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Subgeometrically Ergodic Autoregressions

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  • Meitz, Mika
  • Saikkonen, Pentti

Abstract

In this paper, we discuss how the notion of subgeometric ergodicity in Markov chain theory can be exploited to study stationarity and ergodicity of nonlinear time series models. Subgeometric ergodicity means that the transition probability measures converge to the stationary measure at a rate slower than geometric. Specifically, we consider suitably defined higher-order nonlinear autoregressions that behave similarly to a unit root process for large values of the observed series but we place almost no restrictions on their dynamics for moderate values of the observed series. Results on the subgeometric ergodicity of nonlinear autoregressions have previously appeared only in the first-order case. We provide an extension to the higher-order case and show that the autoregressions we consider are, under appropriate conditions, subgeometrically ergodic. As useful implications, we also obtain stationarity and $\beta $ -mixing with subgeometrically decaying mixing coefficients.

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  • Meitz, Mika & Saikkonen, Pentti, 2022. "Subgeometrically Ergodic Autoregressions," Econometric Theory, Cambridge University Press, vol. 38(5), pages 959-985, October.
    • Handle: RePEc:cup:etheor:v:38:y:2022:i:5:p:959-985_6
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    1. Fokianos, Konstantinos & Rahbek, Anders & Tjøstheim, Dag, 2009. "Poisson Autoregression," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1430-1439.
    2. Meitz, Mika & Saikkonen, Pentti, 2008. "Ergodicity, Mixing, And Existence Of Moments Of A Class Of Markov Models With Applications To Garch And Acd Models," Econometric Theory, Cambridge University Press, vol. 24(5), pages 1291-1320, October.
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    1. Mika Meitz & Pentti Saikkonen, 2022. "Subgeometrically ergodic autoregressions with autoregressive conditional heteroskedasticity," Papers 2205.11953, arXiv.org, revised Apr 2023.

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