0.1 Long Memory Processes (ARFIMA)
Standard stationary ARMA processes have ACF decaying geometrically (\(\rho(h) \sim c\,r^h\)), while ARIMA with \(d\in\{1,2,\dots\}\) is fully non-stationary. Many empirical series — notably absolute/squared asset returns, realized volatility, and river-flow/hydrology data — show ACF decaying much more slowly, hyperbolically (\(\rho(h)\sim c\,h^{2d-1}\)), a signature of long memory (long-range dependence) intermediate between short-memory stationarity and unit-root non-stationarity.
Extend the differencing operator \((1-B)^d\) to non-integer \(d\) via the binomial series \[ (1-B)^d = \sum_{k=0}^\infty \binom{d}{k}(-B)^k, \qquad \binom{d}{k} = \frac{d(d-1)\cdots(d-k+1)}{k!}. \] \(\{X_t\}\) is \(ARFIMA(p,d,q)\) if \(\phi(B)(1-B)^dX_t = \theta(B)Z_t\) for a stationary invertible \(ARMA(p,q)\) driving equation. Key regimes: - \(d \in (0, 0.5)\): stationary with long memory (ACF not summable, \(\sum_h|\rho(h)|=\infty\), spectral density diverges at frequency \(0\)), - \(d=0\): short memory (\(ARMA\)), - \(d \in (-0.5,0)\): stationary “anti-persistent”/intermediate memory, - \(d \geq 0.5\): non-stationary (nests integer-\(d\) ARIMA at \(d=1\)).
Estimation of \(d\). Semi-parametric log-periodogram (Geweke-Porter-Hudak) regression using the low-frequency behavior of the periodogram; or parametric (Whittle) maximum likelihood jointly with \(ARMA\) parameters.
Connection to volatility. Squared/absolute return series typically show long memory empirically (slowly decaying ACF), motivating both direct \(ARFIMA\)/FIGARCH models of volatility and reduced-form approximations that mimic long memory with parsimonious short-memory structure, notably the HAR-RV model (which stacks daily/weekly/monthly averages rather than fractionally differencing).