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REVIEW 4 major objections 4 minor 45 references

Forecasting Intraday Volume in Equity Markets with Machine Learning

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that 15-minute equity trading volume is highly predictable once order-book features, cross-stock pooling, and nonlinear models are combined.

desk verdict Good empirical work on a neglected execution problem, but the headline predictability gain is conflated with same-day information, and the numbers need a careful audit. read the letter →

arxiv 2505.08180 v1 pith:7Z7VF4DK submitted 2025-05-13 q-fin.CP q-fin.ST

Mihai Cucuringu , Kang Li , Chao Zhang This is my paper · ORCID
classification q-fin.CPq-fin.ST
keywords intradaytradingvolumemachinelearningcommonalitylimitorderbookVWAPexecutionschedulingout-of-sampleR2S&P500
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

This paper tries to establish that the next 15-minute trading volume of a U.S. equity is highly predictable, and that the predictability is exploitably large. Starting from the standard component multiplicative error model with an out-of-sample $R^2$ of 0.265, the authors add high-frequency limit-order-book predictors, share information across stocks through clustered or universal models, and switch from linear to nonlinear learners; the best configuration, a DeepLOB-style network trained on all stocks, reaches $R^2=0.624$. They then show that these forecasts cut VWAP replication tracking error by 28.7% on average across five sampled stocks and bring passive-order fill ratios closer to what a perfect-volume oracle would achieve. The paper's claim is that intraday volume is no longer a hard-to-forecast univariate time series once this information is used.

What carries the argument

The load-bearing object is the feature-and-pooling stack built around the CMEM decomposition. Each bin is represented by 54 predictors: seven CMEM components (daily, periodic, non-periodic, and their products) plus high-frequency auxiliary predictors such as buy and sell notional, numbers of trades and shares, and compound sums over the previous two bins, eight bins, the same intraday interval, and the full day. These features are fed into models under three schemes: single-asset (SAM), cluster-of-similar-assets (CAM, with PCA plus K-means++ on volume or feature correlations), and universal pooled (UAM). The final stage is DeepLOBv, a convolutional-Inception-LSTM network that extracts nonlinear temporal and cross-sectional structure from the features. The machinery works by giving a flexible learner enough same-day and cross-sectional information to override the fixed U-shape that limits the benchmark.

What would settle it

Re-estimate Table 3 under static prediction, forecasting all 26 bins of day $t+1$ from data up to day $t$ for every machine-learning model, and compare the resulting out-of-sample $R^2$ with CMEM's static value of 0.240; if the ML advantage largely disappears, the dynamic-updating information is the real driver.

Watch

Extended reading notes

Core claim

The central discovery is a measured progression in out-of-sample $R^2$ for 15-minute volume forecasts on 469 S&P 500 constituents. The CMEM benchmark scores 0.265; adding CMEM's decomposition components as linear features gives a Ridge model 0.379; adding the auxiliary order-book predictors raises Ridge to 0.491; and nonlinear models with the same features reach 0.532 for XGBoost and 0.566 for DeepLOBv in per-stock training. Pooling stocks through a universal model lifts XGBoost to 0.622 and DeepLOBv to 0.624, while a clustered model gives Ridge 0.557 and XGBoost 0.600. The authors attribute the gains to three mechanisms: auxiliary predictors correct CMEM's rigid U-shaped pattern on days when volume deviates from it, cross-sectional pooling exploits commonality in volume across stocks, and nonlinear learners capture herding, overreaction, and spillover effects that linear models miss. The headline comparison uses dynamic one-bin-ahead prediction, where forecasts consume volume realized earlier the same day.

Load-bearing premise

The headline improvement is measured with one-bin-ahead forecasts that use volume already realized on the target day, so if the models were required to forecast the whole day before it opens, the gain over CMEM, whose static $R^2$ is only reported as 0.240, could be much smaller.

Editorial extensions

If this is right

  • If the headline $R^2$ holds, execution desks can schedule 15-minute child orders with substantially more confidence than CMEM allows, because the model explains over 60% of out-of-sample variation rather than about a quarter.
  • Cross-sectional commonality is economically usable: training one nonlinear model on all stocks outperforms per-stock training, and cluster-level training outperforms the linear universal model, so volume forecasts can be improved without per-stock manual tuning.
  • Auxiliary predictors are a large lever on their own: adding them to a Ridge model raises $R^2$ from 0.379 to 0.491, so the information content of order-book features is valuable even before nonlinearity or pooling is introduced.
  • The VWAP application directly converts forecast skill into a cost metric: the best model reduces tracking error by an average of 28.7% on the five sampled stocks, with the largest single-stock drop being 57.0% for AEP.
  • Because the best model uses cross-stock pooled features, the method should also deliver stable forecasts for stocks that have short or noisy individual volume histories, a benefit of the commonality finding.

