Mixing and Moment Properties of Various Garch and Stochastic Volatility Models

Abstract

This paper first provides some useful results on a generalized random coefficient autoregressive model and a generalized hidden Markov model. These results simultaneously imply strict stationarity, existence of higher order moments, geometric ergodicity, and β-mixing with exponential decay rates, which are important properties for statistical inference. As applications, we then provide easy-to-verify sufficient conditions to ensure β-mixing and finite higher order moments for various linear and nonlinear GARCH(1,1), linear and power GARCH(p,q), stochastic volatility, and autoregressive conditional duration models. For many of these models, our sufficient conditions for existence of second moments and exponential β-mixing are also necessary. For several GARCH(1,1) models, our sufficient conditions for existence of higher order moments again coincide with the necessary ones in He and Terasvirta (1999, Journal of Econometrics 92, 173–192).

Learning Near-Optimal Policies with Bellman-Residual Minimization Based Fitted Policy Iteration and a Single Sample Path

“…Note that if X t is β-mixing then the hidden-Markov process {(X t , (A t , R t ))} is also β-mixing with the same rate (see, e.g., the proof of Proposition 4 by Carrasco and Chen (2002) for an argument that can be used to prove this). Our next assumption concerns the average concentrability of the futurestate distribution.…”

Testing for high-dimensional white noise using maximum cross-correlations

“…See Section 2.6.1 of Fan & Yao (2003) and the references within. In fact stationary generalized autoregressive conditional heteroskedasticity models with finite second moments and continuous innovation distributions are also β-mixing with exponentially decaying β k ; see Proposition 12 of Carrasco & Chen (2002). If we only require sup t sup 1≤i≤p pr(|ε i,t | > x) = O{x −2(ν+ ) } for any x > 0 in Condition 2 and β k = O{k −ν(ν+ )/(2 ) } in Condition 3 for some ν > 2 and > 0, we can apply Fuk-Nagaev type inequalities to construct the upper bounds for the tail probabilities of the statistics for which our testing procedure still works for p diverging at some polynomial rate of n. We refer to Section 3.2 of arXiv:1410.2323 for the implementation of Fuk-Nagaev type inequalities in such a scenario.…”

Extremal memory of stochastic volatility with an application to tail shape inference

“…If f40 and gþdo1, or f ¼ 0 and gþdr1, then fs t g is geometrically aÀmixing (Boussama, 1998;Carrasco and Chen, 2002;Meitz and Saikkonen, 2008), hence fs t g is L p -NED on itself as a geometrically aÀmixing base for any p 40.…”

“…Nelson, 1990;Ghysels et al, 1995;Giraitis et al, 2000;Basrak et al, 2002a;Carrasco and Chen, 2002;Davidson, 2004;Meddahi and Renault, 2004), only recently have volatility extremes been considered (de Haan et al, 1989;Chernick et al, 1991;Basrak et al, 2002a, b;Mikosch and Stȃricȃ, 2000;Klüppelberg and Lindner, 2008;Davis and Mikosch, 2009a, b). A large class of stationary GARCH processes, for example, have marginal distribution tails that decay according to a power law, and exhibit extremal clustering.…”