FIGARCH

Definition

FIGARCH (Fractionally Integrated GARCH) is a conditional-variance model that produces long memory in volatility — a slow, hyperbolic rate of decay in the influence of past squared innovations — occupying the middle ground between covariance-stationary GARCH (geometric decay) and IGARCH (a unit root in variance, i.e. infinite persistence). Introduced by Baillie, Bollerslev, and Mikkelsen (1996), it is the volatility analogue of an ARFIMA process for the mean: it takes the IGARCH variance equation and replaces the first-difference operator $(1-L)$ with the fractional differencing operator $(1-L)^d$ for a fractional order $0<d<1$.

Key Ideas

• Between GARCH and IGARCH. GARCH$(1,1)$ with $\alpha+\beta<1$ has geometrically decaying volatility memory; IGARCH ($\alpha+\beta=1$) has infinitely persistent shocks. FIGARCH interpolates: $d=0$ recovers stable GARCH, $d=1$ recovers IGARCH, and $0<d<1$ gives hyperbolic decay.
• Long memory in variance. The infinite-order ARCH weights decline at a slow polynomial (hyperbolic) rate, matching the empirically observed persistence of autocorrelations in squared and absolute returns (Ding–Granger–Engle 1993).
• Mean-reverting forecasts. Despite long memory, the cumulative impulse-response weights of a shock on the optimal multi-step variance forecast tend to zero — so long-horizon forecasts mean-revert, unlike IGARCH where they grow linearly with the horizon.
• Strictly but not covariance stationary. Because $\lambda(1)=1$, the unconditional second moment is infinite (as in IGARCH); the process is strictly stationary and ergodic but not weakly stationary.
• "Spurious IGARCH." Fitting a standard GARCH to FIGARCH data drives the estimated persistence toward unity, so the widespread empirical IGARCH finding in high-frequency returns may be a misdiagnosis of long memory rather than a true unit root in variance.

How It Works

Write the IGARCH$(p,q)$ model as $\phi(L)(1-L)\varepsilon_t^2=\omega+[1-\beta(L)]v_t$, where $v_t=\varepsilon_t^2-\sigma_t^2$ is the mean-zero variance innovation and $\phi(L)=[1-\alpha(L)-\beta(L)](1-L)^{-1}$. The FIGARCH$(p,d,q)$ process replaces the integer difference with a fractional one: $$\phi(L)\,(1-L)^d\,\varepsilon_t^2=\omega+[1-\beta(L)]\,v_t,\qquad 0<d<1,$$ where $(1-L)^d$ has the binomial/hypergeometric expansion $\sum_{k\ge 0}\frac{\Gamma(k-d)}{\Gamma(k+1)\Gamma(-d)}L^k$. Rearranging gives the infinite-order ARCH representation $$\sigma_t^2=\omega\,[1-\beta(1)]^{-1}+\lambda(L)\,\varepsilon_t^2,\qquad \lambda(L)=\lambda_1 L+\lambda_2 L^2+\cdots,$$ in which every weight $\lambda_k$ must be nonnegative to guarantee $\sigma_t^2>0$ almost surely (general conditions, as for Nelson–Cao GARCH, are checked case-by-case). Estimation is by approximate (conditional) maximum likelihood, which — unlike ARFIMA estimation of $d$ for the mean — is $\sqrt{T}$-consistent and asymptotically normal, and performs well at financial sample sizes in simulation.

Why It Matters

• Resolves a diagnostic puzzle. It reframes the near-universal "integrated variance" (IGARCH) finding in daily asset returns as very possibly spurious long memory, cautioning against reading $\alpha+\beta\approx 1$ as a genuine unit root.
• More realistic pricing implications. IGARCH implies extreme dependence of long-dated option/futures prices on current conditions; FIGARCH's mean-reverting-but-slow forecasts are more consistent with observed pricing behavior.
• Long-memory-in-volatility program. FIGARCH launched a family of fractionally integrated volatility models (e.g. Bollerslev–Mikkelsen's FIEGARCH for stock-market volatility), connecting the GARCH literature to fractional integration and to the realized-volatility evidence on persistent but finite volatility dependence.

Open Questions

• General, easily checkable nonnegativity conditions on the ARCH$(\infty)$ weights $\lambda_k$ remain elusive.
• The infinite unconditional variance complicates interpretation and some inference; whether a covariance-stationary long-memory volatility model is preferable is debated.
• Distinguishing true long memory from structural breaks / regime shifts in volatility is difficult — both can generate slowly decaying sample autocorrelations.

Related

• GARCH
• Long Memory and Fractional Integration
• Markov-Switching GARCH
• Realized Volatility
• Stylized Facts of Asset Returns
• Baillie-Bollerslev-Mikkelsen (1996) Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity
• Richard Baillie
• Tim Bollerslev