Reading between the lines

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

  • A natural extension the paper leaves implicit is to test the same feature stack in the static setting, where all 26 bins must be forecast before the day opens; the result would separate genuine next-day skill from the real-time updating advantage.
  • The commonality result suggests the clustered and universal training schemes could transfer to smaller or less liquid universes where individual volume series are too noisy for per-stock models, because pooled training acts as a regularizer.
  • The matching-engine evaluation assumes the parent order does not move prices; in practice, larger order sizes would erode the fill-ratio benefit, so the 28.7% tracking-error reduction and fill-ratio improvements are likely upper bounds for real trading.
  • Because the sample covers only July through December 2017, an out-of-period replication under different volatility and liquidity regimes would show whether the $R^2=0.624$ result is stable or period-specific.
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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 / 4 minor

Summary. The paper proposes a machine-learning framework for forecasting 15-minute intraday trading volumes of U.S. equities. Using LOBSTER order-book data for 469 S&P 500 constituents from July to December 2017, the authors augment CMEM decompositions with high-frequency features, train linear models (OLS/LASSO/Ridge), XGBoost, and a DeepLOB-based neural network under three schemes (single-asset, clustered, universal), and evaluate out-of-sample R2. They report raising R2 from 0.265 (CMEM) to 0.624 (UAM DeepLOBv) and further find that using these forecasts in VWAP replication reduces tracking error by 28.7% on average and improves passive-order fill ratios relative to CMEM.

Significance. If the reported forecasting gains survive a fair static-versus-dynamic comparison, the paper would provide substantial evidence that intraday volume is highly predictable given order-book features and cross-sectional commonality, and it would offer a practical execution tool. The study is carefully structured: genuine out-of-sample evaluation against an external benchmark, an explicit comparison of three training schemes, per-date and per-sector stability analysis, and a concrete economic application with a limit-order-book matching engine. The qualitative ordering across models in Table 3 is internally consistent, and the feature-importance analysis adds transparency. The dynamic-evaluation confound in the headline R2 comparison is, however, a load-bearing unresolved issue.

major comments (4)
  1. [Section 5.1, Table 3, Appendix A.2] The comparison underpinning the headline result is confounded by the evaluation setting. Static out-of-sample R2 is reported only for the CMEM benchmark (0.240), while every machine-learning model in Table 3 is evaluated in the dynamic one-bin-ahead setting (0.265 for CMEM). Because the dynamic setting uses same-day realized volume through the past_2 and past_8 operations (Section 3.3, Table 2), the reported gain from 0.265 to 0.624 combines model improvement with the value of conditioning on already-realized bins of the target day. Please report static R2 for all models (or, at minimum, for the UAM DeepLOBv model) under identical feature sets, and restate the abstract's predictability claim in light of that comparison.
  2. [Section 5.1] The out-of-sample R2 statistic is never defined. The paper reports means and standard deviations across dates, which suggests a per-day computation, but the formula, the benchmark used in the denominator, and the pooling convention are all unspecified. Without a precise definition (e.g., 1 - MSE_pooled/Var or a per-day average), the reader cannot determine what R2 = 0.624 measures or compare it with values in the literature. Please add the exact definition, ideally as an equation in Section 5.
  3. [Section 5.1; Appendix A.5] Several prose statements contradict the tables. Section 5.1 states that 'in UAM, the R2 of the linear model is 0.549,' but Table 3 reports UAM RIDGE as 0.459. Appendix A.5 states that CNN+LSTM achieves R2 of 0.420 and MLP+LSTM 0.408, while Table 8 reports 0.566 and 0.521, respectively. Please correct these numbers and scan the paper for other such inconsistencies.
  4. [Section 6 and Appendix B] The economic value claims (tracking-error reduction and fill-ratio advantage) rest on the matching-engine replay of Frey et al. (2023), which assumes best-quote passive orders, converts unfilled quantity to market orders at the end of each bin, and ignores the price impact of the parent order. These assumptions are strong for a 1% participation rate; the paper should state them as limitations and, ideally, provide a sensitivity analysis, e.g., with alternative participation rates or queue-position assumptions, before claiming that 'precise intraday forecasting offers substantial advantages.'
minor comments (4)
  1. [Section 2; Section 3.1; Section 4.3.2] There are several typographical errors, including 'and and employing' in Section 2, 'thecross-sectional' in Section 3.1, and the heading 'Nonliner Models' in Section 4.3.2; these should be corrected throughout.
  2. [Appendix A.2, Equations (13) and (14)] The notation in Equations (13) and (14) is garbled: '\volumt,i(s)' and '\volume(d)t,i' should be written with consistent hat notation, e.g., \hat{v}^{(s)}_{t,i} and \hat{v}^{(d)}_{t,i}, and the definitions clarified.
  3. [Figure 8 and surrounding text] The caption of Figure 8 and the body text disagree on whether the baseline is CMEM or the Rolling Mean (RM); please align the description to avoid ambiguity about which benchmark the 'advantage' is measured against.
  4. [Section 7, Conclusion] The claim to 'pioneer the identification of commonalities within intraday volume forecasting' is too strong given that the training schemes are drawn from Djanga et al. (2023) and commonality is a theme in Zhang et al. (2024); please temper this statement or clarify the specific new contribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the headline R2 gain is an out-of-sample comparison against an external benchmark, and the ML forecasts are not constructed from the target they predict.

