Goodness of fit tests for high-dimensional linear models

TL;DR

This paper introduces Residual Prediction (RP) tests for assessing goodness of fit in linear models, applicable in both low and high-dimensional settings, using residuals and prediction error proxies.

Contribution

The paper proposes a novel RP testing framework that leverages regression residuals and prediction error proxies, with methods for critical value estimation in high dimensions.

Findings

RP tests perform well in simulations for low and high-dimensional models.

RP tests compare favorably with existing methods for variable significance.

RP tests can detect various model misspecifications like heteroscedasticity and nonlinearity.

Abstract

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.