A two-variable closed-form model beats a twenty-lag VAR on return prediction for all ten stocks in this sample. Against the twenty-lag neural network, it wins on seven. That compression is the result that matters.

Turning network output into two equations

Naviglio and Lillo use trade-time LOBSTER NASDAQ data from June 2024. Their five large-tick names, BAC, INTC, CSCO, CMCSA and PFE, have mean spreads from 1.16 to 1.20 ticks. The five small-tick names, AMZN, AAPL, GILD, TSLA and NVDA, range from 1.70 to 6.27. Each state contains the last p pairs of log return and signed volume, with p = 20. One-step-ahead R2 is compared across a linear VAR, a feed-forward network (dense 64-64-32, tanh, MSE, Adam), and a parametric specification inferred from the network's SHAP attributions. Those attributions divide each prediction into contributions from individual inputs.

The network improves on the VAR for returns, though the size and direction vary by stock. NVDA rises from 0.0166 to 0.0454, TSLA from 0.0433 to 0.0906, and INTC from 0.0814 to 0.1217. GILD goes the other way, with 0.0429 from the network against 0.0499 from the VAR. For signed volume, the network wins across all five small-tick stocks. The plain VAR stays ahead for BAC (0.0215 against 0.0299), CSCO (0.0273 against 0.0328) and PFE (0.0091 against 0.0183).

SHAP attribution shows how little the network uses deeper lags. Across the four prediction blocks in both tick buckets, normalized first-lag importance accounts for roughly 0.4 to 0.6 of total Shapley magnitude. Conditioning those attributions on sign(r_{t-1}) and sign(v_{t-1}) reveals two forms. Signed volume is sign-preserving and saturating. The previous return acts as a switch: at r_{t-1} = 0, its contribution follows v_{t-1}; once |r_{t-1}| becomes large enough, the contribution reverses. The authors express both forms with tanh terms and estimate them by nonlinear least squares. In the fitted parametric specification, the reversal occurs at |r|* = (1/lambda) arctanh(A/B). A softplus reparametrization imposes B > A during calibration.

The fitted one-lag model produces 0.0934 for BAC, versus 0.0494 from the VAR and 0.0591 from the network. PFE comes in at 0.0845 against 0.0356 and 0.0367. CSCO reaches 0.1156 against 0.0764 and 0.0982. The network remains ahead on returns only for INTC (0.1217 against 0.1065), AAPL (0.0951 against 0.0858) and TSLA (0.0906 against 0.0826).

Signed volume favors the one-lag form more broadly. It beats the VAR for every stock, including BAC at 0.0623 against 0.0299. It also exceeds the network for all five large-tick names. For PFE, the comparison is 0.0478 against the network's 0.0091.

The multi-lag version applies the same shapes across lags, with geometric decay. On volume it beats the network for all ten stocks. Most gains over the one-lag version appear among the small-tick names: AAPL reaches 0.1552 against the one-lag 0.1421 and the network's 0.1372, while NVDA reaches 0.1141 against the one-lag 0.0967 and the network's 0.1068. Extra return lags contribute almost nothing. BAC moves to 0.0945 from 0.0934, while TSLA and NVDA remain unchanged at 0.0826 and 0.0498.

Why the first lag does the work

The authors present this interpretation as a predictive reduced form. They explicitly describe the attributions as model explanations rather than causal effects. Contemporaneous signed volume stays outside the return equation and enters the residuals instead. Appendix A argues that adding v_t directly would combine its predictable component with its innovation.

They next project return residuals on volume residuals nonparametrically. Using the neural network's residuals, R2_phi averages 0.162 for large-tick stocks and 0.179 for small-tick stocks. The per-ticker ranges are 0.117 to 0.205 and 0.116 to 0.266. These values refer to the return variation left unexplained by the lagged model.

With residuals from the multi-lag parametric model, the same correction reduces correlation from 0.289 to 0.0113 (LT) and from 0.317 to 0.0104 (ST). Correlation with absolute residuals remains 0.096 and 0.031, leaving a volatility-level relationship. Since contemporaneous order flow is excluded, the headline R2 figures cannot be compared with a Hasbrouck- or TIM-style specification that includes it.

Replication still requires judgment

The parametric equations are fully specified.

They can be written down and fitted directly. Judgment enters through preprocessing and training, where several choices remain unspecified. We did not find epochs, batch size, learning rate or a stopping rule for the network. The paper says the data are normalized for training, but does not state whether the scaling statistics use only the training half. Its quantile filter at [0.005, 0.995] leaves both the affected series and the sample portion used to compute the quantiles unclear. The trim is small, yet it removes precisely the extreme order-flow events where saturation in the impact function should matter.

The DeepExplainer background sample is mentioned without its size, even though attributions depend on that sample. Binned surfaces use a 150-average-observation retention rule. Neither the bin count nor the KDE bandwidth used to cluster returns is provided. The paper defines R2 as the share of actual variance explained by predictions and gives the R2_phi formula. It does not identify the mean used for the out-of-sample denominator. The choice matters when the reported values span 0.0091 (PFE volume, DNN) to 0.1589 (TSLA volume, multi-lag).

One unresolved issue matters most. The authors select the functional forms after inspecting SHAP surfaces, then fit and score those forms on data from the same month and the same ten tickers. The exercise uses a single chronological 50/50 split, with no validation set described. Lag selection follows a similar route: Appendix B plots test-set R2 against lag order, leading to p = 20. The authors argue that larger lags bring no systematic improvement, which they interpret as evidence that fine tuning of memory length does not drive the conclusions. That defense goes only so far. It leaves the specification search behind the tanh switch uncovered.

The reported figures support the compression claim. Generalization remains untested, as the authors acknowledge in the conclusion by listing longer periods, more assets and different regimes as open questions. Every result is one-step-ahead R2 in trade time. We found no PnL, no spread or fee assumption, and no Sharpe. In event time, the reduced model's return R2 ranges from 0.0498 (NVDA) to 0.1156 (CSCO). Those figures do not establish what remains after crossing the spread.

We could not test any of this ourselves. The model requires the midpoint immediately before each visible-limit-order execution and a buy/sell tag for every print. We hold one-minute OHLCV, which cannot reconstruct signed order flow.

Freeze the parameters on June 2024, apply the same two equations to a different quarter and a wider cross-section, and test whether the advantage over the VAR survives.