Tail behaviour of ββ-TARCH models

It is now common knowledge that the simple quadratic ARCH process has a regularly varying tail even when generated by a normally distributed noise, and the tail behaviour is well-understood under more general conditions as well. Much less studied is the case of ββ-ARCH-type processes, i.e. when the conditional variance is a 2β2β-power function with 0<β<10<β<1. Opting for a little more generality and allowing for asymmetry, we consider threshold ββ-ARCH models, driven by noises with Weibull-like tails. (Special cases include the Gaussian and the Laplace distributions.) We show that the process generated has an approximately Weibull-like tail, too, albeit with a different exponent: 1−β1−β times that of the noise, in the sense that the tail can be bounded from both sides by Weibull distributions of this exponent but slightly different constants. The proof is based on taking an appropriate auxiliary sequence and then applying the general result of Klüppelberg and Lindner (2005) for the tail of infinite MA sequences with light-tailed innovations.