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From my reading about the potential outcomes framework (POF) and structural causal models (SCM), I understand that both perspectives have been shown to be equivalent but take different starting points. In particular, the POF takes as a starting point the potential outcomes (+some model) and relates these via the observation rule to observed outcomes. In contrast, the SCM perspective defines a model based on the observed outcomes from which the potential outcomes can then be derived. Let us consider a linear model from both perspectives.

The POF defines a model of $Y(0)$ and $Y(1)$ and uses the observation rule to link these to the observed outcome $Y$.

$Y(0)=a_0 + U(0)$

$Y(1)=Y(0)+\tau$ (this implies a constant treatment effect of $\tau$)

$Y=XY(1)+(1-X)Y(0)$

The latter can be rearranged to $Y=Y(0)+(Y(1)-Y(0))X$ and after plugging in the equations for $Y(0)$ and $Y(1)$ we get

$Y=a_0+U(0)+((Y(0)+\tau)-Y(0))X$ which can be simplified to $Y=a_0+\tau X+U(0)$

Redefining $a_0 = b_0$, $\tau=b_1$, and $U(0)=U$ we get the following

$Y=b_0+b_1X+U$

Would it be ok to say that we here went from the potential outcomes to a linear SCM, as each term in the last equation has a clear structural/causal interpretation?

If so, my question would be how we can go from the SCM to the potential outcomes as is preferred by Pearl? I guess one would start from a SCM

$X=U_X$

$Y=b_0+b_1X+U_Y$

Now from the "Fundamental Law of Counterfactuals" $Y_x(u)=Y_{M_x}(u)$ potential outcomes are defined in models where $X$ is set to $x$ by replacing the respective equation. Thus

$Y_0(u) = b_0+U_Y$ and $Y_1(u) = b_0+b_1+U_Y$

Written in the above notation and using the alternative definitions from above we would have

$Y(0)=a_0+U_Y$ and $Y(1)=Y(0)+\tau$

I have deliberately not replaced $U_Y$ with $U(0)$, as here I'm not really sure about it.

  1. Is the interpretation of $U(0)$, because it has potential outcome notation: "All factors other than $X$ that affect $Y(0)$" or alternatively "All factors other than $X$ that affect $Y$ in the absence of $X$"?

  2. The interpretation of $U_Y$ according to Pearl et al. (2016, p.81) would be "factors ... that influence $Y$ and are not themselves affected by $X$." Is this mathematically and in meaning equivalent to $U(0)$?

A follow-up question would be how to do the same in the scenario of heterogeneity in individual treatment effects. In this scenario, an error term $U(1)$ is added for the equation $Y(1)$, resulting in the end in a more complicated error term $U(0) + (U(1)-U(0))X$ in the model for the observed outcome. How would this be derived from a (linear) SCM?

My goal in asking these questions is to better reconcile my understanding of the "different perspectives". Thanks for your help!

curious
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    related: https://stats.stackexchange.com/questions/377004/regression-and-causality-in-econometrics – markowitz Oct 13 '20 at 12:17
  • Thanks, markowitz! I will have a detailed look at this and see whether it helps me answer my question. Many of your questions/answers on this site, have already helped me a lot to understand things better. – curious Oct 13 '20 at 14:23
  • Related, you may be interested in reading single world intervention graphs (SWIG). SWIG unite the SCM with POF https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.644.1881&rep=rep1&type=pdf – pzivich Oct 19 '20 at 13:45
  • Hi, pzivich. Thanks for pointing me to SWIGS. I read about these in the Heran and Robins textbook. But actually, the use of SWIGS leads to the same underlying question for me. In the translation from SCM or SWIGS to POF the only variables that receive potential outcome notation are consequences of the treatment variable, i.e. the outcomes (Y(0), Y(1)), or intermediate outcomes (e.g., mediators). In the econometrics literature on the POF (where the first perspective from above originates from) also the error terms have potential outcome notation (i.e., U(0), U(1)). How can I make sense of this? – curious Oct 20 '20 at 16:19

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