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Stability of nonlinear AR‐GARCH models

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  • Mika Meitz
  • Pentti Saikkonen

Abstract

. This article studies the stability of nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a nonlinear autoregression of order p [AR(p)] with the conditional variance specified as a nonlinear first‐order generalized autoregressive conditional heteroskedasticity [GARCH(1,1)] model. Conditions under which the model is stable in the sense that its Markov chain representation is geometrically ergodic are provided. This implies the existence of an initial distribution such that the process is strictly stationary and β‐mixing. Conditions under which the stationary distribution has finite moments are also given. The results cover several nonlinear specifications recently proposed for both the conditional mean and conditional variance, and only require mild moment conditions.

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  • Mika Meitz & Pentti Saikkonen, 2008. "Stability of nonlinear AR‐GARCH models," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(3), pages 453-475, May.
    • Handle: RePEc:bla:jtsera:v:29:y:2008:i:3:p:453-475
    DOI: 10.1111/j.1467-9892.2007.00562.x
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    1. 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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    2. Kim, Minjo & Lee, Sangyeol, 2016. "Nonlinear expectile regression with application to Value-at-Risk and expected shortfall estimation," Computational Statistics & Data Analysis, Elsevier, vol. 94(C), pages 1-19.
    3. Murat Midiliç, 2020. "Estimation of STAR–GARCH Models with Iteratively Weighted Least Squares," Computational Economics, Springer;Society for Computational Economics, vol. 55(1), pages 87-117, January.
    4. Chou, Ray Yeutien & Cai, Yijie, 2009. "Range-based multivariate volatility model with double smooth transition in conditional correlation," Global Finance Journal, Elsevier, vol. 20(2), pages 137-152.
    5. Meitz, Mika & Saikkonen, Pentti, 2010. "A note on the geometric ergodicity of a nonlinear AR-ARCH model," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 631-638, April.
    6. Jungsik Noh & Sangyeol Lee, 2016. "Quantile Regression for Location-Scale Time Series Models with Conditional Heteroscedasticity," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(3), pages 700-720, September.
    7. Mika Meitz & Pentti Saikkonen, 2008. "Stability of nonlinear AR‐GARCH models," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(3), pages 453-475, May.
    8. Jiti Gao & Maxwell King, 2011. "A New Test in Parametric Linear Models against Nonparametric Autoregressive Errors," Monash Econometrics and Business Statistics Working Papers 20/11, Monash University, Department of Econometrics and Business Statistics.
    9. Meitz, Mika & Saikkonen, Pentti, 2011. "Parameter Estimation In Nonlinear Ar–Garch Models," Econometric Theory, Cambridge University Press, vol. 27(6), pages 1236-1278, December.
    10. Díaz-Hernández, Adán & Constantinou, Nick, 2019. "A multiple regime extension to the Heston–Nandi GARCH(1,1) model," Journal of Empirical Finance, Elsevier, vol. 53(C), pages 162-180.
    11. Christis Katsouris, 2024. "Robust Estimation in Network Vector Autoregression with Nonstationary Regressors," Papers 2401.04050, arXiv.org.
    12. Isao Ishida & Virmantas Kvedaras, 2015. "Modeling Autoregressive Processes with Moving-Quantiles-Implied Nonlinearity," Econometrics, MDPI, vol. 3(1), pages 1-53, January.
    13. Murat Midilic, 2016. "Estimation Of Star-Garch Models With Iteratively Weighted Least Squares," Working Papers of Faculty of Economics and Business Administration, Ghent University, Belgium 16/918, Ghent University, Faculty of Economics and Business Administration.
    14. Pedersen, Rasmus Søndergaard, 2017. "Robust inference in conditionally heteroskedastic autoregressions," MPRA Paper 81979, University Library of Munich, Germany.
    15. Sandberg, Rickard, 2016. "Trends, unit roots, structural changes, and time-varying asymmetries in U.S. macroeconomic data: the Stock and Watson data re-examined," Economic Modelling, Elsevier, vol. 52(PB), pages 699-713.
    16. Iglesias, Emma M. & Linton, Oliver, 2009. "Estimation of tail thickness parameters from GJR-GARCH models," UC3M Working papers. Economics we094726, Universidad Carlos III de Madrid. Departamento de Economía.
    17. Theis Lange & Anders Rahbek & Søren Tolver Jensen, 2011. "Estimation and Asymptotic Inference in the AR-ARCH Model," Econometric Reviews, Taylor & Francis Journals, vol. 30(2), pages 129-153.
    18. Hill, Jonathan B., 2015. "Robust Generalized Empirical Likelihood for heavy tailed autoregressions with conditionally heteroscedastic errors," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 131-152.

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    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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