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.
Figures & tables
Figure 1: Schematic representation of non parametric model. Input data are processed through multiple hidden layers, each applying a nonlinear transformation, to produce the final output. A single function fθ maps the input data to both return and volume forecasts, allowing the network to learn joint representations and capture complex nonlinear dependencies between the two variables.
ticker
mean spread (ticks)
large tick
BAC
1.16
INTC
1.16
CSCO
1.18
CMCSA
1.19
PFE
1.20
small tick
AMZN
1.70
Table 1: Classification of the stocks considered in the analysis, together with their mean spread measured in ticks.
ticker
Rout2(r) VAR
Rout2(r) DNN
Rout2(v) VAR
Rout2(v) DNN
large tick
BAC
0.0494
0.0591
0.0299
0.0215
INTC
0.0814
0.1217
0.0198
0.0288
CSCO
0.0764
0.0982
0.0328
0.0273
CMCSA
0.0604
0.0859
0.0350
0.0486
PFE
0.0356
0.0367
0.0183
0.0091
small tick
AMZN
0.0246
0.0341
0.0685
0.0705
Table 2: Out-of-sample R2 for the VAR and DNN models at lag p=20 , separately for returns r and signed volumes v . Stocks are grouped according to the large-tick/small-tick classification used in Table 1 .
Figure 3: Results for Equations ( 13 ) and ( 14 ) for small tick (ST) stocks.
Figure 4: Results for Equations ( 13 ) and ( 14 ) for large tick (LT) stocks.
Figure 5: Average binned Shapley contributions for small-tick (ST) stocks for the volume’s term.
Figure 6: Average binned Shapley contributions for ST stocks for the return’s term.
Bucket
Ticker
Rϕ2
LT
BAC
0.163
LT
CMCSA
0.117
LT
CSCO
0.148
LT
INTC
0.178
LT
PFE
0.205
LT
Average
0.162
Table 3: Structural shock decomposition by ticker.
Figure 7: Average binned residual impact function ϕ^(utv) for LT stocks (left) and ST stocks (right). The function maps order-flow innovations into return innovations after removing the lagged predictable component.
Figure 8: Fitted one-lag parametric response functions for large-tick stocks. The top row reports the response of r^t to rt−1 and vt−1 , while the bottom row reports the response of v^t to rt−1 and vt−1 .
Figure 9: Fitted one-lag parametric response functions for small-tick stocks. The top row reports the response of r^t to rt−1 and vt−1 , while the bottom row reports the response of v^t to rt−1 and vt−1 .
Ticker
Rout2(r)
Rout2(v)
VAR
DNN
Reduced
VAR
DNN
Reduced
large tick
BAC
0.0494
0.0591
0.0934
0.0299
0.0215
0.0623
INTC
0.0814
0.1217
0.1065
0.0198
0.0288
0.0644
CSCO
0.0764
0.0982
0.1156
0.0328
0.0273
0.0735
CMCSA
0.0604
0.0859
0.1020
0.0350
0.0486
0.0820
PFE
0.0356
0.0367
0.0845
0.0183
0.0091
0.0478
Table 4: Out-of-sample R2 for the VAR, DNN, and reduced parametric models, separately for returns r and signed volumes ( v ). The VAR and DNN models are estimated at lag p=20 , while the reduced parametric model is the one-lag specification obtained from the Shapley-based reduced-form analysis. Stocks are grouped according to the large-tick/small-tick classification used in Table 1 .
Ticker
Rout2(r)
Rout2(v)
VAR
DNN
Reduced
Multi-lag
VAR
DNN
Reduced
Multi-lag
large tick
BAC
0.0494
0.0591
0.0934
0.0945
0.0299
0.0215
0.0623
0.0623
INTC
0.0814
0.1217
0.1065
0.1073
0.0198
0.0288
0.0644
0.0651
CSCO
0.0764
0.0982
0.1156
0.1168
0.0328
0.0273
0.0735
0.0745
CMCSA
0.0604
0.0859
0.1020
0.1031
0.0350
0.0486
0.0820
0.0828
PFE
0.0356
0.0367
0.0845
0.0855
0.0183
0.0091
0.0478
0.0479
Table 5: Out-of-sample R2 for the VAR, DNN, one-lag reduced parametric model, and multi-lag parametric model, separately for returns r and signed volumes v . The VAR and DNN models are estimated with p=20 lags. The reduced parametric model is the one-lag specification obtained from the Shapley-based reduced-form analysis, while the multi-lag parametric model uses p=20 lags with a shared nonlinear shape and exponential lag decay. Stocks are grouped according to the large-tick/small-tick classification used in Table 1 .
