Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 221 / 4 / a / i

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- Solution i

Solution

By consistency of potential outcomes and conditional exchangeability,

E [Y (z) ∣ X] = E [Y (z) ∣ Z = z, X] = E [Y ∣ Z = z, X] .

Taking conditional expectations given

X_{1} = x_{1}

and subtracting the cases

z = 1

and

z = 0

proves the outcome-regression identification formula.

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