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Explainable Deep Learning for Price-Trade Dynamics: From Black-Box Forecasts to Effective Parametric Models

Manuel Naviglio, Fabrizio Lillo

arXiv:2609.06085v1q-fin.TRcs.AIphysics.data-an

TL;DR

Linear models capture price–trade dynamics with interpretable but restrictive functional forms, while neural networks offer flexibility but can be difficult to interpret. The paper uses explainability to extract nonlinear dependencies and build a parsimonious model that outperforms the linear VAR and performs comparably to the neural network.

  • Problem

    Linear models may miss nonlinear, state-dependent price–trade relationships, while neural-network predictions can be difficult to connect to economically meaningful mechanisms.

    Source: §1 · p. 2, §2.1 · p. 5

  • Method

    The authors use neural networks for structural discovery, apply Shapley-based explainability to identify functional forms, and translate them into a SHAP-inspired nonlinear parametric model.

    Source: §1 · pp. 2–3, §7 · p. 33

  • Results

    The reduced parametric model outperforms the linear VAR and achieves predictive performance comparable to the deep neural network, with some cases performing even better.

    Source: §7 · p. 33

  • Takeaways & Limitations

    The results support using explainability to translate neural-network dependencies into parsimonious, interpretable models that retain most of the network’s predictive power.

    Source: §7 · p. 33

    Models calibrated only on public market data may produce nearly linear impact profiles with limited post-trade reversion, unlike profiles observed for real metaorders.

    Source: §2.1 · p. 5

Abstract

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Understanding the joint dynamics of prices and trades is central to market microstructure, where returns and order flow interact through nonlinear and state-dependent mechanisms. Linear models are interpretable but may miss these effects, while deep neural networks improve forecasting at the cost of transparency. We use neural networks as tools for structural discovery rather than only for prediction. A deep feed-forward network is trained on high-frequency returns and signed volumes for large- and small-tick stocks and compared with a linear VAR benchmark. The neural network improves predictive performance, especially for returns, revealing nonlinear dependencies beyond the linear specification. Using Shapley-based explainability, we show that the dominant contributions are concentrated at the most recent lags. Model-implied responses are consistent with conditional averages reconstructed from the data. Unlike empirical averages, however, the neural-network decomposition isolates individual regressor contributions to the aggregate dependence. Lagged signed volume generates sign-preserving and saturating effects, consistent with nonlinear price impact and order-flow persistence. Lagged returns act as state variables: when the previous trade does not move the price, the model predicts continuation in the direction of past order flow, whereas non-zero returns generate attenuation or reversal. Building on these findings, we introduce a parsimonious SHAP-inspired nonlinear parametric model. It reproduces the main return-volume dependencies, outperforms the linear VAR benchmark, and achieves performance comparable to the neural network. A multi-lag extension captures residual longer-memory effects while preserving interpretability. Overall, explainability offers a route from black-box prediction to economically meaningful parametric models of price and trade dynamics.

1 Introduction

The paper uses explainable neural networks to identify nonlinear price–trade relationships and translate them into interpretable parametric models. Its reduced model retains predictive performance comparable to the neural network while outperforming the linear VAR benchmark.

1 Introduction

  • Linear models are interpretable but restrict functional form, whereas deep learning can capture nonlinear dependencies yet often remains a black box.This leaves the economic meaning of learned features difficult to assess.Source: §1 · p. 2
  • The authors use neural networks for structural discovery, applying Shapley-based explainability to decompose predictions and reconstruct lagged return and signed-volume dependencies.They treat explainability as a way to identify candidate structural relations, not only as a diagnostic for a predictor.Source: §1 · p. 2
  • Lagged signed volume produces sign-preserving, saturating responses, while lagged returns act as state variables that modulate order-flow effects and can generate attenuation or reversal.These patterns are stable across assets, with some differences between large-tick and small-tick stocks.Source: §1 · pp. 2–3
  • When the previous trade does not move the price, the return contribution tends to preserve the sign of lagged signed volume; a non-zero previous return brings attenuation or reversal.Model-implied predictive surfaces also agree qualitatively with conditional averages computed from the data.Source: §1 · pp. 2–3
  • The reduced SHAP-inspired model outperforms the linear VAR and performs comparably to the neural network, with greater stability across lag orders.It represents saturating lagged-volume dependence and a state-dependent correction involving lagged returns and prior order-flow sign.Source: §1 · p. 3

