Asymptotic behavior of the Laplacian quasi-maximum likelihood estimator of affine causal processes

We prove the consistency and asymptotic normality of the Laplacian Quasi-Maximum Likelihood Estimator (QMLE) for a general class of causal time series including ARMA, AR($\mathrm{\infty }$), GARCH, ARCH($\mathrm{\infty }$), ARMA-GARCH, APARCH, ARMA-APARCH,…, processes. We notably exhibit the advantages (moment order and robustness) of this estimator compared to the classical Gaussian QMLE. Numerical simulations confirms the accuracy of this estimator.