A new Granger causality measure for eliminating the confounding influence of latent common inputs

Takashi Arai

REVIEW 3 major objections 5 minor 24 references

A new Granger causality measure for eliminating the confounding influence of latent common inputs

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A new Granger causality measure removes the confounding influence of latent common inputs and keeps its F-distribution under the null.

desk verdict A genuine twist on partial Granger causality, but the central F-claim rests on an unproven generated-regressor step and the abstract overstates the robustness. read the letter →

arxiv 1908.03867 v1 pith:56M7FWXG submitted 2019-08-11 stat.ME physics.data-anq-bio.QM

Takashi Arai This is my paper · ORCID

classification stat.MEphysics.data-anq-bio.QM MSC 62M1062F03

keywords GrangercausalitylatentcommoninputspartialF-distributionautoregressivemodelspuriousconfoundingmultipletesting

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

The paper proposes a Granger causality test meant to stay valid when two observed time series share a latent common input that reaches one series with a delay. The author argues that ordinary Granger causality mistakes such delayed common inputs for directed interaction, producing spurious links that become more certain as sample size grows. The new measure removes the noise correlation between source and target by conditioning on the source series' own autoregressive residual, which acts as a proxy for the common input. The central technical claim is that the resulting F statistic approximately follows the F-distribution under the null hypothesis of no interaction, and numerical experiments support this and show the test can detect genuine interactions once the sample is large enough. A practical lag-selection procedure based on BIC and Bonferroni-corrected multiple tests is included for cases where model orders are unknown.

What carries the argument

The load-bearing object is the conditional-variance reparameterization of the noise covariance: writing $\varepsilon_t=\eta\,\xi_{t-l_\eta}+\omega_t$ isolates $\tau^2=\Sigma_{xx}-\Sigma_{xy}\Sigma_{yy}^{-1}\Sigma_{yx}$, the variance of the target noise after the correlated component with the source noise is removed. The estimation machinery is two-step: first fit the source series $y_t$ by its own autoregressive model, compute residuals $\hat{\xi}_t$, then include those residuals in the regression for the target $x_t$ and test the interaction coefficients $c$ with the scaled deviance difference that yields the F statistic in Eq. (42). This construction ensures the test statistic is nonnegative and gives it a known null distribution under the paper's assumptions.

What would settle it

Simulate the symmetric feedback model $x_t=a x_{t-1}+c y_{t-1}+\varepsilon_t$, $y_t=b y_{t-1}+d x_{t-1}+\xi_t$ with delayed correlated noise, run the proposed F-test, and compare the null histogram and false-positive rate to the nominal F-distribution; if they depart, the no-feedback assumption is essential.

Watch

Extended reading notes

Core claim

At the paper's core is the claim that delayed latent common inputs—noise shared by $x_t$ and $y_t$ at lag $l_\eta$—make ordinary Granger causality report a directed interaction that is not there, and that the spurious link can be removed by conditioning on the source series. The author reparameterizes the noise as $\varepsilon_t=\eta\,\xi_{t-l_\eta}+\omega_t$, so that $\tau^2=\Sigma_{xx}-\Sigma_{xy}\Sigma_{yy}^{-1}\Sigma_{yx}$ is the part of the target's noise uncorrelated with the source's noise. Assuming the source $y_t$ receives no feedback from the target $x_t$, the residuals from fitting $y_t$ alone estimate $\xi$, and including them in the target regression makes the F statistic of Eq. (42) approximately $F$-distributed under the null $c=0$. Numerical experiments show the statistic tracks the null distribution when no interaction exists and separates from it under a true interaction, while ordinary Granger causality detects spurious interactions with high probability as the sample grows.

Load-bearing premise

The load-bearing premise is that the source series receives no feedback from the target, so its autoregressive residual can be estimated from the source alone and used as a clean proxy for the latent common input.

Editorial extensions

If this is right

In the asymmetric setting, practitioners can use standard F-distribution tables for the new statistic, with degrees of freedom $(p_1-p_0,\,T-l_{\max}-p_1)$, without bootstrapping.
When latent common inputs arrive with delay, ordinary Granger causality will keep producing spurious directed links as sample size grows, whereas the new procedure keeps its false-positive rate under control.
The proposed BIC-plus-Bonferroni procedure detects true directed interactions with accuracy approaching one as the sample grows, at the cost of lower power than ordinary Granger causality at small samples.
The method applies to latent common inputs whose effect appears as cross-correlated AR noise; latent variables that create autocorrelated noise are explicitly outside its scope and require a moving-average or state-space treatment.

