ARFIMA: Fractionally Integrated ARMA Model
Also known as: fractionally integrated ARMA, long-memory time series model, ARFIMA / FIGARCH, fractional differencing model, Kesirli Bütünleşik ARMA (ARFIMA / FIGARCH)
ARFIMA is a time series model that captures long-memory behaviour using a fractional differencing parameter d, generalising the integer differencing of ARIMA. It was introduced by Granger and Joyeux (1980) and formalised by Hosking (1981) to describe series whose autocorrelations decay slowly rather than abruptly.
Key highlights
- Captures long-memory (slowly decaying) dependence that integer-differenced ARIMA models cannot represent.
- The single fractional parameter d gives a parsimonious, interpretable measure of persistence.
- Bridges the gap between stationary short-memory ARMA (d = 0) and a unit-root process (d = 1), avoiding both under- and over-differencing.
Intuition
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How it works
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When to use it
Use ARFIMA when you have a long continuous time series (at least about 100 observations) whose autocorrelations decay slowly, signalling long memory rather than a short-memory ARMA process or a unit root. It is appropriate when the estimated d lies in the (0, 0.5) range, so the series is stationary but persistent; this should be confirmed with a GPH or ELW estimate before fitting. It is common for financial volatility, macroeconomic series, and hydrological data. If d is 0.5 or above the series is non-stationary, and at d equal to 1 it reduces to an ordinary integer-differenced ARIMA.
Strengths & limitations
- Captures long-memory (slowly decaying) dependence that integer-differenced ARIMA models cannot represent.
- The single fractional parameter d gives a parsimonious, interpretable measure of persistence.
- Bridges the gap between stationary short-memory ARMA (d = 0) and a unit-root process (d = 1), avoiding both under- and over-differencing.
- Requires a long series (about 100 observations or more) to estimate d reliably.
- Only valid for stationary long memory in the (0, 0.5) range; if d ≥ 0.5 the series is non-stationary and ARFIMA no longer applies.
- The short-run ARMA components must be modelled separately, and estimates of d can be sensitive to the chosen estimator and bandwidth.
Common pitfalls
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Applications
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Frequently asked
What does the fractional parameter d mean?
d measures the degree of long memory. When d = 0 the series is a standard short-memory ARMA process; as d rises toward 0.5 the series becomes increasingly persistent while remaining stationary; at d = 1 it is an ordinary integer-differenced (ARIMA) series. Values of d in (0, 0.5) define the genuine long-memory case.
How is long memory detected before fitting ARFIMA?
Long memory shows up as autocorrelations that decay slowly (hyperbolically) rather than exponentially. The memory parameter d is estimated from the low-frequency part of the spectrum using the GPH (Geweke–Porter-Hudak) or exact local Whittle (ELW) estimator; a d significantly above zero supports an ARFIMA specification.
How is ARFIMA different from ARIMA?
ARIMA differences a series a whole number of times to make it stationary, which can over- or under-correct slowly decaying series. ARFIMA allows a fractional differencing amount d, so it can match long-memory persistence directly. ARIMA is the special case where d is an integer.
What if my estimate of d is 0.5 or larger?
Then the series is not covariance stationary and ARFIMA does not apply in the long-memory sense. At d ≥ 0.5 the variance is no longer finite in the usual way, and at d = 1 you should use a fully differenced ARIMA model instead.
Sources
- 1.Granger, C. W. J. & Joyeux, R. (1980). An Introduction to Long-Memory Time Series Models and Fractional Differencing. Journal of Time Series Analysis, 1(1), 15–29.
- 2.Hosking, J. R. M. (1981). Fractional Differencing. Biometrika, 68(1), 165–176.
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Cite this page
ScholarGate. (2026, June 1). ARFIMA Model. ScholarGate. https://scholargate.app/econometrics/arfima-model