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Near-integrated GARCH sequences

Istvan Berkes·Lajos Horvath·Piotr Kokoszka
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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_kε_k, σ_k^2=ω+αy_{k-1}^2+βσ_{k-1}^2 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\leq 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.

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Istvan Berkes, Lajos Horvath, Piotr Kokoszka. 2005-03-24. Near-integrated GARCH sequences. https://doi.org/10.1214/105051604000000783

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