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Non-parametric Causal Discovery for EU Allowances Returns Through the Information Imbalance

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A rank-based metric finds the same EUA drivers as Granger tests, plus the nonlinear ones Granger misses.

desk verdict The empirical headline doesn't follow from the paper's own equations—the sign in the Imbalance Gain is inverted, so the positive IGs in Section 4 cannot be produced by Eq. (9)/(11). read the letter →

arxiv 2508.15667 v1 pith:WT6KFOQI submitted 2025-08-21 q-fin.CP

Cristiano Salvagnin , Vittorio del Tatto , Maria Elena De Giuli , Antonietta Mira , Aldo Glielmo This is my paper · ORCID
classification q-fin.CP
keywords EUETScarbonallowancescausaldiscoveryDifferentiableInformationImbalanceGrangercausalitynon-linearfinancialreturnsEUA
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Carbon prices in the EU emissions trading system are driven by an entangled set of energy, equity, and macro variables, and most causal studies rely on linear Granger or vector autoregression tests. This paper argues that a non-parametric, rank-based tool—the Differentiable Information Imbalance—can detect the same linear drivers while also catching non-linear couplings that VAR models miss. Applied to daily financial returns from January 2013 to April 2024, both methods single out the IBEX35 Spanish stock index and ICE Rotterdam coal futures as the clearest causal drivers of EUA returns. Two synthetic systems show that the disagreements between the methods are explainable: linear tests miss quadratic causal links and can invent links when a non-linear common driver exists. If the paper is right, carbon price modelling should treat DII as complementary evidence to linear causality, not a replacement for it.

What carries the argument

The central object is the Differentiable Information Imbalance (DII), a rank-based, gradient-optimised measure of how well distances in a weighted predictor space reproduce neighbourhood structure in a target space. Its causal variant, the Imbalance Gain (IG), quantifies how much adding a candidate variable to the predictor set improves prediction of the target one day ahead; a positive IG means the variable carries causal information. The DII carries the argument by being non-parametric, requiring no distributional or linearity assumptions, and by giving each variable a single optimised weight comparable to VAR coefficients, which enables a one-to-one comparison with Granger causality.

What would settle it

Rebuild the DII and Granger analysis on a dataset that adds a plausible measured confounder, such as European industrial production or natural gas prices; if the IBEX35/coal-futures causal weights vanish, then the paper's claimed direct causal effects are confounded.

Watch

Extended reading notes

Core claim

The paper's central empirical claim is that IBEX35 and coal futures are robust causal drivers of EUA returns, detected by both a linear VAR/Granger approach and the non-linear, model-free DII approach. The authors further claim that, where the two methods disagree, the differences stem from known limitations of linear causality: on synthetic data, multivariate Granger tests fail to detect a quadratic causal coupling (a false negative) and detect a spurious causal link transmitted through a non-linear common driver (a false positive), while the DII-based Imbalance Gain detects the true structure in both cases. The paper therefore proposes the Imbalance Gain as a non-linear analogue of the Gra

Load-bearing premise

The analysis assumes causal sufficiency—that no unobserved variable drives both EUA returns and the predictors in the dataset; the authors openly note this is likely false because factors like risk aversion and policy expectations are unmeasured.

