 Solution

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

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

G

supplies variation in an exposure

B

that can be used for causal inference about its effect on

Y

. Its three core requirements are:

Instrument relevance: $G$ changes the distribution of $B$ ; in a simple linear model this requires a nonzero first-stage association.
Instrumental-variable independence: $G$ is independent of unmeasured common causes of $B$ and $Y$ , or independent of the relevant potential outcomes conditional on specified measured covariates.
Exclusion restriction: interventions on $G$ affect $Y$ only through $B$ , with no other causal pathway. In potential outcomes notation, $Y (b, g) = Y (b)$ .

Relevance, independence, and exclusion are the core instrumental-variable assumptions. They do not by themselves identify an arbitrary population-average effect. For a binary instrument and treatment, instrumental-variable monotonicity and the usual consistency in causal inference assumptions can identify a local average treatment effect; stronger effect-homogeneity or structural-model assumptions are needed for other effect interpretations.

 New to topics? Read the docs here!