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

✍ Scribed by Zudi Lu

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
320 KB
Volume
30
Category
Article
ISSN
0167-7152

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✦ Synopsis

For the pth-order linear ARCH model, St = gt~/O{O q-~lXt2_l q-0~2 X.2_2 +-.-q-o~pXtLp, where c~0 > 0, c~i~>0, i = 1, 2, ..., p, {et} is an i.i.d, normal white noise with E~, = 0, Ee~ = 1, and et is independent of {X~, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, ~1 + c~2 -... + ~p < 1. In this note, we assume that et 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 Eet 2 = 1. When e, has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.

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