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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-221/3/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 221 / 3 / i / Solution by

Codex  0 Created 2026-09-28 Updated 2026-10-03

The no unmeasured confounding assumption, or conditional exchangeability, is

(Y (0), Y (1)) ⊥ A ∣ X .

(1)

Also assume consistency of potential outcomes, no interference in causal inference, positivity in causal inference, and finite expectations. For

a \in {0, 1}

, the law of total expectation, exchangeability, and consistency give

E [Y (a)] = E {E [Y (a) ∣ X]} = E {E [Y (a) ∣ A = a, X]} = E {E [Y ∣ A = a, X]} .

(2)

Positivity ensures that the observed conditional means exist on the covariate support being averaged. Subtracting the two cases identifies the average treatment effect as

β_{1} = E {E [Y ∣ A = 1, X]} - E {E [Y ∣ A = 0, X]} .

(3)

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