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REVIEW 4 major objections 5 minor 58 references

Time-Varying Home Field Advantage in Football: Learning from a Non-Stationary Causal Process

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read DYNAMO, a local M-estimator for non-stationary causal discovery, recovers time-varying causal graphs from a single trajectory and attributes Premier League home-field advantage to crowd-dependent referee bias.

desk verdict DYNAMO is a genuine methodological advance for time-varying causal structure learning, but the sports application's referee-bias claims are not identified by the stated assumptions. read the letter →

arxiv 2506.11399 v1 pith:IVCBGDYO submitted 2025-06-13 stat.AP stat.ML

Minhao Qi , Hengrui Cai , Guanyu Hu , Weining Shen This is my paper · ORCID
classification stat.APstat.ML MSC 62M1062F1262G05
keywords causalstructurelearningnon-stationarytimeserieslocallystationaryprocesseslocalM-estimationhomefieldadvantagerefereebiasexpectedgoalsEnglishPremierLeague
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

DYNAMO is a causal discovery method built on the idea that a non-stationary process can be approximated locally by stationary processes, so the causal graph at each minute can be estimated from nearby observations. The paper establishes that this estimator identifies and consistently recovers the time-varying causal structure without faithfulness assumptions or constant-variance Gaussian noise assumptions. Applied to minute-level event data from two English Premier League seasons, one played behind closed doors and one with spectators, the recovered graphs show team-specific home-field advantages whose pathways run through referee bias. The claim is that crowd support changes officiating, not just player output, and that the learned structures predict expected goals better than static baselines. A sympathetic reader would care because this offers a template for causal questions in any setting where the data-generating mechanism evolves smoothly over time.

What carries the argument

The machinery is the locally stationary causal process model paired with a kernel-weighted local M-estimator. For each rescaled time $\tau = t/T$, a kernel $K_h$ with bandwidth $h$ weighs observations within $\tau \pm h$, and the causal parameters $\theta(\tau)$ minimize $L_t(\vartheta) = (Th)^{-1} \sum_l \ell(x_l, Y_{l-1}; \vartheta) K((\tau_l - \tau)/h)$. The key identity is a stationary approximation: under the paper's Lipschitz conditions there exists, for each $\tau$, a stationary process $\tilde{x}_t(\tau)$ with the same causal structure $\theta(\tau)$, and the observed process differs from it by $O(T^{-1} + |\tau_l - \tau|)$. That approximation makes the non-stationary loss converge to the stationary loss, which transfers identifiability results from stationary causal models and yields the consistency rate.

What would settle it

A concrete check is to re-estimate the same graphs with opponent-strength ratings, travel distance, and referee identity added to the conditioning set; if the direct edges from opponent yellow cards and fouls to expected goals disappear or flip sign, Assumption 1 fails and the referee-bias claim does not survive. A second check is a placebo test that randomly relabels home and away in the aggregated series; if DYNAMO still finds home-advantage edges at the same rate, the recovery is not picking up a real match process.

Watch

Extended reading notes

Core claim

The central claim is that a single realization of a locally stationary causal process $x_t = f(\mathrm{Pa}(x_t), \epsilon_t; \theta(t/T))$ contains enough local information to identify and consistently estimate the entire evolving causal structure $G_t$, provided the map from time to parameters is smooth in a Lipschitz sense. DYNAMO minimizes a kernel-weighted M-estimation loss, using the NOTEARS continuous acyclicity constraint for linear graphs and the NTS-NOTEAR neural-network loss for nonlinear graphs; the theory states that as $T h \to \infty$ with $T h^7 \to 0$, the estimated parameters converge to the true ones. On the data side, the application claims that the causal edges from opponent yellow cards and opponent fouls to expected goals, interpreted as referee bias, are time-varying and team-specific, and that their patterns differ sharply between the closed-door 2020-2021 season and the spectator season 2021-2022. This is the paper's evidence that home-field advantage is not a single league-wide constant but a moving set of causal relationships.

Load-bearing premise

The load-bearing premise is that, after taking home-minus-away differences averaged over each team's 38 matches, the measured in-match statistics have no unobserved confounders and behave like a single realization of a locally stationary causal process; if omitted factors such as opponent strength, fatigue, or referee assignment drive both the statistics and expected goals, the referee-bias edges are not causal.

