# Difference-in-difference robust to heterogeneous treatment effect - Gendron-Carrier et al. specification

I am trying to extend the results of Gendron-Carrier et al. (2022) article published in the American Economic Journal : Applied Economics which is about the effect of subway opening on pollution.

I want to use an estimation method robust to heterogeneous treatment effects (meaning that the effect of subway opening on air pollution may have a different trend throughout the different cities where it is implemented).

I cannot understand if the recent robust estimator proposed by De Chaisemartin and d'Hautfoeuille (2020) which is for two-way fixed estimation can be used in my case.

The specification of Gendron-Carrier is as follows :

$$AOD_{it} = \beta_i + \alpha_1D_{it} + \gamma'X_{it} + \epsilon_{it}$$

Where $$AOD$$ is a measure of air pollution, $$D_{it}$$ is a dummy variable equal to 1 when the subway has opened 18 months ago, $$X_{it}$$ is a set of controls consisting of year-by-continent indicators to flexibly account for regional trends in AOD, and city-by-calendar month (1–12) indicators to capture seasonality in pollution patterns as well as climate controls. $$\epsilon_{it}$$ is the error term of the equation.

Do you think the De Chaisemartin and d'Hautfoeuille applies in this case?

Is this a two-way fixed effects estimation?

In our case how can we apply an estimation method robust to heterogeneous treatment effects?

## 1 Answer

My advice is to do a stacked diff-in-diff. For each treated unit, randomly assign a control unit. This forms a group, $$g$$. There will be several such groups.

$$y_{igt} = \sum_{s}\beta_sT_{ig}d_{gs}+\lambda_i +\lambda_{gt}+\varepsilon_{igt}$$

where $$\beta_s$$ is the effect of being $$s$$ periods after (or before) treatment. $$T_{ig}$$ is a binary variable that individual $$i$$ in group $$g$$ was treated. $$d_{gs}$$ is a binary variable for being $$s$$ time periods from treatment for group $$g$$, $$\lambda_i$$ are individual FE, $$\lambda_{gt}$$ are fixed effects for group and being $$t$$ time periods from treatment.

Gardner 2021 shows the stacked diff-in-diff returns a weighted average of the ATT for each treated unit.

I personally find this much more desirable than the De Chaisemartinand d'Hautfoeuille or Sun and Abraham estimators because it is straightforward and similar in spirit to baseline diff-in-diff.