full rationale

No load-bearing step reduces a claimed forecast to its own inputs. The R2 values in Table 3 are produced by training on rolling historical windows and testing on held-out periods; the dynamic one-bin-ahead features (past_2, past_8, daily and intraday sums) are lagged realized quantities that precede the target bin, so the target is not contained in the feature vector. The CMEM components used as regressors are filtered or forecast components computed from information available before the target bin; using them as inputs is model stacking, not fitting the target into the features. The SAM/CAM/UAM training schemes are attributed to Djanga et al. (2023), but their contribution here is empirical and is evaluated against the external CMEM benchmark and an oracle in the VWAP exercise, so the self-citation is provenance rather than the logical load of the argument. The matching-engine results rely on the published Frey et al. (2023) simulator rather than on an unverified theorem imported from the same authors. The static/dynamic asymmetry flagged by a skeptic is a genuine comparability limitation: Section 5.1 reports static R2 only for CMEM (0.240) while all ML rows are dynamic, so the headline 0.265-to-0.624 gain may overstate ex-ante predictability. But an unfair or incomplete comparison is not a by-construction equivalence; no equation in the paper makes the ML prediction identical to an input or to the benchmark output. Accordingly, there are no circular steps and the circularity score is 0.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central results rest on a moderate stack of tuned hyperparameters, none reported numerically, and on two paper-specific evaluation choices: dynamic prediction for the forecasting comparison and a matching-engine replay for the economic analysis. The clustering pipeline adds hand-chosen thresholds (EVR 0.8, L=10, D=10 days) that are tested only partially in Appendix A.7. No new physical or theoretical entities are introduced; DeepLOBv is a modified network architecture rather than an invented entity. The most consequential unstated input is the dynamic-prediction protocol, which makes the headline R2 comparison asymmetric relative to the static CMEM number reported in Section 5.1.

free parameters (6)
  • Ridge/LASSO regularization lambda = Not reported
    Tuned on a validation set per Section 4.3.1, but no values, grid, or selection criterion are given, so the reported R2 is conditional on an unspecified tuning protocol.
  • XGBoost hyperparameters = Not reported
    Tree count, depth, and learning rate are omitted; the XGB gains in Table 3 depend on these choices.
  • DeepLOBv architecture hyperparameters = Not reported
    CNN kernel sizes, Inception widths, LSTM hidden size, learning rate, and early stopping patience are not specified, per Section 4.3.2 and Appendix A.5.
  • CAM cluster count L = 10
    K-means++ is run with L=10 for the headline CAM results; Appendix A.7 tests 5, 20, and 50 clusters with R2 between 0.596 and 0.600.
  • PCA explained-variance threshold = 0.8
    The number of PCA eigenvectors M is chosen at cumulative explained variance 0.8 in the CAM scheme, per Appendix A.7, Table 9.
  • Lookback window and feature lags = D=10 days; lags 2 and 8 bins
    The 260-bin training window and the past_2 and past_8 compound features are selected from the ACF analysis in Sections 3.3 and 4.2.1.
assumptions (5)
  • domain assumption Intraday volume follows a stable U/J-shaped seasonal pattern within the sample.
    The time-of-day and intraday-interval features, the CMEM periodic component, and Figure 1 all rely on this regularity; a regime shift in the six-month 2017 window would weaken the commonality results.
  • domain assumption LOBSTER best-10-level data reconstructs the order book faithfully.
    All predictors and the matching-engine replay use LOBSTER data (Section 3.1); reconstruction errors or missing auction logic would propagate into every table and figure.
  • domain assumption Cross-sectional commonality in volume is a stable, exploitable signal.
    The CAM and UAM schemes assume that pooling across stocks improves forecasts; support comes from Chordia et al. (2000) and the authors' own prior volatility work, not from a structural model here.
  • ad hoc to paper Dynamic one-bin-ahead prediction is the appropriate evaluation mode.
    Adopted in Section 5.1 for all models, with static R2 reported only for CMEM (0.240); this choice injects same-day realized volume into the features and is not tested for the ML models.
  • ad hoc to paper The matching-engine replay approximates real execution outcomes.
    Fill ratios in Section 6 assume historical order replay, best-quote passive placement, and end-of-bin market-order conversion without modeling the parent order's price impact, per Appendix B.

