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

TL;DR

This paper proposes a simple, general diagnostic test based on higher-order least squares to assess the fit of linear causal models, effectively identifying confounded covariates even in high-dimensional settings.

Contribution

It introduces a novel higher-order least squares method for partial goodness of fit testing in linear causal models, capable of handling high-dimensional data and confounding.

Findings

The test effectively distinguishes confounded from non-confounded covariates.

The method is valid for high-dimensional datasets.

It provides a practical tool for causal model diagnostics.

Abstract

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.