Instrumental Variable for Moderator (Causality)

I only know of instrumental variables that causally influence the independent variable and, through the independent variable, are associated with the outcome (making it possible to assume the causal effect of the IV on the DV).

However, is it also possible to use an instrumental variable to investigate the causal role of a moderator? Do you know any examples of papers (preferably social science papers) that did something similar?

Thank you!

• It will be possible. But I don't know of any examples (I thought about shift-share instruments but they seem different). So I guess that in this setting the moderator variable is included as an interaction term with the other explanatory variable? May 30 '20 at 12:15
• Why would you want to? A moderator is, by definition, not a confounder. You don't have a backdoor path in your diagram. There are quite a few methods of analyzing moderation, especially in counterfactual analysis. I would recommend you look up Chapter 4 in Causal Inference in Statistics: A Primer, by Pearl, Glymour, and Jewell. May 30 '20 at 14:40

$$X$$ is the explanatory variable, $$Y$$ the effect, and $$T$$ the mediator. We have an "instrumental variable" $$U.$$ If you want the true causal effect of $$T$$ on $$Y,$$ then we must view $$X$$ as a confounder, because $$T\leftarrow X\to Y$$ is a backdoor path from $$T$$ to $$Y.$$ You can just use the backdoor adjustment formula, swapping the roles of the variables, thus: $$P(Y=y|\operatorname{do}(T=t))=\sum_x P(Y=y|T=t,X=x)\,P(X=x).$$ This would get the true causal effect of $$T$$ on $$Y.$$ $$U$$ is actually unnecessary in this diagram, if you want the true causal effect of $$T$$ on $$Y.$$