Can I control one channel (mediator) to examine the impact of another channel?

[Qingfeng Liu]

I read papers discussing "bad controls", but they do not discuss cases like this:

Suppose shock X impacts Y through two mediators/channels (A and B).

Let's imagine the following Directed Acyclic Graph (DAG):

X→𝐴→Y and X→𝐵→Y. And A and B are not correlated.

If I just want to examine the second channel (B). Is it right to control for A in the regression to study the impact of X on Y through the channel B:

Y=𝛼+𝛽X+𝝺A+𝝴

[Erik Ruzek]

Yes, it is right to control for it, IMO. Using the backdoor criterion, you can see that one can get from Y to X via the two mediating paths (X --> A --> Y and X-->B-->Y). If you want to examine solely the mediating effect of B, you need to close the backdoor that runs through A. That would mean adjusting for it in your model of choice. See this primer on mediation by Chi et al. (2022) that describes a multiple mediation situation that mimics yours. Note that they end up fitting two mediation models and utilize the mediation package in R. You can grab the related Stata user-written programs medeff and medsens and the accompanying Stata Journal article by Hicks & Tingley (2011).