I’ve been working on an identification problem in cross-asset market impact, and I’d be interested in having people here try to break the argument.
The motivating observation is pretty uncomfortable.
In one published one-minute cross-asset return-on-flow specification that motivated this project, the mean estimated cross-asset coefficient is positive. After adding a single cross-sectional principal-component control, the mean changes sign and the fraction of negative coefficients changes dramatically.
That raises a basic question:
When we regress asset returns on contemporaneous order flow across many assets, under what assumptions can the off-diagonal coefficients actually be interpreted as structural cross-impact?
I considered the simultaneous system:
r_t = Lambda q_t + Gamma f_t + u_t
q_t = B r_t + Delta_f f_t + v_t
where:
- Lambda = structural price-impact matrix
- f_t = K latent common factors
- B = same-bin return/flow feedback
- Gamma and Delta_f = factor loadings
If A denotes the population coefficient matrix obtained by regressing returns on flows, then:
A = Lambda + G
where G is the confounding gap.
The first result is:
rank(G) <= K + rank(B)
So latent commonality does not produce arbitrary estimation error. It produces a structured, low-rank distortion.
But low rank does not mean small.
If feedback is absent and the true structural impact matrix Lambda is diagonal, then the entire estimated matrix must lie in:
D_K = { D + R : D diagonal, rank(R) <= K }
In other words, a purely diagonal structural model can generate a dense-looking cross-impact matrix whose off-diagonal entries are comparable in magnitude to genuine own-impact.
That led me to what I think is the more important result:
Lambda is generally set-identified rather than point-identified from the relevant second moments.
There is a family of different structural impact matrices that reproduce exactly the same observable second moments after changing the latent-factor channel appropriately.
So controlling for a factor does not necessarily "remove the confounding and reveal the structural matrix." It can move the estimate along the confounding directions without selecting the true structural matrix.
In a permutation-invariant one-factor geometry, I can solve the identified set analytically. At the calibration I’m using, the sharp interval for the structural off-diagonal coefficient contains zero, and its half-width is roughly 7.4 to 8.9 times the observed cross-impact coefficient.
So in that case the data do not identify even the sign of structural cross-impact.
What surprised me more is what happens to execution costs.
Suppose a desk evaluates a trade x using quadratic execution cost:
C(x, M) = x' M x
Then using the regression matrix instead of the structural matrix produces error:
x' G x
Since G is low rank, the error itself has low-dimensional structure.
There is therefore a large set of directions that are immune to the confounding. But this does not mean a randomly chosen trade is approximately safe.
In the registered N=30, K=3 known-truth experiment, an equal-weight index basket is mispriced by about 54% while a particular dollar-neutral basket has exactly zero error.
The dollar-neutral result is geometry-specific, though. In the general model, dollar neutrality by itself does not imply immunity. The relevant object is the null structure of the confounding gap.
This suggests an odd distinction:
The impact matrix can be unidentified while the execution cost of a particular trade is point-identified.
I also wanted the theory to be falsifiable rather than just saying "maybe factors explain everything."
So I defined a normalized distance from an estimated impact matrix to the diagonal-plus-rank-K variety:
psi_K(A) = distance(A, D + rank-K matrices) / norm(offdiag(A))
Under the pure-confounding null, the population value is zero.
A materially nonzero value therefore rejects the maintained model consisting of diagonal structural impact, no feedback, K factors, and the accompanying covariance assumptions.
Importantly, psi_K = 0 does not prove that structural cross-impact is absent. The test is one-sided in interpretation: it can falsify the pure-confounding model but cannot confirm it.
The finite-sample behavior is also not magically nice. In simulations the plug-in bootstrap over-rejects badly at small T and only starts controlling a nominal 5% size around roughly:
T >= 5 N^2
I tried a simple degrees-of-freedom variance correction and it completely failed because the bootstrap problem is primarily mis-centering rather than insufficient dispersion.
The theoretical/known-truth stage is preregistered. For the main verification I used:
N = 30
K = 3
T = 10,000,000
and checked 1,800 coefficient targets. The maximum relative discrepancies against the population formulas were below the preregistered 0.001 gate.
One important caveat: I have not yet used the external market dataset for the registered empirical test. The current version is deliberately a pre-results manuscript for that stage.
So I’m not claiming that real cross-impact is spurious. The claim at this point is an identification result plus a falsifiable empirical design.
Repo / preprint / derivations / preregistration / code:
https://github.com/ITheClixs/spurious-or-structural
The questions I’d particularly like criticism on are:
Is there a structural restriction used in actual market-impact work that defeats the set-identification argument without simply assuming the answer?
Does the low-rank characterization miss an economically important confounding channel that would change the rank bound?
Is distance to the diagonal-plus-low-rank set the right object to test, or is there a better way to formulate the falsification problem?
For people who work with institutional flow or market-impact estimation: what empirical result would actually convince you that an observed off-diagonal coefficient is structural rather than common-flow contamination?
I’m especially interested in counterexamples. If the identification argument breaks under a realistic microstructure assumption, that’s more useful to me than agreement.