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

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