By <consistency of potential outcomes> and <conditional exchangeability>,
$$
\mathbb E[Y(z)\mid X]
=\mathbb E[Y(z)\mid Z=z,X]
=\mathbb E[Y\mid Z=z,X].
$$
Taking conditional expectations given $X_1=x_1$ and subtracting the cases $z=1$ and $z=0$ proves the outcome-regression identification formula.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-221/4/a/i/solution