Near-integrated GARCH sequences

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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=σₖεₖ, σₖ²=ω+αyₖ₋₁²+βσₖ₋₁² for which the sum α+βapproaches 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 γ:=α+β-1. We find assumptions under which the solutions exhibit increasing oscillations and show that these oscillations grow approximately like a power function if γ≤ 0 and exponentially if γ>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.