  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
carried out to assess the performance of these estimators, and a real example
on the world crude oil price is given.
