<p>My advice is to do a stacked diff-in-diff. For each treated unit, randomly assign a control unit. This forms a group, <span class="math-container">$g$</span>. There will be several such groups.</p>
<p><span class="math-container">$$y_{igt} = \sum_{s}\beta_sT_{ig}d_{gs}+\lambda_i +\lambda_{gt}+\varepsilon_{igt}$$</span></p>
<p>where <span class="math-container">$\beta_s$</span> is the effect of being <span class="math-container">$s$</span> periods after (or before) treatment. <span class="math-container">$T_{ig}$</span> is a binary variable that individual <span class="math-container">$i$</span> in group <span class="math-container">$g$</span> was treated. <span class="math-container">$d_{gs}$</span> is a binary variable for being <span class="math-container">$s$</span> time periods from treatment for group <span class="math-container">$g$</span>, <span class="math-container">$\lambda_i$</span> are individual FE, <span class="math-container">$\lambda_{gt}$</span> are fixed effects for group and being <span class="math-container">$t$</span> time periods from treatment.</p>
<p><a href="https://jrgcmu.github.io/2sdd_current.pdf" rel="nofollow noreferrer">Gardner 2021</a> shows the stacked diff-in-diff returns a weighted average of the ATT for each treated unit.</p>
<p>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.</p>

CC BY-SA 4.0
2022-06-28T07:13:35.417Z
2022-06-28T07:13:35.417Z
Michael Gmeiner
<p>I am trying to extend the results of <a href="https://www.aeaweb.org/articles?id=10.1257/app.20180168" rel="nofollow noreferrer">Gendron-Carrier et al. (2022) article</a> published in the <em>American Economic Journal : Applied Economics</em> which is about the effect of subway opening on pollution.</p>
<p>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 <em>trend</em> throughout the different cities where it is implemented).</p>
<p>I cannot understand if the <a href="https://www.aeaweb.org/articles?id=10.1257/aer.20181169" rel="nofollow noreferrer">recent robust estimator proposed by De Chaisemartin and d'Hautfoeuille (2020)</a> which is for two-way fixed estimation can be used in my case.</p>
<p>The specification of Gendron-Carrier is as follows :</p>
<p><span class="math-container">$AOD_{it} = \beta_i + \alpha_1D_{it} + \gamma'X_{it} + \epsilon_{it}$</span></p>
<p>Where
<span class="math-container">$AOD$</span> is a measure of air pollution, <span class="math-container">$D_{it}$</span> is a dummy variable equal to 1 when the subway has opened 18 months ago, <span class="math-container">$X_{it}$</span> 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. <span class="math-container">$\epsilon_{it}$</span> is the error term of the equation.</p>
<p>Do you think the <a href="https://www.aeaweb.org/articles?id=10.1257/aer.20181169" rel="nofollow noreferrer">De Chaisemartin and d'Hautfoeuille</a> applies in this case?</p>
<p>Is this a two-way fixed effects estimation?</p>
<p>In our case how can we apply an estimation method robust to heterogeneous treatment effects?</p>

<econometrics><applied-econometrics><regression><environmental-economics><difference-in-difference>
Difference-in-difference robust to heterogeneous treatment effect - Gendron-Carrier et al. specification
Galactus
# Difference-in-difference robust to heterogeneous treatment effect - Gendron-Carrier et al. specification

(Asked by: username_0 on 2022-06-27)

I am trying to extend the results of [Gendron-Carrier et al. (2022) article](https://www.aeaweb.org/articles?id=10.1257/app.20180168) 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)](https://www.aeaweb.org/articles?id=10.1257/aer.20181169) 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](https://www.aeaweb.org/articles?id=10.1257/aer.20181169) 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?

**Tags:** econometrics, applied-econometrics, regression, environmental-economics, difference-in-difference


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## 1 answers

### Answer by username_1 on 2022-06-28. Score: 1

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](https://jrgcmu.github.io/2sdd_current.pdf) 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.


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