Appendix figures & tables13 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 10: Predictive performance of VAR model of Eq. ( 3 ) as a function of the lag order p for BAC. The panels report the in-sample and out-of-sample ( R2 ) as functions of the lag order. BAC is shown as a representative example, since the same qualitative behavior is observed across the other stocks in the dataset.
Figure 11: Lag-dependent predictive performance for AAPL using the neural network model. The left panel reports the in-sample and out-of-sample ( R2 ) for returns as a function of the lag order, while the right panel shows the corresponding results for signed volumes. AAPL is shown as a representative example, as similar qualitative patterns are observed across all stocks.
Figure 12: Normalized lag importance for large-tick stocks (LT). Each curve represents the average fraction of total Shapley magnitude explained by lag ℓ .
Figure 13: Normalized lag importance for small-tick stocks (ST). Each curve represents the average fraction of total Shapley magnitude explained by lag ℓ .
Figure 14: Empirical nonlinear relationships for large-tick (LT) stocks.
Figure 15: Empirical nonlinear relationships for small-tick (ST) stocks.
Figure 16: VAR-implied binned predictive surfaces for small-tick (ST) stocks. The panels are organized as in Figure 3 : predicted returns and signed volumes are represented as functions of either rt−1 or vt−1 , conditional on the sign regime of the other lagged variable.
Figure 17: VAR-implied binned predictive surfaces for large-tick (LT) stocks. The panels are organized as in Figure 4 : predicted returns and signed volumes are represented as functions of either rt−1 or vt−1 , conditional on the sign regime of the other lagged variable.
Figure 18: Average Shapley profiles for large tick (LT) stocks for the volume term. Top panels refer to the return prediction r^t , while bottom panels refer to the signed-volume prediction v^t . Left panels show the contribution of the conditioning variable vt−1 when the horizontal axis is vt−1 . Right panels show the contribution of the conditioning variable vt−1 when the horizontal axis is rt−1 . Shaded bands denote ± one cross-sectional standard error, while histograms report the average empirical support.
Figure 19: Average Shapley profiles for large tick (LT) stocks for the return term. Panels are organized as in Figure 18 .
Figure 20: Evolution of the in-sample and out-of-sample R2 as a function of the maximum lag p for the GILD ticker. The improvement after the first lag is limited and the predictive performance rapidly saturates, consistently with a short-memory nonlinear dependence structure.
Figure 21: Estimated structural projection ϕ^(utv) for the multi-lag parametric model. The function is estimated through binned conditional averages and then averaged separately over large-tick and small-tick stocks.
Bucket
Ticker
Corr. before
Corr. after
Spearman before
Spearman after
Corr. sq.
Corr. abs.
LT
BAC
0.301
0.0121
0.361
-0.0214
0.0034
0.111
LT
CMCSA
0.250
0.0083
0.294
-0.0347
-0.0123
0.049
LT
CSCO
0.292
0.0109
0.348
-0.0221
0.0075
0.107
LT
INTC
0.271
0.0119
0.323
0.0027
0.0175
0.126
LT
PFE
0.331
0.0135
0.423
0.0136
-0.0152
0.086
LT
Average
0.289
0.0113
0.369
-0.0108
0.0002
0.096
Appendix
Table 6: Residual structural decomposition for the multi-lag parametric model using the binned estimator of ϕ . The last row of each bucket reports equal-weight averages across tickers.
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Bryan Kelly, Boris Kuznetsov, Semyon Malamud +1
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