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Unlike a standard causal model with A = Treatment, X = Confounder, and Y = Outcome:

enter image description here

a difference-in-differences (DiD) model is concerned with estimating the Average Treatment Effect on the Treated (ATT) $= E(Y_1-Y_0|A=1)$.

Hence we're interested in the causal effects of confounders on the trends in the outcome over time (pre and post periods) between the treatment and control groups. (See see Daw & Hatfield (2018) and Zeldow & Hatfield (2021))

Therefore, for the purposes of drawing a DiD causal diagram is it as simple as replacing Y with ATT:

enter image description here

or perhaps $Y_{post} - Y_{pre}$?

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  • $\begingroup$ Personally, I don't think these graphs are as effective as intended because they stress the variables considered, while a Diff-in-Diff is all about the design/strategy (comparing Treated & Control, Before & After). In particular, these graphs totally lack the time dimension. I do prefer graphs as the one in Figure 5.2.1 in 'Mostly Harmless Econometrics', of the type reported in Wikipedia too under "Difference in differences". $\endgroup$
    – Alessandro
    Commented May 4, 2022 at 16:57
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    $\begingroup$ @Alessandro After some hunting I found a DiD causal diagram on the Health Policy Data Science Lab website (diff.healthpolicydatascience.org/#confounding) - scroll down a little. They split the outcome variable Y into Y1 and Y2 for the pre and post periods, so that a confounder is causally linked by arrows to both Y1 and Y2. $\endgroup$
    – RobertF
    Commented May 4, 2022 at 18:45
  • $\begingroup$ Very interesting, thanks for sharing!! $\endgroup$
    – Alessandro
    Commented May 4, 2022 at 18:56

1 Answer 1

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enter image description here

Here, the outcome of interest is the difference between the pre- and post-treatment period, $Y1 - Y0$. This difference is influenced by the treatment, unobserved factors $U$, and observed covariates $X$. The dashed arrow between $U$ and $A$ indicates a statistical dependency between the variables, but where we remain agnostic to the precise causal mechanism. For example, in the minimum wage example, $U$ might be the average income in restaurant’s neighbourhood, which is dependent on the state, and hence also the treatment.

Source:https://github.com/probml/pml2-book/releases/tag/2022-10-16

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    $\begingroup$ Thank you - this book looks like a good reference. Now I've read three different sources that describe three different ways to represent the outcome in a diff-in-diff causal diagram: as either (1) $Y_1 - Y_0$, or (2) $Y_1$ and $Y_0$ excluding $Y_1 - Y_0$ (Zeldow & Hatfield), or (3) all three quantities $Y_1 - Y_0$, $Y_1$ and $Y_0$ linked in the same diagram, with $Y_0$ and $Y_1$ having deterministic relationships with $Y_1 - Y_0$ (Pearl). The question is which one is correct? $\endgroup$
    – RobertF
    Commented Oct 28, 2022 at 4:07

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