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ASYMPTOTIC INFERENCE FOR AR MODELS WITH HEAVY-TAILED G-GARCH NOISES

Published online by Cambridge University Press:  03 November 2014

Rongmao Zhang  and
Shiqing Ling
Rongmao Zhang*
Affiliation:
Zhejiang University
Shiqing Ling
Affiliation:
Hong Kong University of Science and Technology
*
*Address correspondence to Rongmao Zhang, Zhejiang University, Hangzhou, 310027, China; e-mail: rmzhang@zju.edu.cn
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Abstract

It is well known that the least squares estimator (LSE) of an AR(p) model with i.i.d. (independent and identically distributed) noises is n1/αL(n)-consistent when the tail index α of the noise is within (0,2) and is n1/2-consistent when α ≥ 2, where L(n) is a slowly varying function. When the noises are not i.i.d., however, the case is far from clear. This paper studies the LSE of AR(p) models with heavy-tailed G-GARCH(1,1) noises. When the tail index α of G-GARCH is within (0,2), it is shown that the LSE is not a consistent estimator of the parameters, but converges to a ratio of stable vectors. When α ε [2,4], it is shown that the LSE is n1–2/α-consistent if α ε (2,4), logn-consistent if α = 2, and n1/2 / logn-consistent if α = 4, and its limiting distribution is a functional of stable processes. Our results are significantly different from those with i.i.d. noises and should warn practitioners in economics and finance of the implications, including inconsistency, of heavy-tailed errors in the presence of conditional heterogeneity.

Type
MISCELLANEA
Information
Econometric Theory , Volume 31 , Issue 4 , August 2015 , pp. 880 - 890
Copyright
Copyright © Cambridge University Press 2014 

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Footnotes

We thank Ms. Alice Cheng for her editing comments and three referees, the co-editor Giuseppe Cavaliere, and the editor Peter C.B. Phillips for their very helpful and professional comments. Zhang’s research was supported by NSFC grants 11371318 and 11171074, the Fundamental Research Funds for the Central Universities, and Scientific Research Fund of Zhejiang Provincial Education Department (Y201009944). Ling’s research was supported by the Hong Kong Research Grants Council (Grants HKUST641912, 603413 and FSGRF12SC12).

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