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TEST FOR ZERO MEDIAN OF ERRORS IN AN ARMA–GARCH MODEL

Published online by Cambridge University Press:  09 June 2021

Yaolan Ma,
Mo Zhou,
Liang Peng and
Rongmao Zhang
Yaolan Ma
Affiliation:
North Minzu University
Mo Zhou
Affiliation:
Zhejiang University
Liang Peng
Affiliation:
Georgia State University
Rongmao Zhang*
Affiliation:
Zhejiang University Minnan Normal University
*
Corresponding address: Zhejiang University and Minnan Normal University, China; e-mail: rmzhang@zju.edu.cn.
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Abstract

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Because the ARMA–GARCH model can generate data with some important properties such as skewness, heavy tails, and volatility persistence, it has become a benchmark model in analyzing financial and economic data. The commonly employed quasi maximum likelihood estimation (QMLE) requires a finite fourth moment for both errors and the sequence itself to ensure a normal limit. The self-weighted quasi maximum exponential likelihood estimation (SWQMELE) reduces the moment constraints by assuming that the errors and their absolute values have median zero and mean one, respectively. Therefore, it is necessary to test zero median of errors before applying the SWQMELE, as changing zero mean to zero median destroys the ARMA–GARCH structure. This paper develops an efficient empirical likelihood test without estimating the GARCH model but using the GARCH structure to reduce the moment effect. A simulation study confirms the effectiveness of the proposed test. The data analysis shows that some financial returns do not have zero median of errors, which cautions the use of the SWQMELE.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 38 , Issue 3 , June 2022, pp. 536 - 561
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

We thank three reviewers for helpful comments, which improved the presentation. Peng’s research was partly supported by the Simons Foundation and the NSF grant of DMS-2012448. Ma’s research was partly supported by the First-Class Disciplines Foundation of Ningxia (No. NXYLXK2017B09), the Natural Science Foundation of Ningxia (No. 2019AAC03130), NSSFC (No. 20BTJ026), and General Scientific Research Project of North Minzu University (No. 2018XYZSX06). Zhang’s research was partly supported by NSFC (No. 11771390/11371318), ZPNSFC (No. LZ21A010002), Ten Thousand Talents Plan of Zhejiang Province (No. 2018R52042), and the Fundamental Research Funds for the Central Universities.

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