Self-weighted LSE and residual-based QMLE of ARMA-GARCH models

This paper studies the self-weighted least squares estimator (SWLSE) of the ARMA model with GARCH noises. It is shown that the SWLSE is consistent and asymptotically normal when the GARCH noise does not have a finite fourth moment. Using the residuals from the estimated ARMA model, it is shown that the residual-based quasi-maximum likelihood estimator (QMLE) for the GARCH model is consistent and asymptotically normal, but if the innovations are asymmetric, it is not as efficient as that when the GARCH process is observed. Using the SWLSE and residual-based QMLE as the initial estimators, the local QMLE for ARMA-GARCH model is asymptotically normal via an one-step iteration. The importance of the proposed estimators is illustrated by simulated data and five real examples in financial markets.