# A Local Projections Approach to Difference-in-Differences Event Studies

*Arindrajit Dube, Daniele Girardi, Alan M Taylor*
Link
Video
Video

## Abstract

Recent applied microeconometrics research proposes various difference-in-differences (DiD) estimators for the problem of dynamic heterogeneous treatment effects. We show that the problem can be resolved by the local projection (LP) estimators of the sort used in applied macroeconometrics. Our proposed LP-DiD estimator provides an overarching toolkit with several advantages. First, the method is clear, simple, easy to compute, and transparent and flexible in its handling of treated and control units. Second, it is quite general, including its ability to control for pre-treatment values of the outcome and of other covariates, as under conditional common trends. Third, the LP-DiD can nest other estimators, providing a framework that is not only rigorous but also encompassing. The LP-DiD estimator does not suffer from the negative weighting problem, and indeed can be implemented with any weighting scheme the investigator desires. Simulations demonstrate the good performance of the LP-DiD estimator in common settings. Two empirical applications illustrate how LP-DiD addresses the bias of conventional fixed effects estimators, leading to potentially different results.

The LP-DiD estimator is implemented via OLS via either sample restriction or including interactions against whether or not a particular unit is an unclean control at each point in time. Unclean controls are those units which are treated at some point in time other than the current period.

The LP-DiD estimator regresses future outcomes on a $Δ$treatment indicator, $Δ$outcome lags, $Δ$ covariate lags, and time fixed effects. When the full sample is used, lagged outcomes, covariates, and time fixed effects are interacted with the unclean control indicator and the indicator is additional included as a regressor.

Makes the note referred to in @hoyosTariffsGrowthHeterogeneous that you don’t need unit fixed effects for panel local projections with long differenced outcome variables:

In applications of the local projections estimator, it is common to employ the long difference $Δ_{k}y_{it}≡y_{i,t+k}–y_{i,t–1}$ on the left side of the estimating equation, especially when $y_{it}$ is expressed in logs. A reason is that $100×β_{k}$ can then be interpreted as an approximate percentage change in the outcome at time t + k due to treatment at time t, facilitating interpretation of effect sizes. ==

This transformation also has the advantage of mechanically removing unit-specific fixed effects.==

The dropping of unclean controls doesn’t lead to as much data loss as you might think. It’s not that those future periods aren’t included. They are included as the dependent variable in the samples where the treatment originally happened.