Spurious or Structural?

Low-Rank Confounding and Partial Identification of Cross-Asset Price Impact

Mehmet Demir Güven †

Department of Computer Science, ETH Zürich

† Independent research. ETH Zürich did not fund, sponsor, approve, or endorse this work. The affiliation records the author's status as a student only, and the views expressed are the author's alone.

Preprint · Version 0.3 · 12 August 2026 · CC BY 4.0

Evidence status — 12 August 2026. G0 and G1 have passed. G2 is open: the scientific design and document authority are registered, but executable resource admission is incomplete. No registered G2 resource benchmark, validation panel, research draw, external market data, training sample, holdout, or economic evaluation has been accessed.

Current preprint: PDF · LaTeX source · license notice

Abstract

Cross-asset return-on-flow coefficients are routinely read as entries of a structural price-impact matrix. That reading is not merely noisy; it is generically unidentified, and this project characterises exactly how.

In a simultaneous $N$-asset system with $K$ latent factors and same-bin feedback $B$, the gap between the population regression coefficient and the structural impact matrix has rank at most $K+\mathrm{rank}(B)$. When the structural matrix is diagonal and feedback is absent, the entire estimated cross-impact matrix is confined to the diagonal-plus-rank-$K$ set, yet its off-diagonal entries can be as large as genuine own-impact. Because the gap is a free low-rank object, the structural matrix is set-identified rather than point-identified from second moments. In the permutation-invariant one-spike geometry the sharp identified interval is available in closed form, and at published one-minute commonality statistics it contains zero. The same rank structure makes the consequence tractable: the execution-cost error is a rank-$K$ quadratic form, so it vanishes on an immune subspace of dimension at least $N-K$ while almost every other trade is mispriced. An equal-weight index basket is mispriced by 54% of its true cost while a dollar-neutral basket is mispriced by exactly nothing, and a dollar-neutral trade has a point-identified execution cost even though the matrix behind it does not. Finally the restriction is refutable: a scale-free departure statistic, with a bootstrap null that controls size above roughly $5N^2$ observations and is reported as invalid below.

The derivations are verified in a preregistered known-truth experiment with $N=30$, $K=3$, and $T=10^7$. The empirical premise test is registered here but not reported.

Keywords: cross-impact; order-flow imbalance; latent factors; partial identification; low-rank structure; preregistration.

1. The Motivating Puzzle

Capponi and Cont regress one-minute returns on the order-flow imbalance of every stock in a large cross-section and report a mean cross-asset coefficient of +0.032, with 23.09% of coefficients negative. Adding a single cross-sectional principal-component control moves the mean to −0.039 and the negative share to 84.46%.

One control flips the sign of average estimated cross-impact. Either the uncontrolled regression was contaminated and the controlled one reveals the truth, or the control absorbed real impact. This project's answer is that neither reading is identified, and it makes that precise.

Structural impact matrices enter execution-cost models, liquidity stress tests, and manipulation constraints, so the ambiguity is not academic.

2. Simultaneous System

Let returns $r_t$, flows $q_t$, and shocks $u_t,v_t$ lie in $\mathbb{R}^N$, with latent factors $f_t\in\mathbb{R}^K$:

$$ r_t = \Lambda q_t + \Gamma f_t + u_t, \qquad q_t = B r_t + \Delta_f f_t + v_t. $$

With $L=I_N-B\Lambda$, $H=L^{-1}$, $P=H(B\Gamma+\Delta_f)$, $U=HB$, and $V=H$, the flow reduced form is $q_t=Pf_t+Uu_t+Vv_t$.

Theorem 1 — Pseudo-true cross-impact matrices

$$ \mathrm{plim},\widehat\Lambda_{\mathrm{OLS}} =\Lambda+\Gamma\Sigma_fP^\top\Sigma_{qq}^{-1} +\Sigma_uU^\top\Sigma_{qq}^{-1}. $$

Controlling for $h_t=f_t+\varepsilon_t$ replaces $\Sigma_f$ with the residual factor covariance $R_f=\Sigma_f-\Sigma_f(\Sigma_f+\Sigma_\varepsilon)^{-1}\Sigma_f$ and $\Sigma_{qq}$ with $Q_h=PR_fP^\top+U\Sigma_uU^\top+V\Sigma_vV^\top$.

