We consider a class of M-estimators of the parameters of a GARCH (p,q) model. These estimators involve score functions and, for adequate choices of the score functions, are asymptotically normal under milder moment assumptions than the usual quasi maximum likelihood, which makes them more reliable in the presence of heavy tails. We also consider weighted bootstrap approximations of the distributions of these M-estimators and establish their validity. Through extensive simulations, we demonstrate the robustness of these M-estimators under heavy tails and conduct a comparative study of the performance (bias and mean squared errors) of various score functions and the accuracy (confidence interval coverage rates) of their bootstrap approximations. In addition to the GARCH (1, 1) model, our simulations also involve higher-order models such as GARCH~(2, 1) and GARCH~(1,~\!2) which so far have received relatively little attention in the literature. We also consider the case of order-misspecified models. Finally, we use our M-estimators in the analysis of two real financial time series fitted with GARCH (1, 1) or GARCH (2, 1) models.

翻译：我们认为GARCH(p,q)模型参数的测算器是一类测算器。这些测算器涉及得分函数,并且为了适当选择得分函数,在比通常的准最大可能性更温和的假设下,这些测算器在比通常的准最大可能性更温和的假设下是基本正常的,这使它们在有重尾巴的情况下更加可靠。我们还考虑到这些测算器分布的加权靴子陷阱近似值并确立其有效性。通过广泛的模拟,我们展示了这些测算器在重尾巴下的稳健性,并对各种得分函数的性能(比值和平均正方差差)以及其靴状近似率的准确性(信心间隔率)进行了比较研究。除了GARCH(1,1)模型外,我们的模拟还涉及诸如GRCH~(2,1)和GRCH~(1, ⁇ 2)等更高级的测序模型,这些模型迄今为止在文献中很少受到注意。我们还考虑了定序模型的情况。最后,我们用我们的测测算器分析两个实际财务时间序列(2,GAR1,或1)的模型。