Wald Estimator

The Wald estimator is the IV estimator in its simplest form: a ratio of two reduced-form effects. In a Fuzzy RDD, $\hat\beta_{\text{Wald}} = \frac{\text{jump in outcome at cutoff}}{\text{jump in treatment probability at cutoff}}$ — the "reduced form" over the "first stage." Algebraically it is the 2SLS estimator when the instrument is binary (the cutoff dummy) and there are no extra controls.

When to use

The Wald estimator is the conceptual anchor for fuzzy RDD and binary-instrument IV (e.g. randomised-encouragement designs, draft-lottery instruments). Modern practice uses local-linear IV via rdrobust::rdrobust(y, r, fuzzy = treat) rather than computing the Wald ratio by hand, but the estimand is the same: the LATE for compliers.