Abstract :
Fractionally integrated time series, which have become an important modeling tool
over the last two decades, are obtained by applying the fractional filter (1−L)−d = Σ∞n =0 bn Ln 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.
