Long Memory and Fractional Integration

Definition

Long memory (long-range dependence) refers to a stochastic process whose autocorrelations decay hyperbolically rather than geometrically, so that their sum diverges and the spectral density is unbounded at frequency zero. The fractional difference parameter $d$ (equivalently, Hurst exponent $H = d + \tfrac{1}{2}$ ) quantifies the degree of dependence: $d = 0$ is short memory, $0 < d < \tfrac{1}{2}$ is stationary long memory, $d = \tfrac{1}{2}$ is the boundary, and $d = 1$ is the $I(1)$ unit root. The autoregressive fractionally integrated moving average (ARFIMA) class nests all of these as special cases.

Key Ideas

Long memory generalizes the $I(0)$ / $I(1)$ dichotomy: a series can be integrated of any real order $d$ .
In the time domain, autocorrelations decay as $j^{2H-2}$ (hyperbolically); in the frequency domain, the spectral density diverges as $|\lambda|^{1-2H}$ near zero.
ARFIMA(p,d,q) provides the standard parametric model; semi-parametric estimators (Geweke-Porter-Hudak (GPH), local Whittle) estimate $d$ robustly without fully specifying the autoregressive moving average (ARMA) part.
Fractional cointegration: two $I(d)$ series may share a common trend such that their linear combination is $I(d-b)$ for some $0 < b \leq d$ .
Structural breaks and regime switching can generate hyperbolic autocorrelation decay in finite samples, mimicking genuine long memory (Diebold-Inoue 2001, Granger-Hyung 2004); disentangling the two is an active research challenge.

How It Works

Formal Definitions

Time domain (McLeod-Hipel 1978): a stationary process has long memory if $\sum_{j=-n}^{n} |\rho_j| \to \infty$ as $n \to \infty$ ; equivalently $\rho_j / (c_\rho j^{2H-2}) \to 1$ for some $0 < c_\rho < \infty$ and $H \in (\tfrac{1}{2}, 1)$ .

Frequency domain: spectral density $f(\lambda) \sim c_f |\lambda|^{1-2H}$ as $\lambda \to 0$ , i.e., an unbounded pole at the origin. Both definitions are connected via the Hurst exponent $H$ ; long memory occurs when $H \in (\tfrac{1}{2}, 1)$ .

Fractional White Noise

$y_t = (1-L)^{-d} \varepsilon_t$ where $L$ is the lag operator, $d = H - \tfrac{1}{2}$ , and $\varepsilon_t$ is white noise (Granger 1980, Hosking 1981). Properties:

Stationary for $d < \tfrac{1}{2}$ ; invertible for $d > -\tfrac{1}{2}$ .
Autocorrelations proportional to $j^{2d-1}$ for large $j$ (hyperbolic decay for $d > 0$ ).
Spectral density proportional to $|\lambda|^{-2d}$ near zero.

ARFIMA(p,d,q)

$\phi(L)(1-L)^d (y_t - \mu) = \theta(L) \varepsilon_t$ introduced by Granger and Joyeux (1980) and Hosking (1981), where $\phi(L)$ and $\theta(L)$ have all roots outside the unit circle and $\varepsilon_t$ is white noise.

Covariance stationary for $|d| < \tfrac{1}{2}$ .
Mean-reverting for $d < 1$ .
For $d > \tfrac{1}{2}$ : infinite variance (but finite-sample implementations use autoregressive integrated moving average (ARIMA) initial conditions).
ARFIMA(0,d,0) reduces to fractional white noise; ARFIMA(p,0,q) reduces to ARMA.

GPH Log-Periodogram Estimator (Geweke-Porter-Hudak 1983)

Regress the log-periodogram on log-frequency near zero: $\log I(\omega_j) = a + b \log(\omega_j) + u_j$ for $j = 1,\ldots,m$ , where $\omega_j = 2\pi j/T$ are Fourier frequencies. The ordinary least squares (OLS) estimate $-\hat{b}/2$ gives $\hat{d}$ ; asymptotically normal with standard error $\pi(24m)^{-1/2}$ . Bandwidth $m = \sqrt{T}$ is the common rule of thumb. A larger $m$ reduces standard error but induces bias (the log-periodogram approximation holds only near zero); a smaller $m$ reduces bias but inflates variance. Robinson (1995a) refines the estimator to improve small-sample properties.

