# Testing for time-varying properties under misspecified conditional mean and variance

## TL;DR

This paper investigates the performance of statistical tests for time-varying properties under misspecified models, highlighting the effectiveness of bootstrap methods for mean tests but limitations for variance tests when models are misspecified.

## Contribution

It demonstrates that bootstrap methods improve size accuracy for mean property tests under misspecification, but not for variance tests, revealing robustness differences.

## Key findings

- Bootstrap improves size accuracy for mean tests under misspecification.
- Variance tests suffer from size distortions even with bootstrap.
- Mean property tests are robust to variance misspecification.

## Abstract

This study examines statistical performance of tests for time-varying properties under misspecified conditional mean and variance. When we test for time-varying properties of the conditional mean in the case in which data have no time-varying mean but have time-varying variance, asymptotic tests have size distortions. This is improved by the use of a bootstrap method. Similarly, when we test for time-varying properties of the conditional variance in the case in which data have time-varying mean but no time-varying variance, asymptotic tests have large size distortions. This is not improved even by the use of bootstrap methods. We show that tests for time-varying properties of the conditional mean by the bootstrap are robust regardless of the time-varying variance model, whereas tests for time-varying properties of the conditional variance do not perform well in the presence of misspecified time-varying mean.