Long-Memory Models (ARFIMA, FIGARCH)
Also known as: ARFIMA, FIGARCH, fractionally integrated models, fractional integration, Uzun Hafıza Modelleri (ARFIMA, FIGARCH)
Long-memory models are fractional-integration methods that capture genuine long memory through a hyperbolically decaying autocorrelation structure. ARFIMA, introduced by Granger and Joyeux (1980), models long memory in return series, while FIGARCH, introduced by Baillie, Bollerslev and Mikkelsen (1996), captures long memory in volatility series; the parameter d measures the degree of fractional integration.
Key highlights
- Captures genuine long memory through a hyperbolically (rather than geometrically) decaying autocorrelation structure that short-memory ARMA/GARCH models cannot reproduce.
- A single fractional parameter d spans the gap between stationary I(0) and non-stationary I(1) behaviour instead of forcing a 0-or-1 choice.
- Fractional and short-memory dynamics can be combined, so ARMA mean structure or GARCH volatility structure is preserved alongside the long-memory component.
Intuition
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How it works
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When to use it
Use long-memory models for continuous financial time series of at least about 200 observations that display slowly decaying, persistent autocorrelation. ARFIMA suits return or level series with long memory in the mean, while FIGARCH suits volatility series with persistent conditional heteroskedasticity. The integration order should fall in 0 < d < 1 (stationary for d < 0.5), d is estimated by R/S analysis or GPH, FIGARCH requires β + δ < 1 for stability, and short-memory ARMA components can be combined alongside the fractional part. They are not appropriate for short series or for purely short-memory data where a standard ARMA or GARCH model already fits.
Strengths & limitations
- Captures genuine long memory through a hyperbolically (rather than geometrically) decaying autocorrelation structure that short-memory ARMA/GARCH models cannot reproduce.
- A single fractional parameter d spans the gap between stationary I(0) and non-stationary I(1) behaviour instead of forcing a 0-or-1 choice.
- Fractional and short-memory dynamics can be combined, so ARMA mean structure or GARCH volatility structure is preserved alongside the long-memory component.
- Requires a long series (at least about 200 observations) to estimate the fractional order d reliably.
- Estimation is sensitive to the chosen d-estimator (R/S versus GPH) and to contamination from structural breaks, which can masquerade as long memory.
- FIGARCH is only well behaved when the stability condition β + δ < 1 holds; outside it the model is invalid.
Common pitfalls
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Applications
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Frequently asked
What does the parameter d mean?
d is the fractional order of integration. It measures how persistent the memory is: 0 < d < 0.5 gives stationary long memory, while 0.5 ≤ d < 1 gives non-stationary long memory. A standard ARMA model corresponds to d = 0 and a unit-root I(1) process to d = 1.
How is d estimated?
d is typically estimated before the ARMA part, using rescaled-range (R/S) analysis or the GPH log-periodogram regression. The fractional differencing operator (1−L)^d is then applied and the remaining short-memory parameters are fitted by maximum likelihood.
What is the difference between ARFIMA and FIGARCH?
ARFIMA models long memory in the level or mean of a series (for example returns), whereas FIGARCH models long memory in the conditional variance (volatility). FIGARCH additionally requires the stability condition β + δ < 1.
How much data do I need?
At least about 200 observations. Estimating a fractional integration order reliably needs a long series, because long memory is a property of the slow, distant decay of the autocorrelations.
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.Baillie, R. T., Bollerslev, T. & Mikkelsen, H. O. (1996). Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 74(1), 3-30.
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ScholarGate. (2026, June 1). Long-Memory Models. ScholarGate. https://scholargate.app/finance/long-memory-models