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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-207/3/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 207 / 3 / c / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

Under the displayed causal directed acyclic graphs, with nonzero effects along the relevant arrows and no exact cancellation of the effects on

B

, the valid instrumental variables are scenarios (i), (ii), and (iv). The path arguments are:

In (i), $G \to B$ supplies instrument relevance, and every directed path from $G$ to $Y$ goes through $B$ . The unmeasured $U$ does not cause $G$ , so instrumental-variable independence holds.
In (ii), $G \to A \to B$ supplies instrument relevance. Although $A$ is unmeasured, every directed path from $G$ to $Y$ still passes through $B$ . The path $G \to A \leftarrow U \to Y$ is blocked at its collider $A$ without conditioning, so it does not violate instrumental-variable independence.
In (iii), $G \to C \to Y$ bypasses $B$ , violating the exclusion restriction. There is also no directed effect of $G$ on $B$ , so instrument relevance is absent in this graph.
In (iv), both $G \to B$ and $G \to A \to B$ affect $B$ , and all directed paths to $Y$ still pass through $B$ . The unconditioned collider $G \to A \leftarrow U$ blocks the route to $U$ , so both instrumental-variable independence and the exclusion restriction hold. Conditioning on $A$ could introduce collider bias, but $A$ is not being conditioned on here.
In (v), $G \to C \to Y$ bypasses $B$ , so the exclusion restriction fails despite the additional $G \to B$ arrow.
In (vi), $G \to U \to Y$ bypasses $B$ , and $G$ affects the unmeasured common cause $U$ . 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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