There is a serious and long-standing restriction in the literature on heavy-tailed phenomena in that moment conditions, which are unrealistic, are almost always assumed in modelling such phenomena. Further, the issue of stability is often insufficiently addressed. To this end, we develop a comprehensive statistical inference for an asymmetric generalized autoregressive conditional heteroskedasticity model with standardized non-Gaussian symmetric stable innovation (sAGARCH) in a unified framework, covering both the stationary case and the explosive case. We consider first the maximum likelihood estimation of the model including the asymptotic properties of the estimator of the stable exponent parameter among others. We then propose a modified Kolmogorov-type test statistic for diagnostic checking, as well as those for strict stationarity and asymmetry testing. We conduct Monte Carlo simulation studies to examine the finite-sample performance of our entire statistical inference procedure. We include empirical examples of stock returns to highlight the usefulness and merits of our sAGARCH model.

翻译：长期以来，关于厚尾现象的文献中存在一个严重且持久的限制，即在对此类现象建模时几乎总是假设不切实际的矩条件。此外，稳定性问题往往未能得到充分探讨。为此，我们在一个统一框架内，为具有标准化非高斯对称稳定分布新息的非对称广义自回归条件异方差模型（sAGARCH）开发了一套全面的统计推断方法，涵盖了平稳情形与爆炸情形。我们首先考虑了模型的最大似然估计，包括稳定指数参数估计量的渐近性质等。随后，我们提出了一种改进的Kolmogorov型检验统计量用于诊断检验，以及用于严格平稳性和非对称性检验的统计量。我们进行了蒙特卡洛模拟研究，以检验我们整个统计推断程序在有限样本下的性能。最后，我们通过股票收益的实证案例展示了sAGARCH模型的实用性和优势。