2 Price and trades models

The paper models returns and signed volumes as jointly dependent on lagged observations, comparing linear and nonlinear specifications. It separately analyzes residual contemporaneous dependence so that lagged prediction and instantaneous order-flow impact are not conflated.

2 Price and trades models

  • Price-and-trade dynamics are represented as recursive relationships between past prices and volumes; the price equation is expressed in returns to ensure stationarity.The variables are indexed in trade time, with signed volume representing order flow.Source: §2 · pp. 3–4

2.1 Parametric Linear models

  • The VAR benchmark models price returns and signed volume using linear lag dependencies, a tractable and interpretable structure with restrictions on nonlinear, state-dependent effects.The residual terms can encode contemporaneous dependence between returns and order flow.Source: §2.1 · p. 5

2.2 Non parametric Non Linear models

  • The nonlinear benchmark is a deep feed-forward network that maps lagged returns and volumes jointly to one-step-ahead forecasts, without restricting their effects to be linear or additive.The same network function produces both return and volume predictions.Source: §2.2 · pp. 5–6, Figure 1 · p. 6

2.3 Residual Contemporaneous Dependence

  • The forecasting models use lagged information only; the authors then examine residual dependence to separate predictable dynamics from contemporaneous order-flow innovation.Under a recursive interpretation, trade-time order-flow innovation precedes quote revision, and a nonlinear function can represent its instantaneous price impact.Source: §2.3 · p. 7

3 Data

The analysis uses NASDAQ limit-order-book data, selecting visible limit-order executions and measuring price at the pre-execution midpoint. The sample covers five small-tick and five large-tick stocks in June 2024.

3 Data

  • The study uses LOBSTER data, combining transaction messages with the order-book state after each transaction, and focuses on visible limit-order executions.Prices are measured as the midpoint immediately before each selected execution.Source: §3 · p. 8
  • Orders at the same timestamp and sign are aggregated, the first and last 30 minutes are removed, and observations outside the 0.005–0.995 quantile range are filtered.The DNN training data are normalized, and the first half of the sample is used for training while the second half is reserved for out-of-sample testing.Source: §3 · p. 8
  • The June 2024 sample contains five small-tick and five large-tick stocks, classified using their mean spreads measured in ticks.Large-tick stocks generally have spreads near one tick, while small-tick stocks typically have spreads of several ticks.Source: §3 · p. 8, Table 1 · p. 8

4 Forecasting Performance of the Models

Out of sample, the DNN generally improves on the VAR, with the clearest gains for returns and for signed volumes in small-tick stocks. Predictive patterns differ by tick regime, and both models indicate that useful forecasting information is concentrated in relatively recent lags.

Lag choice

  • For the VAR, out-of-sample R2 stabilizes as lag order increases; the DNN shows a similar overall pattern but fluctuates after the first lags.The authors use p = 20 to include short-memory dependencies without unnecessarily long inputs or excessive parameters.Source: §4 · p. 9

Performances

  • At lag p = 20, the DNN generally improves on the VAR, with higher return R2 for almost all stocks and gains for signed volumes in all small-tick stocks.For large-tick signed volumes, the models are closer and VAR remains competitive in some cases.Source: Table 2 · p. 9, §4 · p. 9
  • Returns are more predictable than signed volumes for large-tick stocks, while signed volumes are more predictable than returns for small-tick stocks.The text relates the large-tick pattern to price discreteness and queue depletion, and the small-tick pattern to more persistent signed order flow.Source: §4 · pp. 9–10
  • Because the DNN generally outperforms VAR, the authors next use explainability to investigate which additional dependencies the nonlinear model captures.They ask whether the network refines linear effects or learns dependencies absent from the linear benchmark.Source: §4 · p. 10

5 Explaining the Neural Network

This section uses SHAP explanations, response-surface comparisons, and structural-shock analysis to examine how the neural network represents nonlinear price–volume dependencies, including the contributions of lagged returns and signed volume and a contemporaneous order-flow channel.

