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 Incoming links: Average treatment effect on the treated

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 221 / 2 / i / Solution 2026-09-28

The no unmeasured confounding assumption is the conditional exchangeability statement

(Y (0), Y (1)) ⊥ A ∣ X .

Also assume consistency of potential outcomes, no interference between units, and positivity in causal inference, in particular

P (A = 0 ∣ X) > 0

on the covariate support of treated units. Then

E [Y (0) ∣ A = 1] = E {E [Y (0) ∣ X, A = 1] ∣ A = 1} = E {E [Y (0) ∣ X, A = 0] ∣ A = 1} = E {μ (X) ∣ A = 1} .

The first treated potential outcome equals the observed treated mean by consistency. Moreover,

E {μ (X) ∣ A = 1} = \frac{E [ A μ ( X )]}{P ( A = 1 )} = \frac{E [ π ( X ) μ ( X )]}{E [ π ( X )]} .

Subtracting proves the displayed identification formula for the average treatment effect on the treated.

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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 207 / 1 / g / Solution Created 2026-09-24 Updated 2026-09-25

The average treatment effect is more relevant than the average treatment effect on the treated because the contemplated policy changes treatment status for schools that are presently non-UTC, not merely the 43 schools that selected into existing UTC status. More precisely, the ideal target would be the average effect among the candidate schools the minister might convert; if that population is represented by all eligible schools, it is the ATE. An ATT from existing UTCs answers a narrower historical question and may not transport to the policy targets.

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