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

Jianqing Fan, Lei Qi, Dacheng Xiu

Research output: Contribution to journalComment/debatepeer-review

72 Scopus citations


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 ηf 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.

Original languageEnglish (US)
Pages (from-to)178-191
Number of pages14
JournalJournal of Business and Economic Statistics
Issue number2
StatePublished - Apr 3 2014

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Social Sciences (miscellaneous)
  • Economics and Econometrics
  • Statistics, Probability and Uncertainty


  • Heavy-tailed error
  • Quasi-likelihood
  • Three-step estimator


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