Limit theory and robust evaluation methods for the extremal properties of GARCH(p, q) processes

Fabrizio Laurini, Paul Fearnhead, Jonathan A. Tawn

Abstract: Generalized autoregressive conditionally heteroskedastic (GARCH) processes are widely used for modelling financial returns, with their extremal properties being of interest for market risk management. For GARCH( $$p,q$$ p , q ) processes with $$\max (p,q) = 1$$ max ( p , q ) = 1 all extremal features have been fully characterised, but when $$\max (p,q)\ge 2$$ max ( p , q ) ≥ 2 much remains to be found. Previous research has identified that both marginal and dependence extremal features of strictly stationary GARCH( $$p,q$$ p , q ) processes are determined by a multivariate regular variation property and tail processes. Currently there are no reliable methods for evaluating these characterisations, or even assessing the stationarity, for the classes of GARCH( $$p,q$$ p , q ) processes that are used in practice, i.e., with unbounded and asymmetric innovations. By developing a mixture of new limit theory and particle filtering algorithms for fixed point distributions we produce novel and robust evaluation methods for all extremal features for all GARCH( $$p,q$$ p , q ) processes, including ARCH and IGARCH processes. We investigate our methods’ performance when evaluating the marginal tail index, the extremogram and the extremal index, the latter two being measures of temporal dependence.