Editorial extensions

If this is right

  • Team-level home advantage can be decomposed into minute-by-minute causal pathways, so a club could target specific match phases or opponent habits rather than treating home advantage as one number.
  • Referee bias is not constant across match phases; official training and evaluation could focus on the phases where crowd pressure shifts decisions.
  • The same estimator applies to any non-stationary observational time series where a single long trajectory is available, not just football.
  • Because DYNAMO's learned graphs improve goal prediction over static baselines, the paper implies that causally informed features carry predictive value beyond raw expected-goal aggregates.
  • The empty-stadium versus full-stadium seasons provide a natural benchmark: the crowd's effect on home advantage can be read off the changing referee-bias edges.

Reading between the lines

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

  • The natural next test is to run DYNAMO on matches with partial stadium capacity or neutral venues; if referee-bias edges track crowd density continuously, the crowd's causal role is confirmed, whereas if only home/away status matters, the mechanism is not crowd volume.
  • The paper aggregates each team's 38 matches into one home-minus-away series, so the team-specific graphs are season averages; applying the method to a single match would be a much harder test and would clarify whether the graphs capture within-match process or between-match selection.
  • If the claim transfers beyond sports, the same local-M machinery could estimate time-varying causal structures in economics or epidemiology where stationarity fails, but the unconfoundedness assumption will usually be harder to defend outside event data with rich controls.
  • The referee-bias result suggests a record-level falsification: compare estimated officiating edges with actual referee reports or disciplinary actions; if decisions recorded as errors do not correlate with the estimated bias edges, the causal label is in question.
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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

4 major / 5 minor

Summary. The paper proposes DYNAMO, a kernel-weighted local M-estimator that extends NOTEARS-style structure learning to locally stationary causal processes, with identifiability and consistency theorems, simulation comparisons against several benchmarks, and an application to two English Premier League seasons. The application constructs minute-level home-minus-away difference series per team, estimates time-varying graphs for within-match variables, and interprets edges from opponent fouls and yellow cards to expected goals as evidence of referee bias. It also claims that the learned causal structures improve goal prediction accuracy compared with existing methods.

Significance. If the theoretical results hold, the methodological core is a useful contribution: DYNAMO localizes a flexible class of DAG learners and supplies consistency guarantees in the locally stationary setting, and the simulation study is unusually broad, covering linear and nonlinear models, varying node counts and time lengths, faster changes, more lags, and non-Gaussian noise. The comparisons against DYNOTEARS, NTS-NOTEARS, PCMCI+, and CD-NOD are meaningful. The real-data claims, however, are substantially stronger than the evidence: the causal referee-bias interpretation rests on unconfoundedness and on treating cross-match averaged series as a causal process, and the prediction improvement is demonstrated only through an in-sample reconstruction. These gaps affect the paper's central applied claims rather than its local methodological machinery.

