Extremal Dependence-Based Specification Testing of Time Series

Abstract

We propose a specification test for conditional location–scale models based on extremal dependence properties of the standardized residuals. We do so comparing the left-over serial extremal dependence—as measured by the pre-asymptotic tail copula—with that arising under serial independence at different lags. Our main theoretical results show that the proposed Portmanteau-type test statistics have nuisance parameter-free asymptotic limits. The test statistics are easy to compute, as they only depend on the standardized residuals, and critical values are likewise easily obtained from the limiting distributions. This contrasts with some extant tests (based, e.g., on autocorrelations of squared residuals), where test statistics depend on the parameter estimator of the model and critical values may need to be bootstrapped. We show that our tests perform well in simulations. An empirical application to S&P 500 constituents illustrates that our tests can uncover violations of residual serial independence that are not picked up by standard autocorrelation-based specification tests, yet are relevant when the model is used for, for example, risk forecasting.

Keywords:

• Location–scale models
• Serial extremal dependence
• Specification test
• Tail copula
• Time series

Supplementary Materials

The appendix contains the verification of Assumptions 3 and 4 for APARCH models (Appendix A), the proofs of Theorems 1–2 (Appendices B and C), the proof of Theorem 3 (Appendix D), additional simulations (Appendix E), and theoretical results along with simulations for a test with an automatic choice of D (Appendix F). Furthermore, the supplementary material contains the R code to reproduce the simulation study and the empirical application.

Disclosure Statement

The author reports there are no competing interests to declare.

Acknowledgments

The author is indebted to the Editor Christian Hansen, the Associate Editor, and three anonymous referees for their detailed comments, that significantly improved the quality of the article. The author is also grateful to seminar participants at CREST and Erasmus University Rotterdam for valuable suggestions, in particular Christian Francq, Jeroen Rombouts, Jean-Michel Zakoïan and Chen Zhou. Finally, the author would like to thank Christoph Hanck and Till Massing for their careful reading of an earlier version of this manuscript.

Notes

1 The data-generating process is taken from Francq and Thieu (2019, sec. 3.1), who obtained the parameters in (3) from fitting an APARCH–X model to Boeing returns.

2 The second-to-top panel of Figure 6 in Appendix E.2 of the supplementary materials shows a representative trajectory of z_t .

Additional information

Funding

This work was supported by the German Research Foundation (DFG) under grant HO 6305/1-1.