Asymmetric GARCH modelling without moment conditions

Abstract: 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.