On some mixing properties of ARCH and time-varying ARCH processes
There exists very few results on mixing for nonstationary processes.
However, mixing is often required in statistical inference for
nonstationary processes, such as time-varying ARCH (tvARCH) models.
In this paper,
bounds for the mixing rates of a stochastic process are derived
in terms the conditional densities of the process. These bounds
are used to obtain the strong and 2-mixing rates of the nonstationary
time-varying ARCH$(p)$ process and ARCH$(\infty)$ process. It is shown that
the mixing rate of time-varying ARCH$(p)$ process is geometric, whereas
the bounds on the mixing rate of the ARCH$(\infty)$ process depends on the rate of decay of
the ARCH$(\infty)$ parameters.
These results are generalised to consider mixing rates for
moving averages of ARCH$(\infty)$ random variables.