ARFIMA(p, d, q) model

Slowly but surely we have finally reached ARFIMA model! Taking such small step (adding letter F to the acronym) took a lot of effort, but it was worth it. Why? Well, this exercise has allowed me to get a glimpse at fractional calculus and develop some intuition with this tool.

ARFIMA(0, d, 0) is also one of the ways to generate fractional Gaussian noise (abbr. fGn). It is fractional derivative of order \( d \) of the ordinary Gaussian white noise. fGn is important in our field as the integral of fGn is fractional Brownian motion (abbr. fBm) one of the stochastic processes, which encodes true long range memory. Relationship between Hurst index of the generated fGn is given by:

\begin{equation} H = d + \frac{1}{2} . \end{equation}

Confusingly if we integrate fGn with Hurst index \( H \) we obtain fBm with the same Hurst index (this confusion is somewhat resolved by MF-DFA method).

Using the app below observe that for \( d > 0 \) noise appears to be somewhat persistent - there are clumps of similar values grouped together as if the process would be not fGn, but fBm. For \( d < 0 \) noise becomes anti-persistent - it rapidly changes value and sign.