For fixed $\beta$, conditional least squares is minimized by
$$
g_\beta(X)=\mathbb E[Y-\beta A\mid X]
=\mu(X)-\beta e(X).
$$
Therefore the remaining objective is
$$
\mathbb E\left[
\{Y-\mu(X)-\beta(A-e(X))\}^2
\right],
$$
whose <normal equation> gives
$$
\beta_3
=\frac{\mathbb E[(A-e(X))(Y-\mu(X))]}
{\mathbb E[(A-e(X))^2]}.
$$
Let
$$
\tau(X)=\mathbb E[Y(1)-Y(0)\mid X]
$$
be the <conditional average treatment effect>. Since $A$ is binary, conditional exchangeability implies
$$
\operatorname{Cov}(A,Y\mid X)
=e(X)\{1-e(X)\}\tau(X),
$$
and $\operatorname{Var}(A\mid X)=e(X)\{1-e(X)\}$. Hence
$$
\beta_3
=\frac{\mathbb E[e(X)\{1-e(X)\}\tau(X)]}
{\mathbb E[e(X)\{1-e(X)\}]}.
$$
Thus $\beta_3$ is the <overlap-weighted average treatment effect>. It weights covariate strata by the <overlap weight> and generally differs from the ordinary ATE $\beta_1=\mathbb E[\tau(X)]$ when treatment effects are heterogeneous and overlap varies with $X$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 221 / 3 / iii / Solution

The overlap-weighted average treatment effect averages the <conditional average treatment effect> with weight proportional to $e(X)\{1-e(X)\}$, where $e(X)$ is the <propensity score>. It emphasizes covariate strata in which both treatments are plausible.

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 Incoming links: Overlap-weighted average treatment effect