Local Whittle Estimator (Robinson 1995b)

Minimizes a frequency-domain approximate Gaussian log-likelihood averaged over a neighborhood of zero: $Q(d) = \log\!\left(m^{-1} \sum_{j=1}^{m} \lambda_j^{2d} I(\lambda_j)\right) - (2d)\, m^{-1} \sum_{j=1}^{m} \log \lambda_j$ Asymptotically normal and more efficient than GPH. Extended to non-stationary $d > \tfrac{1}{2}$ by Velasco-Robinson (2000).

R/S and Modified R/S Statistics

R/S (Hurst 1951): range of partial-sum deviations from mean, divided by sample standard deviation: $Q_n = \frac{R_n}{S_n} = \frac{\max_{1\leq k\leq n} \sum_{j=1}^{k}(X_j - \bar{X}_n) - \min_{1\leq k\leq n} \sum_{j=1}^{k}(X_j - \bar{X}_n)}{S_n}$ Under long memory, $Q_n / n^H$ converges to a constant $> \tfrac{1}{2}$ . Under short memory, $H = \tfrac{1}{2}$ . Modified R/S (MR/S) (Lo 1991): replaces $S_n$ with a Newey-West long-run standard deviation estimate $\sigma(q) = \bigl[c_0 + 2\sum_{j=1}^{q} w_j(q) c_j\bigr]^{1/2}$ , where $w_j(q) = 1 - j/(q+1)$ are Bartlett weights. Controls for short-memory autocorrelation and heteroscedasticity; the standard R/S statistic is biased toward detecting long memory in the presence of short-memory dependence.

FIGARCH: Long Memory in Conditional Variance (Baillie-Bollerslev-Mikkelsen 1996)

The GARCH(p,q) (generalized autoregressive conditional heteroskedasticity) conditional variance equation can be rewritten as an ARMA in $\varepsilon_t^2$ : $\{1 - \alpha(L) - \beta(L)\}\varepsilon_t^2 = \omega + \{1-\beta(L)\}v_t$ When $\{1-\alpha(L)-\beta(L)\}$ has a unit root the process is integrated GARCH (IGARCH; infinite persistence). Fractionally integrated GARCH (FIGARCH) replaces that unit root with a fractional difference: $\phi(L)(1-L)^d \varepsilon_t^2 = \omega + \{1-\beta(L)\}v_t, \quad 0 < d < 1$ yielding impulse-response weights in the conditional variance that decay hyperbolically at rate $j^{d-1}$ rather than exponentially. Properties:

$d = 0$ : reduces to stable GARCH (geometric decay).
$d = 1$ : reduces to IGARCH (unit root, infinite persistence).
$0 < d < 1$ : intermediate — shocks to variance are persistent but eventually die out; conditional variance is covariance nonstationary but finite-variance impulse responses. Applied to daily USD exchange rates by Baillie-Bollerslev-Mikkelsen (1996): estimated $\hat d \approx 0.4$ – $0.6$ , substantially better log-likelihood than GARCH and more parsimonious than IGARCH. Caveat: The original FIGARCH parameterization can generate negative impulse-response weights for some parameter combinations, which violates non-negativity of the conditional variance. Later modifications (e.g., truncated FIGARCH, HYGARCH) address this constraint.

Fractional Cointegration

If $y_t \sim I(d)$ and $x_t \sim I(d)$ , they are fractionally cointegrated of order $(d,b)$ if $z_t = y_t - \beta x_t \sim I(d-b)$ for $0 < b < d$ . The error correction representation (Granger 1986): $\Phi(L)(1-L)^d y_t = -\gamma\bigl(1-(1-L)^b\bigr) z_{t-1} + c(L) \varepsilon_t$ Testing for fractional cointegration reduces to testing for fractional integration in the estimated residuals. Dittmann (2004) proposes a three-step error correction model (ECM) estimator for fractionally cointegrated systems.

Sign Tests for d (Delgado-Velasco 2005)

Standard tests for the long-memory parameter (Tanaka 1999 score test, Dickey-Fuller for the unit root boundary $d=1$ ) require finite variance innovations and lose all power under stable or heavy-tailed errors. Delgado and Velasco (2005) construct tests based only on the signs of fractionally filtered residuals, which are Rademacher ( $\pm 1$ equiprobable) under any symmetric continuous $H_0$ distribution.

