Neither Forever Nor Fleeting: FIGARCH and the Long Memory of Volatility

By the mid-1990s, volatility modelling had painted itself into a corner with only two doors.

Door one: standard GARCH. Volatility shocks decay quickly. Today's panic fades geometrically, like a hot cup of coffee cooling toward room temperature. Within a few weeks, it is gone from your forecast.

Door two: IGARCH. Volatility shocks never decay at all. Today's panic becomes a permanent fixture of every forecast from now until the heat death of the universe.

Neither door matched what people actually saw in market data. Volatility shocks clearly last much longer than plain GARCH allows. But "forever" is obviously too much. The truth was somewhere in the middle, and there was no door for the middle. In 1996, Richard Baillie, Tim Bollerslev and Hans Ole Mikkelsen built one, and called it FIGARCH.

The problem: real volatility forgets, but very slowly

Look at the correlation between today's market jumpiness and the jumpiness of a day some distance in the past. Then push that distance further and further out. What you find in real markets is striking: the correlation stays positive and meaningful even hundreds of trading days back. It fades, but it fades at a crawl.

A standard GARCH model cannot reproduce this. Its memory decays geometrically, which sounds slow but is actually brutally fast: after enough steps, geometric decay makes the correlation effectively zero. So GARCH says the link should have vanished long before the data says it did.

IGARCH goes the other way and says the correlation should never fade at all. Also wrong.

This awkward "slow but not infinite" decay has a name in statistics: long memory.

The key idea via analogy: the volume dial that never quite returns

Think of the three models as three different ways a room cools down after you switch off a heater.

• GARCH is a well-insulated room with a strong air conditioner. Turn off the heater and within an hour you are back to normal. Yesterday's heating is irrelevant today.
• IGARCH is a room in a vacuum. Turn off the heater and the room stays hot forever. Nothing ever leaks out.
• FIGARCH is a room with a tiny crack under the door. Turn off the heater and the heat does leak out, genuinely and permanently. But it leaks out slowly, and if you come back next month you can still feel a faint warmth. Come back next year and there is a trace of it still.

Mathematically, the authors got this by borrowing an idea from the study of long memory in the level of time series and applying it to the variance. Instead of forcing the memory parameter to be either 0 (shocks fade fast) or 1 (shocks are permanent), they let it be a fraction, something like 0.4. Hence "fractionally integrated." That fraction is what you estimate from the data, and it controls exactly how leisurely the decay is.

The payoff is that FIGARCH nests both of the old doors as special cases. Set the fraction to zero and you get GARCH back. Set it to one and you get IGARCH. Anything in between is new territory, and it is in that territory that real financial data seems to live.

Why it mattered

• It matched a genuine, robust feature of the data. The extremely slow decay of volatility correlation is one of the most reliable stylised facts in finance, showing up in stocks, currencies, commodities and bonds. FIGARCH was the first widely adopted model built specifically to reproduce it.
• It changed long-horizon forecasts. If shocks fade slowly rather than quickly, then your volatility forecast for six months out should still carry information from today's shock. That matters enormously for pricing long-dated options and for setting capital at long horizons, where GARCH forecasts collapse to the unconditional average too quickly.
• It gave "persistence" a dial instead of a switch. Before FIGARCH, the persistence debate was binary and slightly silly. Afterward, it became a quantitative question with a number attached.
• It fed directly into modern practice. The insight that volatility has components operating on many different timescales, which is what long memory really amounts to, is the seed of the multi-horizon models that came later, most notably Corsi's HAR model.

The honest limitations

• Long memory may be an illusion created by structural breaks. This is the serious objection, and it has never gone away. If the average level of volatility genuinely shifts from time to time (a new regulatory regime, a new market structure, a crisis), then a model that assumes a constant average will misread those level shifts as extremely slow-decaying memory. Long memory and occasional structural breaks can look almost identical in finite samples, and telling them apart is genuinely hard.
• It is painful to estimate. Fractional integration means the model technically depends on the entire history of the series, so in practice you have to truncate it. Fitting is slower, fiddlier and more prone to numerical trouble than plain GARCH.
• The theory has gaps. The statistical properties of FIGARCH, including whether the process is well behaved in the ways one would like, have been the subject of ongoing and sometimes uncomfortable debate.
• Better fit does not guarantee better forecasts. FIGARCH describes the in-sample autocorrelation of volatility beautifully. Whether it consistently beats a simple GARCH out of sample is much less clear, and horse races have often found the improvement modest at best.
• Simpler models capture most of the benefit. Corsi's HAR model, published later, reproduces long-memory-like behaviour by just averaging volatility over daily, weekly and monthly windows. It is far easier to fit and forecasts at least as well. Much of the practical world has voted for the simpler approximation.

The one-line takeaway

Baillie, Bollerslev and Mikkelsen replaced the crude either-or choice between "volatility shocks fade fast" and "volatility shocks last forever" with a dial, letting the data decide how slowly the market's memory of a crisis actually leaks away, and the data said: very slowly indeed.