Linear model: potential outcome framework vs. structural causal model

Question

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!

Answer 1

While reading “Judea, P. (2009). Causal inference in statistics: An overview. Statistics Surveys, 3, 96-146. https://doi.org/10.1214/09-SS057”, I search for more straightforward examples to illustrate the two perspectives, and I appreciated the example provided in this question. I’d like to share my thoughts on $U_Y$ and U(0), and I welcome any corrections.

Regarding the second point, I believe that $U_Y$ in SCM and U(0) in POF share the same interpretation (or meaning). As Pearl often reminds the readers, it's crucial to be clear about the assumptions beyond the mathematical formulas. The formula for the Potential Outcome Framework (POF) implies that X and U(0) are the only two random variables that cause/influence Y. The Structural Causal Model (SCM) also makes this assumption. (Another assumption is the independence between X and $U_Y$, which goes beyond the mathematical linear model but can be represented in the causal graph in SCM. It seems this assumption is not that straightforward in POF but could be derived?)

Regarding the first point, if we substitute the value of Y(0) to the equation for Y(1), we will see that U(0) affects Y(1). So joinly with the assumptions bear in mind, we can interpret U(0) as “All factors other than X that affect Y in the absence of X.”