28 research outputs found
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 {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.Comment: Published at http://dx.doi.org/10.1214/105051604000000783 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org