Total effect versus direct effect when you control a mediator
In a causal system, a treatment's total effect on the outcome is . Of that, flows indirectly through a mediator (treatment moves , moves the outcome), and the remaining is a direct effect that does not pass through .
If you control for in your regression, what does the treatment coefficient estimate, and how far is it from the total effect you intended to measure?
Your answer
This one is open-ended. Work it through, then check your reasoning against the full solution.