(g,t)=\frac{1}{N_g}\sum _{i:G_i=g}[Y_{it}-Y_{i,g-1}]-\frac{1}{N_{{\mathcal{G}}_{comp}}}\sum _{i:G_i\in {\mathcal{G}}_{comp}}[Y_{it}-Y_{i,g-1}]$$

then aggregate by event time $l$ with researcher-chosen non-negative weights (eq. 9):

$$AT{T}_l^w=\sum _g{w}_{g}ATT(g,g+l)$$

Evaluating the same construction at $l<0$ gives placebo event-study coefficients. This is exactly the event-study aggregation in Callaway & Sant’Anna, where the compositional-change caveat and the “balanced” version are developed. Roth et al.’s recommendation (§4.6): with a common treatment date TWFE event studies are fine; “in contexts with staggered timing, we recommend plotting estimates of $AT{T}_l^w$.”

A common framework

Rambachan & Roth (§2.1-2.2) stress that the $\beta =\tau +\delta$ structure is estimator-agnostic: it holds for canonical DiD, Callaway–Sant’Anna or Sun–Abraham event-study aggregates (with ${\delta }_r={\sum }_g{w}_g{\delta }_{g,g+r}$), IV event studies, and covariate-adjusted DiD. Roth (2022, Remark 1) makes the same point from the inference side — all yield $\sqrt{N}(\hat{\beta }-\beta ){\to }_d\mathcal{N}(0,\Sigma )$ — so that “the issues surrounding pretesting are distinct from those related to the interpretation of TWFE models under heterogeneity.” Fixing the staggered-timing problem does not fix the pre-trend problem.

Examples

Marketing reading. A retailer launches a loyalty programme in 40 of 200 stores in week 0. The event-study coefficients ${\hat{\beta }}_r$ on weekly sales trace (i) leads $r<0$: were programme stores already trending differently? (ii) $r=1,2,\dots$: ramp-up and plateau of the effect; (iii) if the programme is withdrawn, the decay — the quasi-experimental analogue of an adstock/carryover curve. If the rollout is staggered across regions and late regions get an improved programme (cohort heterogeneity), dynamic TWFE leads can be non-zero because of contamination; use $AT{T}_l^w$.

import pandas as pd, statsmodels.formula.api as smf
 
# common treatment date: leads/lags interacted with treated-group dummy, ref = last pre-period
df["rel"] = df["week"] - T0                     # rel = 0 is the last pre-period
df["rel_b"] = df["rel"].clip(-8, 12)            # bin endpoints (imposes constant effects beyond)
m = smf.ols("y ~ C(store) + C(week) + C(rel_b, Treatment(reference=0)):treated", df) \
       .fit(cov_type="cluster", cov_kwds={"groups": df["store"]})
# staggered timing: do NOT use this; use Callaway-Sant'Anna (R `did`: aggte(type = "dynamic"))
# keep m.params and the FULL covariance m.cov_params() -- needed for pretrends / HonestDiD

Published example (Rambachan & Roth §6.2): Benzarti & Carloni’s event study of the 2009 French restaurant VAT cut on log profits rejects ${\beta }_{pre}=0$ ($p<0.01$) because of a 2006–07 wiggle, yet post-treatment coefficients are far larger than any lead. A binary pre-test says “fail”; the right question is how large a post-period violation is plausible given the observed leads — see Honest DiD - Sensitivity to Parallel Trends Violations.

Connections

• Differences-in-Differences — the static parent design; “leads and lags” are introduced there as a robustness check.
• Difference-in-Differences with Multiple Time Periods - Overview, Group-Time Average Treatment Effects, Aggregating Group-Time Effects, Identifying Assumptions for Staggered DiD — the heterogeneity-robust construction of event-study coefficients.
• Simultaneous Inference via Multiplier Bootstrap — uniform bands for the event-study path.
• Fixed-Effects Model — the TWFE regression being extended.
• Carryover Effects and Distributed Lags — post-treatment ${\beta }_r$ are a nonparametric lag distribution for a one-off “pulse” treatment.
• Bayesian Difference in Differences — the Bayesian treatment of the static design; a smoothness prior on the ${\beta }_r$ path is the natural Bayesian event study and mirrors the ${\Delta }^{SD}$ restriction of Honest DiD.
• Synthetic Difference-in-Differences - Overview — SDID replaces the single reference period with data-driven time weights.

See Also

• Pre-Trend Testing and Its Pitfalls
• Honest DiD - Sensitivity to Parallel Trends Violations
• Doubly-Robust Estimands for ATT(g,t)
• Standard Errors and Clustering