Why Squaring Returns Was a Mistake: Ding, Granger and Engle

Here is a question so basic that almost nobody had thought to ask it: when we say volatility clusters, what exactly is the quantity that clusters?

The standard answer, baked into ARCH and GARCH and everything descended from them, is: squared returns. Take today's return, square it, and that is your measure of how violent today was. Volatility clustering means today's squared return predicts tomorrow's.

But why squared? Honestly, because of mathematical convenience. Variance is defined as an average of squared deviations, so squaring is what the theory hands you. Nobody had checked whether the data prefers squares.

In 1993, Zhuanxin Ding, Clive Granger and Robert Engle checked. The answer was no.

The problem: convenience is not evidence

The authors took a very long run of daily US stock index returns, over sixty years of it, and did something almost embarrassingly simple. Instead of just looking at squared returns, they looked at the absolute return raised to a power, and let the power be anything: 0.25, 0.5, 1, 1.5, 2, 3.

For each choice of power, they asked: how strong and how long-lasting is the correlation between today's value and the value some days back? In other words, which transformation of returns has the most memory?

If the standard theory were right, the answer would be power 2, the squared return. It was not.

The key idea via analogy: tuning a radio to find the clearest signal

Think of the power as a dial on a radio. As you turn it, you tune into a different transformation of the return series, and you listen for how much predictable structure is coming through.

Ding, Granger and Engle turned the dial and listened. What they found is that the signal is strong across a whole band of settings, but it peaks at a surprising place: right around power 1. That is the plain absolute return, no squaring at all. Just "how big was today's move, ignoring direction."

And when you tune to that setting, something remarkable appears. The correlation between today's absolute return and the absolute return of days long past is positive, meaningful, and stubbornly slow to fade. It is still visibly there hundreds and even a thousand trading days back. That is far, far longer than any standard GARCH model can produce.

Two findings, then, and both are important:

The market's memory is much longer than anyone's model allowed. This is the "long memory" of the title, and it became one of the canonical stylised facts of financial data.
The memory shows up most clearly in absolute returns, not squared returns. Squaring, it turns out, actually destroys some of the structure by over-weighting the biggest days. The standard modelling choice was throwing away signal.

The new model: let the data pick the power

Having shown that the power matters, the authors did the obvious thing: they built a model where the power is a parameter you estimate, rather than a number you assume.

Their model, usually called APARCH (asymmetric power ARCH), does two things at once. It lets the data choose which transformation of returns drives volatility, and it also allows the asymmetry that Nelson and GJR had been arguing about, so that bad news moves volatility more than good news.

It is a genuinely flexible framework, and a long list of earlier models turn out to be special cases of it. Fix the power at two and remove the asymmetry, and you are back to plain GARCH. Fix the power at one, and you get a model in absolute returns. The old models become points inside a bigger space, and the data gets to pick where in that space to sit.

Why it mattered

It established long memory in volatility as a fact to be reckoned with. This paper, more than any other, is why researchers accepted that market volatility has a memory measured in years, not weeks. That observation drove the development of FIGARCH, of long-memory stochastic volatility models, and eventually of the HAR framework.
It questioned a foundational modelling choice. "Model squared returns" was an unexamined default. The paper made it a decision, and showed the default was probably the wrong one.
It anticipated modern practice. Practitioners today routinely use measures based on absolute moves rather than squared moves, partly because they are more robust to outliers. The empirical case for that starts here.
The power is a robustness dial. Squaring gives one enormous day the same weight as a hundred merely bad ones. Using a lower power tames that, which is why lower-power volatility measures often behave better in the presence of fat tails.

The honest limitations

The long-memory-versus-breaks debate is unresolved. As with FIGARCH, the very slow decay they document could reflect genuine long memory, or it could reflect a volatility level that occasionally shifts to a new regime. Over sixty years of data covering the Depression, the war, the 1970s and 1987, structural change is not exactly a far-fetched hypothesis. Distinguishing the two is notoriously hard.
The flexible model is harder to fit. More parameters, including one in an exponent, means a trickier likelihood surface and slower, less stable estimation. In practice many people still reach for GARCH or GJR.
The optimal power is not universal. The "around one" finding comes from a particular index over a particular sixty-year span. On other assets and other periods the best power moves around.
Flexibility invites overfitting. A model that can nest a dozen others will always fit better in sample. Whether it forecasts better out of sample is a separate question, and the evidence there is mixed.

The one-line takeaway

Ding, Granger and Engle asked whether volatility models should really be built on squared returns, discovered that the market's memory is longest and clearest in plain absolute returns, and in the process documented a volatility memory so long, stretching over years, that it broke every model then in use.