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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.

LetandBecause $U$ contains all exposure-outcome confounding and $Z$ is independent of $U$, the exclusion restriction and the law of total expectation giveLikewise,The no confounder-instrument interaction assumption says $\delta (U)=\delta$ is constant. Relevance gives $\delta \ne 0$, so the Wald ratio isthe overall average treatment effect.