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

The no unmeasured confounding assumption is the conditional exchangeability statementAlso assume consistency of potential outcomes, no interference between units, and positivity in causal inference, in particular $\mathbb{P}(A=0\mid X)>0$ on the covariate support of treated units. ThenThe first treated potential outcome equals the observed treated mean by consistency. Moreover,Subtracting proves the displayed identification formula for the average treatment effect on the treated.

For discrete $X$,Perturbing the marginal mass $p(x)$ contributes $\pi (X)\mu (X)-\beta$. Perturbing $\pi (x)=\mathbb{E}[A\mid X=x]$ contributesTheir sum is $A\mu (X)-\beta$. Finally, the influence function ofisMultiplication by the derivative $p(x)\pi (x)$ of $\beta$ with respect to $\mu (x)$ and summation over $x$ givesAdding the three contributions yields the claimed mean-zero influence curve

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.