# Higher-order least squares: assessing partial goodness of fit of linear causal models

arXiv: https://arxiv.org/abs/2109.14544 · HTML: https://mlsanity.com/paper/2109.14544/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2021.

## Built on by (most cited first, 4 of 4)

- [Assessing the overall and partial causal well-specification of nonlinear additive noise models](https://mlsanity.com/paper/2310.16502/) (2023)
- [Detecting Unobserved Confounders: A Kernelized Regression Approach](https://mlsanity.com/paper/2601.00200/) (2026)
- [Causal discovery under mean independence and linearity](https://mlsanity.com/paper/2605.04381/) (2026)
- [Linear Regression Beyond Covariance: Latent Slope Spectra from Higher-Order Cumulants](https://mlsanity.com/paper/2610.17772/) (2026)

## Abstract (arXiv)

We introduce a simple diagnostic test for assessing the overall or partial goodness of fit of a linear causal model with errors being independent of the covariates. In particular, we consider situations where hidden confounding is potentially present. We develop a method and discuss its capability to distinguish between covariates that are confounded with the response by latent variables and those that are not. Thus, we provide a test and methodology for partial goodness of fit. The test is based on comparing a novel higher-order least squares principle with ordinary least squares. In spite of its simplicity, the proposed method is extremely general and is also proven to be valid for high-dimensional settings.