A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model

Zudi Lu

doi:10.1016/S0167-7152(95)00233-2

A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model

Zudi Lu

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

16 Citations (Scopus)

Abstract

For the pth-order linear ARCH model, X_t = ε_t√α₀ + α₁X²_t-1 + α₂X²_t-2 + ··· + α_pX²_t-p, where α₀ > 0, α_i ≥ 0, i = 1, 2, ..., p, {ε_t is an i.i.d. normal white noise with Eε_t = 0, Eε²_t = 1, and ε_t, is independent of {X_s, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, α₁ + α₂ + ··· + α_p < 1. In this note, we assume that ε_t has the probability density function p(t) which is positive and lower-semicontinuous over the real line, but not necessarily Gaussian, then the geometric ergodicity of the ARCH(p) process is proved under Eε²_t = 1. When ε_t has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.

Original language	English
Pages (from-to)	305-311
Journal	Statistics and Probability Letters
Volume	30
Issue number	4
DOIs	https://doi.org/10.1016/S0167-7152(95)00233-2
Publication status	Published - 15 Nov 1996
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

Arch model
Conditional heteroscedasticity
Geometric ergodicity
Markov process
Nonlinear time series

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title = "A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model",

abstract = "For the pth-order linear ARCH model, Xt = εt√α0 + α1X2t-1 + α2X2t-2 + ··· + αpX2t-p, where α0 > 0, αi ≥ 0, i = 1, 2, ..., p, \{εt is an i.i.d. normal white noise with Eεt = 0, Eε2t = 1, and εt, is independent of \{Xs, s < t\}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, α1 + α2 + ··· + αp < 1. In this note, we assume that εt has the probability density function p(t) which is positive and lower-semicontinuous over the real line, but not necessarily Gaussian, then the geometric ergodicity of the ARCH(p) process is proved under Eε2t = 1. When εt has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.",

keywords = "Arch model, Conditional heteroscedasticity, Geometric ergodicity, Markov process, Nonlinear time series",

author = "Zudi Lu",

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 = "1996",

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language = "English",

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T1 - A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model

AU - Lu, Zudi

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 - 1996/11/15

Y1 - 1996/11/15

N2 - For the pth-order linear ARCH model, Xt = εt√α0 + α1X2t-1 + α2X2t-2 + ··· + αpX2t-p, where α0 > 0, αi ≥ 0, i = 1, 2, ..., p, {εt is an i.i.d. normal white noise with Eεt = 0, Eε2t = 1, and εt, is independent of {Xs, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, α1 + α2 + ··· + αp < 1. In this note, we assume that εt has the probability density function p(t) which is positive and lower-semicontinuous over the real line, but not necessarily Gaussian, then the geometric ergodicity of the ARCH(p) process is proved under Eε2t = 1. When εt has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.

AB - For the pth-order linear ARCH model, Xt = εt√α0 + α1X2t-1 + α2X2t-2 + ··· + αpX2t-p, where α0 > 0, αi ≥ 0, i = 1, 2, ..., p, {εt is an i.i.d. normal white noise with Eεt = 0, Eε2t = 1, and εt, is independent of {Xs, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, α1 + α2 + ··· + αp < 1. In this note, we assume that εt has the probability density function p(t) which is positive and lower-semicontinuous over the real line, but not necessarily Gaussian, then the geometric ergodicity of the ARCH(p) process is proved under Eε2t = 1. When εt has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.

KW - Arch model

KW - Conditional heteroscedasticity

KW - Geometric ergodicity

KW - Markov process

KW - Nonlinear time series

UR - https://www.scopus.com/pages/publications/0030588766

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U2 - 10.1016/S0167-7152(95)00233-2

DO - 10.1016/S0167-7152(95)00233-2

M3 - RGC 21 - Publication in refereed journal

SN - 0167-7152

VL - 30

SP - 305

EP - 311

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

IS - 4

ER -

A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model

Abstract

Bibliographical note

Research Keywords

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