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IMPULSE RESPONSES OF FRACTIONALLY INTEGRATED PROCESSES WITH LONG MEMORY
Published online by Cambridge University Press: 08 April 2010
Abstract
Fractionally integrated time series, which have become
an important modeling tool over the last two
decades, are obtained by applying the fractional
filter to a weakly
dependent (short memory) sequence. Weakly dependent
sequences are characterized by absolutely summable
impulse response coefficients of their Wold
representation. The weights
bn decay at the rate
nd−1 and
are not absolutely summable for long memory models
(d > 0). It has been believed
that this rate is inherited by the impulse responses
of any long memory fractionally integrated model. We
show that this conjecture does not hold in such
generality, and we establish a simple necessary and
sufficient condition for the rate
nd−1 to be
inherited by fractionally integrated processes.
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- Copyright © Cambridge University Press 2010
Footnotes
This research was partially supported by NSF grants DMS-0804165 and DMS-0931948 at Utah State University. We thank Stanley Williams for suggesting Example 2.1. We thank three referees and Professor P.C.B. Phillips for comments that led to substantive improvements of our results and Professor Giuseppe Cavaliere for efficiently handling this submission.
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