The tail of the stationary distribution of an autoregressive process with ARCH(1) errors

Milan Borkovec; Claudia Klüppelberg

doi:10.1214/aoap/1015345401

The tail of the stationary distribution of an autoregressive process with ARCH(1) errors

Milan Borkovec, Claudia Klüppelberg

Chair of Mathematical Statistics

Research output: Contribution to journal › Article › peer-review

74 Scopus citations

Abstract

We consider the class of autoregressive processes with ARCH(1) errors given by the stochastic difference equation X_n = αX_n-1 + √ β + λX²_n-1ε_n, n ∈ ℕ, where (ε_n)_n∈ℕ i.i.d. random variables. Under general and tractable assumptions we show the existence and uniqueness of a stationary distribution. We prove that the stationary distribution has a Pareto-like tail with a well-specified tail index which depends on α, λ and the distribution of the innovations (ε_n)_n∈ℕ. This paper generalizes results for the ARCH(1) process (the case α = 0). The generalization requires a new method of proof and we invoke a Tauberian theorem.

Original language	English
Pages (from-to)	1220-1241
Number of pages	22
Journal	Annals of Applied Probability
Volume	11
Issue number	4
DOIs	https://doi.org/10.1214/aoap/1015345401
State	Published - Nov 2001

Keywords

ARCH model
Autoregressive process
Geometric ergodicity
Heavy tail
Heteroscedastic model
Markov process
Recurrent Harris chain
Regular variation
Tauberian theorem

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10.1214/aoap/1015345401

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title = "The tail of the stationary distribution of an autoregressive process with ARCH(1) errors",

abstract = "We consider the class of autoregressive processes with ARCH(1) errors given by the stochastic difference equation Xn = αXn-1 + √ β + λX2n-1εn, n ∈ ℕ, where (εn)n∈ℕ i.i.d. random variables. Under general and tractable assumptions we show the existence and uniqueness of a stationary distribution. We prove that the stationary distribution has a Pareto-like tail with a well-specified tail index which depends on α, λ and the distribution of the innovations (εn)n∈ℕ. This paper generalizes results for the ARCH(1) process (the case α = 0). The generalization requires a new method of proof and we invoke a Tauberian theorem.",

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N2 - We consider the class of autoregressive processes with ARCH(1) errors given by the stochastic difference equation Xn = αXn-1 + √ β + λX2n-1εn, n ∈ ℕ, where (εn)n∈ℕ i.i.d. random variables. Under general and tractable assumptions we show the existence and uniqueness of a stationary distribution. We prove that the stationary distribution has a Pareto-like tail with a well-specified tail index which depends on α, λ and the distribution of the innovations (εn)n∈ℕ. This paper generalizes results for the ARCH(1) process (the case α = 0). The generalization requires a new method of proof and we invoke a Tauberian theorem.

AB - We consider the class of autoregressive processes with ARCH(1) errors given by the stochastic difference equation Xn = αXn-1 + √ β + λX2n-1εn, n ∈ ℕ, where (εn)n∈ℕ i.i.d. random variables. Under general and tractable assumptions we show the existence and uniqueness of a stationary distribution. We prove that the stationary distribution has a Pareto-like tail with a well-specified tail index which depends on α, λ and the distribution of the innovations (εn)n∈ℕ. This paper generalizes results for the ARCH(1) process (the case α = 0). The generalization requires a new method of proof and we invoke a Tauberian theorem.

KW - ARCH model

KW - Autoregressive process

KW - Geometric ergodicity

KW - Heavy tail

KW - Heteroscedastic model

KW - Markov process

KW - Recurrent Harris chain

KW - Regular variation

KW - Tauberian theorem

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DO - 10.1214/aoap/1015345401

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JO - Annals of Applied Probability

JF - Annals of Applied Probability

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The tail of the stationary distribution of an autoregressive process with ARCH(1) errors

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Keywords

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