Testing the Square-Root Law of Market Impact in AAPL

Summary

This study tests whether market impact follows the square-root law using anonymous order flow for Apple stock. It reconstructs metaorders from Nasdaq TotalView-ITCH events across 178 trading days, then estimates impact against daily volatility and traded volume while fixing the impact exponent at one-half. The estimated raw coefficient is 0.69, with a bias-corrected estimate of 0.34.

The evidence includes comparisons with linear and logarithmic impact models, tests across reconstruction settings, and weekly walk-forward recalibrations. Shuffling trade signs removes directional impact, while scrambling event order makes the square-root relationship fail, supporting the role of signed and temporally structured order flow. The study also reports persistent order flow alongside diffusive price behavior, patterns associated with theoretical explanations of the law. Its scope is limited to one large-cap U.S. equity and a specific sample period; metaorders are inferred without broker-tagged parent orders, so results depend on the reconstruction method.

Key ideas

• The study infers metaorders from anonymous order-level market data rather than broker identifiers.
• It calibrates price impact to the square root of metaorder size relative to daily volume.
• Model comparisons favor the square-root specification over linear and logarithmic alternatives.
• Shuffling trade signs or event chronology weakens the observed impact relationship.
• The evidence comes from one U.S. stock and may depend on metaorder reconstruction choices.

Tags

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


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









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 $I/σ_D = c\,(Q/V_D)^{1/2}$ 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 $β= 1.54 \pm 0.15$ -- both consistent with the worldwide cross-section. A direct model comparison decisively prefers the square-root form over linear ($Δ{\rm AIC}=22$) 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 ($γ=0.66$) 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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.