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In the following DAG:

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Can I use IV1 as an instrument for exposure?

In the this video at 4:26 the teacher explains a principle of "exclusivity" for instrumental variables.

Cutoff causes outcome only through program

I wonder whether I got that concept right

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You can use IV1 as an instrument, if you also adjust for Covariate, with some reservations. Since you are conditioning on Covariate, there is no open confounding path from IV1 to Outcome and the coefficient of IV1 in the regression Outcome~IV1+Covariate estimates the part of the effect of IV1 acting via Exposure. A regression of Exposure on IV1 estimates the effect of IV1 on Exposure,and the ratio gives the effect of Exposure on Outcome.

Among the reservations: in general, there will be an interaction between Covariate and Exposure so there's no such thing as the effect of Exposure there to be estimated. Also, any measurement error in Covariate will mean that adjusting for Covariate isn't completely effective in blocking that path and the exclusion restriction will be violated.

There's an example like this on the Wikipedia page, which also shows how the answer varies with changes in the causal graph.

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  • $\begingroup$ "A regression of Exposure on IV1 estimates the effect of IV1 on Exposure, and the ratio gives the effect of Exposure on Outcome." Shouldn't it be a regression of Exposure on IV1 and covariate, instead of just on IV1? $\endgroup$
    – Chiii Zhang
    Commented May 10, 2021 at 23:14
  • $\begingroup$ I don't think so. According to the graph, there's no confounding in the regression of exposure on IV1, just in the regression of outcome on IV1. $\endgroup$
    – Thomas Lumley
    Commented May 10, 2021 at 23:52

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