The limit distribution of the quasi-maximum likelihood estimator (QMLE) for parameters in
the ARMA-GARCH model remains an open problem when the process has infinite 4th
moment. We propose a self-weighted QMLE and show that it is consistent and
asymptotically normal under only a fractional moment condition. Based on this estimator, the
asymptotic normality of the local QMLE is established for the ARMA model with GARCH
(finite variance) and IGARCH errors. Using the self-weighted and the local QMLEs, we …

This paper investigates the asymptotic theory of the quasi-maximum exponential likelihood
estimators (QMELE) for ARMA–GARCH models. Under only a fractional moment condition,
the strong consistency and the asymptotic normality of the global self-weighted QMELE are
obtained. Based on this self-weighted QMELE, the local QMELE is showed to be
asymptotically normal for the ARMA model with GARCH (finite variance) and IGARCH
errors. A formal comparison of two estimators is given for some cases. A simulation study is …