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An ARCH model without intercept

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  • Hafner, Christian M.
  • Preminger, Arie

Abstract

While theory of autoregressive conditional heteroskedasticity (ARCH) models is well understood for strictly stationary processes, some recent interest has focused on the nonstationary case. In the classical model including a positive intercept parameter, the volatility process diverges to infinity at least in probability, and it has been shown that no consistent estimator of the full parameter vector, including intercept, exists. This paper considers a nonstationary ARCH model which arises by setting the intercept term to zero. Unlike nonstationary ARCH models with positive intercept, this model includes the interesting case of log volatility following a random walk, which is called the stability case. For the ARCH(1) model without intercept, the paper derives asymptotic theory of the maximum likelihood estimator and proposes a test of the stability hypothesis. Numerical evidence illustrates the finite sample properties of the maximum likelihood estimator and the stability test.

Suggested Citation

  • Hafner, Christian M. & Preminger, Arie, 2015. "An ARCH model without intercept," Economics Letters, Elsevier, vol. 129(C), pages 13-17.
    • Handle: RePEc:eee:ecolet:v:129:y:2015:i:c:p:13-17
    DOI: 10.1016/j.econlet.2015.01.029
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    References listed on IDEAS

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    1. Christian Francq & Jean-Michel Zakoïan, 2008. "Can One Really Estimate Nonstationary GARCH Models ?," Working Papers 2008-06, Center for Research in Economics and Statistics.
    2. Kristensen, Dennis & Rahbek, Anders, 2005. "ASYMPTOTICS OF THE QMLE FOR A CLASS OF ARCH(q) MODELS," Econometric Theory, Cambridge University Press, vol. 21(5), pages 946-961, October.
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    4. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    5. Weiss, Andrew A., 1986. "Asymptotic Theory for ARCH Models: Estimation and Testing," Econometric Theory, Cambridge University Press, vol. 2(1), pages 107-131, April.
    6. 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.
    7. 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.
    8. Christian Francq & Jean‐Michel Zakoïan, 2012. "Strict Stationarity Testing and Estimation of Explosive and Stationary Generalized Autoregressive Conditional Heteroscedasticity Models," Econometrica, Econometric Society, vol. 80(2), pages 821-861, March.
    9. Nelson, Daniel B., 1990. "Stationarity and Persistence in the GARCH(1,1) Model," Econometric Theory, Cambridge University Press, vol. 6(3), pages 318-334, September.
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    Cited by:

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    5. Li, Dong & Zhang, Xingfa & Zhu, Ke & Ling, Shiqing, 2018. "The ZD-GARCH model: A new way to study heteroscedasticity," Journal of Econometrics, Elsevier, vol. 202(1), pages 1-17.

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    More about this item

    Keywords

    Nonstationarity; Volatility; Lyapunov exponent; Random walk;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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