Impulse Responses of Fractionally Integrated Processes With Long Memory

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 = ∑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 b n decay at the rate n d −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 n d −1 to be inherited by fractionally integrated processes.