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We propose a consistent estimator for the noise mean  m  of the GARCH process based on Laplace ( m ,1) errors. Also, we prove the consistency and asymptotic normality for the Quasi-Maximum Likelihood Estimator of the GARCH model based on the Laplace residuals with any mean value for the residuals. Numerical simulations confirm the accuracy of the estimators.

Standard Laplace quasi-maximum likelihood estimator for GARCH processes