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Finite Sample Theory of QMLE in ARCH Models with Dynamics in the Mean Equation

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  • Emma M. Iglesias
  • Garry D. A. Phillips

Abstract

. We provide simulation and theoretical results concerning the finite‐sample theory of quasi‐maximum‐likelihood estimators in autoregressive conditional heteroskedastic (ARCH) models when we include dynamics in the mean equation. In the setting of the AR(q)–ARCH(p), we find that in some cases bias correction is necessary even for sample sizes of 100, especially when the ARCH order increases. We warn about the existence of important biases and potentially low power of the t‐tests in these cases. We also propose ways to deal with them. We also find simulation evidence that when conditional heteroskedasticity increases, the mean‐squared error of the maximum‐likelihood estimator of the AR(1) parameter in the mean equation of an AR(1)‐ARCH(1) model is reduced. Finally, we generalize the Lumsdaine [J. Bus. Econ. Stat. 13 (1995) pp. 1–10] invariance properties for the biases in these situations.

Suggested Citation

  • Emma M. Iglesias & Garry D. A. Phillips, 2008. "Finite Sample Theory of QMLE in ARCH Models with Dynamics in the Mean Equation," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(4), pages 719-737, July.
    • Handle: RePEc:bla:jtsera:v:29:y:2008:i:4:p:719-737
    DOI: 10.1111/j.1467-9892.2008.00582.x
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    References listed on IDEAS

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    1. Emma Iglesias & Garry Phillips, 2011. "Small Sample Estimation Bias in GARCH Models with Any Number of Exogenous Variables in the Mean Equation," Econometric Reviews, Taylor & Francis Journals, vol. 30(3), pages 303-336.
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    1. Kiviet, Jan F. & Phillips, Garry D.A., 2012. "Higher-order asymptotic expansions of the least-squares estimation bias in first-order dynamic regression models," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3705-3729.
    2. Mazur Błażej & Pipień Mateusz, 2018. "Time-varying asymmetry and tail thickness in long series of daily financial returns," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 22(5), pages 1-21, December.
    3. Liu-Evans Gareth D. & Phillips Garry D. A., 2012. "Bootstrap, Jackknife and COLS: Bias and Mean Squared Error in Estimation of Autoregressive Models," Journal of Time Series Econometrics, De Gruyter, vol. 4(2), pages 1-35, November.
    4. Ana Maria Herrera & Pinar Ozbay, 2005. "A Dynamic Model of Central Bank Intervention," Working Papers 0501, Research and Monetary Policy Department, Central Bank of the Republic of Turkey.
    5. Maixé-Altés, J. Carles & Iglesias, Emma M., 2009. "Domestic monetary transfers and the inland bill of exchange markets in Spain (1775-1885)," Journal of International Money and Finance, Elsevier, vol. 28(3), pages 496-521, April.
    6. Yong Bao, 2013. "On Sample Skewness and Kurtosis," Econometric Reviews, Taylor & Francis Journals, vol. 32(4), pages 415-448, December.

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