Estimate the <propensity score> $\pi$ and the untreated outcome regression $\mu$, preferably with <cross-fitting> when flexible methods are used, and set
$$
\widehat\beta
=\frac1n\sum_{i=1}^n\left[
(1-A_i)\frac{\widehat\pi(X_i)}{1-\widehat\pi(X_i)}
\{Y_i-\widehat\mu(X_i)\}
+A_i\widehat\mu(X_i)
\right].
$$
With $n_1=\sum_iA_i$, the resulting <one-step estimator> of the ATT is
$$
\widehat\tau_{\mathrm{ATT}}
=\frac1{n_1}\sum_iA_iY_i-\frac{\widehat\beta}{n_1/n}.
$$
Equivalently,
$$
\widehat\tau_{\mathrm{ATT}}
=\frac1{n_1}\sum_i\left[
A_i\{Y_i-\widehat\mu(X_i)\}
-(1-A_i)\frac{\widehat\pi(X_i)}{1-\widehat\pi(X_i)}
\{Y_i-\widehat\mu(X_i)\}
\right].
$$
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.

Solution

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

A one-step estimator starts from a plug-in estimate and adds the empirical mean of an estimated influence function. Under suitable nuisance-estimation and remainder conditions, this correction removes the plug-in estimator's first-order bias and produces an asymptotically linear estimator.


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