Write $G:=\mathrm{plim},\widehat\Lambda_{\mathrm{OLS}}-\Lambda$ for the confounding gap. Everything below concerns its structure.

3. The Confounding Gap Is Low Rank

Theorem 2 — Rank of the confounding gap

$$ \mathrm{rank}(G);\le;K+\mathrm{rank}(B). $$

The factor channel passes through a $K$-dimensional bottleneck and the feedback channel through the column space of $B$; rank is subadditive. The bound is attained generically.

$\mathrm{rank}(B)$	Observed $\mathrm{rank}(G)$	Bound
0	3	3
1	4	4
2	5	5
30	30	33

Corollary — Spurious cross-impact is confined, but not small

If $\Lambda$ is diagonal and $B=0$, the population coefficient matrix lies in $\mathcal{D}_K={D+R : D \text{ diagonal},\ \mathrm{rank}(R)\le K}$.

This bounds the shape, not the size. At the registered fixture a strictly diagonal truth induces spurious off-diagonals reaching 0.2207 against genuine own-impact spanning 0.2061 to 0.3953. Confounding does not perturb a cross-impact matrix; it manufactures one of realistic magnitude out of nothing.

Proposition 3 — The structural matrix is set-identified

With $B=0$, $\Sigma_f=I_K$, and $W=\Sigma_{qq}^{-1}\Delta_f$, any $\Lambda = A - \Gamma W^\top$ satisfying the positive-semidefiniteness constraints reproduces the observed second moments exactly. Cross-impact is a partial identification problem; a factor control moves the estimate along the confounding directions rather than out of the identified set.

Proposition 4 — Sharp interval in the one-spike geometry

In the registered permutation-invariant geometry the gap collapses to a single constant added to every entry, and

$$ \Lambda_{\mathrm{off}}\in \left[A_{\mathrm{off}}-\tfrac{T}{N},; A_{\mathrm{off}}+\tfrac{T}{N}\right], \qquad T^2=\frac{(r_1-q_1a_1^2)(q_1-q_0)}{q_1q_0}. $$

At the source-matched calibration ($N=30$, $s_q=0.2827$, $s_r=0.32$, $d=0.29$):

$o$	$A_{\mathrm{off}}$	Half-width	Identified interval	Contains 0
0.0029	0.010554	0.094306	$[-0.083753,\ 0.104860]$	yes
0.0046	0.012254	0.090420	$[-0.078166,\ 0.102673]$	yes

The identified half-width is 7.4 to 8.9 times the observed coefficient it is meant to pin down. Under the stated conventions the structural off-diagonal is not identified even in sign.

A bisection over the exact positive-semidefiniteness frontier reproduces the closed form to relative error below $10^{-10}$.

Definition — A refutable diagnostic

$$ \psi_K(\widehat A)= \frac{\min_{D,\ \mathrm{rank}(R)\le K}\lVert \widehat A-D-R \rVert_F} {\lVert \widehat A-\mathrm{diag}(\widehat A) \rVert_F}. $$

Its population value is zero under pure confounding, so a materially nonzero $\psi_K$ rejects the maintained pure-confounding null under the stated factor budget. That null bundles diagonal $\Lambda$, $B=0$, exactly $K$ factors, the covariance restrictions, linearity, and stationarity — rejection falsifies the conjunction, and genuine cross-impact is one explanation among several. This still inverts the conventional reading: off-diagonal magnitude carries no information, only departure from $\mathcal{D}_K$ does.

Two caveats are stated up front. $\psi_K=0$ does not prove diagonality — the test refutes, it does not confirm. And $\psi_K$ depends on the assumed factor count, so a sweep is mandatory:

Assumed $K$	1	2	3	4	6	10
$\psi_K$	0.6396	0.3748	0.0391	0.0377	0.0328	0.0250

The elbow at the true $K=3$ is a practical way to read the factor count off the coefficient matrix itself.

The full derivation and proofs are in CONFOUNDING_RANK_AND_PARTIAL_ID.md, registered before implementation as amendment A028.

4. What the Failure Costs

Identification failure only matters if it changes a decision. A desk pays $C(x,M)=x^\top M x$ to execute trade $x$.

