Quasi-Maximum Likelihood Estimation of GARCH Models With Heavy-Tailed Likelihoods
针对GARCH模型中非高斯极大似然估计因密度设定错误导致的不一致问题，提出一种三步拟极大似然方法，通过识别未知尺度参数实现一致估计，并在厚尾创新下比高斯方法更有效。
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.