Autoregressive Fractionally Integrated Moving Average - ARFIMA(0,d,0)

ARFIMA(0,d,0)

The simplest autoregressive fractionally integrated model, ARFIMA(0,d,0), is, in standard notation,

where this has the interpretation

ARFIMA(0,d,0) is similar to fractional Gaussian noise (fGn): with d = H−½, their covariances have the same power-law decay. The advantage of fGn over ARFIMA(0,d,0) is that many asymptotic relations hold for finite samples. The advantage of ARFIMA(0,d,0) over fGn is that it has an especially simple spectral density—

f(λ) = (1/2π) (2sin(λ/2))−2d

—and it is a particular case of ARFIMA(p,d,q), which is a versatile family of models.