Theorem 6 — The cost error is low rank

$C(x,A)-C(x,\Lambda)=x^\top G x$, so by Theorem 2 the error vanishes on a subspace of dimension at least $N-K-\mathrm{rank}(B)$. In the fixture that immune subspace has dimension 27, and a trade drawn from it has cost error $5.2\times10^{-17}$.

The immune set has measure zero. Of 200,000 randomly drawn trades, every one carried nonzero cost error. Low rank does not mean most trades are safe; it means a desk can construct safety if it knows the subspace. Since $x^\top Gx = x^\top\mathrm{sym}(G)x$, the economically relevant object is the symmetric part, whose rank is $2K$ generically, and the exact immune set is a quadric cone rather than a subspace.

In the one-spike geometry the error is exactly $g(\mathbf 1^\top x)^2$:

Trade	True cost	Error	Relative
Equal-weight index	0.4234	+0.2296	+54.23%
Random unit vector	0.2950	+0.0160	+5.42%
Dollar-neutral pair	0.2854	0	0.00%

Corollary — Cost can be identified where the matrix is not

In the one-spike geometry, the identified cost interval has half-width $(T/N)(\mathbf 1^\top x)^2$, so a dollar-neutral trade has a degenerate interval. Its execution cost is point-identified even though the impact matrix is set-identified with an interval containing zero. An unidentified parameter need not mean an unidentified cost.

The qualifier matters. Dollar-neutrality confers immunity only because the one-spike confounding direction is the equal-weight direction. In a general geometry a trade with $\mathbf 1^\top x=0$ to machine precision still carries a −18.24% cost error at our fixture. The condition that actually confers immunity is membership of the null space of $G$ (Theorem 6); one-spike merely makes that null space easy to name.

Minimax-cost execution

For a desk that must take exposure, the schedule minimising worst-case cost subject to $c^\top x=q$ is $x^*(\pi)=\tfrac12\lambda M(\pi)^{-1}c$ with $M(\pi)=A_s+\pi\mathbf 1\mathbf 1^\top$ at $\pi=T/N$. It matches a 20,000-point grid search exactly.

Target	Worst-case improvement	Exposure naive → robust
Index-like	0.00%	+1.000 → +1.000
Neutral	0.00%	0.000 → 0.000
General	3.11%	−0.0663 → −0.0130

Robustness buys nothing when the constraint already pins factor exposure. Both degenerate cases were predicted before implementation.

5. Is the Restriction Refutable? A Test

$\psi_K$ needs a critical value to be usable. A parametric plug-in bootstrap supplies one, with the factor count chosen by a rule fixed before use.

$T$	$T/N^2$	Realised size (nominal 0.05)
500	0.56	0.267
1,000	1.11	0.127
2,000	2.22	0.100
5,000	5.56	0.040

Power at $T=5000$: 0.070, 0.270, 0.870, 1.000 against perturbations of 0.05, 0.10, 0.20, 0.40.

The test is valid only above roughly $T=5N^2$ and over-rejects severely below. The cause is diagnosed: the plug-in null is refitted to the noisy estimate, so the bootstrap centres too low. A degrees-of-freedom variance inflation was tried and fails completely — it drives size to exactly zero and removes all power, because the bias is in the centre of the null distribution, not its scale. It is registered as not adopted.

6. Completed Known-Truth Verification

The G1 derivation was frozen before simulation code and random-number access. One master draw was split into 100 immutable shards of 100,000 observations each, publishing only mergeable sufficient statistics, for $T=10^7$ observations at $N=30$ and $K=3$.

Quantity	Verified value
Assets / factors / observations	30 / 3 / 10,000,000
Reported coefficient targets	1,800
Uncontrolled OLS maximum relative discrepancy	0.0005639467093140219
Proxy-controlled maximum relative discrepancy	0.0005123714186295689
Preregistered gate threshold	0.001
Targets inside simultaneous intervals	1,800 / 1,800
Interval method	Student-t, Bonferroni 95% FWER

Replay reused all validated checkpoints and reproduced the summary, estimates, and success marker byte for byte. G1 is closed; the frozen draw must not be rerun.