major comments (4)
  1. [Section 3.1 and Appendix D] The causal interpretation of the EPL application is not identified. Assumption 1 rules out unobserved confounders, but the input series defined in Appendix D as X_i^t = (2/N_i) sum_j X^{ij}_t (I(loc_j=1) - I(loc_j=0)) averages 38 matches per minute, so a 'shock' to this series at minute t is cross-match sampling variation rather than a structural intervention within a single match. OF and OY are endogenous match-state variables: a team falling behind commits more fouls and also concedes higher XG, while opponent strength, red cards, tactical setup, and fatigue are common drivers of both OF/OY and XG. Sections 5.1 and 5.3 label direct OF->XG and OY->XG edges as 'referee bias' (Figure 4), but such edges can arise from confounding even when referees are perfectly unbiased. The robustness checks in Appendix D.1 and the in-sample MSE in Table 1 do not control for these drivers; the supplement's own 'Limitation' section calls for more granular referee-bias data such as free kicks and penalties, underscoring that OF and OY are not direct measures of referee decisions.
  2. [Section 5.2, Table 1] The claim of improved goal prediction accuracy is not supported by Table 1. DYNAMO minimizes squared error against XG using contemporaneous match variables, and the table then compares its fitted values with actual goals, while the baseline XG does not see those contemporaneous variables. This is an in-sample reconstruction comparison, not a predictive evaluation. No holdout matches or time-based split are used, and no confidence intervals accompany the reported MSE values. An out-of-sample evaluation, or a comparison using only information available at prediction time, is required before the abstract's prediction-improvement claim can stand.
  3. [Section 3.5, Theorem 1, and Supplement C.4.2] The abstract's claim of identifiability 'without imposing additional assumptions' is overstated. Theorem 1 is derived from Lemma C.5, which for additive-noise and post-nonlinear stationary models invokes Assumptions C.6 (causal minimality) and C.7 (well-defined density), and for Gaussian DAGs relies on the equal-error-variance condition of Peters and Buehlmann (2014). These conditions should be stated as assumptions of Theorem 1 in the main text rather than deferred to the supplement, since they are load-bearing for the identifiability result.
  4. [Section 5.2 and Appendix D.2] The application-level conclusions in Figure 4 are asserted without uncertainty quantification. Each team-season series has T = 90 time points and roughly d = 16 variables, and local estimation uses bandwidths around 0.5-0.9, so each local fit sees only a small fraction of the 90 observations; the asymptotic regime of Theorem 2 (T h -> infinity, T h^7 -> 0) is not directly informative at this sample size. Edges are thresholded at 0.05, but no confidence intervals, bootstrap stability, or false-discovery measures are reported. The second- and third-best bandwidth comparisons in Appendix D.1 show visual agreement but do not quantify the uncertainty of the reported time-varying structures.
minor comments (5)
  1. [Section 3.1] Assumption 1 is titled 'Unconfoundness'; the standard spelling is 'Unconfoundedness'.
  2. [Section 3.5 and Theorem 2] Theorem 2 refers to 'the loss in Eq. (1)' but the objective is Eq. (3); the same equation reference is used incorrectly in the consistency statement.
  3. [Table 1 and Appendix D.1] The header 'DYNOTEAR' is inconsistent with 'DYNOTEARS' elsewhere, and Appendix D.1 contains the typo 'DYNOTEAERS'.
  4. [Figure 4 and Supplement Figures] The captions say 'Orange notes represent contemporary variables'; 'notes' should be 'nodes'.
  5. [Algorithm 3] Bandwidth selection randomly divides the time series into folds; for dependent locally stationary data this can place adjacent observations in different folds and overstate effective sample size, so a blocked or leave-one-out scheme consistent with Richter and Dahlhaus (2019) would be safer.

Circularity Check

1 steps flagged · score 6.0 of 10

The goal-prediction validation in Table 1 is an in-sample fit of the very XG series DYNAMO is trained on, so the headline accuracy claim reduces by construction; the referee-bias conclusions rest on an unverified unconfoundedness assumption rather than on circular reasoning.

  1. fitted input called prediction [Section 5.2 (Model Validation), Table 1; Eq. (4)]
    "Third, we use our DYNAMO model to predict the expected goals (XG) at each minute and compare these predictions to the actual goals scored. ... L^{lr}_t(W_t,A_t)=\frac{1}{Th}\sum_l \|x_l-W_t^\top x_l-A_t^\top Y_{l-1}\|_2^2 K_h(\tau_l-\tau_t)+\lambda_1\|W_t\|+\lambda_2\|A_t\|+\frac{\alpha}{2}H(W_t)^2+\rho H(W_t)."

    The 'predictions' in Table 1 are the fitted values of the XG coordinate of x_l under the same loss (4) that DYNAMO minimizes: the objective includes \|x_l - W_t^\top x_l - A_t^\top Y_{l-1}\|_2^2, so the XG component is the training target, and contemporaneous variables (TS, OF, OY, etc.) are available as regressors. The quasi-k-fold CV in Eq. (6) is used only to choose the bandwidth h, not to produce held-out goal predictions; the table compares these in-sample reconstructions of a 38-match home-minus-away averaged XG series against 'actual goals scored' and against the fixed 'original XG' baseline, which does not use the same contemporaneous covariates. The gap therefore measures fitting flexibility, not predictive or causal validity.