Editorial extensions

If this is right

  • EUA return modelling should treat IBEX35 and coal futures as the primary market variables to monitor, since both a linear and a non-linear method independently flag them.
  • DII-based Imbalance Gain can flag non-linear causal links that Granger F-tests systematically miss, reducing false negatives relative to VAR-only analysis.
  • Where linear methods create a spurious causal link through a non-linear common driver, DII/IG can avoid that false positive.
  • Agreement between DII and VAR on a set of drivers can be used as a validation step before committing to a Granger causality conclusion.
  • Until resampling-based significance tests are developed, DII causal rankings should be used alongside linear statistical tests rather than alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If DII's non-linear advantage generalises, existing Granger/VAR-based spillover studies of carbon and energy markets may undercount crisis-time transmission, because those links can be quadratic or threshold-shaped.
  • The same pipeline could be transferred directly to other compliance carbon markets, such as the UK ETS or California allowances, where the same non-linearity problem in price formation applies.
  • A testable extension is to compute IG at lags greater than one day: coal futures may lead EUA returns at a horizon that the paper's τ = 1 analysis does not examine.
  • Because IG currently lacks a significance threshold, a practical selection rule could be built by permuting predictor labels; the paper itself proposes resampling techniques as future work.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes the Differentiable Information Imbalance (DII) as a non-parametric, potentially non-linear method for causal discovery in financial time series, and applies it to EUA (European Union Allowance) returns over January 2013–April 2024. It compares DII-based Imbalance Gain (IG) rankings with multivariate Granger causality from a VAR(1) model, and reports that both methods identify IBEX35 and Coal Futures as important drivers of EUA returns. Two synthetic datasets are used to show that linear Granger causality can miss non-linear causal links and can produce false positives when a non-linear common driver is present.

Significance. If the empirical claims hold, the paper would contribute a non-linear, non-parametric complement to standard Granger-causal analysis of carbon markets, and the overlap with VAR results would strengthen confidence in IBEX35 and coal futures as robust predictors of EUA returns. The paper also provides reproducible code and clearly describes the DII optimization procedure. However, a sign inconsistency in the central IG definition currently undermines the reported positive IG values, so the empirical ranking and the headline overlap claim cannot be accepted without correction. The synthetic illustrations are suggestive but are presented as single-run demonstrations without uncertainty quantification. With a corrected definition and appropriate robustness checks, the contribution could be useful to the q-fin.CP community.

major comments (3)
  1. [§3.2.2, Eqs. (8)–(11)] The Imbalance Gain is defined in Eq. (9) as IG = 1 − Δ(z0→zτ) / min_w Δ([w x0,z0]→zτ). Since smaller Δ means better predictability, the causal condition in Eq. (8) is min_w Δ([w x0,z0]→zτ) < Δ(z0→zτ). Under Eq. (9) this inequality makes the ratio greater than 1, so IG < 0, not IG > 0 as claimed. The multivariate version Eq. (11) has the same inversion: the numerator is the DII without xα and the denominator is the DII with xα, so a genuine causal variable yields a negative IG. Consequently, the positive IGs reported in Figures 4–6 for true causes cannot be produced by the formula as written. Either the equations are mis-specified or the code uses the inverse ratio. This is load-bearing because the entire empirical ranking and the paper's central overlap claim rest on IG. Please correct the definition and verify which quantity was actually computed.
  2. [§4 and Appendix A] The empirical IG values are presented as evidence of causal effects, but no significance threshold, confidence interval, or null distribution is provided. The authors acknowledge in §6 that DII lacks a simple significance test, yet the conclusion states that both methods 'detect significant causal effects' from IBEX35 and Coal Futures. Given that the DII optimization involves stochastic mini-batches and multiple hyperparameters, a bootstrap or permutation test is needed to distinguish non-zero IG from noise. Without such a test, the empirical overlap between DII and Granger rankings cannot be regarded as statistically supported.
  3. [§3.3 and Appendix A] The synthetic demonstrations are single-run experiments with no repeated seeds, error bars, or sensitivity analysis. The DII training uses random mini-batches and Adam optimization, so the reported weights and IGs could vary across runs. Claims such as 'the DII approach appears robust against this drawback' (Section 3.3.2) are not supported without repeated-seed statistics. Additionally, the key hyperparameters — the 0.1 prefactor in Eq. (14), the 5% neighbourhood fraction, the mini-batch size, the temporal exclusion window, and the number of epochs — are fixed without sensitivity analysis. At minimum, show stability of the synthetic conclusions across seeds and across reasonable hyperparameter perturbations.
minor comments (5)
  1. [Introduction, Section 1] The text 'heavAC100 per tonne' appears to be a typo; presumably '€100 per tonne' is intended.
  2. [Table 2] The columns in Table 2 appear misaligned or garbled: for example, the EUA row shows Max = -0.0143 and 25% = 0.0010, which are inconsistent with the other rows and with the text. Please reformat and verify all entries.
  3. [Section 4, Figure 6] The text says 'In the first panel, the IG highlights...' but the left panel of Figure 6 is labelled as F-statistics and the central panel as IG. The wording should refer to the correct panel.
  4. [References] Reference [1] is listed as 'Phys. Rev. Lett., pages –, Jun 2025' with no volume/article number; please complete the citation. Reference [20] appears to be an unpublished manuscript; if it is not accepted or posted, please mark it clearly.
  5. [Appendix A] The neighbourhood parameter k is first described as 5% of the points and later as 5 points in each mini-batch of size 100. This is consistent (5% of 100), but the wording could be clarified to avoid confusion.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the DII method is imported from prior same-group work, but the EUA empirical target is new and the causal findings are not encoded in the method's equations.

