Fifty configurations per stock-year give a researcher plenty of room to hit a chosen target. Goliath and Gebbie aim at two targets separately, and the answers conflict. The conflict is the paper's real contribution.
A disclosure comes first. The paper studies JSE-listed equities through signed trade prints, using mid-quotes from either side of every execution. We have US equity minute bars, without quote or trade-print data. We ran nothing, and nothing below is a replication.
How the tape is partitioned
A metaorder is a sequence of buy or sell orders from one agent, divided into same-sign child orders. Lillo, Mike and Farmer proposed that trade-sign long memory arises because the lengths of these runs follow a power law. Write P(L) ∝ L^(−α−1) for α > 1. Sign autocorrelation then falls as C(τ) ∝ τ^(−γ), where γ = α − 1. Sato and Kanazawa tested the relation quantitatively with account-level order-splitting data from the Tokyo Stock Exchange, and it held. Anonymous tape leaves everyone else unable to identify which trades belong to the same parent.
Maitrier, Loeper and Bouchaud offered a workaround. Every trade on the public tape is assigned to one of N synthetic traders, while chronological order is retained. Participation weights come from a power law P(f) ∝ f^−δ. Runs of same-signed trades assigned to the same trader become metaorders. The resulting partition is claimed to reproduce the established impact regularities.
Goliath and Gebbie apply this construction to 239 stocks from 1 January 2023 to 31 December 2025. Their universe consists of the top 250 JSE-listed companies by market capitalisation as of 13 March 2026. They use Level 1 trade-by-trade prints from BMLL, the market-data vendor whose Data Lab hosts the JSE tape, together with mid-prices immediately before and after each trade. The first and last ten minutes of each stock-day are removed, leaving 09:10 to 16:50.
Their Version 1 implementation took its parameters from Maitrier et al. without calibrating them to JSE data. Version 2 performs that calibration. For each stock and year, the authors search N ∈ {5, 10, 20, 30, 40, 50, 100, 500, 1000, 1500} and δ ∈ {1.5, 2, 3, 4, 5} traders per year. This produces 50 unique configurations, evaluated under two scores.
The first score, e_M, rewards simultaneous agreement with three impact targets and penalises the variance of each estimate. Those targets are a Q/V_D exponent 0.5, a ϕ exponent 0.5, and decay exponent β = (1−γ)/2, with λ = 1 and equal weights η_i = 1. The second score is e_LMF, defined as |α̂ − γ̂ − 1| / (γ̂ + 1). It measures the in-sample distance from the LMF relation.
Selection by e_M wipes out the LMF relation. Median γ, binned against α − 1, stays roughly 0.5 under both the NLLS and spectral estimator of γ.
Flat means independent.
Selection by e_LMF makes the relation appear immediately. For Gold Fields (GFI), the discrepancy reported for 2023, 2024 and 2025 is 0.00 in every year. The authors state the value of this result plainly. They call it "tautological since we specifically chose configurations that would result in the best" γ ≈ α − 1 relation, adding that possible overfitting may be involved.
Their partial defence rests on aggregation. Each stock year may fall short of a perfect γ = α − 1 relation, they write, while the combined result gives an almost perfect relation. Yet choosing each stock-year on e_LMF is exactly the procedure likely to generate that near-perfect aggregate line. The aggregate supplies no independent evidence.
The paper also says: "Neither objective independently identifies the synthetic partitions as the latent metaorders." Its abstract divides the same conclusion into two parts. Aggregate impact stylised facts alone cannot identify LMF-consistent order splitting. The LMF-targeted exercise shows compatibility inside the reconstruction class instead of providing an independent test.
The headline result is negative, and the authors reached it themselves. Good.
The prior does the work
Algorithm 1 and Algorithm 2 are described closely enough to implement. The paper lays out the trade-to-trader mapping, cumulative-probability draw, chronological constraint and run definition. Its estimation machinery is unusually explicit for this literature as well.
The ACF uses an uncentred sign-product estimator computed by FFT. At lag τ, the lagged sum is divided by N_ε − τ. The fitting range is selected by the highest log-log R², provided at least 10,000 points remain. Spectral γ is estimated from the lowest 15% of positive frequencies, with γ̂ = s + 1.
All the discretion enters through the prior. Neither N nor δ has an external anchor. The authors acknowledge that the number of active traders is unknown and changes from day to day for the same stock, yet N remains fixed within each stock-year. Growthpoint (GRT) shows how little the fitted value resembles a stable participant count. Minimising e_M gives N = 100 in 2023, N = 1500 in 2024 and N = 100 in 2025. One stock jumps fifteen-fold in a year, then returns to its earlier value.
