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Quasi-Maximum Likelihood Estimation of GARCH Models With Heavy-Tailed Likelihoods

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  • Jianqing Fan
  • Lei Qi
  • Dacheng Xiu

Abstract

The non-Gaussian maximum likelihood estimator is frequently used in GARCH models with the intention of capturing heavy-tailed returns. However, unless the parametric likelihood family contains the true likelihood, the estimator is inconsistent due to density misspecification. To correct this bias, we identify an unknown scale parameter η f that is critical to the identification for consistency and propose a three-step quasi-maximum likelihood procedure with non-Gaussian likelihood functions. This novel approach is consistent and asymptotically normal under weak moment conditions. Moreover, it achieves better efficiency than the Gaussian alternative, particularly when the innovation error has heavy tails. We also summarize and compare the values of the scale parameter and the asymptotic efficiency for estimators based on different choices of likelihood functions with an increasing level of heaviness in the innovation tails. Numerical studies confirm the advantages of the proposed approach.

Suggested Citation

  • Jianqing Fan & Lei Qi & Dacheng Xiu, 2014. "Quasi-Maximum Likelihood Estimation of GARCH Models With Heavy-Tailed Likelihoods," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(2), pages 178-191, April.
    • Handle: RePEc:taf:jnlbes:v:32:y:2014:i:2:p:178-191
    DOI: 10.1080/07350015.2013.840239
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