# Homogeneity in the instrument-treatment association is not sufficient for the Wald estimand to equal the average causal effect for a binary instrument and a continuous exposure

arXiv: https://arxiv.org/abs/2107.01070 · HTML: https://mlsanity.com/paper/2107.01070/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jul 2021.

## Built on by (most cited first, 2 of 2)

- [Instrumental Variables in Causal Inference and Machine Learning: A Survey](https://mlsanity.com/paper/2212.05778/) (2022)
- [Confounder Balancing for Instrumental Variable Regression with Latent Variable](https://mlsanity.com/paper/2211.10008/) (2022)

## Abstract (arXiv)

Background: Interpreting instrumental variable results often requires further assumptions in addition to the core assumptions of relevance, independence, and the exclusion restriction. Methods: 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. Results: 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 non-linear on the additive scale. For a binary exposure, the exposure-outcome effect is necessarily additive linear, so the homogeneity condition is sufficient. Conclusions: 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.