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Strict stationarity testing and estimation of explosive ARCH models

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  • Francq, Christian
  • Zakoian, Jean-Michel

Abstract

This paper studies the asymptotic properties of the quasi-maximum likelihood estimator of ARCH(1) models without strict stationarity constraints, and considers applications to testing problems. The estimator is unrestricted, in the sense that the value of the intercept, which cannot be consistently estimated in the explosive case, is not fixed. A specific behavior of the estimator of the ARCH coefficient is obtained at the boundary of the stationarity region, but this estimator remains consistent and asymptotically normal in every situation. The asymptotic variance is different in the stationary and non stationary situations, but is consistently estimated, with the same estimator, in both cases. Tests of strict stationarity and non stationarity are proposed. Their behaviors are studied under the null assumption and under local alternatives. The tests developed for the ARCH(1) model are able to detect non-stationarity in more general GARCH models. A numerical illustration based on stock indices is proposed.

Suggested Citation

  • Francq, Christian & Zakoian, Jean-Michel, 2010. "Strict stationarity testing and estimation of explosive ARCH models," MPRA Paper 22414, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:22414
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    References listed on IDEAS

    as
    1. Jensen, Søren Tolver & Rahbek, Anders, 2004. "Asymptotic Inference For Nonstationary Garch," Econometric Theory, Cambridge University Press, vol. 20(6), pages 1203-1226, December.
    2. Shiqing Ling & Michael McAleer, 2001. "On Adaptive Estimation in Nonstationary ARMA Models with GARCH Errors," ISER Discussion Paper 0548, Institute of Social and Economic Research, Osaka University.
    3. Linton, Oliver & Pan, Jiazhu & Wang, Hui, 2010. "Estimation For A Nonstationary Semi-Strong Garch(1,1) Model With Heavy-Tailed Errors," Econometric Theory, Cambridge University Press, vol. 26(1), pages 1-28, February.
    4. Søren Tolver Jensen & Anders Rahbek, 2004. "Asymptotic Normality of the QMLE Estimator of ARCH in the Nonstationary Case," Econometrica, Econometric Society, vol. 72(2), pages 641-646, March.
    5. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    6. Drost, F.C. & Klaassen, C.A.J. & Werker, B.J.M., 1994. "Adaptive estimation in time-series models," Discussion Paper 1994-88, Tilburg University, Center for Economic Research.
    7. Francq, Christian & Zakoïan, Jean-Michel, 2006. "Mixing Properties Of A General Class Of Garch(1,1) Models Without Moment Assumptions On The Observed Process," Econometric Theory, Cambridge University Press, vol. 22(5), pages 815-834, October.
    8. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
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    Cited by:

    1. Aknouche, Abdelhakim & Al-Eid, Eid M. & Hmeid, Aboubakry M., 2011. "Offline and online weighted least squares estimation of nonstationary power ARCH processes," Statistics & Probability Letters, Elsevier, vol. 81(10), pages 1535-1540, October.
    2. Abdelhakim Aknouche & Eid Al-Eid, 2012. "Asymptotic inference of unstable periodic ARCH processes," Statistical Inference for Stochastic Processes, Springer, vol. 15(1), pages 61-79, April.
    3. Christian Bucio-Pacheco & Miriam Sosa-Castro & Francisco Reyes-Zarate, 2023. "Volatilidad dinamica en el sector bancario en Mexico: evidencia DCC-GARCH vs Copula-GARCH," EconoQuantum, Revista de Economia y Finanzas, Universidad de Guadalajara, Centro Universitario de Ciencias Economico Administrativas, Departamento de Metodos Cuantitativos y Maestria en Economia., vol. 20(2), pages 69-93, Julio-Dic.

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    More about this item

    Keywords

    ARCH model; Inconsistency of estimators; Local power of tests; Nonstationarity; Quasi Maximum Likelihood Estimation;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics

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