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.