Near-integrated GARCH sequences

Details Near-integrated GARCH sequences Near-integrated GARCH sequences

2005-03-24

arxiv.org/abs/math/0503520v1

Annals of Applied Probability 2005, Vol. 15, No. 1B, 890-913

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051604000000783 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051604000000783

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=\sigma_k\epsilon_k, \sigma_k^2=\omega +\alpha y_{k-1}^2+\beta \sigma_{k-1}^2 for which the sum \alpha +\beta 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 \gamma :=\alpha +\beta -1. We find assumptions under which the solutions exhibit increasing oscillations and show that these oscillations grow approximately like a power function if \gamma \leq 0 and exponentially if \gamma >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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