Tail behaviour of -TARCH models

Péter Elek (1) , László Márkus (1)

1 Department of Probability Theory and Statistics

It is now common knowledge that the simple quadratic ARCH process has 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 -ARCH-type processes, i.e. when the conditional variance is a -power function with . Being a little more general and allowing for asymmetry, we consider threshold -ARCH models, driven by noises with Weibull-like tails. (Special cases include the Gaussian or the Laplace distributions.) We show that the generated process has approximately Weibull like tail, too, albeit with different exponent: 1- 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.

Mots clés: ARCH-type model; Conditional heteroscedasticity; Extreme value theory; Tail behaviour

Domaines: Bio-informatique [q-bio.QM]; Bio-Informatique, Biologie Systémique [q-bio.QM]

Péter Elek, László Márkus. Tail behaviour of -TARCH models. Statistics and Probability Letters, 2010, ⟨10.1016/j.spl.2010.07.020⟩. ⟨hal-00691789⟩