full rationale

The derivation of the DYNAMO estimator itself is largely self-contained: identifiability is reduced to external identifiability results for stationary Gaussian/non-Gaussian, additive-noise, and post-nonlinear models, and the consistency proof invokes generic locally stationary process bounds plus base-learner rates. The simulation studies generate data from the same structural equation model in Eq. (1) with known ground truth, so they are not circular. The genuine circular step is the model-validation claim in Section 5.2 and Table 1: Eq. (4) minimizes squared error of the full vector x_l, including the XG coordinate, using contemporaneous variables as regressors; the 'predictions' compared with 'actual goals scored' are in-sample fitted values from the same 38-match smoothed series, with no held-out split for the comparison. Thus the abstract's 'improve goal prediction accuracy' claim reduces to a fit of the series being predicted. Separately, the referee-bias interpretation of OF-to-XG and OY-to-XG edges is not circular but is an identification risk: it requires Assumption 1 (no unobserved confounders) on an aggregate home-minus-away series and treats endogenous foul and card counts as pure referee inputs. The supplement's own limitation section concedes that more granular referee-bias data such as free kicks and penalty kicks would offer additional insights, underscoring that OF and OY are coarse proxies. These are correctness concerns rather than circularity, so they do not by themselves raise the score beyond 6. The self-citations (Price et al. 2022 for preprocessing; Cai et al. 2023 for a Gaussian-DAG rate) are not load-bearing after the design argument for differencing and the availability of external identifiability theorems.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central method rests on standard causal discovery assumptions (unconfoundedness, acyclicity), locally stationary process assumptions, and two additional identifiability assumptions in the appendix. No new physical entities are posited. The application additionally assumes that averaging home-away differences across 38 matches produces a valid 90-point causal time series.

free parameters (4)
  • Kernel bandwidth h = Per team and time point, roughly 0.4 to 0.9
    Selected by quasi-k-fold cross-validation on the same data (Algorithm 3). It controls the effective local sample size and directly influences the estimated graphs.
  • Sparsity penalties lambda1, lambda2 = 0.05 for linear DYNAMO; 0.005/0.01 for nonlinear DYNAMO
    Chosen as fixed defaults, not tuned on the real data. They shape the sparsity of the estimated DAGs and can materially change reported edges.
  • Edge threshold = 0.05
    Edges with standardized influence below 0.05 are removed. This post-hoc threshold affects all reported graphs in the application and is not derived from the data.
  • Lagged period L = 1 (with a robustness check for L=2)
    Chosen by inspecting the magnitude of estimated lagged weights, which is a data-dependent model selection step.
assumptions (6)
  • domain assumption Assumption 1: No unobserved confounders
    Stated in Section 3.1. Very strong for football match data where team tactics, injuries, fatigue, and referee assignment are unobserved. Load-bearing for causal interpretation.
  • domain assumption Assumption 2: The graph G_t is acyclic at each time t
    Stated in Section 3.1. Forbids instantaneous feedback loops, which may be violated in match processes where passes, shots, and referee decisions influence each other within a minute.
  • ad hoc to paper Assumption 3: Locally Stationary Causality (Lipschitz and contraction conditions)
    Introduced in Section 3.2. This is the key structural assumption borrowed from locally stationary process theory; it limits the time-variation of the causal mechanism.
  • ad hoc to paper Assumption C.6: Causal Minimality
    Introduced in Appendix C.4.2. Needed for identifiability of additive noise and post-nonlinear models, but not stated in the main text.
  • domain assumption Assumption C.7: Well-defined density and smoothness
    Introduced in Appendix C.4.2. Requires absolute continuity and finite log-likelihood; needed for the identifiability results.
  • domain assumption Home-away differencing yields a valid locally stationary causal process of length 90
    Appendix D constructs a single time series per team as the average difference between home and away match statistics. This treats the aggregate as a causal process realization, which is a strong modeling assumption.

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Cite this review

Pith. "Pith review of Time-Varying Home Field Advantage in Football: Learning from a Non-Stationary Causal Process." pith.science (2026). https://pith.science/paper/IVCBGDYO

@misc{pith2026250611399,
  author       = {Pith},
  title        = {Pith review of: Time-Varying Home Field Advantage in Football: Learning from a Non-Stationary Causal Process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IVCBGDYO}},
  note         = {Machine review of arXiv:2506.11399}
}
read the original abstract

In sports analytics, home field advantage is a robust phenomenon where the home team wins more games than the away team. However, discovering the causal factors behind home field advantage presents unique challenges due to the non-stationary, time-varying environment of sports matches. In response, we propose a novel causal discovery method, DYnamic Non-stAtionary local M-estimatOrs (DYNAMO), to learn the time-varying causal structures of home field advantage. DYNAMO offers flexibility by integrating various loss functions, making it practical for learning linear and non-linear causal structures from a general class of non-stationary causal processes. By leveraging local information, we provide theoretical guarantees for the identifiability and estimation consistency of non-stationary causal structures without imposing additional assumptions. Simulation studies validate the efficacy of DYNAMO in recovering time-varying causal structures. We apply our method to high-resolution event data from the 2020-2021 and 2021-2022 English Premier League seasons, during which the former season had no audience presence. Our results reveal intriguing, time-varying, team-specific field advantages influenced by referee bias, which differ significantly with and without crowd support. Furthermore, the time-varying causal structures learned by our method improve goal prediction accuracy compared to existing methods.

