TY - JOUR
T1 - The tail of the stationary distribution of an autoregressive process with ARCH(1) errors
AU - Borkovec, Milan
AU - Klüppelberg, Claudia
PY - 2001/11
Y1 - 2001/11
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
UR - http://www.scopus.com/inward/record.url?scp=0035564935&partnerID=8YFLogxK
U2 - 10.1214/aoap/1015345401
DO - 10.1214/aoap/1015345401
M3 - Article
AN - SCOPUS:0035564935
SN - 1050-5164
VL - 11
SP - 1220
EP - 1241
JO - Annals of Applied Probability
JF - Annals of Applied Probability
IS - 4
ER -