Let $e(x)=\mathbb P(A=1\mid X=x)$ and let $\widehat e$ be a consistent estimate. The <inverse-probability-weighted estimator of the average treatment effect> is
$$
\boxed{
\widehat\tau_{\rm IPW}
=\frac1n\sum_{i=1}^n\left\{
\frac{A_iY_i}{\widehat e(X_i)}-
\frac{(1-A_i)Y_i}{1-\widehat e(X_i)}
\right\}}.
$$
Under the exchangeability, consistency, and positivity conditions in part ii, and consistent estimation of the <propensity score>, its probability limit is
$$
\mathbb E[Y(1)]-\mathbb E[Y(0)],
$$
so it consistently estimates the <average treatment effect>. It does not require the additive outcome-regression model used to interpret the ordinary-least-squares coefficient.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 221 / 4 / b / iii / Solution

{title2=$\widehat{\operatorname{ATE}}_{\rm IPW}$}

For binary treatment, outcomes $Y_i$, and estimated <propensity score> $\widehat e(X_i)$, the inverse-probability-weighted estimator is
$$
\widehat{\operatorname{ATE}}_{\rm IPW}
=\frac1n\sum_{i=1}^n\left\{
\frac{A_iY_i}{\widehat e(X_i)}-
\frac{(1-A_i)Y_i}{1-\widehat e(X_i)}
\right\}.
$$
It is consistent under <conditional exchangeability>, <consistency of potential outcomes>, <positivity in causal inference>, and a consistent propensity-score estimator.

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