Three Averages Beat Everything: Corsi's HAR Model

If you want to forecast tomorrow's volatility, here is a model. Take three numbers:

Yesterday's realized volatility.
The average realized volatility over the last week.
The average realized volatility over the last month.

Run an ordinary linear regression of tomorrow's volatility on those three numbers. That is the entire model. You can fit it in one line of code with the same least-squares routine you would use for a first-year statistics homework.

This is Fulvio Corsi's HAR model, and the uncomfortable fact for the volatility modelling profession is that it is very hard to beat. Twenty years of increasingly baroque volatility models, and one of the best forecasters anyone has found is three moving averages and a regression.

The problem: long memory was real, but the models for it were awful

Everyone by then agreed on the empirical fact. Volatility has an extremely long memory: today's turbulence is still faintly detectable in the data a year later. Standard GARCH cannot produce this, because its memory decays too fast.

The available fixes were technically heavy. Fractionally integrated models like FIGARCH could generate long memory, but they are difficult to estimate, numerically fragile, theoretically contentious, and honestly rather hard to explain to anyone. The mathematics of fractional integration is not something you want in your production risk system if you can avoid it.

Corsi asked a practical question: can we get the behaviour of long memory without the machinery of long memory?

The key idea via analogy: three kinds of trader in the same market

Corsi's motivation is economic, not statistical, and it is what makes the model feel right rather than merely convenient.

Markets are not populated by one type of participant. They contain at least three, and they operate on wildly different clocks:

The day traders and market makers. Their horizon is minutes to hours. They care about what happened this morning. Yesterday is ancient history.
The portfolio managers and swing traders. Their horizon is days to weeks. They rebalance weekly. They care about how the last week went.
The pension funds, insurers and long-term allocators. Their horizon is months to years. They adjust slowly and care about the general level of market turbulence over the last month or quarter.

Now here is the key observation, which Corsi borrows from the "heterogeneous market" tradition: these groups do not just have different horizons, they influence each other asymmetrically. When the long-horizon players get nervous and start moving size, the short-horizon players see the turbulence and become cautious too. Volatility cascades down from long horizons to short ones. But a burst of frantic day-trading does not much perturb a pension fund's outlook. The cascade is one-directional.

So the volatility a day trader faces is driven by their own recent activity plus the volatility being generated by the slower players above them. Which suggests: to forecast tomorrow, you need the recent short-horizon volatility, the medium-horizon volatility, and the long-horizon volatility. Daily, weekly, monthly.

That is the HAR model. It is not a statistical trick, it is a picture of who is in the market.

Why it works

Adding together components that decay at three different speeds produces an aggregate that decays very slowly, in a way that closely mimics true long memory. It is not mathematically identical to long memory. Corsi is completely upfront about this: the title says "approximate," and he explicitly notes the model does not have real long-memory properties.

But when he simulated from it, the model reproduced the things that matter: the extremely slow decay of volatility correlation, the fat tails of returns, the way volatility looks similar at different timescales. And when he forecast out of sample, it performed remarkably well.

The reason practitioners love it is a stack of practical virtues:

It is a linear regression. No likelihood maximisation, no optimiser that fails to converge, no numerical fragility.
It is trivial to extend. Want to add a jump component? Add a column. Want to add implied volatility, or overnight returns, or a leverage term? Add a column. The whole subsequent literature of HAR variants exists because extending it is so easy.
It is interpretable. The three coefficients tell you how much of tomorrow's volatility comes from short, medium and long horizon influences.

Why it mattered

It became the default benchmark. Any new realized-volatility forecasting model, including every machine learning approach, is now expected to justify itself against HAR. Many fail.
It made long memory practical. The theoretical debate about whether volatility truly has long memory rumbles on. HAR sidesteps it entirely by producing something that behaves like long memory without claiming to be it.
It gave a family of models a foundation. HAR-J, which splits out jumps. HAR-CJ. HAR with leverage effects. HAR with implied volatility. Almost all applied realized-volatility work builds on this template.
It is a lesson in modelling philosophy. A simple, economically motivated approximation beat a mathematically exact but fragile alternative. That happens more often in quantitative finance than the literature likes to admit.

The honest limitations

It is an approximation and it says so. The model does not truly possess long memory. At very long forecast horizons the approximation degrades, and the choice of daily, weekly, monthly is essentially arbitrary. Why not fortnightly and quarterly? No deep reason.
It inherits every weakness of realized volatility. The model's inputs are measured with error, contaminated by microstructure noise, and blind to overnight moves. Garbage in, garbage out applies.
It is linear and symmetric out of the box. The basic version does not capture the leverage effect, does not distinguish jumps, and cannot react to a regime break. You can bolt these on, and people do, but the plain model is naive about them.
It reacts slowly to sudden shifts. Because monthly and weekly averages move sluggishly, HAR can be slow to acknowledge that the world has changed, for example at the start of a crisis.
It needs high-frequency data. No intraday prices, no realized volatility, no HAR.

The one-line takeaway

Corsi noticed that markets contain traders working on daily, weekly and monthly clocks who influence each other from the top down, turned that observation into a three-term linear regression on yesterday's, last week's and last month's volatility, and produced a forecasting model so simple and so effective that it remains the benchmark every fancier model must beat.