M-estimation in GARCH models without higher order moments
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We consider a class of M-estimators of the parameters of the GARCH models which are asymptotically normal under mild assumptions on the moments of the underlying error distribution. Since heavy-tailed error distributions without higher order moments are common in the GARCH modeling of many real financial data, it becomes worthwhile to use such estimators for the time series inference instead of the quasi maximum likelihood estimator. We discuss the weighted bootstrap approximations of the distributions of M-estimators. Through extensive simulations and data analysis, we demonstrate the robustness of the M-estimators under heavy-tailed error distributions and the accuracy of the bootstrap approximation. In addition to the GARCH (1, 1) model, we obtain extensive computation and simulation results which are useful in the context of higher order models such as GARCH (2, 1) and GARCH (1, 2) but have not yet received sufficient attention in the literature. Finally, we use M-estimators for the analysis of three real financial time series fitted with GARCH (1, 1) or GARCH (2, 1) models.
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