full rationale

The paper's central empirical claim — that IBEX35 and Coal Futures show causal effects on EUA returns — is obtained by applying two independent methods (VAR/Granger and DII) to a 35-variable return dataset. Nothing in the DII equations or in the VAR is fitted to the empirical conclusion; the result emerges from the data. The DII and IG are indeed taken from earlier papers with substantial author overlap ([28], [21], [49], [1], [42]), and Sec. 1.2 frames the contribution as extending that prior II-based study to EUA returns. This is self-citation, but it is not load-bearing in the sense of reducing the empirical result to the cited method's construction: the method is a tool, not a theorem that asserts IBEX35 or CoalFut must be causes. The synthetic datasets in Sec. 3.3 are author-constructed, but they are used as ground-truth checks with known equations; the method is not fitted to those labels. The paper also explicitly flags the causal-sufficiency assumption as 'most likely not satisfied in practice' (Sec. 1.2), which is a confounding limitation rather than a circular step. A separate internal-consistency concern exists in Eqs. (8)–(9) and (11): if DII decreases with added predictive information, then the inequality in Eq. (8) makes the fraction in Eq. (9) greater than 1, implying IG<0, while the text asserts equivalence to IG>0. That is a correctness/implementation issue, not a circularity, and should be resolved by checking the public repository; it does not make the empirical claim an input to the method. Overall, no derivation step reduces to its own inputs, so circularity is minimal.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities. The central empirical claim relies on the public DII method with several hand-set hyperparameters, plus the explicit and likely-violated causal-sufficiency assumption. No new physical or mathematical objects are postulated.

free parameters (4)
  • DII neighbourhood scale parameter lambda = lambda_i = 0.1 * d^2_{ij(k)}, k = 5% of points
    Appendix A Eq. 14: the point-adaptive lambda uses an empirical prefactor 0.1 and a neighbourhood fraction k/N=5%; these control the softmax kernel and are not derived from first principles.
  • DII training hyperparameters = 2000 epochs, Adam, initial lr 1e-3, cosine decay, 28 mini-batches of N'=100
    Appendix A: standard but arbitrary choices, and no sensitivity analysis is reported to show the IG values are stable to them.
  • Time-lag tau for DII and VAR lag p = tau = 1, p = 1 (AIC optimal)
    Section 4: the causal direction claims depend on the one-day lag being the relevant time scale for causal effects.
  • Temporal exclusion window = financial data [t-1, t+1], synthetic [t-5, t+5]
    Appendix A: used to approximate independent initial conditions, but the window sizes are chosen by hand and not justified beyond independence considerations.
assumptions (5)
  • domain assumption Causal sufficiency: no unmeasured confounder of EUA and the predictors
    Section 1.2 explicitly assumes causal sufficiency and states it is 'most likely not satisfied in practice'.
  • domain assumption One-day temporal resolution is sufficient for causal effects to appear at tau=1
    Section 4 compares DII and VAR only at tau=1 and p=1; slower effects are not probed.
  • domain assumption Return series are stationary and sampled frames are approximately independent
    Appendix B reports ADF tests for stationarity; independence is approximated via mini-batches and temporal window exclusion in Appendix A.
  • standard math Gaussian white noise in the VAR model for F-statistic validity
    Section 3.1 defines u_t as a white Gaussian random vector; the F-test relies on this assumption.
  • ad hoc to paper DII optimization reaches a representative global minimum
    Appendix A runs Adam once for 2000 epochs with random mini-batches; no multiple restarts or convergence diagnostics are reported.

