> [!tldr] > An ARFIMA Model $\mathrm{ARFIMA}(p,d,q)$ is a generalization of [[ARIMA Models|ARIMA models]] where the degree of differencing $d$ is not restricted to integers, so the differenced time series is defined as a binomial expansion: $\begin{align*} W_{t}&= (1-B)^{d}X_{t}\\ &= \left[ 1-dB+\frac{d(d-1)}{2!}B^{2}- \cdots \right]X_{t} \end{align*}$ ### Stationarity An ARFIMA model is stationary if $|d| < 0.5$, and since differencing an ARFIMA process reduces its $d$ by 1, any ARFIMA process with $d > 0$ can be made stationary by differencing. ARFIMA models with $d \in (0, 0.5)$ are particularly relevant, being stationary and **long-memory models**, where the ac.f. decay slowly. ```R fold library("fracdiff") library("dplyr") arfima <- fracdiff.sim arfima_data <- arfima(n = 2000, ar = 0.3, ma = c(-0.3,0.5), d = 0.4)$series dev.off() par(mfrow=c(2,1), mai=c(1,4,1,1)/4, oma=c(2,0,0,0), bg=NA) plot(arfima_data, type="l", ylab="Series Value", main="ARFIMA(1, 0.4, 2) Process") acf(arfima_data, ylab = "Autocorr.", main="") ``` ![[ARFIMA0.4ACF.png#invert]] More precisely, long-memory models have the ac.f.s $\rho(k)$ satisfy $\sum_{k}|\rho(k)| = \infty$, and ARFIMA models with $d \in (0,0.5)$ have $\rho(k) =O(k^{2d-1})$, satisfying the definition.