Near-integrated GARCH sequences

Abstract

Motivated by regularities observed in time series of returns on speculative assets, we develop an asymptotic theory of GARCH(1,1) processes {y_k} defined by the equations y_k=σ_kɛ_k, σ_k²=ω+αy_k−1²+βσ_k−1² for which the sum α+β 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 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.