Test statistic for $H_0$ : $d = d_0$

$T_n(d_0) = n\sum_{j=1}^{n-1} \frac{1}{j} \tilde\gamma_{nj}(d_0), \qquad \tilde\gamma_{nj}(d_0) = \frac{1}{n}\sum_{t=j+1}^n S_t(d_0) S_{t-j}(d_0)$

where $S_t(d_0) = \text{sign}(\hat\varepsilon_t(d_0))$ and $\hat\varepsilon_t(d_0) = (1-L)^{d_0} u_t$ .

Under $H_0$ , $\{S_t(d_0)\}$ are i.i.d. $\pm 1$ regardless of the innovation distribution, so $T_n$ has an exact distribution computable by Monte Carlo from $2^n$ equi-probable sign sequences. Theorem 1 proves $T_n$ is locally most powerful (LMP) among sign tests: it maximizes the derivative of power with respect to $d$ at $d_0$ .

Asymptotic distribution (slow convergence):

$n^{-1/2} T_n(d_0) \overset{d}{\to} \mathcal{N}\!\left(0,\, \frac{\pi^2}{6}\right) \quad \text{under H}_0$

but convergence is very slow — at $n=2000$ the empirical size is still only 4.8% vs. nominal 5%. Exact Monte Carlo critical values are always needed in practice.

Composite ARFIMA test (Section 4): replace unknown ARMA parameters with $\sqrt{n}$ -consistent robust estimates (least absolute deviations (LAD) or sign-based), then form the projected score statistic $L_n^p \xrightarrow{d} \chi^2(1)$ . The test remains valid under infinite-variance innovations, which rule out standard maximum likelihood estimation (MLE).

Monte Carlo comparison ( $H_0$ : $d=1$ ; alternatives $d = 1-a/\sqrt{n}$ ):

Test	$\mathcal{N}(0,1)$	$t_4$	$t_2$	$t_1$ (Cauchy)
Sign (exact)	8.3%	9.1%	10.7%	16.2%
Dickey-Fuller	6.4%	6.3%	5.5%	3.2%
Tanaka	6.0%	5.6%	4.3%	1.6%

(Power at $n=400$ , $a=0.5$ , 5% level. DF and Tanaka use asymptotic critical values; sign test uses exact.)

Sign test power increases with heavier tails while DF and Tanaka lose power — under Cauchy errors DF and Tanaka are essentially useless (below nominal size), while the sign test maintains validity.

Practical guidance: Use the sign test as a robustness check when innovation distributions may be fat-tailed or when infinite variance is plausible (finance, telecommunications, hydrology). Always use exact or Monte Carlo quantiles rather than the normal approximation.

Interface with Structural Breaks

Diebold-Inoue (2001) show that stochastic regime switching (Markov-switching, random-level-shift, and permanent-break models) satisfies the variance-of-partial-sums definition of long memory for any $d > 0$ in finite samples. Granger-Hyung (2004) show a linear process with occasional mean shifts generates sample autocorrelations consistent with ARFIMA(0,d,0). Conversely, Hillebrand (2004) shows ignored GARCH regime changes cause estimated $\alpha+\beta \to 1$ , the variance analogue. These results mean any finding of long memory should be preceded by a structural break test; see Structural Break Testing.

Why It Matters

ARFIMA is more parsimonious than multiple unit-root or highly-parameterized AR models for describing gradual decay in autocorrelation.
The $d$ parameter has direct economic content: $d$ close to $\tfrac{1}{2}$ implies slow mean reversion (long business cycle persistence in inflation, real interest rates); $d$ near $0$ implies rapid decay.
Semi-parametric estimators (GPH, local Whittle) are robust to ARMA misspecification, making them practical screening tools.
Fractional cointegration allows for weaker long-run equilibria than classical I(1) cointegration, appropriate when adjustment is extremely slow.
Long memory in variance (FIGARCH, long-memory stochastic volatility) has implications for option pricing and risk management beyond standard GARCH models.

Open Questions

No consensus on optimal bandwidth $m$ for GPH/local Whittle; data-driven selection rules exist but have not converged.
Distinguishing genuine long memory from breaks or regime switching in finite samples remains unresolved; see Structural Break Testing.
Panel and multivariate generalizations are less mature than univariate tests.
Bayesian inference for ARFIMA models is computationally demanding; Doornik-Ooms (2003) exact MLE requires $O(n^2)$ computation.