Abstract
Estimating the orders of the autoregressive fractionally integrated moving average (ARFIMA) model has been a long-standing problem in time series analysis. This paper tackles this challenge by establishing the consistency of the Bayesian information criterion (BIC) for ARFIMA models with independent errors. Since the memory parameter of the model can be any real number, this consistency result is valid for short memory, long memory and nonstationary time series. This paper further extends the consistency of the BIC to ARFIMA models with conditional heteroscedastic errors, thereby extending its applications to encompass many real-life situations. Finite-sample implications of the theoretical results are illustrated via numerical examples.
Funding Statement
Huang and Ing’s research was supported by grant 109-2118-M-007-007-MY3 from the Ministry of Science and Technology, Taiwan.
Chan’s research was supported, in part, by the General Research Fund of HKSAR-RGC-GRF Nos. 14308218 and 14307921, HKSAR-RGCCRF:CityU8/CRG/12G, and the Theme-based Research Scheme of HKSAR-RGC-TBS T32-101/15-R.
Chen’s research was supported by the National Natural Science Foundation of China under Contract No. 12001444, and the MOE (Ministry of Education in China) Project of Humanities and Social Sciences (20YJC910001).
Acknowledgments
We would like to thank the Editor, an Associate Editor and an anonymous referee for their critical comments and thoughtful suggestions, which led to an improved version of this paper. Further, we would like to thank Professor Ruey S. Tsay for his insightful comments and encouragement on an earlier draft of this paper.
Citation
Hsueh-Han Huang. Ngai Hang Chan. Kun Chen. Ching-Kang Ing. "Consistent order selection for ARFIMA processes." Ann. Statist. 50 (3) 1297 - 1319, June 2022. https://doi.org/10.1214/21-AOS2149
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