What is the type of the left variable if this is not an instrument or conditional instrument? Is it just a covariate? Moderator?
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[EDIT (HT: Noah)] Your mystery variable is an instrumental variable since it satisfies the exclusion criteria of an instrumental variable, even though you wouldn't necessarily have any reason to do 2SLS. As Noah said in his comment, conditioning on the instrument will decrease the precision of the estimated causal effect.
We can also definitely call it a pre-treatment variable, since it causes the exposure/treatment.
answered Mar 16, 2022 at 14:25
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1$\begingroup$ It's an instrument regardless of whether there is unmeasured confounding; an instrument is defined by the exclusions restriction, which is met. Conditioning on it in a sample will decrease the precision of the estimated causal effect. $\endgroup$– NoahCommented Mar 16, 2022 at 14:31
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$\begingroup$ @Noah Ah, I see. I'll edit. Thanks! $\endgroup$ Commented Mar 16, 2022 at 14:34