The assumed family creates another judgment call. Power-law participation is chosen as more realistic than homogeneous participation, after which δ is selected from a five-point grid alongside N. The run-length exponent α is estimated through powerlaw.Fit, using a package-selected L_min. As the paper states, the committed implementation uses no separate bootstrap goodness-of-fit test. The power-law form for run lengths is therefore assumed, fitted and passed into the relation under examination.
We did not find a holdout year or cross-validation for the chosen (N, δ) pairs. Each stock-year is calibrated on that same stock-year.
The objectives also treat uncertainty differently. Every estimate in e_M carries a variance penalty. No equivalent penalty appears in e_LMF because the variance of the α estimate is unmeasured. For GFI, e_M is 0.66/0.60/0.55 when selection uses e_M. Under e_LMF selection, it rises to 0.66/0.69/0.63. None of this change is variance-adjusted.
Square-root impact cannot choose between them
Across the pooled stock universe, the impact curve follows the square-root law for Q/V_D ≥ 10^−2 under either selection criterion. The pooled e_M and e_LMF curves are visually similar. Sign balance is similar too: 24,596,453 buys versus 24,561,605 sells under e_M, and 25,529,711 versus 25,497,441 under e_LMF. Average Q/V_D agrees to three digits in both cases.
The two configurations yield indistinguishable aggregate impact curves while giving wholly different answers for the LMF exponent.
Pooling hides a great deal. In the stock-year tables, GFI has a fitted Q/V_D exponent of 0.04 in each of the three years under e_M, far from the 0.5 target. GRT records 0.15, 0.02, 0.13. Changing the criterion does not repair the stock-level fit. Under e_LMF selection, GFI comes in at 0.09, 0.11 and 0.02. These are still the best configurations among the 50 searched.
The SQL term is evidently overwhelmed by the other two targets in the aggregate loss. A square-root-like pooled curve coexists with stock-level exponents an order of magnitude away. The paper reports both facts.
For Q/V_D below 10^−2, the pooled curve flattens above the square-root prediction under both criteria. The authors test the obvious explanation and reject it. Under e_M, 39.53% of buy metaorders and 39.41% of sells occupy this region. The subset has the same average Q/V_D for each sign, 3.47×10^−4. They instead link the plateau to Harvey and co-authors' evidence that small JSE transactions produced greater-than-expected price impact after the September 2013 fee restructuring. This plateau is the paper's most interesting empirical detail. It has no connection to LMF.
Profile diagnostics show the price of targeting the LMF fit. Under e_M, the execution-profile exponent is 0.802 ± 0.044 for GFI and 0.704 ± 0.046 for GRT, already above the theoretical 0.5. Selecting on e_LMF pushes GFI farther away, to 0.896 ± 0.042, while GRT moves closer at 0.677 ± 0.038.
Decay exponents shift in the opposite direction. Under e_M, GFI has β = 0.159 ± 0.002 and GRT has 0.177 ± 0.003. Both lie near the 0.2 that Brokmann and co-authors report for real metaorders. Under e_LMF, those estimates fall to 0.146 ± 0.003 and 0.167 ± 0.002, producing more linear decay. GRT's average metaorder duration also falls from about 100 minutes to about 32. The tape and stock are unchanged, yet the alternative partition creates parent orders three times shorter.
Only those two tickers receive detailed stock-level diagnostics. The remaining 237 enter the pooled curves and boxplots.
What would test it?
Account or broker identifiers would provide the clean test. Sato and Kanazawa had them; this dataset does not. An out-of-sample exercise is the next option: estimate (N, δ) using 2023 and 2024, then measure the LMF discrepancy in 2025. The jumps in N between adjacent years for a single stock make that test informative whichever result emerges. Harder still would be external labels for parent orders, even on a subsample, against which the synthetic partition could be checked.
The method requires signed trade prints and mid-quotes surrounding every execution. Minute aggregation erases the child-order sequence being partitioned, leaving our data unable to test the mechanism. Transfer to other liquid equity markets remains plausible. We do not possess the necessary trade-level and quote-level data.
One construction choice also deserves attention from anyone extending the work. The 239-name universe comes from the top 250 by market capitalisation as of 13 March 2026, then is applied retrospectively across the 2023 to 2025 sample. The impact exponents are estimated on survivors. A return study would suffer more from that choice.
This is candid calibration work ending in a clean negative result. Reproducing all four impact regularities gives no purchase on the LMF exponent relation, while forcing the LMF relation reduces profile accuracy. The evidence covers 49,158,058 synthetic metaorders selected under e_M and 51,027,152 selected under e_LMF, spanning three years and 239 names. I would revise my view if the same grid, estimated on two years and then frozen, still placed γ near α − 1 in the third.