# Average causal effect estimation via instrumental variables: the no simultaneous heterogeneity assumption

arXiv: https://arxiv.org/abs/2010.10017 · HTML: https://mlsanity.com/paper/2010.10017/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Oct 2020.

## Built on by (most cited first, 3 of 3)

- [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)
- [A Partial Initialization Strategy to Mitigate the Overfitting Problem in CATE Estimation with Hidden Confounding](https://mlsanity.com/paper/2501.08888/) (2025)

## Abstract (arXiv)

Background: Instrumental variables (IVs) can be used to provide evidence as to whether a treatment X has a causal effect on an outcome Y. Even if the instrument Z satisfies the three core IV assumptions of relevance, independence and the exclusion restriction, further assumptions are required to identify the average causal effect (ACE) of X on Y. Sufficient assumptions for this include: homogeneity in the causal effect of X on Y; homogeneity in the association of Z with X; and no effect modification (NEM). Methods: We describe the NO Simultaneous Heterogeneity (NOSH) assumption, which requires the heterogeneity in the X-Y causal effect to be mean independent of (i.e., uncorrelated with) both Z and heterogeneity in the Z-X association. This happens, for example, if there are no common modifiers of the X-Y effect and the Z-X association, and the X-Y effect is additive linear. We illustrate NOSH using simulations and by re-examining selected published studies. Results: When NOSH holds, the Wald estimand equals the ACE even if both homogeneity assumptions and NEM (which we demonstrate to be special cases of - and therefore stronger than - NOSH) are violated. Conclusions: NOSH is sufficient for identifying the ACE using IVs. Since NOSH is weaker than existing assumptions for ACE identification, doing so may be more plausible than previously anticipated.