An instrument needs three properties: relevance (it genuinely moves treatment, testable via the first-stage F statistic), independence (it is as good as randomly assigned), and the untestable exclusion restriction (no path to the outcome except through treatment). The Wald estimator divides the instrument’s effect on the outcome by its effect on uptake; two-stage least squares generalizes the ratio to continuous instruments and covariates.
The fine print is the estimand: with heterogeneous effects and a monotonicity assumption (the instrument pushes no one away from treatment), IV recovers the local average treatment effect, the average among compliers whose treatment the instrument actually flipped. In industry the most practical instrument is a randomized encouragement, such as an email nudging users toward an optional feature, or an old A/B test that happened to move uptake of the thing you now care about.