Reading between the lines

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

A practical diagnostic the paper implies but does not develop: the delay structure of the noise, not the mere presence of correlation, determines when ordinary Granger causality goes wrong, so checking whether cross-correlation at nonzero lags exists could flag data sets needing the new test.
The two-step residual-substitution idea should extend to any asymmetric pair of autoregressive processes, including nonlinear or point-process versions where one channel's 'residual' plays the role of the common input; the paper lists point-process extension as future work.
In symmetric systems the residual from $y$ alone is contaminated by feedback from $x$, so a testable extension would fit both directions, compare the two one-sided residuals, or iterate the estimation until the residuals stabilize.
Because the proposed test has lower power than ordinary Granger causality at small samples, it may be most useful as a confirmatory filter after ordinary Granger causality flags candidates, rather than as a replacement in small-sample studies.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version: 1908.03867 is worth a careful read, but it's a conditional result, not the general fix the abstract implies. The core idea is a real improvement over partial Granger causality. Guo et al. remove the noise correlation between target and conditioning variable; Arai points out that the correlation causing spurious directed interactions is between target and source, so he conditions on the source's estimated noise. Concretely, estimate y's AR model, take the residual ξ̂ as a proxy for the latent common input, and test the coefficient c in x_t = Σ a_i x_{t-i} + Σ c_i y_{t-i} + η ξ̂_{t-l} + ω_t. If ξ were observed, the likelihood-ratio F-test is textbook and the paper's derivation is standard. The numerical experiments are honest: the null distribution is approximately F, and power grows with sample size. The paper also proposes a practical BIC/Bonferroni procedure and explicitly lists its limitations: latent variables are out of scope and the network structure is assumed asymmetric.

That last limitation is the soft spot. The whole two-step estimator depends on y being a univariate AR. If x→y feedback exists, the first-stage residual for y contains the x-driven component and is not a consistent proxy for the common input; the likelihood factorization in Eq. (26) fails and the F statistic in Eq. (42) need not hold under the null. The paper does acknowledge the asymmetric assumption in Secs. II and IV, but the abstract and title claim robustness to latent common inputs without that caveat. Also, the central distributional claim with estimated residuals is assumed rather than derived — the generated-regressor problem is waved off as an approximation, and the econometric residual-inclusion literature isn't cited. Simulations are all within the assumed model family, so they can't challenge the boundary.

None of this kills the paper. The idea is clear and could be useful for applied researchers who can credibly assume an asymmetric network. But it's not a general-purpose tool, and the missing proof (or at least a serious treatment of the estimated-residual step) is a genuine gap. No code and no data, though the simulations are easy to reproduce. I'd send it to peer review with a referee who knows the generated-regressor literature. A Monte Carlo under symmetric feedback should be requested, and the abstract should carry the asymmetry caveat.

Referee Report

3 major / 5 minor

Summary. The paper proposes a new Granger causality measure intended to be robust to latent common inputs with delayed effects. The model is a two-variable AR system with an asymmetric interaction (y→x only) and a lagged cross-noise correlation between ε_t and ξ_{t-1}. The author reparametrizes the noise, conditions on the source AR innovation, estimates the innovation in a first step from a univariate AR fit to y, and constructs an F statistic for testing c=0. The paper claims that this statistic approximately follows an F distribution under the null, verifies this by simulation, and develops a BIC-based stepwise procedure for the case where the lags are unknown. Numerical experiments compare the proposed method with normal Granger causality and report that the proposed method avoids spurious interactions under the simulated latent-common-input scenarios.

