<p>I'm not sure but ATE doesn't seem to be identified under the parallel trends assumption for the potential <em>untreated</em> outcomes because ATU isn't. Here is why.</p>
<p>First, let us show that ATU should be identified in order for ATE to be identified when ATT is. This part is straightforward because
<span class="math-container">\begin{align}
ATE &amp;= E(y^1 - y^0)\\
&amp;= E(y^1-y^0|d=1)P(d=1) + E(y^1-y^0|d=0) P(d=0)\\
&amp;= ATT\cdot P(d=1) + ATU \cdot P(d=0),
\end{align}</span>
where ATT is identified by DID (as OP mentions, under the parallel trends assumption for <span class="math-container">$y^0$</span>) and <span class="math-container">$P(d=1)$</span> and <span class="math-container">$P(d=0)$</span> are obviously identified. So ATE is identified if and only if ATU is identified unless <span class="math-container">$P(d=0)\ne 0$</span>, which is true. Thus, the question is whether ATU is identified by a DID regression.</p>
<p>For this second question, we have
<span class="math-container">$$ATU = E(y^1 - y^0|d=0) = E(y^1|d=0) - E(y|d=0),$$</span>
where <span class="math-container">$E(y|d=0)$</span> is obviously identified. Without assumptions on <span class="math-container">$E(y^1|g,t)$</span>, it is impossible to identify <span class="math-container">$E(y^1|d=0)$</span>. Therefore, with parallel trends only for <span class="math-container">$y^0$</span>, ATU remains unidentified, and correspondingly ATE is unidentified.</p>
<p>What, then, happens if parallel trends are assumed for the potential <em>treated</em> outcomes <span class="math-container">$y^1$</span>? Well, in that case, ATU should be identified but ATT and ATE aren't.</p>
<p>The bottom line is that identifying ATE requires some assumptions on both <span class="math-container">$y^0$</span> and <span class="math-container">$y^1$</span>. This is my reasoning. What would you think?</p>

CC BY-SA 4.0
2024-03-22T12:41:00.213Z
2024-03-22T12:41:00.213Z
chan1142
<p>I understand that the coefficient of the interaction term in a standard diff-in-diff model is the average treatment effect on the treated (ATT). But I was wondering how/when we can derive the average treatment effect (ATE) from a diff-in-diff regression.</p>

<econometrics><difference-in-difference>
Diff-in-diff and ATE vs ATT
Econ Wanderer
# Diff-in-diff and ATE vs ATT

(Asked by: username_0 on 2024-03-22)

I understand that the coefficient of the interaction term in a standard diff-in-diff model is the average treatment effect on the treated (ATT). But I was wondering how/when we can derive the average treatment effect (ATE) from a diff-in-diff regression.

**Tags:** econometrics, difference-in-difference

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## 1 answers

### Answer by username_1 on 2024-03-22. Score: 0

I'm not sure but ATE doesn't seem to be identified under the parallel trends assumption for the potential *untreated* outcomes because ATU isn't. Here is why.

First, let us show that ATU should be identified in order for ATE to be identified when ATT is. This part is straightforward because
\begin{align}
ATE &= E(y^1 - y^0)\\
&= E(y^1-y^0|d=1)P(d=1) + E(y^1-y^0|d=0) P(d=0)\\
&= ATT\cdot P(d=1) + ATU \cdot P(d=0),
\end{align}
where ATT is identified by DID (as OP mentions, under the parallel trends assumption for $y^0$) and $P(d=1)$ and $P(d=0)$ are obviously identified. So ATE is identified if and only if ATU is identified unless $P(d=0)\ne 0$, which is true. Thus, the question is whether ATU is identified by a DID regression.

For this second question, we have
$$ATU = E(y^1 - y^0|d=0) = E(y^1|d=0) - E(y|d=0),$$
where $E(y|d=0)$ is obviously identified. Without assumptions on $E(y^1|g,t)$, it is impossible to identify $E(y^1|d=0)$. Therefore, with parallel trends only for $y^0$, ATU remains unidentified, and correspondingly ATE is unidentified.

What, then, happens if parallel trends are assumed for the potential *treated* outcomes $y^1$? Well, in that case, ATU should be identified but ATT and ATE aren't.

The bottom line is that identifying ATE requires some assumptions on both $y^0$ and $y^1$. This is my reasoning. What would you think?


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