The impulse response function of the long memory GARCH process

In this article we derive convenient representations for the cumulative impulse response function of the long memory GARCH(p, d, q) (LMGARCH) process. Our results extend the results in Baillie et al. (1996) [Baillie, R.T., Bollerslev, T., Mikkelsen, H.O. 1996. Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 74, 3-30.] on the first order LMGARCH. Using the derived impulse response functions we compare the persistence of shocks to the conditional variance in various GARCH models of interest such as stable, integrated and LMGARCH.