5.1 The Shapley decomposition

  • SHAP decomposes predictions into feature-level contributions, separating the roles of lagged returns, signed volumes, and more distant history beyond what empirical conditional averages reveal.For the linear VAR, these contributions reduce to coefficients multiplied by input deviations from their background means; neural-network deviations can expose dependencies beyond that linear pattern.Source: §5 · p. 10, §5.1 · pp. 11–12

5.2 Shapley values

  • SHAP contributions concentrate at the first lag, with order flow producing sign-preserving effects and lagged returns revealing short-term reversal in return predictions.For predicted volume, lagged returns relate more symmetrically to the magnitude of past price changes, consistent with a link between volatility and trading activity.Source: Figure 2 · p. 13, §5.2 · pp. 13–14

5.3 Model-Implied Predictive Surface

  • The surface analysis varies the most recent return–volume pair while retaining the full-lag fitted model, then compares its predictions with empirical responses.This comparison tests whether the learned dependencies appear in observed outcomes rather than arising from model flexibility alone.Source: §5.3 · p. 14, §5.3.1 · p. 16

5.3.1 Aggregated conditional surfaces plots results

  • Lagged signed volume has a sign-preserving, saturating effect on predicted returns and volumes, while the previous return modulates the strength of that response.Small-tick stocks show sharper transitions; large-tick stocks are smoother and more constrained. Small previous returns favor continuation of order-flow pressure, whereas larger returns attenuate or may reverse it.Source: §5.3.1 · pp. 16–17, Figure 3 · p. 17, Figure 4 · p. 18
  • Nonlinearities and sharper transitions are more pronounced in small-tick stocks, while large-tick stocks exhibit smoother, more constrained dynamics.The reported interpretation is that tick size limits price adjustments in large-tick assets.Source: Figure 3 · p. 17, §5.3.1 · pp. 17–18, Figure 4 · p. 18
  • Neural-network conditional surfaces qualitatively reproduce the main empirical dependencies, though large-tick profiles show stronger asymmetry than the empirical surfaces.The comparison is a consistency check; the VAR surfaces remain essentially linear and cannot reproduce the nonlinear aggregate dependencies observed in the data.Source: §5.3.1 · pp. 18–19

5.3.2 Shap values results

  • For small-tick stocks, the return contribution preserves the sign of subsequent signed volume after small price changes but reverses it after larger absolute returns.The return contribution also retains residual dependence on lagged signed volume.Source: §5.3.2 · p. 21
  • When lagged signed volume moves no price, the return contribution reinforces order-flow persistence; with a non-zero lagged return, it tends to reverse future signed volume and return.The same qualitative condition-dependent pattern appears for large-tick stocks, whose non-starred profiles show stronger asymmetry and more evident mean reversion than small-tick profiles.Source: §5.3.2 · p. 21
  • The Shapley decomposition separates regressor contributions that aggregate response plots cannot isolate, and non-flat cross-profiles reveal nonlinear modulation by the other variable.In the volume term, dependence on lagged returns is approximately flat, so separation in aggregate profiles is likely captured by a residual return-term contribution.Source: §5.3.2 · pp. 20–21