Significance. If the approximate F claim were rigorously established, the measure would be a useful alternative to partial Granger causality: it targets the source-target noise correlation rather than the target-conditioning correlation, avoids the negative-test-statistic issue, and the simulations are suggestive. The author is honest about the two-step approximation and lists limitations in the conclusions. However, the central distributional result is an unproven assumption, and the robustness claim is explicitly restricted to asymmetric networks; these gaps prevent acceptance in the current form.

major comments (3)

[§II.D and §IV] The validity of the two-step estimator in Eqs. (35)–(36) and (45)–(46), and hence of the F test in Eq. (42), depends critically on the asymmetric-network assumption stated in §II.B. If the true system contains x→y feedback, the univariate AR fit to y is misspecified, ξ̂ is not a consistent estimate of the structural innovation ξ, and the conditioning argument underlying the F statistic does not hold. The manuscript acknowledges this only in the concluding future-work sentence, while the title and abstract claim robustness to latent common inputs without this caveat. The claims should be scoped explicitly, or the method should be extended to symmetric networks.
[§II.D, Eq. (42)] The statement that F approximately follows the F distribution after substituting estimated residuals is assumed, not derived. Standard GLM deviance theory does not cover generated regressors; the text below Eq. (34) says 'we assume that the test statistic obeys the F-distribution approximately even if we substitute ξ̂_t.' The numerical support in §III.A is limited to one null model (model (b)) with known lags, and it does not vary ρ, the AR coefficients, or the lag structure. A theoretical justification, or a substantially broader simulation study (including misspecified lags, varying feedback strengths, and a range of ρ values), is needed to support the central claim.
[§III.B] The proposed practical procedure performs sequential tests with BIC-based lag selection and a Bonferroni correction, but no analysis is given for the familywise error rate of the full sequential procedure. Figure 6 reports accuracy for specific simulation settings; it does not establish control of false positives across the multiple testing decisions. The text should either provide a formal FWER statement or report simulation results that specifically give false-positive rates under the null for the entire procedure.

minor comments (5)

[Eq. (37)] The second summation is written as ∑ c_i x_{t-i}; it should be ∑ c_i y_{t-i} to match the model specification and all subsequent equations.
[Eq. (29)] The summation in Eq. (29) starts at t=2, whereas the likelihood in Eq. (27) and the final statistic in Eq. (42) start at t=3 or lmax+1; the index ranges should be made consistent.
[Fig. 4] The caption states sample sizes 300, 1,000, and 3,000, but the panel labels read T=75, T=300, and T=3000; this mismatch should be corrected.
[Fig. 6] The accuracy labels on the bars, e.g., '0.9710.914', are printed without separation, making the reported values difficult to read.
[Sec. II.A] The text cites Ref. [19] for the negative-value criticism of partial Granger causality but does not mention the later multivariate extension's treatment of the same issue; a brief remark would help place the contribution in the literature.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the confounder removal is a definitional construction, and the F-distribution claim is an explicitly approximate assumption verified numerically, not a fitted input renamed as a prediction.

full rationale

Walking the derivation chain, I find no circular step. The paper proposes a new Granger causality measure by reparameterizing the noise covariance: Eq. (16) writes epsilon_t = eta * xi_{t-1} + omega_t, and Eq. (22) shows that conditioning on y_t yields a conditional model for x_t whose mean includes eta*(y_{t-1} - b*y_{t-2}), i.e., the lagged y-residual. Thus including the estimated residual as a regressor removes the delayed latent common input by construction. This is the method itself, not a hidden reuse of the target claim. The paper is explicit that the F-distribution statement with estimated residuals is an assumption: 'we assume that the test statistic obeys the F-distribution approximately even if we substitute xi_hat_t' (Sec. II.D, near Eqs. (35)-(36)), and it is checked numerically under the null model. This is an honest approximate validation, not a fitted parameter renamed as a prediction. There is no load-bearing self-citation: the author is the sole author and does not cite himself. The acknowledged asymmetry restriction, stated in Sec. II.B and Sec. IV, is a correctness boundary rather than circularity; the paper identifies symmetric-network extension as future work. Therefore no specific reduction of a claimed result to its inputs is exhibited, and the numerical demonstrations, while confined to the assumed model family, are standard validation rather than circular reasoning.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The method does not introduce new physical entities. Its load-bearing assumptions are the single-lag noise correlation structure, the asymmetric network, and the approximate F-distribution of the test statistic with generated regressors.

free parameters (2)

lag of noise correlation lη = selected by BIC (searched 0 to 6 in simulations)
The delay between the common input's effect on y and x is unknown in practice and chosen via BIC in the practical procedure (Sec. III.B).
lag of directed interaction lc = selected by BIC (searched 0 to 6 in simulations)
The number of past y terms that predict x is unknown and selected by BIC before the test (Sec. III.B).

