Mixing properties of ARCH and time-varying ARCH processes

By Piotr Fryzlewicz, S. Rao. 2011

53 citations recorded by the source corpus.

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 $\operatorname {ARCH}(p)$ process and $\operatorname {ARCH}(\infty)$ process. It is shown that the mixing rate of the time-varying $\operatorname {ARCH}(p)$ process is geometric, whereas the bound on the mixing rate of the $\operatorname {ARCH}(\infty)$ process depends on the rate of decay of the $\operatorname {ARCH}(\infty)$ parameters. We note that the methodology given in this paper is applicable to other processes.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e09764ebfcd7403d2d9223ffd5dfd27ff0f30a32

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Citations in the corpus

Nonparametric regression for locally stationary time series (2012, background): “It is, for example, fulfilled for the class of bounded densities fε whose first derivative f ′ ε is bounded, satisfies ∫ |zf ′ε(z)|dz <∞ and declines monotonically to zero for values |z| > C for some constant C > 0; see also Section 3 in Fryzlewicz and Subba Rao [11] who work with assumptions closel”
Detecting gradual changes in locally stationary processes (2013, background): “A wide variety of locally stationary processes can be shown to be mixing under appropriate conditions; see for example Fryzlewicz & Subba Rao [12] and Vogt [27].”