5.4 The structural shocks

  • Average R2ϕ is 0.162 for LT stocks and 0.179 for ST stocks, indicating meaningful contemporaneous order-flow explanatory power after lagged predictors are removed.Ticker-level R2ϕ ranges from 0.117 to 0.205 for LT stocks and from 0.116 to 0.266 for ST stocks.Source: §5.4 · p. 22, Table 3 · p. 23
  • The residual impact function is increasing, sign-preserving, and saturating in both tick-size groups, with a steeper response near zero that flattens in the tails.LT curves span a wider innovation range and reach larger return-residual magnitudes, while ST curves are more concentrated around zero and saturate more smoothly.Source: Figure 7 · p. 23, §5.4 · p. 23

6 From Explanations to a Parametric Model

This section develops a parsimonious, interpretable parametric model from Shapley-isolated contributions, with a one-lag form capturing much of the predictive structure and a multi-lag extension incorporating longer memory. Because performance gains from additional lags quickly saturate, the extension uses exponential decay to represent longer-lag information without separate nonlinear parameters for each lag.

6.1 One-lag model construction

  • The one-lag model combines sign-preserving, saturating signed-volume effects with a return-state correction that reinforces order-flow direction for small prior returns but attenuates or reverses it for large returns.When the prior return is zero, the contribution preserves the sign of lagged signed volume; nonzero returns shift the effect toward reversal, with approximate symmetry in the small-tick specification.Source: §6.1 · pp. 25–26

6.2 Calibration of the SHAP-Inspired Parametric Model

  • Calibration qualitatively recovers the main neural-network and empirical return–volume dependencies in both tick-size groups, though asymmetries remain outside the parsimonious baseline.The authors leave asymmetric corrections as a possible extension.Source: §6.2 · p. 27
  • For returns, one-lag out-of-sample R² beats VAR across stocks, rising from 0.0494 to 0.0934 for BAC, 0.0764 to 0.1156 for CSCO, and 0.0356 to 0.0845 for PFE.The reduced model often matches or outperforms the 20-lag DNN, particularly for large-tick stocks.Source: §6.2 · pp. 28–29, Table 4 · p. 30

6.3 Multi-lag extension

  • The multi-lag extension shares nonlinear shape parameters across lags and applies geometric decay to lag-specific strengths, incorporating longer memory without a separate nonlinear function for every lag.This restriction is motivated by the concentration of predictive contribution at early lags and helps avoid fitting lag-specific noise.Source: §6.3 · pp. 30–31
  • The multi-lag parametric model improves on VAR for returns and signed volumes; return gains over one lag are modest, while volume gains are larger and its R² exceeds the DNN’s for every stock.Additional lags provide limited residual information for return forecasting, but contribute more to persistent signed-volume dynamics.Source: §6.3 · p. 31
  • Performance gains saturate after the first few lags, supporting a shared-shape, exponentially decaying multi-lag model that captures residual longer-lag information without separate nonlinear parameters for each lag.Table 5 reports out-of-sample R2 comparisons for returns and signed volumes across the VAR, DNN, one-lag reduced model, and 20-lag multi-lag model.Source: §6.3 · p. 31, Table 5 · p. 32

7 Conclusions

The reduced SHAP-inspired nonlinear model outperforms the linear VAR and matches or sometimes exceeds the neural network, while retaining interpretable price–volume mechanisms. The analysis also finds that predictive contributions concentrate at recent lags and that extending the model captures residual longer memory with rapidly saturating gains.

