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I'm struggling to understand the difference between Event Studies and Difference-in-Difference regressions. Both seem to have a discreet event that is assumed to cause some change of interest, subject to controls, and both seem to have leads and lags of the event to ensure that the event or change is well-isolated.

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There are differences:

Differences-in-Differences: is very specific term for a regression, which regressed the difference in treatment status on differences in outcomes. It is a specific type of model. Formally, diff-in-diff estimator is given as:

$$\hat{\delta} = (y_{11}-y_{12}) - (y_{21}-y_{22}) $$

where $y_{it}$ is average outcome for group $i$ at time $t$.

Event Study: It is more broader term that encompasses wide range of possible methodologies. For example, this introduction to event studies just uses average of cumulative average of abnormal returns, post event (although what is abnormal is defined based on pre-treatment period).

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DiD and Event Study are two different "model-based" research designs ("model-based" means the estimand is identified using assumptions, see Paul Goldsmith-Pinkham's slides). Comparing to DiD, Event Study allows for estimating dynamic effects, and provides a way to (partially) test the parallel trends assumption. Also note that research designs are not estimators, and thus not "regressions". There can have different estimators for one research design, and for some estimators, Event Study turns out to be doing multiple "DiD"s and then aggregating the estimates (see Pedro Sant’Anna's 4-hours video on DiD and Event Study).

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My understanding is that the difference is largely semantic.

The term 'Event Study' or 'Event Study Design' originates in Finance, not in Economics. It can be used differently in Finance papers than it is in Economics.

However, within Economics it is common to treat Event Studies as a subset of Dynamic Difference in Difference methods (which, in turn, are a subset of Difference in Difference methods), in which the treatment is staggered.

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