Under the displayed causal directed acyclic graphs, with nonzero effects along the relevant arrows and no exact cancellation of the effects on , the valid instrumental variables are scenarios (i), (ii), and (iv). The path arguments are:
- In (i), supplies instrument relevance, and every directed path from to goes through . The unmeasured does not cause , so instrumental-variable independence holds.
- In (ii), supplies instrument relevance. Although is unmeasured, every directed path from to still passes through . The path is blocked at its collider without conditioning, so it does not violate instrumental-variable independence.
- In (iii), bypasses , violating the exclusion restriction. There is also no directed effect of on , so instrument relevance is absent in this graph.
- In (iv), both and affect , and all directed paths to still pass through . The unconditioned collider blocks the route to , so both instrumental-variable independence and the exclusion restriction hold. Conditioning on could introduce collider bias, but is not being conditioned on here.
- In (v), bypasses , so the exclusion restriction fails despite the additional arrow.
- In (vi), bypasses , and affects the unmeasured common cause . Thus the exclusion restriction and the stated instrumental-variable independence requirement fail.
An upstream intermediate variable need not invalidate an instrumental variable if its routes to the outcome all pass through the target exposure; an outcome intermediate reached without that exposure does invalidate it.
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