Baillie-Bollerslev-Mikkelsen (1996) Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity

Summary

This paper introduces the FIGARCH (Fractionally Integrated GARCH) class of conditional-variance models, filling the gap between covariance-stationary GARCH — whose shocks to volatility decay geometrically — and IGARCH, in which shocks are infinitely persistent (a unit root in variance). By replacing the first-difference operator $(1-L)$ in the IGARCH variance equation with the fractional differencing operator $(1-L)^d$, $0<d<1$, FIGARCH produces a slow hyperbolic rate of decay in the influence of lagged squared innovations — the volatility analogue of an ARFIMA process for the mean. The authors show (unlike ARFIMA for the mean) that the approximate MLE of the FIGARCH parameters is $T^{1/2}$-consistent, verify this in a simulation study, and demonstrate that fitting a standard GARCH to FIGARCH data spuriously produces near-integrated (IGARCH-like) estimates. A daily Deutschmark–U.S. dollar exchange-rate application favors FIGARCH over both stable GARCH and IGARCH. (Published as OCR of the scanned Journal of Econometrics 74(1): 3–30 article.)

Key Claims

• The I(0)/I(1) dichotomy is too restrictive for variances. Just as the mean literature moved beyond the "knife-edge" I(0)-vs-I(1) distinction via fractionally integrated I(d) processes (Granger–Joyeux 1980; Hosking 1981), conditional variances need a long-memory alternative to the GARCH-vs-IGARCH extremes. Long memory in squared/absolute returns is documented (Ding–Granger–Engle 1993; Dacorogna et al. 1993).
• Definition. Starting from Engle–Bollerslev's (1986) IGARCH$(p,q)$ written as $\phi(L)(1-L)\varepsilon_t^2=\omega+[1-\beta(L)]v_t$ (with $v_t=\varepsilon_t^2-\sigma_t^2$), the FIGARCH$(p,d,q)$ process replaces $(1-L)$ with the fractional differencing operator: $\phi(L)(1-L)^d\varepsilon_t^2=\omega+[1-\beta(L)]v_t$, $0<d<1$. Equivalently the conditional variance has an infinite-order ARCH representation $\sigma_t^2=\omega[1-\beta(1)]^{-1}+\lambda(L)\varepsilon_t^2$, whose weights must all be nonnegative for $\sigma_t^2>0$ a.s.
• Hyperbolic decay, but forecasts mean-revert. For $0<d<1$ the ARCH$(\infty)$ weights decay at a slow hyperbolic rate (long memory), yet the cumulative impulse-response weights of a volatility shock on the optimal multi-step variance forecast eventually tend to zero — a mean-reversion property shared with weakly stationary GARCH and not with IGARCH (whose weights tend to a nonzero constant, giving forecasts that grow linearly in the horizon).
• Infinite unconditional variance. Because the hypergeometric function gives $\lambda(1)=1$ (as $F(-d,1,1;1)=0$), the $\omega>0$ term plays the same role as in IGARCH and the second moment of the unconditional distribution of $\varepsilon_t$ is infinite — so FIGARCH is strictly stationary but not covariance stationary.
• $T^{1/2}$-consistent MLE. Unlike the $I(d)$ mean case (where estimating $d$ is delicate), the approximate (conditional) MLE of the FIGARCH parameters is argued to be $\sqrt{T}$-consistent and asymptotically normal; a Monte Carlo study shows the approximate MLE and its asymptotic standard errors perform well at financial sample sizes.
• Spurious IGARCH. When the true DGP is FIGARCH but a standard GARCH$(1,1)$ is estimated, the fitted autoregressive parameter is driven very close to unity — i.e. one finds IGARCH. The authors conjecture that the widespread empirical "IGARCH" finding in high-frequency asset returns may be spurious, an artifact of long memory rather than a genuine unit root in variance; IGARCH is a poor diagnostic for distinguishing integration from long memory.
• Empirical evidence (Deutschmark–U.S. dollar, daily, 13 Mar 1979 – Dec 1992). The standard GARCH$(1,1)$ estimate of $\alpha+\beta$ sits at essentially unity (apparent IGARCH); the FIGARCH$(1,d,\cdot)$ fit yields a fractional differencing parameter significantly strictly between 0 and 1, delivering clear rejections of both the stable-GARCH ($d=0$) and IGARCH ($d=1$) nulls. The FIGARCH impulse-response weights are more economically realistic than either the rapid GARCH decay or the infinite IGARCH persistence.

Concepts Introduced or Extended

• FIGARCH
• GARCH
• Long Memory and Fractional Integration

Entities Mentioned

• Richard Baillie
• Tim Bollerslev
• Hans Ole Mikkelsen

Quotes

"The conditional variance of the process implies a slow hyperbolic rate of decay for the influence of lagged squared innovations."

"The apparent widespread IGARCH property so frequently reported with high-frequency asset pricing data may well be spurious ... the IGARCH process provides a poor diagnostic for distinguishing between integrated, as opposed to long-memory, formulations of the conditional variance process."

My Take

FIGARCH's lasting contribution is conceptual as much as technical: it reframed the ubiquitous "IGARCH" finding as very likely a misdiagnosed long-memory phenomenon, the volatility twin of the spurious-unit-root problem in the mean. The construction is elegant — literally the ARFIMA idea transplanted from the mean equation into the ARCH$(\infty)$ variance weights — and the $\sqrt{T}$-consistency result is what made it usable in practice. The awkward edge is that FIGARCH shares IGARCH's infinite unconditional variance and lacks fully general nonnegativity conditions on the ARCH weights, which is why later work (including component and other long-memory-in-volatility specifications, and Bollerslev–Mikkelsen's FIEGARCH) refined the idea. For this wiki it is the direct bridge between fractional integration and GARCH, and the natural companion to the realized-volatility evidence that daily-return volatility is strongly, but not infinitely, persistent.