Testing exogeneity in the functional linear regression model

TL;DR

This paper introduces a new bootstrap-based test for exogeneity in functional linear regression models, addressing limitations of traditional Hausman-type tests and demonstrating improved robustness and performance through simulations.

Contribution

It develops a novel test statistic for exogeneity in functional linear regression, with proven asymptotic properties and bootstrap consistency, enhancing robustness over existing methods.

Findings

Bootstrap methods outperform traditional tests in finite samples.

The proposed test is robust to regularization parameter choices.

Simulations confirm the test's effectiveness and consistency.

Abstract

We propose a novel test statistic for testing exogeneity in the functional linear regression model. In contrast to Hausman-type tests in finite dimensional linear regression setups, a direct extension to the functional linear regression model is not possible. Instead, we propose a test statistic based on the sum of the squared difference of projections of the two estimators for testing the null hypothesis of exogeneity in the functional linear regression model. We derive asymptotic normality under the null and consistency under general alternatives. Moreover, we prove bootstrap consistency results for residual-based bootstraps. In simulations, we investigate the finite sample performance of the proposed testing approach and illustrate the superiority of bootstrap-based approaches. In particular, the bootstrap approaches turn out to be much more robust with respect to the choice of the regularization parameter.