Hosking, J. R. M. (1981).
Fractional differencing.
Biometrika, 68, 165-176.
Abstract.
The family of autoregressive integrated moving-average processes, widely
used in time series analysis, is generalized by permitting the degree of
differencing to take fractional values. The fractional differencing
operator is defined as an infinite binomial series expansion in powers
of the backward-shift operator. Fractionally differenced processes
exhibit long-term persistence and antipersistence; the dependence
between observations a long time span apart decays much more slowly with
time span than is the case with the more commonly studied time series
models. Long-term persistent processes have applications in economics
and hydrology; compared to existing models of long-term persistence,
the family of models introduced here offers much greater flexibility in
the simultaneous modelling of the short-term and long-term behaviour of
a time series.
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