7 Conclusions

  • Across five large-tick and five small-tick stocks, the neural network generally improves on the VAR, especially for return prediction.This comparison suggests relevant nonlinear dependencies are not fully captured by the linear specification.Source: §7 · p. 32
  • Model-implied predictive surfaces are qualitatively consistent with empirical conditional averages, after which Shapley decomposition isolates individual regressor contributions.This connects the learned response patterns to regularities in the data while separating contributions that aggregate conditional averages do not isolate.Source: §7 · p. 32
  • Lagged signed volume has sign-preserving, saturating effects, while lagged returns condition order-flow responses: zero prior returns favor continuation and non-zero returns favor attenuation or reversal.The state-dependent return contribution is organized by prior signed-volume sign, with the qualitative structure appearing in both tick-size regimes.Source: §7 · p. 33
  • After lagged nonlinear predictability is removed, a sizeable instantaneous order-flow channel remains in return residuals for both large- and small-tick stocks.The estimated residual impact function is increasing, sign-preserving, and saturating.Source: §7 · p. 33
  • The reduced parametric model outperforms the linear VAR and matches the neural network, sometimes achieving higher predictive performance.The authors attribute much of the neural network’s advantage to a small set of stable, interpretable nonlinear mechanisms.Source: §7 · p. 33
  • The multi-lag extension improves on the one-lag model, with clearest gains in the signed-volume equation, but gains saturate rapidly as maximum lag increases.Its structured lag decay gives more stable dependence on lag length than the unconstrained neural-network benchmark.Source: §7 · p. 33
  • The proposed approach translates explainability-derived dependencies into explicit parametric forms that can be calibrated and compared, and may be used in simulation or structural market-microstructure models.The authors leave richer limit-order-book state variables and tests across longer periods, more assets, and different market regimes as open directions.Source: §7 · pp. 33–34

A Residual Contemporaneous Dependence and Structural Identification

The appendix separates lagged forecasting from contemporaneous structural dependence by modeling lagged information first and then analyzing residual dependence. Under a recursive ordering, signed-volume innovations may affect return innovations contemporaneously, but not vice versa within the same trade-time event.

A Residual Contemporaneous Dependence and Structural Identification

  • Directly inserting contemporaneous signed volume is not the cleanest way to separate lagged predictability from instantaneous impact; the reduced-form residuals provide the latter dependence.The residual formulation treats the unexpected component of order flow as the contemporaneous channel.Source: A Residual Contemporaneous Dependence… · pp. 36–37
  • Directly conditioning a nonlinear return model on contemporaneous signed volume can mix predictable order-flow effects with the innovation, preventing a unique separation of lagged and instantaneous components.The authors note that this mixing can make an apparent instantaneous effect partly reflect the predictable component.Source: A Residual Contemporaneous Dependence… · p. 37
  • The proposed alternative estimates a reduced-form model using lagged information only, then analyzes residual dependence to assess contemporaneous impact.Residual dependence between unexpected order flow and unexpected price changes indicates a remaining instantaneous relation.Source: A Residual Contemporaneous Dependence… · pp. 37–38
  • Under nonlinear structural representations, recovering shocks requires identifying restrictions because the mapping from reduced-form innovations to structural shocks is not characterized by a constant impact matrix.The paper cites nonlinear structural VAR work on this identification requirement [25,26].Source: A Residual Contemporaneous Dependence… · p. 38
  • The structural interpretation imposes a recursive ordering in which order-flow innovations can affect return innovations contemporaneously, but not vice versa within the same trade-time event.The ordering reflects the paper’s premise that signed volume is observed at the event before the associated quote revision.Source: A Residual Contemporaneous Dependence… · pp. 36, 38
  • The paper distinguishes forecasts using information through t −1 from predictions conditioned on signed volume observed before the associated quote revision.Only the lagged-information forecast is used for comparing predictive performance across the VAR, nonlinear parametric model, and neural network.Source: A Residual Contemporaneous Dependence… · p. 38

B Lag-order dependence of forecasting performance

Increasing lag order initially improves forecasting, but performance shows no robust out-of-sample gains after the short lags. The analysis supports using p = 20 without relying on a finely tuned memory length.

