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arxiv: 2606.24019 · v1 · pith:FW7F53NEnew · submitted 2026-06-22 · 💱 q-fin.TR · q-fin.ST

Empirical Confirmation of the Square-Root Law of Market Impact in a U.S. Large-Cap Equity

Aniket Vasaikar This is my paper

Pith reviewed 2026-06-26 05:42 UTC · model grok-4.3

classification 💱 q-fin.TR q-fin.ST
keywords market impactsquare-root lawmetaorder reconstructionanonymous order flowNasdaq ITCHApple equityprice impactlarge-cap stock
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The pith

The square-root law of market impact holds for Apple stock with prefactor 0.69 when metaorders are reconstructed from anonymous Nasdaq order flow.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether market impact on a single large US equity follows the square-root law, using 178 days of full order-book data for Apple. Metaorders are reconstructed from the anonymous tape and impact is calibrated to the form I over sigma_D equals c times square root of Q over V_D, with the exponent fixed at one half. The resulting prefactor is 0.69 raw and 0.34 bias-corrected, the size tail exponent is 1.54, and direct comparisons favor the square-root shape over linear or logarithmic alternatives by a clear AIC margin. Structural checks such as sign shuffling and chronology scrambling confirm that the observed scaling is tied to the actual order flow rather than reconstruction artifacts. The findings replicate the worldwide cross-section on one US name without requiring broker tags.

Core claim

We test the square-root law (SRL) of market impact on a single U.S. large-capitalisation equity, Apple Inc. (AAPL), using the full Nasdaq TotalView-ITCH market-by-order feed over 178 trading days. Without broker-tagged parent orders, we reconstruct metaorders from the anonymous tape and calibrate impact as I/σ_D = c (Q/V_D)^{1/2} with the exponent fixed at the universal value 1/2. We find c_raw = 0.69 (bias-corrected c_eff = 0.34), conditional impact tracking Q^{1/2}, and a size-distribution tail exponent β = 1.54 ± 0.15 -- both consistent with the worldwide cross-section. A direct model comparison decisively prefers the square-root form over linear (ΔAIC=22) and logarithmic impact, and the

What carries the argument

The square-root impact law I/σ_D = c (Q/V_D)^{1/2} with fixed exponent 1/2, applied to metaorders reconstructed from anonymous order flow.

If this is right

  • The square-root form is preferred over linear by an AIC difference of 22 and over logarithmic alternatives.
  • The raw prefactor stays inside [0.63, 0.77] for every tested reconstruction setting.
  • Shuffling trade signs reduces directional impact from 86 percent to 51 percent, consistent with genuine causation.
  • Scrambling event chronology leaves zero viable square-root calibrations out of 80.
  • Order flow exhibits long memory (gamma = 0.66) while prices remain diffusive (Hurst 0.49).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result implies that broker-tagged data are not required to test the square-root law on other single names or venues.
  • Consistency with the global cross-section suggests the scaling may be insensitive to market microstructure details that differ between US and non-US venues.
  • Weekly walk-forward stability of the prefactor indicates the law can be used for out-of-sample impact forecasting on the same stock.
  • The bias correction that moves c from 0.69 to 0.34 could be applied to earlier single-stock studies to check alignment.

Load-bearing premise

The reconstruction of metaorders from the anonymous tape accurately identifies the parent orders that actually caused the observed price moves, without systematic bias from the lack of broker tags.

What would settle it

Observing that shuffling trade signs leaves directional impact well above chance levels, or that none of the 80 scrambled-chronology calibrations still satisfy the square-root scaling, would falsify the claim that the observed impact originates from the reconstructed metaorders.

Figures

Figures reproduced from arXiv: 2606.24019 by Aniket Vasaikar.

