Mixing properties of ARCH and time-varying ARCH processes

Abstract

There exist very few results on mixing for non-stationary processes. However, mixing is often required in statistical inference for non-stationary processes such as time-varying ARCH (tvARCH) models. In this paper, bounds for the mixing rates of a stochastic process are derived in terms of the conditional densities of the process. These bounds are used to obtain the $\alpha$, 2-mixing and $\beta$-mixing rates of the non-stationary time-varying $\mathrm{ARCH}(p)$ process and $\mathrm{ARCH}(\infty )$ process. It is shown that the mixing rate of the time-varying $\mathrm{ARCH}(p)$ process is geometric, whereas the bound on the mixing rate of the $\mathrm{ARCH}(\infty )$ process depends on the rate of decay of the $\mathrm{ARCH}(\infty )$ parameters. We note that the methodology given in this paper is applicable to other processes.

Published in at http://dx.doi.org/10.3150/10-BEJ270 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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