# Goodness of fit tests for high-dimensional linear models

arXiv: https://arxiv.org/abs/1511.03334 · HTML: https://mlsanity.com/paper/1511.03334/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Nov 2015.

## Built on by (most cited first, 5 of 5)

- [A Unified View of Optimal Kernel Hypothesis Testing](https://mlsanity.com/paper/2503.07084/) (2025)
- [A Simple Unified Approach to Testing High-Dimensional Conditional Independences for Categorical and Ordinal Data](https://mlsanity.com/paper/2206.04356/) (2022)
- [GRASP: A Goodness-of-Fit Test for Classification Learning](https://mlsanity.com/paper/2209.02064/) (2022)
- [The Debiased Score Test: Hunt-and-test for Semiparametric Hypotheses](https://mlsanity.com/paper/2607.28861/) (2026)
- [SPARK: A General Goodness-of-Fit Assessment via Residual Projection](https://mlsanity.com/paper/2609.37705/) (2026)

## Abstract (arXiv)

In this work we propose a framework for constructing goodness of fit tests in both low and high-dimensional linear models. We advocate applying regression methods to the scaled residuals following either an ordinary least squares or Lasso fit to the data, and using some proxy for prediction error as the final test statistic. We call this family Residual Prediction (RP) tests. We show that simulation can be used to obtain the critical values for such tests in the low-dimensional setting, and demonstrate using both theoretical results and extensive numerical studies that some form of the parametric bootstrap can do the same when the high-dimensional linear model is under consideration. We show that RP tests can be used to test for significance of groups or individual variables as special cases, and here they compare favourably with state of the art methods, but we also argue that they can be designed to test for as diverse model misspecifications as heteroscedasticity and nonlinearity.