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Small Sample Estimation Bias in GARCH Models with Any Number of Exogenous Variables in the Mean Equation


  • Emma Iglesias
  • Garry Phillips


In this article we show how bias approximations for the quasi maximum likelihood estimators of the parameters in Generalized Autoregressive Conditional Heteroskedastic (GARCH)(p, q) models change when any number of exogenous variables are included in the mean equation. The approximate biases are shown to vary in an additive and proportional way in relation to the number of exogenous variables, and they do not depend on the moments of the regressors under the correct specification of the model. This suggests a rule of thumb in testing for misspecification in GARCH models. We also extend the theoretical bias approximations given in Linton (1997) for the GARCH(1, 1). Because the expressions are not in closed form, we concentrate in detail, and for simplicity of interpretation, on the ARCH(1) model. At each stage, we check our theoretical results by simulation and generally, we find that the approximations are quite accurate for sample sizes of at least 50. We find that the biases are not trivial in some circumstances and we discuss how the bias approximations may be used, in practice, to reduce the bias. We also carry out simulations for the GARCH(1,1) model and show that the biases change as predicted by the approximations when the mean equation is augmented. Finally, we illustrate the usefulness of our approach for U.S. monthly inflation rates.

Suggested Citation

  • 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.
  • Handle: RePEc:taf:emetrv:v:30:y:2011:i:3:p:303-336
    DOI: 10.1080/07474930903562551

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    Cited by:

    1. Stelios Arvanitis & Antonis Demos, "undated". "A Class of Indirect Inference Estimators: Higher Order Asymptotics and Approximate Bias Correction (Revised)," DEOS Working Papers 1411, Athens University of Economics and Business, revised 23 Sep 2014.
    2. Iglesias, Emma M., 2006. "Higher-order asymptotic properties of QML in [beta]-ARCH and [mu]-ARCH models," Economics Letters, Elsevier, vol. 93(2), pages 261-266, November.
    3. Emma M. Iglesias & Garry D.A. Phillips, 2004. "Multivariate Arch Models: Finite Sample Properties Of Ml Estimators And An Application To An Lm-Type Test," Working Papers. Serie AD 2004-09, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
    4. Bao, Yong & Ullah, Aman, 2004. "Bias of a Value-at-Risk estimator," Finance Research Letters, Elsevier, vol. 1(4), pages 241-249, December.
    5. Antonis Demos & Dimitra Kyriakopoulou, 2018. "Finite Sample Theory and Bias Correction of Maximum Likelihood Estimators in the EGARCH Model," DEOS Working Papers 1802, Athens University of Economics and Business.


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