B Lag-order dependence of forecasting performance

  • The analysis uses lagged returns and signed volumes to predict target variables and compares in-sample and out-of-sample R2 across lag orders.The purpose is to test whether longer memory yields systematic predictive gains and to check the robustness of the main-text choice p = 20.Source: B Lag-order dependence of forecasting… · p. 38
  • For the VAR, increasing p initially improves in-sample and out-of-sample R2, but out-of-sample performance stabilizes after the first few lags across stocks.Figure 10 shows BAC as a representative example; at larger lag orders, out-of-sample performance can even decline slightly.Source: B Lag-order dependence of forecasting… · p. 39, Figure 10 · p. 39
  • For the neural network, test R2 fluctuates after the first lags rather than improving monotonically, with no structurally privileged lag order.Figure 11 presents AAPL as a representative example, with similar qualitative patterns across stocks.Source: B Lag-order dependence of forecasting… · p. 39, Figure 11 · p. 39
  • Across models, relevant predictive memory is concentrated at relatively short horizons, with no robust out-of-sample improvement from increasing p further.Neural-network oscillations are consistent with weak long-lag signals interacting with nonlinear flexibility and finite-sample estimation, not sharply optimal lag lengths.Source: B Lag-order dependence of forecasting… · pp. 40–41, Figure 13 · p. 41

C Shapley Lag-Importance Profiles

Shapley-based lag-importance profiles place most explanatory contribution on the most recent return-volume pair. Volume contributions generally persist longer than return contributions, with some differences between tick-size regimes.

C Shapley Lag-Importance Profiles

  • Across all prediction blocks and both tick-size regimes, the first lag dominates the reported profiles, suggesting that most nonlinear structure learned by the neural network is concentrated in the most recent return–volume pair.Figures 12 and 13 report the resulting profiles for LT and ST stocks.Source: C Shapley Lag-Importance Profiles · p. 42
  • Lagged signed volumes generally retain importance longer than lagged returns, especially for predicting signed volumes, while return contributions decay more rapidly.This pattern is consistent with order-flow persistence and return effects concentrated at very short horizons.Source: C Shapley Lag-Importance Profiles · p. 42
  • The profiles differ by tick regime: large-tick stocks show more gradual volume-lag contributions, while small-tick stocks show more persistent return contributions to future signed volumes.For small-tick stocks, the effect of lagged volumes on returns is more concentrated at the first lag.Source: C Shapley Lag-Importance Profiles · p. 42
  • The lag profiles motivate focusing predictive-surface analysis on the first lag and extending the parametric model with decaying lag weights.The profiles are constructed from mean absolute Shapley values, normalized within ticker and prediction block before averaging across tickers.Source: C Shapley Lag-Importance Profiles · pp. 41–42

D Empirical surfaces and validation

The section compares model-implied predictive surfaces with conditional averages reconstructed from observed returns and signed volumes. Their qualitative agreement supports the interpretation that the nonlinear patterns reflect empirical regularities.

D Empirical surfaces and validation

  • The plotted shaded bands represent cross-sectional uncertainty across assets, not confidence intervals at the observation level.Only bins with at least 150 average observations are retained in the final plots, while histograms remain complete.Source: D Empirical surfaces and validation · pp. 42–43
  • Unlike empirical conditional averages, the Shapley decomposition attributes the model prediction additively to returns, signed volumes, and remaining lagged inputs.The comparison uses four lag-one response-explanatory pairings and also conditions on the sign regime of the other lagged variable.Source: D Empirical surfaces and validation · p. 43
  • Empirical curves are formed from bin-wise response averages and then interpolated and averaged across tickers for each tick-size group.Signed volumes use uniform ticker-specific bins; returns use data-driven clusters because price discreteness concentrates returns in local clusters.Source: D Empirical surfaces and validation · pp. 42–43
  • Model-implied and data-reconstructed conditional response surfaces agree qualitatively for both large-tick and small-tick stocks, providing a sanity check on the learned nonlinear patterns.Figures 14 and 15 show the empirical nonlinear relationships for large-tick and small-tick stocks.Source: D Empirical surfaces and validation · p. 43

E VAR-implied predictive surfaces

The VAR surfaces recover directional price-impact and order-flow-persistence patterns, but their linear form cannot reproduce the sharper nonlinearities and state dependence in the neural-network surfaces. The comparison also differs across tick-size regimes.

