Standard time series practice offers a binary choice. Either the series is stationary and you model it directly, or it has a unit root and you difference it. The Dickey-Fuller test decides which.
Some series answer neither. Their autocorrelation decays too slowly for stationarity but too quickly for a random walk, and forcing them into either category discards real structure. Fractional integration describes the space in between.
Two Kinds of Decay
The distinction is visible in how the autocorrelation function falls away.
A stationary ARMA process has autocorrelation decaying exponentially: $\rho_k \sim \phi^k$. After a few dozen lags it is indistinguishable from zero, and shocks die out quickly.
| A long memory process has autocorrelation decaying hyperbolically: $\rho_k \sim k^{2d-1}$ for a parameter $d$. This is far slower. Autocorrelations remain measurably positive at lag 100, sometimes lag 1000, and the sum $\sum_k | \rho_k | $ diverges — which is the formal definition of long memory. |
The practical signature is a correlogram that declines steadily without ever quite reaching zero, combined with a Dickey-Fuller test that is ambiguous or rejects weakly.
Fractional Differencing
Ordinary differencing applies the operator $(1-L)$ where $L$ is the lag operator. Differencing twice applies $(1-L)^2$. The integration order $d$ is a whole number by construction.
Fractional integration allows $d$ to be any real value, defining $(1-L)^d$ through its binomial expansion:
\[(1-L)^d = \sum_{k=0}^{\infty} \binom{d}{k}(-L)^k = 1 - dL - \frac{d(1-d)}{2}L^2 - \frac{d(1-d)(2-d)}{6}L^3 - \cdots\]For integer $d$ the series terminates and recovers ordinary differencing. For fractional $d$ it does not terminate: every past observation receives a weight, decaying slowly. That infinite, slowly decaying weighting is precisely what “long memory” means operationally.
The parameter partitions the behaviour:
| $d$ | Behaviour |
|---|---|
| $d = 0$ | Short memory; standard ARMA |
| $d \in (0, 0.5)$ | Stationary with long memory |
| $d \in [0.5, 1)$ | Non-stationary but mean-reverting |
| $d = 1$ | Unit root; random walk |
The range $d \in (0, 0.5)$ is the interesting one: the series is stationary, so standard asymptotics apply, yet shocks persist far longer than any ARMA model would allow.
ARFIMA($p,d,q$) combines fractional differencing with ARMA terms, letting $d$ capture the long-run persistence while $p$ and $q$ handle short-run dynamics.
import numpy as np
def frac_diff_weights(d, n):
"""Binomial expansion weights for (1-L)^d, truncated at n terms."""
w = np.zeros(n)
w[0] = 1.0
for k in range(1, n):
w[k] = w[k - 1] * (k - 1 - d) / k
return w
for d in (0.0, 0.3, 0.5, 1.0):
w = frac_diff_weights(d, 6)
print(f"d={d}: weights {np.round(w, 4)}")
# the filter weights and the process autocorrelation decay at DIFFERENT rates
print("\nfilter weights |w_k| ~ k^(-d-1), at lag 100:")
for d in (0.1, 0.3, 0.45):
print(f" d={d}: {abs(frac_diff_weights(d, 200)[100]):.6f}")
print("process autocorrelation rho_k ~ k^(2d-1), at lag 100:")
for d in (0.1, 0.3, 0.45):
print(f" d={d}: {100.0 ** (2 * d - 1):.5f}")
At $d = 1$ the weights are exactly $[1, -1, 0, 0, \dots]$, which is ordinary first differencing. At $d = 0$ they are $[1, 0, 0, \dots]$, leaving the series untouched. At fractional $d$ the expansion never terminates: every past observation receives a non-zero weight, and that is the operational meaning of long memory.
Two decay rates are easy to confuse here, and the output separates them deliberately. The filter weights fall off as $k^{-d-1}$, so a larger $d$ makes them decay faster. The process autocorrelation falls off as $k^{2d-1}$, so a larger $d$ makes it decay slower — at lag 100, $d = 0.45$ still leaves a correlation around 0.63 while $d = 0.1$ has dropped to 0.03. It is the second quantity that carries the memory; the first is just the filter needed to remove it.
Estimating d
Several approaches exist, differing in robustness and assumptions.
The rescaled range statistic, from Hurst’s work on Nile flooding, estimates the Hurst exponent $H$, related to the differencing parameter by $d = H - 0.5$. It is intuitive and known to be biased in small samples and sensitive to short-run correlation.
The GPH estimator regresses the log periodogram on log frequency near zero, exploiting the fact that the spectral density of a long memory process diverges at the origin like $f(\lambda) \sim \lambda^{-2d}$. It is semiparametric, requiring no model for the short-run dynamics, at the cost of depending on how many frequencies you include.
Exact maximum likelihood on a full ARFIMA specification is efficient when the model is right and sensitive to misspecification when it is not.
A recurring difficulty deserves emphasis: long memory and structural breaks are easy to confuse. A stationary series with occasional level shifts produces slowly decaying sample autocorrelation that mimics long memory closely. Estimating $d$ from such a series yields a significant value that describes breaks rather than persistence. Testing for breaks before concluding long memory is not optional.
Where It Appears
Long memory shows up consistently in a few domains.
Financial volatility is the strongest case. Returns themselves are close to unpredictable, but absolute or squared returns show autocorrelation persisting over hundreds of days — which motivated FIGARCH and related fractionally integrated volatility models.
Hydrology is where the phenomenon was first quantified. Hurst’s analysis of Nile river levels found persistence that standard models could not reproduce, and the Hurst exponent carries his name for that reason.
Network traffic, inflation, and some climate series also display it.
In machine learning, fractional differencing has found a use worth noting: differencing a price series to stationarity destroys almost all of its memory, whereas fractional differencing with the smallest $d$ that achieves stationarity preserves considerably more information while satisfying the model’s requirements.
Practical Cautions
Do not conclude long memory from a slowly decaying correlogram alone — check for breaks and trends first, since both produce the same appearance.
Be careful with forecasts. Long memory implies shocks persist, which makes long-horizon forecasts more sensitive to the estimated $d$ than to anything else in the model. Small errors in $d$ compound over the horizon.
And weigh the cost. ARFIMA is harder to fit, harder to explain, and frequently no better than a well-specified short-memory model at short horizons. It earns its place when the persistence itself is the object of study — as in volatility modelling — rather than when it is a marginal accuracy improvement.
References
- 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.
- Hosking, J. R. M. (1981). Fractional differencing. Biometrika, 68(1), 165-176.
- Geweke, J., & Porter-Hudak, S. (1983). The estimation and application of long memory time series models. Journal of Time Series Analysis, 4(4), 221-238.
- Baillie, R. T. (1996). Long memory processes and fractional integration in econometrics. Journal of Econometrics, 73(1), 5-59.
- Diebold, F. X., & Inoue, A. (2001). Long memory and regime switching. Journal of Econometrics, 105(1), 131-159.
