Event Studies (Dynamic Treatment Effects)

Motivating Example

A publicly traded firm announces that its longtime CEO is stepping down and a new CEO will take over. You want to know: what is the causal effect of CEO succession on stock prices?

You cannot just compare stock prices before and after the announcement — maybe the whole market was going up, or maybe the firm was already in decline. A simple before-after comparison confounds the succession effect with everything else happening at the same time.

What you need is a picture. You want to see what happened to the stock price in the weeks before the announcement (to check whether something was already going on) and in the weeks after (to see the actual effect and how it evolves). Did the stock jump immediately? Did it drift? Did the effect fade or grow?

That time-profile picture is an event study.

A. Overview: What Is an Event Study?

An event study is a regression framework that estimates separate treatment effects for each time period relative to the event. Instead of getting one number ("the treatment effect is 3.2 percentage points"), you get a time profile: the effect at the moment of treatment, one period after, two periods after, and so on — plus coefficients for the periods before treatment that serve as a diagnostic check.

The core idea is elegantly simple. You define event time as the number of periods relative to when the treatment happens. If a state adopts a policy in 2010, then for that state, event time -2 is 2008, event time 0 is 2010, and event time +3 is 2013. You then estimate a coefficient for each event-time indicator.

In finance, event studies have been used since at least Fama et al. (1969).

In the broader social sciences, event-study designs have become the standard way to visualize treatment dynamics in difference-in-differences settings. In many applied fields, an event-study plot is now a standard expectation in DiD papers.

The Static DiD Problem

Recall the canonical 2x2 DiD specification:

This specification gives you a single coefficient $\delta$ that averages the treatment effect across all post-treatment periods. But what if the effect takes three years to materialize? What if it fades after one year? What if it was already trending upward before treatment? The static model hides all of this dynamic information.

Common Confusions

"Which period should I omit as the reference?" The standard practice is to omit the period immediately before treatment (event time -1). This normalization ensures all coefficients are measured relative to the last pre-treatment period. Some researchers omit -2 instead, which is fine — just be consistent and transparent.

"Should I bin the endpoints?" If you have observations far from the event (e.g., event time -10 or +15), you typically bin them: group all observations at or below some threshold into a single indicator (e.g., event time $\leq -6$) and similarly for the right tail. This binning avoids estimating noisy coefficients from small samples at the extremes. But be careful — the binned coefficient averages effects across all those periods, which can be misleading if there are trends in the tails.

"Is an event study different from DiD?" An event study is a DiD. It is the dynamic generalization. The static DiD is a special case where you constrain all post-treatment effects to be the same and all pre-treatment effects to be zero.

B. Identification

The event-study specification is:

where $\alpha_i$ and $\lambda_t$ are unit and time fixed effects, $K_{it}$ is the event time for unit $i$ at calendar time $t$, and the sum runs over all event-time indicators except the reference period $k = -1$.

Identifying assumptions:

1. Parallel trends: In the absence of treatment, treated and control units would have followed the same trajectory. Formally, the counterfactual trend in outcomes absent treatment is the same for treated and control groups: $E[Y_{it}(0) - Y_{i,t-1}(0) \mid D_i = 1] = E[Y_{it}(0) - Y_{i,t-1}(0) \mid D_i = 0]$.

2. No anticipation: The treatment has no effect before it is actually implemented. This condition means $\beta_k = 0$ for all $k < 0$.

The pre-treatment coefficients $\{\beta_k\}_{k < -1}$ are your diagnostic: if they are jointly close to zero, this pattern is consistent with both parallel trends and no anticipation. If they show a clear trend, either parallel trends or no anticipation (or both) may be violated.

C. Visual Intuition

The classic event-study plot has event time on the x-axis and coefficient estimates (with 95% confidence intervals) on the y-axis. The reference period (typically $k = -1$) is normalized to zero.

A good event study has the following pattern: pre-treatment coefficients hover around zero with confidence intervals that include zero, then there is a clear jump at $k = 0$ that persists (or grows, or fades) in the post-treatment periods.

A problematic event study has the following pattern: pre-treatment coefficients trend upward (or downward), suggesting that treated units were already diverging from control units before the treatment. If you see such a pattern, your parallel trends assumption is in trouble.

D. Mathematical Derivation

E. Implementation

1library(fixest)
2
3# Standard event study using fixest
4# i(event_time, ref = -1) creates event-time dummies with k=-1 as reference
5# Unit and year fixed effects are absorbed via | unit_id + year
6# Standard errors are clustered at the state level
7est <- feols(y ~ i(event_time, ref = -1) | unit_id + year,
8           data = df, vcov = ~state)
9
10# Plot event-study coefficients with 95% confidence intervals
11iplot(est, main = "Event Study: Treatment Effect Over Time")
12
13# Sun & Abraham (2021) estimator for staggered treatment timing
14# sunab(cohort_var, time_var) avoids contamination from already-treated units
15# treatment_year = year each unit was first treated (Inf if never treated)
16est_sa <- feols(y ~ sunab(treatment_year, year) | unit_id + year,
17              data = df, vcov = ~state)
18iplot(est_sa)

F. Diagnostics

1. Joint F-test of pre-treatment coefficients. Test whether $\beta_{-2}, \beta_{-3}, \ldots$ are jointly zero. This joint test is more informative than eyeballing individual coefficients. In Stata: testparm on the pre-treatment indicators. In R with fixest: wald(est, "event_time::-").

2. Visual inspection of pre-trends. Even if the F-test does not reject, look at the pattern. A clear monotonic trend that approaches zero at $k = -1$ is concerning even if individual coefficients are not significant.

3. Confidence interval widths. If your pre-treatment confidence intervals are wide enough to include economically meaningful effects, your pre-trend test lacks power. You cannot distinguish "no pre-trend" from "pre-trend we cannot detect."

4. Placebo treatment dates. Artificially reassign treatment to earlier periods and re-estimate. If you still see "effects," something is wrong.

5. Sensitivity to endpoint binning. Re-estimate with different bin cutoffs. If your results change substantially, the specification is fragile.

Interpreting Your Results

Pre-treatment coefficients near zero: This finding supports (but does not prove) parallel trends. Report the joint F-test p-value alongside the visual.

Pre-treatment coefficients trending: This pattern is a red flag. Consider whether the trend is economically meaningful. If it is large relative to your post-treatment effects, your results are not credible without further adjustment (e.g., including unit-specific linear trends, though this introduces its own problems).

Post-treatment effects that grow over time: This dynamic is common and often substantively meaningful. A policy may take time to have its full effect. Report the dynamics, not just the average.

Post-treatment effects that shrink: This pattern could mean the effect is transitory, or that control units are "catching up." Think about the mechanism.

G. What Can Go Wrong

H. Practice

I. Swap-In: When to Use Something Else

• Canonical DiD: When a single treatment-effect estimate is sufficient and the time profile of effects is not of primary interest.
• Staggered DiD estimators: When the goal is a single aggregate ATT across cohorts — modern estimators (Callaway & Sant'Anna, Sun & Abraham) aggregate event-study coefficients into a robust summary.
• Synthetic control: When there is a single treated unit and constructing a counterfactual from donor units is more credible than assuming parallel trends.
• Interrupted time series (ITS): When there is no control group and the design relies entirely on modeling the pre-treatment trend to forecast the counterfactual.

J. Reviewer Checklist