7. Confronting Published Evidence

Because the one-spike gap is a constant added to every entry, a single factor control should shift every cross-coefficient by the same amount. Two implications follow, and they disagree.

Dispersion invariance holds. The reported mean cross coefficient moves from 0.032 to −0.039, a shift of −0.071, while the cross-sectional standard deviation stays at 0.06 and the own-coefficient standard deviation moves only from 0.78 to 0.77. Both are at or within reported precision. This check is unit-free.

The shape implication fails. Under an exactly constant shift the post-control negative fraction should equal the pre-control mass below the shift magnitude, which a Gaussian approximation puts at 0.7422 against a reported 0.8446 — a gap of 0.1024, exceeding the declared 0.05 tolerance.

The failure is reported rather than removed; the registered protocol forbids retuning the one-spike convention to close it. It points at loading heterogeneity beyond one common factor and does not bear on Theorem 2, which is an inequality in $K$.

Both are conditional analytic exhibits at published summary statistics, not estimates of any market's impact matrix.

8. Registered Premise Test

The next gate asks a deliberately narrow question before empirical data is opened:

Can confounding alone produce economically material off-diagonal coefficient error in a transparent model constrained by opened primary-source summaries, even when the estimator receives a favourable factor proxy?

The registered system sets $B=0$ so same-bin feedback cannot explain a positive result, retains $N=30$, uses one permutation-invariant factor, sets proxy reliability to 0.95, and evaluates 17 frozen structural off-diagonal values from 0.0029 to 0.0046. Three smooth estimators bind at every grid point; a six-specification published-protocol reconstruction supplies a separate veto. Passage requires

$$ \left|\widehat\Lambda_{01}-o\right|-0.50|o|>3,SE_{\mathrm{boot}} $$

with 499 shared whole-date bootstrap replicates, after 100 validation superpanels license the exact procedure.

Current status: contract, test-only RNG namespace, pure DGP maps, smooth estimator core, checkpoint/recovery boundary, deterministic paper kernels, and the A027 paper-cache codec are implemented and tested. Executable resource admission and rehearsal are incomplete, so no registered G2 stream is licensed.

9. Gate Status

Gate	Status	Licensed statement
G0 — environment and compute plan	Passed	Reproducible software and bounded compute skeleton
G1 — derivation and known-truth recovery	Passed	Derived population targets recovered under the frozen simulation law
G2 — premise test / kill switch	Open	Design authority and deterministic software evidence only; no G2 result
G3–G7 — data, identification, estimation, validation, holdout	Locked	No market-data, predictive, causal, trading, or economic claim
G8 — final-results paper and release	Locked	This pre-results preprint adds no downstream result

Authoritative live state: STATE.md. Amendments and rejected specifications remain visible in PREREGISTRATION.md, DECISIONS.md, ASSUMPTIONS.md, and SPECIFICATION_LOG.md. Working standards are in RESEARCH_PROTOCOL.md.

10. Reproducibility

uv sync --locked --extra dev
make check        # lint, format, strict types, tests, smoke, drift
make exhibits     # regenerate every manuscript number; fails on drift
make paper        # build the preprint PDF

Fresh local gate for preprint version 0.3:

Check	Result
Ruff lint	Pass
Ruff format	38 files checked
Strict mypy	Pass, 38 source files
Pytest	410 passed
Deterministic G0 demo	64 rows; expected hashes reproduced
Committed-result drift	Pass; no drift

These are software and artifact-consistency checks, not scientific trials, and they do not license a registered G2 stream. Every quantitative value in the manuscript is regenerated by make exhibits; none is transcribed by hand.

Do not run make mc, make g1-benchmark, or any G2 resource, validation, or research entry point without the exact authority recorded in the current gate ledger.

