Autoregressive Fractionally Integrated Moving Average

Basics

In an ARIMA model, the integrated part of the model includes the differencing operator (1 − B) (where B is the backshift operator) raised to an integer power. For example

where

so that

In a fractional model, the power is allowed to be fractional, with the meaning of the term identified using the following formal binomial series expansion

$\begin{align} (1 - B)^d &= \sum_{k=0}^{\infty} \; {d \choose k} \; (-B)^k \\ & = \sum_{k=0}^{\infty} \; \frac{\prod_{a=0}^{k-1} (d - a)\ (-B)^k}{k!}\\ &=1-dB+\frac{d(d-1)}{2!}B^2 -\cdots \, . \end{align}$

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Autoregressive Fractionally Integrated Moving Average - Basics