Testing the Square-Root Law Where Nobody Expected It: A Million Bitcoin Metaorders

Every empirical finding in market microstructure has the same nagging problem. You measure something on US equities, and you have no idea whether you have discovered a law of nature or merely a property of the New York Stock Exchange.

Maybe the square-root law of market impact holds because of something deep about how supply and demand work. Or maybe it holds because of Reg NMS, or because of the specific way institutional brokers slice orders, or because of the particular ecosystem of high-frequency market makers that grew up around Nasdaq. From inside the equity market, you cannot tell the difference.

Jonathan Donier and Julius Bonart found a way to tell. They tested the law on Bitcoin.

The problem: you need a market that shares nothing but the basics

To distinguish a universal law from a local quirk, you want a market that is as different as possible from equities while still being a market where people buy and sell things.

Bitcoin around 2014, which is the period they study, is close to ideal for this. Consider how little it had in common with a stock exchange.

• Essentially no professional market makers. The ecosystem of firms whose business is to continuously quote two-sided prices, which many people believe creates the impact patterns we see in equities, barely existed.
• Almost no statistical arbitrage. The armies of quant funds that keep equity prices efficient and grind predictability out of order flow were not there in any meaningful way.
• No regulation to speak of, no best-execution rules, no fragmented venue routing obligations. None of the institutional machinery of modern equity markets.
• A completely different participant mix. Retail speculators, miners, and enthusiasts, rather than pension funds and asset managers.
• A different asset entirely. No earnings, no dividends, no fundamental value that anyone could agree on.

If the square-root law is an artefact of the institutional structure of equity markets, it has no business showing up here. If it is something deeper, about the fundamental nature of how latent supply and demand meet, it should appear anyway.

And there was a bonus: because of how the exchange they studied worked, the authors could reconstruct over a million individual metaorders, complete parent orders with known owners and known directions. In equities, this kind of data is proprietary and rare. Here it was recoverable.

The key idea via analogy: testing gravity on another planet

If you want to know whether a physical law is universal or just a peculiarity of Earth, you go and test it somewhere with different gravity, different atmosphere, different everything. If the law still holds, you have learned something profound. If it breaks, you have learned you were fooling yourself.

That is what Bitcoin was here. A different planet, same physics question.

The result: the square-root law held. Impact was proportional to the square root of the size of the metaorder, across roughly four decades of order size. In a market with none of the institutions that were supposed to produce the effect.

This is a strong argument that the square-root law is not a consequence of market makers, arbitrageurs, regulation, or the mechanics of institutional order slicing. It comes from something much more basic: the way latent supply and demand are distributed around the current price, which is the mechanism Toth and colleagues had proposed a few years earlier. The theory made a prediction of universality, and the prediction survived a genuinely hostile test.

The paper adds two findings that go beyond simple confirmation.

The square root holds throughout the trajectory, not just at the end. Most studies measure impact by comparing the price before the order started to the price when it finished. Donier and Bonart could track the price all the way through the execution and show that the square-root relationship describes the whole path, not merely the endpoints. That is a much sharper test, and it rules out a class of alternative explanations.

Impact decays, and most of it decays away. After the metaorder finishes, the price largely relaxes back. The authors attempt to split order flow into an informed component, where the trader knew something and the price move sticks, and an uninformed component, where the trader was simply demanding liquidity and the price eventually returns. The uninformed part shows an almost complete long-term decay of impact. That is a significant result: it says the majority of the price move you cause by trading is a temporary rent you pay for immediacy, not a permanent revaluation of the asset.

Why it mattered

• It is the cleanest universality test the impact literature has. Confirming the law in a market that lacks every institutional feature people suspected of causing it is far more informative than confirming it on yet another equity dataset.
• It supported a specific theory over its rivals. The latent order book and criticality explanation predicts universality. Institutional explanations do not. Bitcoin adjudicated between them.
• The full-trajectory result is a genuine methodological advance. Being able to see impact build up during execution, rather than just measuring the endpoints, is a much richer object and a much harder thing to fit by accident.
• The decomposition into informed and uninformed flow is practically important. If most of your impact is temporary, then the true long-run cost of a large trade is much smaller than the peak price move suggests, and execution algorithms that panic about peak impact are optimising the wrong quantity.
• It legitimised crypto as a research laboratory. Crypto markets have the property that academics can obtain data that would be commercially impossible to get in equities. That is genuinely valuable, whatever one thinks of the asset.

The honest limitations

• Bitcoin in 2014 was a strange and immature market. Thin, volatile, dominated by a small number of exchanges of variable reliability, and subject to episodes that would be scandals anywhere else. Whether findings from that environment generalise upward to mature markets is a fair question, even if the direction of the argument here is from crypto to equities rather than the other way.
• Metaorder reconstruction requires assumptions. Grouping individual trades into parent orders is an inference, not an observation. Get the grouping rules wrong and you can manufacture or destroy the very patterns you are looking for.
• The informed and uninformed split is model-dependent. Deciding which portion of order flow was "informed" is not something you can read off the tape. It depends on how you define information and how you attribute subsequent price moves.
• A single exchange in a single period. The dataset comes from one venue over a limited window. Crypto market structure has changed beyond recognition since.
• Confirming the exponent is not the same as confirming the theory. The square root emerging is consistent with the latent order book story, but consistency is not proof, and other mechanisms could in principle produce the same exponent.

The one-line takeaway

Donier and Bonart tested the square-root law of market impact on Bitcoin, a market with none of the market makers, arbitrageurs or regulation that were supposed to cause it, and found the law held anyway across four decades of order size, which is powerful evidence that impact is a universal consequence of how latent supply and demand sit around the price rather than an artefact of any exchange's plumbing.