Near-integrated GARCH sequences

Motivated by regularities observed in time series of returns on speculative assets, we develop an asymptotic theory of GARCH(1,1) processes {yk} defined by the equations yk=.k.k, .k2=.+.yk.12+..k.12 for which the sum .+. approaches unity as the number of available observations tends to infinity. We call such sequences near-integrated. We show that the asymptotic behavior of near-integrated GARCH(1,1) processes critically depends on the sign of .:=.+..1. We find assumptions under which the solutions exhibit increasing oscillations and show that these oscillations grow approximately like a power function if ..0 and exponentially if .>0. We establish an additive representation for the near-integrated GARCH(1,1) processes which is more convenient to use than the traditional multiplicative Volterra series expansion.