Field Note 11 / Research Methods

Why Crypto Move Size Is Hard to Predict: A Heavy-Tail Test

A radar detects an unusual move. The next question is how far it might go. Our latest Klines research tested whether the available features could answer that question—and whether the distribution of outcomes helped explain the difficulty.

The first result was striking: a small share of signals accounted for much of the measured growth. A power-law fit seemed to offer a mathematical explanation. Further checks made that explanation less tidy, and more useful.

What we measured

The initial analysis examined growth to a historical peak for 1,948 filtered signals with positive growth. It did not describe every crypto movement or every candle in our database. Both the filtering and the definition of a peak shape the population being measured.

Within that sample, the largest 1% of observations contributed 15.9% of the sum of measured growth. The largest 10% contributed 51.2%. The coefficient of variation—the sample standard deviation divided by the mean—was 2.06.

Those percentages describe concentration across observations. They are not portfolio returns: there is no capital allocation, executable exit or cost model in that sum. Their practical meaning is that a few large outcomes can strongly influence aggregate summaries.

A power-law fit is a hypothesis, not a verdict

The initial tail fit gave an exponent of 1.98, with a reported 95% bootstrap interval of 1.93–2.03. The selected lower cutoff was 12.1% growth, leaving 975 observations in the fitted tail. The bootstrap used 500 resamples.

Our approach drew on Clauset, Shalizi and Newman's work on power-law distributions in empirical data: estimate the exponent by maximum likelihood and select a cutoff using the Kolmogorov–Smirnov distance. But we did not perform their full bootstrap goodness-of-fit test. An estimated exponent and its interval do not establish that a power law generated the observations.

This distinction mattered in the follow-up. We measured maximum favourable excursion, or MFE, over fixed future windows of 24 hours, 48 hours and seven days. These were different targets and a broader raw-trigger population, so their results should not be treated as a rerun of the same 1,948 observations.

A stretched exponential achieved higher tail log-likelihood than the pure power law on all three horizons. For seven-day MFE, the reported average log-likelihood was −4.0759 for the stretched exponential and −4.1220 for the power law. This was a descriptive model comparison without a formal likelihood-ratio significance test.

The concentration persisted: the top 10% accounted for about half of total measured MFE across the three horizons. The evidence supports taking the tail seriously. It does not justify declaring an exact power law.

Why an exponent near two does not prove prediction is impossible

Power-law exponents require a stated convention. In the follow-up code, the estimator is for a density proportional to x^−α. For an ideal, unbounded distribution under that convention, the mean is finite only when α > 2, and the variance only when α > 3. The thresholds differ if an exponent refers to the survival probability instead of the density.

More fundamentally, a finite dataset cannot establish that the real distribution extends indefinitely with that mathematical form. A fitted stretched exponential has different moment properties. We therefore cannot use the exponent to announce infinite variance as an established market fact.

Nor does a heavy unconditional tail prove that useful conditional forecasts are impossible. Features may still contain information about particular outcomes. Whether they do is a question for held-out tests. Tail concentration is a reason to inspect error distributions and robust summaries; it is not an explanation that excuses every failed model.

The magnitude model failed its held-out test

The follow-up used training observations from November 2025 through March 2026, validation observations from April through June, and 1,934 held-out observations from July through September 2026, with a seven-day embargo. The population consisted of raw radar triggers, rather than the complete production pipeline.

For seven-day MFE, the magnitude model's Spearman correlation fell from 0.170 on validation to 0.002 on the held-out block. Its mean absolute error was 16.22 percentage points, compared with 15.42 for the constant baseline. On all three horizons, the model lost to the baseline on held-out mean absolute error, root mean squared error and median quantile loss.

That supports a specific conclusion: this model, with these features and this evaluation, did not provide a useful improvement in magnitude estimation. It does not establish that no future method could do so.

The held-out months had also appeared in other project analyses, although the report states that this model series was not fitted on them. This limits how strongly we can describe the period as completely unseen research data.

Predicting threshold exceedance gave a narrower result

We also tested a different question: does the price reach a specified favourable excursion within seven days? For the +50% threshold, the held-out ROC-AUC was 0.603. Among the highest-ranked 10% of signals, 12.4% reached that threshold, against a 6.8% rate across the held-out population.

The resulting lift was 1.82, with a reported coin-cluster bootstrap interval of 1.21–2.58. This suggests some ranking information for that target in this sample. However, the calibration slope was only 0.29: the scores were not reliable probabilities as produced.

The result also depended on the question. At the +20% threshold, held-out AUC was 0.515. A combined label requiring at least +20% MFE and adverse excursion no worse than −10% produced AUC 0.543 and lift 1.06. Those excursion summaries do not themselves specify the order in which an executable trade would hit its exit levels.

The full twelve-feature model barely differed from volatility alone on the +50% target: AUC 0.603 versus 0.598. There were no observations in the earliest 0–6% movement phase under the trigger's construction. These results do not demonstrate early detection of a coming large move.

Where Hawkes processes and extreme-value theory fit

Heavy tails invite explanations involving cascades of market activity. Hardiman, Bercot and Bouchaud's Hawkes-process study models arrivals of E-mini mid-price changes and investigates near-critical behaviour. It offers a mechanism worth studying; it does not establish the same mechanism in our crypto signal population.

We have not validated a Hawkes model for these outcomes. Per-bar trade counts would require a method appropriate to aggregated observations, rather than being treated as exact event timestamps. Even a near-critical estimate would not prove that conditional move-size prediction is impossible.

Extreme-value theory offers another direction: modelling exceedances above a threshold, subject to assumptions and checks on threshold choice, dependence and stability. The classification results above are not an extreme-value model, and the theory does not make uncalibrated scores into trustworthy probabilities.

What changes for Klines

The next useful step is to evaluate ranking and calibration on additional time periods, with simple baselines and explicit target definitions. A historical peak remains a label available after the event. It cannot become an input available to the detector at the start.

This connects to our work on defining crypto price movements across scales: a stable measurement base is necessary before assessing what the radar sees. For the wider research context, see the Klines research note and the system's current scope.

The result is a failed magnitude model, a limited threshold-ranking result, and a better account of the tail. None establishes a tradable edge. They do tell us which claims the next experiment needs to test.