Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 221 / 2 / iii / Solution

Estimate the propensity score $\pi$ and the untreated outcome regression $\mu$, preferably with cross-fitting when flexible methods are used, and setWith $n_1={\sum }_i{A}_i$, the resulting one-step estimator of the ATT isEquivalently,This is an augmented inverse-probability-weighted estimator; it is consistent when either the propensity model or the untreated outcome model is correct, subject to the usual regularity and positivity conditions.