assumptions (3)

domain assumption The noise terms ε_t and ξ_t are zero-mean Gaussian white noises with covariance structure var(ε)=σx^2, var(ξ)=σy^2, and cov(ε_t, ξ_{t-1})=ρσxσy at a single lag only.
Assumed in Eqs. (11)-(13); used to justify the reparameterization and the claim that a single lag is sufficient. If the common input creates auto-correlated noise or multiple lags, the method is not applicable.
domain assumption The network structure is asymmetric, with no interaction from x to y.
Stated in Sec. II.B and again in Sec. IV; required for estimating b from y alone without bias and for the factorization of the likelihood.
ad hoc to paper The test statistic with estimated residuals ξ̂ approximately follows the F-distribution.
Assumed after Eqs. (35)-(36): 'we assume that the test statistic obeys the F-distribution approximately even if we substitute ξ̂_t.' This is the central distributional claim, not derived.

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Cite this review

Pith. "Pith review of A new Granger causality measure for eliminating the confounding influence of latent common inputs." pith.science (2026). https://pith.science/paper/56M7FWXG

@misc{pith2026190803867,
  author       = {Pith},
  title        = {Pith review of: A new Granger causality measure for eliminating the confounding influence of latent common inputs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/56M7FWXG}},
  note         = {Machine review of arXiv:1908.03867}
}

abstract

In this paper, we propose a new Granger causality measure which is robust against the confounding influence of latent common inputs. This measure is inspired by partial Granger causality in the literature, and its variant. Using numerical experiments we first show that the test statistics for detecting directed interactions between time series approximately obey the $F$-distributions when there are no interactions. Then, we propose a practical procedure for inferring directed interactions, which is based on the idea of multiple statistical test in situations where the confounding influence of latent common inputs may exist. The results of numerical experiments demonstrate that the proposed method successfully eliminates the influence of latent common inputs while the normal Granger causality method detects spurious interactions due to the influence of the confounder.

Figures

Figures reproduced from arXiv: 1908.03867 by the authors.

**Figure 2.** FIG. 2. Run-sequence plot for each simulated model. Black li [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** FIG. 3. The distributions of test statistics for normal Gran [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. The distributions of test statistics for proposed me [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 5.** Figure 5: FIG. 5. The distributions of test statistics for proposed me [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

**Figure 6.** Figure 6: FIG. 6. Comparison of accuracy in detecting directed intera [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

**Figure 7.** Figure 7: FIG. 7. Comparison of statistical power in detecting direct [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Wiener, The theory of prediction in Modern Mathematics for the Engine er (New York: McGraw-Hill, 1956) pp

N. Wiener, The theory of prediction in Modern Mathematics for the Engine er (New York: McGraw-Hill, 1956) pp. 165–190

work page 1956

[2] [2]

C. W. J. Granger, Econometrica 37, 424 (1969)

1969

[3] [3]

Granger, Journal of Economic Dynamics and Control 2, 329 (1980)

C. Granger, Journal of Economic Dynamics and Control 2, 329 (1980)

work page 1980

[4] [4]

Baccala, K

L. Baccala, K. Sameshima, G. Ballester, and A. C. do Valle , Applied Signal Processing 5 (1998)

work page 1998

[5] [5]

Gourevitch, R

B. Gourevitch, R. L. B. Jeannes, and G. Faucon, Biologica l Cybernetics 95 (2006)

work page 2006

[6] [6]

A. B. Barrett, M. Murphy, M.-A. Bruno, Q. Norhomme, M. Bol y, S. Laureys, and A. K. Seth, Journal of Neuroscience 35, 3293 (2015)

work page 2015

[7] [7]

M. Ding, Y. Chen, and S. L. Bressler, Granger Causality: Basic Theory and Applications to Neurosci ence. In: Schlter S, Winterhalder N., Timmer J., eds. handbook of time series analysis (Wienheim: Wiley, 2006) Chap. 17, pp. 437–460, https://onl inelibrary.wiley.com/doi/pdf/10.1002/9783527609970.ch17

doi:10.1002/9783527609970.ch17 2006

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Y. Chen, S. L. Bressler, and M. Ding, Journal of Neuroscie nce Methods 150, 228 (2006)

work page 2006