Skip to content

ARFIMA Processes

The Autoregressive Fractionally Integrated Moving Average (ARFIMA) process is a generalized version of the ARIMA model. It is specifically designed to model time series that exhibit long-range dependence, where the effects of shocks decay much more slowly than in standard ARMA models.

1. Beyond ARIMA: The Need for Fractional Integration

Standard L11__Non-stationary Time Series models (ARIMA) require a finite integer number of differences (usually \(d=1\) or \(d=2\)) to achieve stationarity. However, they struggle to capture persistence.

  • ARMA/ARIMA: Features short-memory where the ACF drops off exponentially.
  • ARFIMA: Features long-memory where the ACF decays hyperbolically (slowly), allowing the model to "remember" shocks from the distant past.

2. Fractionally Integrated Noise: \(ARFIMA(0, d, 0)\)

A series \(Y_{t}\) is considered fractionally integrated noise if it is a stationary solution to:
$\((1-B)^d Y_{t} = e_{t}\)$
Where \(-0.5 < d < 0.5\) and \(e_{t}\) is white noise with variance \(\sigma_{e}^{2}\).

Infinite Moving Average Representation

We can express the process as an infinite sum of past shocks:
$\(Y_{t} = (1-B)^{-d} e_{t} = \sum_{j=0}^\infty \psi_{j} e_{t-j}\)$
The weights \(\psi_{j}\) are determined using the Gamma function \(\Gamma\):
$\(\psi_{j} = \dfrac{\Gamma(j+d)}{\Gamma(j+1)\Gamma(d)}\)$

3. The Role of the Fractional Parameter \(d\)

The value of \(d\) determines the "memory" and stationarity of the process:

Value of \(d\) Process Characteristic Memory Type
\(0 < d < 0.5\) Persistent / Long Memory Shocks decay very slowly (hyperbolically).
\(d = 0\) ARMA Process Short memory; standard stationarity.
\(-0.5 < d < 0\) Anti-persistent / Intermediate Mean-reverting; negative autocorrelations.
\(d \geq 0.5\) Non-stationary Variance increases with time.
\(d = 1\) ARIMA Process Unit root process (Random Walk).

Stirling's Approximation

Using Stirling’s approximation (\(\Gamma(x) \approx \sqrt{2\pi} x^{x-1/2} e^{-x}\) as \(x \to \infty\)), we can show that for large lags \(k\), the autocorrelation \(\rho_{k}\) is proportional to \(k^{2d-1}\). This power-law decay is exactly what defines a long-memory process.

4. General ARFIMA Structure

The complete \(ARFIMA(p, d, q)\) model combines fractional integration with standard autoregressive and moving average components to capture both short-term dynamics and long-term persistence simultaneously:
$\((1-B)^d \Phi(B) Y_{t} = \Theta(B) e_{t}\)$