Abstract

Heavy tails and stability are two persistent challenges in modelling

financial time series, yet most existing approaches rely on finite-moment assumptions and pay insufficient attention to stability issues. To bridge this gap, we

propose an asymmetric GARCH model with standardized non-Gaussian stable

innovations (sAGARCH), which accommodates infinite variance and even infinite

mean. We establish a comprehensive inference framework for both stationary and

explosive cases, proving the strong consistency and asymptotic normality of the

maximum likelihood estimator, including the tail index parameter. We also discuss multiple estimators for the asymptotic variance. Additionally, we propose a

modified Kolmogorov-type test statistic for diagnostic checking, along with tests

for strict stationarity and asymmetry. Through Monte Carlo simulations with

heavy-tailed innovations, we provide further insight into the finite-sample performance of the intercept estimator. Empirical applications to stock returns further

highlight the usefulness and merits of the proposed sAGARCH model.

Key words and phrases: Heavy tails, Kolmogorov-type test, Maximum likelihood estimation, Nonstationarity, Stable distribution

Information

Preprint No.SS-2025-0271
Manuscript IDSS-2025-0271
Complete AuthorsYuxin Tao, Huan Gong, Dong Li
Corresponding AuthorsHuan Gong
Emailsgonghuan@nudt.edu.cn

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Acknowledgments

The authors are very grateful to the co-editor, the associate editor, and

two anonymous referees for their constructive suggestions and comments,

which led to significant improvement. We would like to thank Professor

Howell Tong for his insightful suggestions. Tao’s research is supported by

the NSFC (No.72503086) and the Natural Science Foundation of Guangdong Province (No.2025A1515010697). Gong’s research is supported by the

NSFC (No.72501294) and the Innovation Research Foundation of NUDT

(No.ZK25-61). Li’s research is supported by the NSFC (No.72471127).

Supplementary Materials

The Supplementary Material contains additional simulation results, empirical analyses on portfolio returns, and proofs of all theorems.

Supplementary materials are available for download.