Recommendation: high-level comparison of formal causal reasoning approaches I am looking for a good article, blog post, or text that situates the key 'high-level' differences and agreements between the kinds of counterfactual formal causal reasoning championed by Pearl, Hernán & Robins, and Glymour (i.e. 'structural causal models'), with, say, Imbens' and Rubin's work with potential outcomes.
I am trying to situate some of the formal causal reasoning stuff in the econometrics world with respect to the counterfactual formal causal inference I am more familiar with (Pearl, etc.) without having to work my way through everything Imbens, Rubin and Angrist ever wrote. :)
Has there been a good compare & contrast written?
 A: Take a look at Section 4 for a comparison of PO and DAG/SCM approaches:

Imbens, Guido W. 2020. "Potential Outcome and Directed Acyclic Graph
Approaches to Causality: Relevance for Empirical Practice in
Economics." Journal of Economic Literature, 58 (4): 1129-79
https://doi.org/10.1257/jel.20191597 (ungated draft)

Pearl's response can be found at Pearl, J. On Imbens’s Comparison of Two Approaches to Empirical Economics. (2020). Academic blog. Causal Analysis in Theory and Practice. Posted January 29, 2020.
A: We have a unifying theory for potential outcomes, graphical models, and structural econometrics. It is based on the theory of structural causal models.
Check out:

*

*Appendix of the Crash Course in Good and Bad Controls.


*On Pearl’s Hierarchy and the Foundations of Causal Inference


*Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press., Chapter 7.


*Pearl, J., & Mackenzie, D. (2018). The Book of Why: The New Science of Cause and Effect (1st ed.). Basic Books., Chapter 8.
These answers may also help:
Which Theories of Causality Should I know?
Is the linearity assumption in linear regression merely a definition of $\epsilon$?
The references above concern a unified mathematical framework for the counterfactual and structural theories of causation.
In practice, of course, each "school" may have different conventions and cultural differences. For instance, certain identification strategies are more popular in economics than epidemiology, or inside  econometrics itself there is a cultural division between "reduced form" and "structural" econometrics, etc.

As for Pearl's answer to Imbens' article, it is simply stating that there's no such thing as a "DAG approach."  The formal mathematical framework is a structural causal model. The DAG is just one tool for partially specifying a structural model, namely, imposing certain types of exclusion and independence restrictions. You can impose as many assumptions as you would like, and monotonicity has nothing special in it. For instance, see how Pearl and I defined monotonicity here. That is why it makes no sense to ask how you would represent monotonicity in the "DAG approach" vs "PO approach." Maybe a better question would be: how can we formally represent monotonicity constraints (or other shape constraints) graphically, in a way that we can leverage such constraints to algorithmically derive new identification results? This is the topic of ongoing research.
PS: it goes without saying that I think Imbens is a great scholar, and it has inspired a lot of my own work as well! The above is just a comment on this specific point.
