Specification Tests for GARCH Processes with Nuisance Parameters on the Boundary

Abstract

This article develops tests for the correct specification of the conditional variance function in GARCH models when the true parameter may lie on the boundary of the parameter space. The test statistics considered are of Kolmogorov-Smirnov and Cramér-von Mises type, and are based on empirical processes marked by centered squared residuals. The limiting distributions of the test statistics depend on unknown nuisance parameters in a nontrivial way, making the tests difficult to implement. We therefore introduce a novel bootstrap procedure which is shown to be asymptotically valid under general conditions, irrespective of the presence of nuisance parameters on the boundary. The proposed bootstrap approach is based on shrinking of the parameter estimates used to generate the bootstrap sample toward the boundary of the parameter space at a proper rate. It is simple to implement and fast in applications, as the associated test statistics have simple closed form expressions. Although the bootstrap test is designed for a data generating process with fixed parameters (i.e., independent of the sample size n), we also discuss how to obtain valid inference for sequences of DGPs with parameters approaching the boundary at the $n^{-1/2}$ rate. A simulation study demonstrates that the new tests: (i) have excellent finite sample behavior in terms of empirical rejection probabilities under the null as well as under the alternative; (ii) provide a useful complement to existing procedures based on Ljung-Box type approaches. Two data examples illustrate the implementation of the proposed tests in applications.

Keywords:

• Bootstrap
• Cramér-von Mises
• Kolmogorov-Smirnov
• Marked empirical process

Supplementary Materials

The supplementary material contains some additional simulation results, the proofs of the main results stated in the article, as well as some auxiliary lemmas.

Disclosure Statement

The authors report there are no competing interests to declare.

Acknowledgments

We thank the Coeditor and two anonymous referees for their important comments which improved the quality of the article. We also thank conference participants at the CFE 2022 conference at King’s College London.

Additional information

Funding

This research was supported by the Danish Council for Independent Research (DSF grant 7015-00028B) and by the Italian Ministry of University and Research (PRIN 2020 grant 2020B2AKFW). Also Rahbek gratefully acknowledges financial support from the Independent Research Fund Denmark (grant no. 0133-00162B).