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Feb 21, 2017 at 15:51 comment added Julian Schuessler @Peter OPs first question implies D potentially affects Y through channels other than the specific M. Then D can not be a valid IV. Certainly also not conditional on the other Ms, if these and Y are also driven by unobserved confounders. Then, even conditional on the other Ms, D and Y will be connected because one conditions on colliders.
Feb 21, 2017 at 14:57 comment added Peter @JulianSchuessler, that is why I started my reply with "For the second question,..."
Feb 20, 2017 at 11:50 comment added Julian Schuessler This does not make any sense. OP explicitly asks how to check whether M mediates the effect of D on Y. Your approach ASSUMES it COMPLETELY mediates, and provides an altnerative approach to identify the ATE of D, which the OP assumes is identified anyways via DiD!
Feb 13, 2017 at 21:04 comment added Peter @greg Slapping myself as I forgot the exclusion restriction. I agree that you could control for the other observed Ms, but you'd be assuming there are no unobserved Ms.
Feb 13, 2017 at 17:41 comment added greg I thought about IV but the problem is the exclusion restriction -- i.e. the assumption that the only link between D and Y goes through M_j. I could control for the other measured M, which partly addresses the problem, but relies on the assumption that there are no other unobserved M. Correct?
Feb 13, 2017 at 15:23 history answered Peter CC BY-SA 3.0