  Long and short memory in economic processes is usually described by the
so-called discrete fractional differencing and fractional integration. We prove
that the discrete fractional differencing and integration are the
Grunwald-Letnikov fractional differences of non-integer order d. Equations of
ARIMA(p,d,q) and ARFIMA(p,d,q) models are the fractional-order difference
equations with the Grunwald-Letnikov differences of order d. We prove that the
long and short memory with power law should be described by the exact
fractional-order differences, for which the Fourier transform demonstrates the
power law exactly. The fractional differencing and the Grunwald-Letnikov
fractional differences cannot give exact results for the long and short memory
with power law, since the Fourier transform of these discrete operators satisfy
the power law in the neighborhood of zero only. We prove that the economic
processes with the continuous time long and short memory, which is
characterized by the power law, should be described by the fractional
differential equations.