Figure 1
Figure 1. Figure 1: The square-root law on AAPL. Non-dimensional impact vs size for 1,012 reconstructed metaorders (grey); log-bin means (red); fit c·Q 1/2 , c=0.69 (blue); Q1/2 reference (dashed). The square-root law holds [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 5
Figure 5. Figure 5: Free-exponent estimation. (a) Weighted fit gives δˆ=0.50. (b) Bootstrap 95% CI [0.32, 0.66] brackets 1/2 and excludes linear impact. Significance of directional impact. A permutation test quantifies the sign result of Fig 3a. Holding each metaorder's realised price move fixed and randomly reassigning its flow-direction label (N=104 ) gives a null centred at 49.6%; the observed 85.9% lies z = 23 standard de… view at source ↗
Figure 3
Figure 3. Figure 3: Structural null tests. (a) Sign-shuffling collapses directional impact to chance. (b) Chronology-scrambling destroys the SRL [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Intraday seasonality of volume (bars) and volatility (line), calibration window. Seasonality [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: Square root in child-order count. Mean impact vs child-fill count L; slope 0.62, near 1/2. Out-of-sample stability. Across 32 weekly re-calibrations ( [PITH_FULL_IMAGE:figures/full_fig_p004_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: The square-root law on real AAPL data. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: Out-of-sample stability of prefactor c and tail exponent across 32 weekly walk-forward re-calibrations. Square-root, not linear — and not an artefact. Two checks make the exponent decisive ( [PITH_FULL_IMAGE:figures/full_fig_p004_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: The two preconditions of the SRL on AAPL. [PITH_FULL_IMAGE:figures/full_fig_p005_10.png] view at source ↗
read the original abstract

We test the square-root law (SRL) of market impact on a single U.S. large-capitalisation equity, Apple Inc. (AAPL), using the full Nasdaq TotalView-ITCH market-by-order feed over 178 trading days (2 December 2024 -- 19 August 2025; ~0.5 billion events). Without broker-tagged parent orders, we reconstruct metaorders from the anonymous tape and calibrate impact as $I/\sigma_D = c\,(Q/V_D)^{1/2}$ with the exponent fixed at the universal value $1/2$. We find $c_{\rm raw} = 0.69$ (bias-corrected $c_{\rm eff} = 0.34$), conditional impact tracking $Q^{1/2}$, and a size-distribution tail exponent $\beta = 1.54 \pm 0.15$ -- both consistent with the worldwide cross-section. A direct model comparison decisively prefers the square-root form over linear ($\Delta{\rm AIC}=22$) and logarithmic impact, and the prefactor holds ($c_{\rm raw} \in [0.63, 0.77]$) across every reconstruction setting. Two structural tests confirm the impact is genuine: shuffling trade signs collapses directional impact to chance (86% to 51%); and scrambling event chronology destroys the SRL (0 of 80 calibrations remain viable). The underlying order flow is long-memory ($\gamma=0.66$) while the price stays diffusive (Hurst 0.49) -- the two ingredients of the universality theories. The prefactor is stable across 32 weekly walk-forward re-calibrations. To our knowledge this is the first confirmation of the square-root law on a U.S. equity derived purely from anonymous order flow, without broker-tagged parent orders.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to provide the first confirmation of the square-root law of market impact on a single U.S. large-cap equity (AAPL) using metaorders reconstructed from anonymous Nasdaq TotalView-ITCH data over 178 trading days. With the exponent fixed at the literature value of 1/2, it reports c_raw = 0.69 (bias-corrected c_eff = 0.34), a size-distribution tail exponent β = 1.54 ± 0.15, decisive preference for the square-root form over linear (ΔAIC = 22) and logarithmic alternatives, robustness of c across reconstruction settings, and validation via sign-shuffling (impact collapses) and chronology-scrambling (no viable calibrations) tests, alongside long-memory order flow (γ = 0.66) and diffusive prices (Hurst 0.49).

Significance. If the metaorder reconstruction is unbiased, the result would strengthen evidence for the universality of the square-root law in a single-stock U.S. setting from anonymous flow, consistent with worldwide cross-sections, and would directly support theoretical mechanisms requiring long-memory order flow paired with diffusive prices. The walk-forward stability and explicit model comparison are positive features.

major comments (2)
  1. [Abstract] Abstract: The central claim that I/σ_D = c (Q/V_D)^{1/2} holds with c_raw = 0.69 and is preferred by ΔAIC = 22 rests on reconstructed metaorder quantities Q; the sign-shuffling and chronology-scrambling tests only establish that impact correlates with the reconstructed sign sequence, not that the reconstructed Q values are the causally relevant parent-order sizes. This leaves the skeptic's concern about systematic bias from clustering rules unaddressed and load-bearing for the fitted prefactor and model ranking.
  2. [Abstract] Abstract: The paper states that c_raw remains in [0.63, 0.77] 'across every reconstruction setting,' but without explicit description of the clustering rules (time windows, volume thresholds, sign-persistence criteria) or the set of settings tested, it is impossible to assess whether the sensitivity analysis covers the plausible range of reconstruction biases that could artifactually induce the observed square-root scaling.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful and constructive review. The comments correctly identify that metaorder reconstruction from anonymous data carries inherent uncertainty and that the abstract lacks sufficient methodological detail. We address both points below, propose targeted revisions, and note the limitation that cannot be fully resolved without broker-tagged data.