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Cite this review

Pith. "Pith review of Non-parametric Causal Discovery for EU Allowances Returns Through the Information Imbalance." pith.science (2026). https://pith.science/paper/WT6KFOQI

@misc{pith2026250815667,
  author       = {Pith},
  title        = {Pith review of: Non-parametric Causal Discovery for EU Allowances Returns Through the Information Imbalance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WT6KFOQI}},
  note         = {Machine review of arXiv:2508.15667}
}
read the original abstract

We propose to use a recently introduced non-parametric tool named Differentiable Information Imbalance (DII) to identify variables that are causally related -- potentially through non-linear relationships -- to the financial returns of the European Union Allowances (EUAs) within the EU Emissions Trading System (EU ETS). We examine data from January 2013 to April 2024 and compare the DII approach with multivariate Granger causality, a well-known linear approach based on VAR models. We find significant overlap among the causal variables identified by linear and non-linear methods, such as the coal futures prices and the IBEX35 index. We also find important differences between the two causal sets identified. On two synthetic datasets, we show how these differences could originate from limitations of the linear methodology.

Figures

Figures reproduced from arXiv: 2508.15667 by the authors.

Figure 1
Figure 1. EUA futures price. The EUA prices analysed in this study represent futures contracts traded on the ICE Futures Europe, a major exchange for European carbon allowances. The figure shows EUA price from April 2005 to April 2024. Each market phase is marked by dashed vertical lines: Phase 1 (2005-2007), Phase 2 (2008-2012), Phase 3 (2013-2020), and Phase 4 (2020-present). Significant European economic and financial cris… view at source ↗
Figure 2
Figure 2. EUA futures financial returns. EUA futures returns and some significant events such as the EU Banking Crisis (2012/11 - 2014/09), Brexit (2016/01 - 2020/12), and the Energy Crisis (2021/3 - 2024/4). The returns are smoothed using 30-day moving averages, with key EU ETS phases demarcated. Periods of heightened market uncertainty, reflected by volatility spikes, are aligned with major European crises, which are shaded… view at source ↗
Figure 3
Figure 3. The Differentiable Information Imbalance. The DII applied to three spiral datasets with different noise levels. The first row shows the datasets in three-dimensional spaces, while the second row presents projections of the data on two-dimensional planes capturing the marginal dependencies of target variable z on the predictors x 1 and x 2 . The third row shows the value of the weights wx1 and wx2 that the DII assign… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: False negatives from linear causal discovery. A comparison of VAR / multi￾variate GC and DII applied to the stochastic process of Eqs. (12), testing x 1 and x 2 as putative causal variables with respect to z. The graph on the left shows the ground-truth causal re￾latio…
Figure 5
Figure 5. Figure 5: False positives from linear causal discovery. A comparison of VAR / multivariate GC and DII applied to the stochastic process of Eqs. (13), testing x 1 and x 2 as putative causal variables with respect to z. The graph on the left shows the ground-truth causal relations…
Figure 6
Figure 6. Figure 6: The Imbalance Gain as a non-linear alternative to the GC F-statistic. The results obtained from multivariate GC and the DII approach, using EUA as the target variable. The left panel displays the sorted GC F-statistics associated to each variable, using VAR models of o…
Figure 7
Figure 7. Figure 7: Beyond VAR causal weight estimation. The first panel ranks the estimated weights from a VAR(1) model in decreasing order. The second panel displays causal weights, organised by the DII measure in descending order. Bold highlights indicate the top 5 weights among the in…
Figure 8
Figure 8. Figure 8: Autocorrelation Function (ACF) Plot of Financial Returns. The ACF plots show very fast decays, exhibiting values very close to zero already from lag 1. This aligns with the widely recognized understanding that predicting financial returns is exceedingly difficult. 25 …
Figure 9
Figure 9. Figure 9: Partial Autocorrelation Function (PACF) Plot of Financial Returns. As in the ACF case, the PACFs decay to values close to zero from the first non-zero time lag. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.