LADE-Based Inference for ARMA Models With Unspecified and Heavy-Tailed Heteroscedastic Noises

Abstract

This article develops a systematic procedure of statistical inference for the auto-regressive moving average (ARMA) model with unspecified and heavy-tailed heteroscedastic noises. We first investigate the least absolute deviation estimator (LADE) and the self-weighted LADE for the model. Both estimators are shown to be strongly consistent and asymptotically normal when the noise has a finite variance and infinite variance, respectively. The rates of convergence of the LADE and the self-weighted LADE are n^{− 1/2}, which is faster than those of least-square estimator (LSE) for the ARMA model when the tail index of generalized auto-regressive conditional heteroskedasticity (GARCH) noises is in (0, 4], and thus they are more efficient in this case. Since their asymptotic covariance matrices cannot be estimated directly from the sample, we develop the random weighting approach for statistical inference under this nonstandard case. We further propose a novel sign-based portmanteau test for model adequacy. Simulation study is carried out to assess the performance of our procedure and one real illustrating example is given. Supplementary materials for this article are available online.

• ARMA(p, q) models
• Asymptotic normality
• G/ARCH noises
• Heavy-tailed noises
• LADE
• Random weighting approach
• Self-weighted LADE
• Sign-based portmanteau test
• Strong consistency

Additional information

Notes on contributors

Ke Zhu

Ke Zhu is Assistant Professor, Institute of Applied Mathematics, Chinese Academy of Sciences, Haidian District, Zhongguancun, Beijing, China (E-mail: [email protected]).

Shiqing Ling

Shiqing Ling is Professor, Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (E-mail: [email protected]). The authors greatly appreciate two anonymous referees, the associate editor, and the editors X. He and D. Ruppert for their very constructive suggestions and comments. This work is supported in part by Hong Kong Research Grants Commission Grants HKUST641912 and HKUST603413, National Natural Science Foundation of China (no. 11201459 and 11371354), the President Fund of the Academy of Mathematics and System Science, Chinese Academy of Sciences (grant 2014-cjrwlzx-zk), and Key Laboratory of RCSDS, Chinese Academy of Sciences.