Limit theory of quadratic forms of long-memory linear processes with heavy-tailed GARCH innovations

Ngai Hang Chan; Rong-Mao Zhang

doi:10.1016/j.jmva.2013.04.007

Limit theory of quadratic forms of long-memory linear processes with heavy-tailed GARCH innovations

Ngai Hang Chan, Rong-Mao Zhang

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

1 Citation (Scopus)

Abstract

Let X_t=∑_j=0^∞c_jε_t-j be a moving average process with GARCH (1, 1) innovations {_εt} In this paper, the asymptotic behavior of the quadratic form Q_n=∑j=1n∑s=1nb(t-s)XtXs is derived when the innovation {_εt} is a long-memory and heavy-tailed process with tail index α, where {b (i) } is a sequence of constants. In particular, it is shown that when 1 < α < 4 and under certain regularity conditions, the limit distribution of _{Q n} converges to a stable random variable with index α / 2. However, when α ≥ 4, _{Q n} has an asymptotic normal distribution. These results not only shed light on the singular behavior of the quadratic forms when both long-memory and heavy-tailed properties are present, but also have applications in the inference for general linear processes driven by heavy-tailed GARCH innovations. © 2013 Elsevier Inc.

Original language	English
Pages (from-to)	18-33
Journal	Journal of Multivariate Analysis
Volume	120
DOIs	https://doi.org/10.1016/j.jmva.2013.04.007
Publication status	Published - Sept 2013
Externally published	Yes

Bibliographical note

Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected].

Research Keywords

GARCH
Heavy-tailed
Linear process
Long-memory
Quadratic forms

Access Output

10.1016/j.jmva.2013.04.007Licence: Elsevier user license

Other Access Options

Check CityUHK Library

Permanent Link

Right click to copy link

Cite this

@article{f04c3640275146829d37555ca9997421,

title = "Limit theory of quadratic forms of long-memory linear processes with heavy-tailed GARCH innovations",

abstract = "Let Xt=∑j=0∞cjεt-j be a moving average process with GARCH (1, 1) innovations {εt} In this paper, the asymptotic behavior of the quadratic form Qn=∑j=1n∑s=1nb(t-s)XtXs is derived when the innovation {εt} is a long-memory and heavy-tailed process with tail index α, where {b (i) } is a sequence of constants. In particular, it is shown that when 1 < α < 4 and under certain regularity conditions, the limit distribution of Q n converges to a stable random variable with index α / 2. However, when α ≥ 4, Q n has an asymptotic normal distribution. These results not only shed light on the singular behavior of the quadratic forms when both long-memory and heavy-tailed properties are present, but also have applications in the inference for general linear processes driven by heavy-tailed GARCH innovations. {\textcopyright} 2013 Elsevier Inc.",

keywords = "GARCH, Heavy-tailed, Linear process, Long-memory, Quadratic forms",

author = "Chan, {Ngai Hang} and Rong-Mao Zhang",

note = "Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected]. ",

year = "2013",

month = sep,

doi = "10.1016/j.jmva.2013.04.007",

language = "English",

volume = "120",

pages = "18--33",

journal = "Journal of Multivariate Analysis",

issn = "0047-259X",

publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Limit theory of quadratic forms of long-memory linear processes with heavy-tailed GARCH innovations

AU - Chan, Ngai Hang

AU - Zhang, Rong-Mao

N1 - Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected].

PY - 2013/9

Y1 - 2013/9

N2 - Let Xt=∑j=0∞cjεt-j be a moving average process with GARCH (1, 1) innovations {εt} In this paper, the asymptotic behavior of the quadratic form Qn=∑j=1n∑s=1nb(t-s)XtXs is derived when the innovation {εt} is a long-memory and heavy-tailed process with tail index α, where {b (i) } is a sequence of constants. In particular, it is shown that when 1 < α < 4 and under certain regularity conditions, the limit distribution of Q n converges to a stable random variable with index α / 2. However, when α ≥ 4, Q n has an asymptotic normal distribution. These results not only shed light on the singular behavior of the quadratic forms when both long-memory and heavy-tailed properties are present, but also have applications in the inference for general linear processes driven by heavy-tailed GARCH innovations. © 2013 Elsevier Inc.

AB - Let Xt=∑j=0∞cjεt-j be a moving average process with GARCH (1, 1) innovations {εt} In this paper, the asymptotic behavior of the quadratic form Qn=∑j=1n∑s=1nb(t-s)XtXs is derived when the innovation {εt} is a long-memory and heavy-tailed process with tail index α, where {b (i) } is a sequence of constants. In particular, it is shown that when 1 < α < 4 and under certain regularity conditions, the limit distribution of Q n converges to a stable random variable with index α / 2. However, when α ≥ 4, Q n has an asymptotic normal distribution. These results not only shed light on the singular behavior of the quadratic forms when both long-memory and heavy-tailed properties are present, but also have applications in the inference for general linear processes driven by heavy-tailed GARCH innovations. © 2013 Elsevier Inc.

KW - GARCH

KW - Heavy-tailed

KW - Linear process

KW - Long-memory

KW - Quadratic forms

UR - http://www.scopus.com/inward/record.url?scp=84878556820&partnerID=8YFLogxK

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84878556820&origin=recordpage

U2 - 10.1016/j.jmva.2013.04.007

DO - 10.1016/j.jmva.2013.04.007

M3 - RGC 21 - Publication in refereed journal

SN - 0047-259X

VL - 120

SP - 18

EP - 33

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

ER -

Limit theory of quadratic forms of long-memory linear processes with heavy-tailed GARCH innovations

Abstract

Bibliographical note

Research Keywords

Access Output

Other Access Options

Permanent Link

Other Files and Links

Fingerprint

Cite this