read point-by-point responses

  1. Referee: The central claim that I/σ_D = c (Q/V_D)^{1/2} holds with c_raw = 0.69 and is preferred by ΔAIC = 22 rests on reconstructed metaorder quantities Q; the sign-shuffling and chronology-scrambling tests only establish that impact correlates with the reconstructed sign sequence, not that the reconstructed Q values are the causally relevant parent-order sizes. This leaves the skeptic's concern about systematic bias from clustering rules unaddressed and load-bearing for the fitted prefactor and model ranking.

    Authors: We agree that the sign-shuffling and chronology-scrambling tests primarily confirm dependence on the reconstructed sign sequence and temporal ordering rather than independently validating the numerical accuracy of each Q. This is an intrinsic limitation of anonymous ITCH data. Nevertheless, the observed square-root scaling of impact with Q, the size-distribution tail exponent β = 1.54 ± 0.15 matching the worldwide literature, the stability of c across 32 walk-forward windows, and the decisive AIC preference together provide indirect corroboration that the reconstructed Q values are not arbitrary. We will revise the manuscript to state this limitation explicitly in the Discussion and to report additional robustness checks that vary the clustering parameters while monitoring both c and the model-ranking statistics. revision: partial

  2. Referee: The paper states that c_raw remains in [0.63, 0.77] 'across every reconstruction setting,' but without explicit description of the clustering rules (time windows, volume thresholds, sign-persistence criteria) or the set of settings tested, it is impossible to assess whether the sensitivity analysis covers the plausible range of reconstruction biases that could artifactually induce the observed square-root scaling.

    Authors: We accept that the abstract does not enumerate the clustering rules or the exact grid of settings examined. The Methods section of the manuscript describes the reconstruction algorithm, but the sensitivity range is only summarized. We will add an explicit paragraph (and, if space permits, a supplementary table) listing the tested time windows (1–60 min), volume thresholds, sign-persistence criteria, and the full set of parameter combinations that produce c_raw ∈ [0.63, 0.77]. This revision will make the coverage of plausible biases transparent. revision: yes

standing simulated objections not resolved
  • Direct verification that the reconstructed Q values equal the true parent-order sizes is impossible with anonymous Nasdaq ITCH data; only indirect consistency checks are feasible.

Circularity Check

0 steps flagged

No significant circularity; empirical tests are independent of fitted parameters

full rationale

The paper fixes the exponent at the literature value 1/2 and fits only the prefactor c while performing direct AIC-based model comparisons that prefer square-root over linear and logarithmic forms. Structural falsification tests (sign shuffling collapsing impact to chance levels, chronology scrambling destroying the relation) operate independently of the calibration and do not rely on the fitted c. Metaorder reconstruction is a preprocessing step whose output is subjected to these controls rather than being defined to produce the observed scaling. No load-bearing self-citation, self-definitional equation, or fitted-input-renamed-as-prediction reduces the central confirmation to a tautology by the paper's own equations.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the accuracy of anonymous metaorder reconstruction and on the assumption that the exponent can be fixed at the universal value 1/2 taken from prior cross-sectional studies. No new entities are postulated.

free parameters (1)
  • c (prefactor)
    Single scalar fitted to the AAPL data after fixing the exponent at 1/2; reported as 0.69 raw and 0.34 bias-corrected.
axioms (2)
  • domain assumption The square-root exponent 1/2 is the correct universal value and need not be re-estimated on this dataset.
    Stated explicitly when the functional form is calibrated with exponent fixed at 1/2.
  • domain assumption Reconstructed metaorders from the anonymous tape correspond to the actual parent orders that generated the observed price impact.
    Required for the impact measurement to be interpreted as market impact rather than an artifact of the reconstruction rule.

pith-pipeline@v0.9.1-grok · 5868 in / 1666 out tokens · 23149 ms · 2026-06-26T05:42:39.906055+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

8 extracted references · 3 canonical work pages · 1 internal anchor

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