When Volatility Shocks Never Fade: Engle and Bollerslev on Persistence

Fit a GARCH model to the stock market. Then to a currency pair. Then to a bond yield. You will notice something eerie: in almost every case, the two main parameters, the one on yesterday's shock and the one on yesterday's volatility, add up to something very close to one. Not 0.7. Not 1.4. Something like 0.98.

That is not a coincidence, and it is not a bug. It is telling you something important about volatility, and in 1986 Engle and Bollerslev were the first to work out what.

The problem: what does it mean when the parameters sum to one?

In an ordinary GARCH model, that sum controls how fast volatility forgets. Think of it as a leak rate.

• If the sum is small, say 0.5, then a shock to volatility drains away fast. A wild day today is basically forgotten in a week.
• If the sum is 0.9, the shock drains slowly. A wild day today still nudges your forecast a month later.
• If the sum is exactly one, the shock never drains away at all. It becomes a permanent part of the forecast, out to every horizon, forever.

The empirical estimates kept landing right on the doorstep of that last case. Engle and Bollerslev asked the obvious question: what if the truth really is one? What does a volatility model look like when shocks are permanent, and can you even use such a model?

The key idea via analogy: a thermostat with no set point

Picture an ordinary GARCH model as a room with a thermostat. Something heats the room up (a shock), but the thermostat keeps pulling the temperature back toward a fixed set point. The set point is the long-run average volatility. No matter what happens, if you wait long enough, the room drifts back to it. That is why a GARCH forecast for two years from now is basically just "average volatility," regardless of what happened this week.

Now remove the set point. Keep the heater, keep the cooler, but delete the target temperature. Now the room has no level to return to. Whatever shock hits it just moves the temperature and stays. That is the integrated GARCH, or IGARCH, model. Volatility wanders. Today's shock still shows up in your forecast for next year at full strength.

Engle and Bollerslev showed you can still fit such a model, still forecast with it, and still make sense of it, even though it has no long-run average variance to anchor to. The mathematics is delicate, which is much of what makes the paper a real contribution rather than an observation, but the intuition is that simple: a volatility process whose shocks are permanent, not temporary.

There is a familiar cousin here. Economists had already argued about whether GDP or stock prices have a "unit root," meaning shocks to the level are permanent rather than temporary. IGARCH is the same idea, but applied to the variance rather than the level. Engle and Bollerslev imported that whole conversation into volatility modelling.

Why it mattered

• It explained a stubborn empirical fact. Everyone was seeing parameters that summed to almost one and nobody had a clean framework for it. This paper gave the phenomenon a name and a model.
• It quietly became an industry standard. The famous RiskMetrics exponentially weighted moving average, the volatility estimator used by thousands of risk desks, is essentially an IGARCH model with the parameters pinned to fixed values. Every time someone computes an EWMA volatility, they are using a descendant of this idea.
• It sharpened the forecasting question. If shocks to volatility are permanent, then long-horizon volatility forecasts should not just collapse to the historical average. That has real consequences for pricing long-dated options and for capital requirements at long horizons.
• It opened the long-memory research programme. Once you can ask "is the shock temporary or permanent," the natural next question is "what if it is somewhere in between?" That question produced fractionally integrated GARCH a decade later.

The honest limitations

• Permanent shocks are probably too strong a claim. Taken literally, IGARCH says a single wild day in 1987 still fully influences your volatility forecast for 2030. Most people find that implausible. Volatility does seem to revert to something in the very long run, just slowly.
• The near-one estimate may be an illusion. A well-known and uncomfortable possibility is that the sum comes out near one not because shocks are permanent, but because the average level of volatility genuinely shifts over time (regime changes, structural breaks) and a GARCH model, unable to represent those shifts, fakes them by cranking persistence up to the maximum. If that is what is happening, IGARCH is fitting an artefact.
• Estimation gets strange at the boundary. A model sitting exactly at the edge of stability has awkward statistical properties, and standard inference is not entirely straightforward.
• Forecasts can be extreme. Without any pull back toward an average, a model that gets a big shock will keep forecasting high volatility for a very long time, which can lead to persistently overcautious risk numbers after a crisis.

The one-line takeaway

Engle and Bollerslev noticed that estimated GARCH models are always sitting right at the edge where volatility shocks stop fading, took that seriously, and built the model of permanently persistent volatility that still lives inside the exponentially weighted risk numbers used on trading desks today.