Evidence map

Artifact	Role
CONFOUNDING_RANK_AND_PARTIAL_ID.md	Rank bound, identified set, sharp interval, diagnostic
THEORY_EXTENSION.md	Six predictions frozen before implementation
identification.py	Probability limits, confounding gap, one-spike bounds
rank_diagnostic.py	The $\psi_K$ statistic and its bootstrap test
execution.py	Cost error, immune subspace, cost interval, minimax schedule
EXECUTION_COST_UNDER_CONFOUNDING.md	A029 execution derivation
PSI_NULL_DISTRIBUTION.md	A030 null distribution and size study
generated/psi_study.json	Committed confirmatory size and power study
exhibits.py	Deterministic generator for every manuscript number
generated/exhibits.json	Committed exhibit values
GATE_G1_PROBABILITY_LIMITS.md	Theorem 1 derivation
results/g1/summary.json	Accepted G1 gate statistic
configs/g2.toml	Hash-sealed S0004 scientific contract
GATE_G2_PREMISE.md	G2 estimands, algorithms, decision rules
G2_SOURCE_AUDIT.md	Primary-source audit of published statistics
data/manifest.json	Zero-external-data manifest

11. Limitations

• Theorem 2 is an inequality, so a high observed rank is uninformative if $K$ is misspecified, and the gap's rank is not separately observable from that of a genuinely low-rank $\Lambda$.
• Proposition 3 assumes $B=0$; with feedback the identified set is larger, not smaller, so the sign non-identification result is conservative in that direction only.
• The closed-form interval is conditional on the one-spike and isotropic-residual conventions, which are declared maximum-entropy choices and not identified features of any exchange.
• $\psi_K$ is an upper bound from a stationary point of an alternating projection and cannot confirm diagonality. Its bootstrap test is valid only above roughly $T=5N^2$, and its size and power were established on a single Gaussian, homoskedastic, serially independent fixture that market data violates. The factor count is assumed, not estimated with a validated rule.
• The execution results use a static one-period impact model with no decay kernel, risk aversion, or timing risk, and support no profitability claim.
• The G1 fixture is Gaussian, known-truth, and large-sample; its dense positive targets do not reproduce the small, sign-sensitive off-diagonals central to the empirical question.
• The published-evidence exercise uses summary statistics carrying no inferential intervals, so it supports consistency statements only.
• Predictive performance, structural identification, market impact, execution savings, transaction costs, capacity, and profitability are unresolved. The present evidence supports no trading rule, deployment claim, or return claim.

12. Selected References

The manuscript carries 45 references. The most directly relevant:

1. Kyle, A. S. (1985). Continuous auctions and insider trading. Econometrica, 53(6), 1315–1335.
2. Hasbrouck, J., & Seppi, D. J. (2001). Common factors in prices, order flows, and liquidity. Journal of Financial Economics, 59(3), 383–411.
3. Cont, R., Kukanov, A., & Stoikov, S. (2014). The price impact of order book events. Journal of Financial Econometrics, 12(1), 47–88.
4. Benzaquen, M., Mastromatteo, I., Eisler, Z., & Bouchaud, J.-P. (2017). Dissecting cross-impact on stock markets. JSTAT, 2017(2), 023406.
5. Capponi, F., & Cont, R. (2020). Multi-asset market impact and order flow commonality. SSRN 3706390.
6. Cont, R., Cucuringu, M., & Zhang, C. (2023). Cross-impact of order flow imbalance in equity markets. Quantitative Finance, 23(10), 1373–1393.
7. Manski, C. F., & Tamer, E. (2002). Inference on regressions with interval data on a regressor or outcome. Econometrica, 70(2), 519–546.
8. Chandrasekaran, V., Parrilo, P. A., & Willsky, A. S. (2012). Latent variable graphical model selection via convex optimization. Annals of Statistics, 40(4), 1935–1967.
9. Miao, W., Geng, Z., & Tchetgen Tchetgen, E. J. (2018). Identifying causal effects with proxy variables of an unmeasured confounder. Biometrika, 105(4), 987–993.

Full bibliography: references.bib.

13. License and Citation

Copyright © 2026 Mehmet Demir Güven. The preprint manuscript, its source, and its original figures are licensed under CC BY 4.0, and are outside the "Software" covered by the repository's MIT license. The MIT license governs the software and its associated documentation.

No arXiv identifier exists yet. Until one is assigned, cite as:

Mehmet Demir Güven (2026). "Spurious or Structural? Low-Rank Confounding and Partial Identification of Cross-Asset Price Impact." Preprint, version 0.3, 12 August 2026.

For a reproducible reference to the repository state, include the commit hash and access date.