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 $α$, 2-mixing and $β$-mixing rates of the non-stationary time-varying ARCH($p$) process and ARCH($∞$) process. It is shown that the mixing rate of the time-varying ARCH($p$) process is geometric, whereas the bound on the mixing rate of the ARCH($∞$) process depends on the rate of decay of the ARCH($∞$) parameters. We note that the methodology given in this paper is applicable to other processes.