Let
$$
\delta(u)=\mathbb E[A\mid Z=1,U=u]
-\mathbb E[A\mid Z=0,U=u]
$$
and
$$
\tau(u)=\mathbb E[Y(1)-Y(0)\mid U=u].
$$
Because $U$ contains all exposure-outcome confounding and $Z$ is independent of $U$, the exclusion restriction and the law of total expectation give
$$
\mathbb E[Y\mid Z=1]-\mathbb E[Y\mid Z=0]
=\mathbb E\{\tau(U)\delta(U)\}.
$$
Likewise,
$$
\mathbb E[A\mid Z=1]-\mathbb E[A\mid Z=0]
=\mathbb E\{\delta(U)\}.
$$
The no <confounder-instrument interaction> assumption says $\delta(U)=\delta$ is constant. Relevance gives $\delta\ne0$, so the Wald ratio is
$$
\frac{\delta\,\mathbb E\tau(U)}{\delta}
=\mathbb E[Y(1)-Y(0)],
$$
the overall <average treatment effect>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 221 / 4 / iii / Solution

A confounder-instrument interaction occurs when an instrument's effect on treatment varies with a variable that also affects the outcome. If the conditional first-stage effect is $\delta(U)=\mathbb E[A\mid Z=1,U]-\mathbb E[A\mid Z=0,U]$, absence of this interaction means that $\delta(U)$ is constant.


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 Incoming links: Confounder-instrument interaction