E VAR-implied predictive surfaces

  • The VAR surfaces capture directional effects: lagged signed volume predicts signed volume in the same direction and returns in the corresponding direction.Figures 16 and 17 show the small-tick and large-tick VAR surfaces; the signed-volume response is approximately monotone in lagged signed volume.Source: E VAR-implied predictive surfaces · pp. 44–45, Figure 16 · p. 46, Figure 17 · p. 47
  • In the VAR, lagged returns affect predictions proportionally, so it cannot reproduce the neural network's attenuation of order-flow continuation at large absolute returns.The VAR responses in lagged returns are approximately linear, with curves separated by the sign of lagged signed volume.Source: E VAR-implied predictive surfaces · p. 45
  • For large-tick stocks, the difference between VAR and neural-network surfaces is less pronounced, consistent with the VAR capturing part of their directional dependence.For small-tick stocks, the VAR recovers average directions but is substantially smoother than the neural network.Source: E VAR-implied predictive surfaces · pp. 45–46
  • Compared with the VAR, the neural network shows sharper transitions, flattening at large absolute signed volume, and stronger state-dependent modulation.The VAR's linear slope and conditioning-related level shifts cannot fully reproduce these patterns.Source: E VAR-implied predictive surfaces · pp. 45–46

F Large tick Shapley Components

Large-tick Shapley profiles broadly resemble small-tick results, but the return contribution is more asymmetric across prior-return states. Lagged volume carries its sign directly, while return contributions vary with prior price moves.

F Large tick Shapley Components

  • For large-tick stocks, lagged signed volume contributes with its own sign, without an added dependence on the previous return.The conditional volume profiles for nonzero previous returns nearly collapse onto the zero-return profile.Source: F Large tick Shapley Components · p. 47, Figure 18 · p. 48
  • When the previous trade leaves price unchanged, the return contribution preserves prior order-flow sign; after a price move, it tends to reverse sign.The reversal pressure applies to both the next return and signed volume.Source: F Large tick Shapley Components · pp. 47–48, Figure 19 · p. 49
  • Large-tick return profiles are more asymmetric across positive and negative prior returns than small-tick profiles, suggesting a stronger price mean-reverting component.Positive and negative previous returns are associated with opposite-signed return contributions.Source: F Large tick Shapley Components · p. 48, Figure 19 · p. 49

G Lag dependence for the parametric multi-lag model

Predictive performance saturates as the maximum lag increases, consistent with short dominant nonlinear memory and only marginal contributions from more distant lags.

G Lag dependence for the parametric multi-lag model

H Structural shocks extraction for the parametric multi-lag model

The binned structural correction removes most contemporaneous signed dependence between return and signed-volume residuals in both tick buckets. Some residual volatility-level dependence may remain.

H Structural shocks extraction for the parametric multi-lag model

  • For GILD, predictive performance rapidly saturates as maximum lag increases, with limited improvement after the first lag.Source: Figure 20 · p. 50
  • The multi-lag structural projection uses binned conditional averages and yields results similar to the neural network, with a larger relation scale for large-tick stocks.The larger scale is consistent with stronger price discreteness and queue-driven price formation in that regime.Source: H Structural shocks extraction for the… · pp. 49–50
  • Spearman correlations fall from about 0.37 to close to zero for LT stocks, and from about 0.42 to slightly negative values for ST stocks.Source: H Structural shocks extraction for the… · p. 50
  • The correction removes the main directional dependence, but residual volatility-level dependence may remain, especially for large-tick stocks.Correlation with absolute residuals remains larger, though moderate; correlation with squared residuals is very small on average.Source: H Structural shocks extraction for the… · p. 50
  • Residual return–volume correlations fall from about 0.29 (LT) and 0.32 (ST) to approximately 10^-2 in both buckets after correction.Source: H Structural shocks extraction for the… · p. 50, Table 6 · p. 51

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