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Mixing properties of ARCH and time-varying ARCHprocesses (technical report)Piotr Fryzlewicz Suhasini Subba RaoAbstractThere exists very few results on mixing for nonstationary processes. However, mixingis often required in statistical inference for nonstationary processes, such as time-varyingARCH (tvARCH) models. In this paper, bounds for the mixing rates of a stochastic pro-cess are derived in terms the conditional densities of the process. These bounds are usedto obtain the α, 2-mixing and β-mixing rates of the nonstationary time-varying ARCH(p)process and ARCH(∞) 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(∞) process de-pends on the rate of decay of the ARCH(∞) parameters. We mention that the methodologygiven in this paper is applicable to other processes.Key words: Absolutely regular (β-mixing) ARCH(∞), conditional densities, time-varying ARCH, strong mixing (α-mixing), 2-mixing.2000 Mathematics Subject Classification: 62M10, 60G99, 60K99.1 IntroductionMixing is a measure of dependence between elements of a random sequence that has a wide rangeof theoretical applications (see Bradley (2007) and below). One of the most popular mixingmeasures is α-mixing (also called strong mixing), where the α-mixing rate of the nonstationarystochastic process {Xt} is defined as a sequence of coefficients α(k) such thatα(k) = supt∈ZsupH∈σ(Xt,Xt−1,...)G∈σ(Xt+k,Xt+k+1,...)|P (G ∩ H) − P (G)P (H)|. (1)1{Xt} is called α-mixing if α(k) → 0 as k → ∞. If {α(k)} decays sufficiently fast to zero ask → ∞, then, amongst other results, it is possible to show asymptotic normality of sums of{Xk} (c.f. Davidson (1994), Chapter 24), as well as exponential inequalities for such sums (c.f.Bosq (1998)). The notion of 2-mixing is related to strong mixing, but is a weaker condition as itmeasures the dependence between two random variables and not the entire tails. 2-mixing is oftenused in statistical inference, for example deriving rates in nonparametric regression (see Bosq(1998)). The 2-mixing rate can be used to derive bounds for the covariance between functions ofrandom variables, say cov(g(Xt), g(Xt+k)) (see Ibragimov (1962)), which is usually not possiblewhen only the correlation structure of {Xk} is known. The 2-mixing rate of {Xk} is defined asa sequence ˜α(k) which satisfies˜α(k) = supt∈ZsupH∈σ(Xt)G∈σ(Xt+k)|P (G ∩ H) − P (G)P (H)|. (2)It is clear that ˜α(k) ≤ α(k). A closely related mixing measure, introduced in Volkonskii andRozanov (1959) is β-mixing (also called absolutely regular). The β-mixing rate of the stochasticprocess {Xt} is defined as a sequence of coefficients β(k) such thatβ(k) = supt∈Zsup{Hj}∈σ(Xt,Xt−1,...){Gj}∈σ(Xt+k,Xt+k+1,...)XiXj|P (Gi∩ Hj) − P (Gi)P (Hj)|, (3)where {Gi} and {Hj} are finite partitions of the sample space Ω. {Xt} is called β-mixing ifβ(k) → 0 as k → ∞. It can be seen that this measure is slightly stronger than α-mixing (sincean upper bound for β(k) immediately gives a bound for α(k); β(k) ≥ α(k)).Despite the versatility of mixing, its main drawback is that in general it is difficult to derivebounds for α(k), ˜α(k) and β(k). However the mixing bounds of some processes are known.Chanda (1974), Gorodetskii (1977), Athreya and Pantula (1986) and Pham and Tran (1985) showstrong mixing of the MA(∞) process. Feigin and Tweedie (1985) and Pham (1986) have showngeometric ergodicity of Bilinear processes (we note that stationary geometrically ergodic Markovchains are geometrically α-mixing, 2-mixing and β-mixing - see, for example, Francq and Zako¨ıan(2006)). More recently, Tjostheim (1990) and Mokkadem (1990) have shown geometric ergodicityfor a general class of Markovian processes. The results in Mokkadem (1990) have been appliedin Bousamma (1998) to show geometric ergodicity of stationary ARCH(p) and GARCH(p, q)processes, where p and q are finite integers. Related results on mixing for GARCH(p, q) processes2can be found in Carrasco and Chen (2002), Liebscher (2005), Sorokin (2006) and Lindner (2008)(for an excellent review) and Francq and Zako¨ıan (2006) and Meitz and Saikkonen (2008) (wheremixing of ‘nonlinear’ GARCH(p, q) processes are also considered). Most of these these resultsare proven by verifying the Meyn-Tweedie conditions (see Feigin and Tweedie (1985) and Meynand Tweedie (1993)), and, as mentioned above, are derived under the premise that the process isstationary (or asymptotically stationary) and Markovian. Clearly, if a process is nonstationary,then the aforementioned results do not hold. Therefore for nonstationary processes, an alternativemethod to prove mixing is required.The main aim of this paper is to derive a bound for (1), (2) and (3) in terms of the densitiesof the process plus an additional term, which is an extremal probability. These bounds can beapplied to various processes. In this paper, we will focus on ARCH-type processes and use thebounds to derive mixing rates for time-varying ARCH(p) (tvARCH) and ARCH(∞) processes.The ARCH family of processes is widely used in finance to model the evolution of returns onfinancial instruments: we refer the reader to the review article of Giraitis et al. (2005) for acomprehensive overview of mathematical properties of ARCH processes, and a list of furtherreferences. It is worth mentioning that H¨ormann (2008) and Berkes et al. (2008) have considereda different type of dependence, namely a version of the m-dependence moment measure, forARCH-type processes. The stationary GARCH(p, q) model tends to be the benchmark financialmodel. However, in certain situations it may not be the most appropriate model, for exampleit cannot adequently explain the long memory seen in the data or change according to shiftsin the world economy. Therefore, recently attention has been paid to tvARCH models (see,for example, Mikosch and St˘aric˘a (2003), Dahlhaus and Subba Rao (2006), Fryzlewicz et al.(2008) and Fryzlewicz and Subba Rao (2008)) and ARCH(∞) models (see Robinson (1991),Giraitis et al. (2000), Giraitis and Robinson (2001) and Subba Rao (2006)). The derivationsof the sampling properties of some of the above mentioned papers rely on quite sophisticatedassumptions on the dependence structure, in particular their mixing properties.We will show that due to the p-Markovian nature of the time-varying ARCH(p) process,


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