Homogeneity in the Instrument-exposure Association and Point Estimation Using Binary Instrumental Variables

Interpreting instrumental variable results often requires further assumptions in addition to the core assumptions of relevance, independence, and the exclusion restriction.

We assess whether instrument-exposure additive homogeneity renders the Wald estimand equal to the average derivative effect (ADE) in the case of a binary instrument and a continuous exposure.

Instrument-exposure additive homogeneity is insufficient for ADE identification when the instrument is binary, the exposure is continuous, and the effect of the exposure on the outcome is nonlinear on the additive scale. For a binary exposure, the exposure-outcome effect is necessarily additive linear, so the homogeneity condition is sufficient.

For binary instruments, instrument-exposure additive homogeneity identifies the ADE if the exposure is also binary. Otherwise, additional assumptions (such as additive linearity of the exposure-outcome effect) are required.