What are long-memory processes and fractional integration in return series?

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AcadiFi · Accepted Answer

A stationary process has long memory when its autocorrelations decay at a hyperbolic rate rho(k) ~ k^(2d-1) for 0 < d < 0.5...

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Fractional integration (Granger and Joyeux, 1980) is defined through the operator (1 - L)^d, expanded as an infinite binomial series. Realized volatility of major equity indices typically has d around 0.4, meaning shocks decay very slowly but the series is still mean-reverting.

At Wrenfield, if you're forecasting one-month-ahead realized volatility, ignoring long memory leads to underreaction in your forecast after a volatility spike. ARFIMA(p,d,q) models explicitly capture this. Estimation uses Whittle likelihood in the frequency domain or local Whittle for semi-parametric d estimation.

Economic implications: long memory in volatility drives persistent VaR elevations, affects option pricing (term-structure of implied vols), and shapes optimal hedging horizons. For AcadiFi candidates, remember log-realized-variance follows a log-normal-like distribution with persistent autocorrelation — the HAR-RV model of Corsi (2009) approximates long memory with a tractable mix of daily, weekly, and monthly lags, and is widely used in practice.

What are long-memory processes and fractional integration in return series?

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