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Cite this review

Pith. "Pith review of Forecasting Intraday Volume in Equity Markets with Machine Learning." pith.science (2026). https://pith.science/paper/7Z7VF4DK

@misc{pith2026250508180,
  author       = {Pith},
  title        = {Pith review of: Forecasting Intraday Volume in Equity Markets with Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Z7VF4DK}},
  note         = {Machine review of arXiv:2505.08180}
}
read the original abstract

This study focuses on forecasting intraday trading volumes, a crucial component for portfolio implementation, especially in high-frequency (HF) trading environments. Given the current scarcity of flexible methods in this area, we employ a suite of machine learning (ML) models enriched with numerous HF predictors to enhance the predictability of intraday trading volumes. Our findings reveal that intraday stock trading volume is highly predictable, especially with ML and considering commonality. Additionally, we assess the economic benefits of accurate volume forecasting through Volume Weighted Average Price (VWAP) strategies. The results demonstrate that precise intraday forecasting offers substantial advantages, providing valuable insights for traders to optimize their strategies.

Figures

Figures reproduced from arXiv: 2505.08180 by the authors.

Figure 1
Figure 1. Intraday Trading Volume. The plot shows the fluctuations in trading volume over a trading day period. The solid line denotes the mean, and the shaded area denotes one standard deviation. The value has been normalized by dividing it by the mean of one day’s trading volume. high intraday volume.9 A structured model is used in our research to characterize this pattern. In addition to the intraday volume pattern, the au… view at source ↗
Figure 2
Figure 2. Clustered Asset Model Diagram. Two inputs are used to obtain correlations: trading volume and features. After generating the correlation matrix by clustering, PCA is employed to compress the information from the matrix. Subsequently, the K-Means++ algorithm is applied for clustering, dividing stocks into multiple groups. 4.2.2 Universal Asset Model ··· Feature1 Matrix 260 X N Feature1 Matrix Stock N [PITH_FULL_IMAG… view at source ↗
Figure 3
Figure 3. Universal Model Diagram. The left matrix is the predictors and the right matrix is the target, with shape to be (N_Bins ∗ N_stocks, N_F eatures) and (N_Bins ∗ N_stocks, 1) respectively. Under the universal asset model scheme, we train a single model on the pooled dataset en￾compassing all stocks, leveraging each stock’s historical features as predictors. The procedure of forecasting is illustrated in [PITH_FULL_IMA… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Diagram of the DeepLOBv architechture. The light yellow rectangles in the figure are the network module. The two Convolution modules are employed to extract information from the features. The Inception module is utilized to combine the information extracted. The LSTM m…
Figure 5
Figure 5. Figure 5: A Sankey diagram illustrates the properties that contribute to improve￾ments in prediction. The marked numbers around the bars are the out-of-sample R2 of each model and the difference in R2 between them. Consequently, by addressing these discrepancies, our model not o…
Figure 6
Figure 6. Figure 6: Comparing stability across different conditions. In the left figure, the solid blue [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Tracking Error Comparison. Comparison of tracking errors between the CMEM and our UAM DeepLOBv model across different tickers. The tracking error is measured in basis points (bp), where 1 bp equals to 0.01%. higher liquidity is associated with improved fill rates. By f…
Figure 8
Figure 8. Figure 8: Comparison of the Relative Fill Ratio. The yellow bars denote the advantage of our proposed model over the CMEM model; the orange bars denote the advantage of the oracle model over the CMEM model. oracle volume. Afterward, we assess the advantages of the latter three m…
Figure 9
Figure 9. Figure 9: Static and Dynamic Prediction. The upper diagram illustrates static prediction, where the left side represents the input from the past several days, specifically from day t to day t − k, and the right side shows the predicted volume for the next day, namely day t + 1. …
Figure 10
Figure 10. Figure 10: shows the autocorrelation analysis of the intraday trading volume, the pattern of which indirectly corroborates the U-shape of the intraday volume [PITH_FULL_IMAGE:figures/full_fig_p031_10.png]
Figure 11
Figure 11. Figure 11: Comparison of correlation matrix heatmaps. The left heatmap is the corre￾lation matrix computed based on trading volume data; the right heatmap is the correlation matrix computed based on the features for forecasting data. The red rectangles indicate correlations with…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.