Figures

Figures reproduced from arXiv: 2506.11399 by the authors.

Figure 1
Figure 1. The time-varying average expected goals of 4 EPL tea [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Stationary approximation of non-stationary causa [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Comparison study of linear and nonlinear non-stati [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: Model results for 4 representative teams, Manchester City, Liverpool, Arsenal and Manchester United. Variables: TP - Total Passes, TS - Total Shots, PA - Pass Accuracy, SA - Shot Accuracy, OY - Opponent’s yellow cards, OF - Opponent’s Fouls. Orange notes represent cont…
Figure 4
Figure 4. Figure 4: figure.4 [PITH_FULL_IMAGE:figures/full_fig_p050_4.png]
Figure 1
Figure 1. Figure 1: The model result for 4 representative teams by DYNOTEAR [PITH_FULL_IMAGE:figures/full_fig_p058_1.png]
Figure 2
Figure 2. Figure 2: The robustness checks for 4 representative teams us [PITH_FULL_IMAGE:figures/full_fig_p059_2.png]
Figure 3
Figure 3. Figure 3: The robustness checks for 4 representative teams us [PITH_FULL_IMAGE:figures/full_fig_p060_3.png]
Figure 4
Figure 4. Figure 4: The robustness checks for 4 representative teams ch [PITH_FULL_IMAGE:figures/full_fig_p061_4.png]
Figure 5
Figure 5. Figure 5: Model results for 4 higher-ranked teams, Chelsea, Leicester City, Tottenham Hotspur, and West Ham United. Variables: TP - Total Passes, TS - Total Shots, PA - Pass Accuracy, SA - Shot Accuracy, OY - Opponent’s yellow cards, OF - Opponent’s Fouls. Orange notes represent…
Figure 6
Figure 6. Figure 6: Model results for 4 middle-ranked teams, Newcastle United, Aston Villa, Wolverhampton Wanderers, and Brighton. Variables: TP - Total Passes, TS - Total Shots, PA - Pass Accuracy, SA - Shot Accuracy, OY - Opponent’s yellow cards, OF - Opponent’s Fouls. Orange notes repr…
Figure 7
Figure 7. Figure 7: Model results for 4 lower-ranked teams, Southampton, Crystal Palace, Everton, and Leeds United. Variables: TP - Total Passes, TS - Total Shots, PA - Pass Accuracy, SA - Shot Accuracy, OY - Opponent’s yellow cards, OF - Opponent’s Fouls. Orange notes represent contempor…
Figure 8
Figure 8. Figure 8: Model results for 7 relegated teams, Burnley, Fulham, West Bromwich, Sheffield United, Watford, Norwich City, and Brentford. Variables: TP - Total Passes, TS - Total Shots, PA - Pass Accuracy, SA - Shot Accuracy, OY - Opponent’s yellow cards, OF - Opponent’s Fouls. Ora…
Figure 9
Figure 9. Figure 9: Comparison studies for running time of different met [PITH_FULL_IMAGE:figures/full_fig_p066_9.png]
Figure 10
Figure 10. Figure 10: Comparison studies for linear and nonlinear nonst [PITH_FULL_IMAGE:figures/full_fig_p066_10.png]
Figure 11
Figure 11. Figure 11: Comparison studies for linear and nonlinear nonst [PITH_FULL_IMAGE:figures/full_fig_p067_11.png]
Figure 12
Figure 12. Figure 12: Comparison studies for linear and nonlinear nonst [PITH_FULL_IMAGE:figures/full_fig_p068_12.png]
Figure 13
Figure 13. Figure 13: Comparison studies for linear and nonlinear nonst [PITH_FULL_IMAGE:figures/full_fig_p069_13.png]
Figure 14
Figure 14. Figure 14: Comparison studies for linear and nonlinear nonst [PITH_FULL_IMAGE:figures/full_fig_p070_14.png]
Figure 15
Figure 15. Figure 15: Comparison studies for linear and nonlinear nonst [PITH_FULL_IMAGE:figures/full_fig_p071_15.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.