Non-Gaussian Quasi Maximum Likelihood Estimation of GARCH Models

The non-Gaussian quasi maximum likelihood estimator is frequently used in GARCH models with intension to improve the efficiency of the GARCH parameters. However, the method is usually inconsistent unless the quasi-likelihood happens to be the true one. We identify an unknown scale parameter that is critical to the consistent estimation of non-Gaussian QMLE. As a part of estimating this unknown parameter, a two-step non-Gaussian QMLE (2SNG-QMLE) is proposed for estimation the GARCH parameters. Without assumptions on symmetry and unimodality of the distributions of innovations, we show that the non-Gaussian QMLE remains consistent and asymptotically normal, under a general framework of non-Gaussian QMLE. Moreover, it has higher efficiency than the Gaussian QMLE, particularly when the innovation error has heavy tails. Two extensions are proposed to further improve the efficiency of 2SNG-QMLE. The impact of relative heaviness of tails of the innovation and quasi-likelihood distributions on the asymptotic efficiency has been thoroughly investigated. Monte Carlo simulations and an empirical study confirm the advantages of the proposed approach.