A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model |
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Authors: | Zudi Lu |
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Institution: | Institute of Systems Science, Academia Sinica, Beijing 100080, People's Republic of China |
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Abstract: | For the pth-order linear ARCH model, , where 0 > 0, i 0, I = 1, 2, …, p, { t} is an i.i.d. normal white noise with E t = 0, E t2 = 1, and t is independent of {Xs, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, 1 + 2 + ··· + p < 1. In this note, we assume that 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 E t2 = 1. When t has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given. |
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Keywords: | Geometric ergodicity Conditional heteroscedasticity ARCH model Nonlinear time series Markov process |
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