Residual Learning in Empirical Asset Pricing

• Existing evidence finds neural-network performance peaks at relatively shallow depth, while simply adding feedforward layers does not deliver greater economic value.Source: §1 · p. 4
• Because identity mappings can extend a well-performing shallow network without changing performance, the authors argue deep models can retain shallow functions yet potentially outperform them.Source: §1 · p. 4
• Prior studies of deeper asset-pricing architectures do not examine residual models across varying depths; this study isolates residual learning as depth increases (Zhang (2022); Lo and Singh (2023); Kelly et al. (2026)).Source: §1 · p. 4
• Residual blocks pair shortcut identity paths that retain earlier representations with residual branches that learn updates; successive layers refine cross-sectional stock rankings.Source: §1 · p. 4
• They build NN+ for low-signal-to-noise financial panel data and asset-pricing training protocols, alongside its residual-learning version.Source: §1 · pp. 4–5
• The authors compare NN with ResNet and tailored NN+ with ResNet+, holding data, preprocessing, training protocol, and portfolio evaluation methods fixed.Source: §1 · p. 5
• 2.07 is the out-of-sample Sharpe ratio for deep ResNet+ value-weighted long-short portfolios, versus 1.92 for shallow ResNet+ and 0.89 for deep NN+.Source: §1 · p. 5
• Deep residual models preserve shallow stock rankings and refine them with depth; their ResNet+ versus NN+ gap survives transaction costs and extreme-cap exclusions and persists across most NBER business cycles.Source: §1 · p. 5

2 Framework and Model Design

• The framework uses information at time t to forecast one-month-ahead excess stock returns, minimizes forecast MSE, and holds fitted parameters fixed for out-of-sample forecasts.The information set includes currently available firm characteristics, macroeconomic predictors, and other signals observed before or at time t.Source: §2 · p. 6

2.1 From Feedforward Networks to Residual Neural Networks

• A residual block carries the current representation forward through a shortcut while its branch learns an update, preserving the existing pricing signal in return forecasting.Unlike a feedforward block, whose learned transformation becomes the next representation, a residual block adds its learned update to the carried input.Source: §2.1 · p. 7, Figure 1 · p. 7
• In the idealized residual recursion, setting later updates to zero lets a deeper model reproduce any shallower specification, nesting shallow models within the deeper class.The recursion starts from a predictor-derived hidden representation and adds a residual update at each block.Source: §2.1 · p. 7

2.2 Asset Pricing Residual Neural Network Specification

• ResNet+ accommodates changing hidden widths with projection paths, retaining feedforward transitions when a width first changes and residual learning for added depth within that stage.Its implementation uses layer normalization, LeakyReLU, dropout, and pre-activation residual branches.Source: §2.2 · p. 8
• NN+ isolates residual learning by matching ResNet+’s overall structure while replacing residual blocks with tailored feedforward blocks.This comparison is designed to separate residual learning from the combined effect of residual learning and the tailored architecture.Source: §2.2 · pp. 8–9

3.1 Data and Preprocessing

• The stock panel covers January 1963 through December 2023, combining 153 firm characteristics and 8 macroeconomic predictors, all lagged one month to forecast next-month stock excess returns.Firm characteristics are rank-normalized; macroeconomic predictors are not. A characteristic missing for every stock in a month is filled with zero, the cross-sectional median after rank normalization.Source: §3.1 · pp. 9–10

3.2 Experimental Design

• The two comparison sets test residual versus feedforward architectures at matched widths and depths, with ResNet+ versus NN+ as the main comparison and specifications prespecified to avoid ex post selection.Models share implementation details other than architecture within each comparison; each specification’s forecasts are averaged across ten random-seed runs before portfolio formation.Source: §3.2 · pp. 11–12

3.3 Long-Short Portfolio Performance

• Value-weighted Sharpe ratios for ResNet+ rise from 1.92 to 1.94 to 2.07 across shallow, medium, and deep groups, while NN+ falls from 1.84 to 1.47 to 0.89.The mean monthly long-short return and volatility of ResNet+ change little across depths; the reported pattern indicates that residual learning improves performance as depth increases, while NN+ deteriorates.Source: Table 2 · p. 14, §3.3 · p. 14
• ResNet+ beats NN+ in 4 of 14 shallow, 17 of 20 medium, and all 40 deep specifications; ResNet also has higher Sharpe ratios in most reported specifications.NN collapses to a constant forecast in some specifications, while residual learning protects it from collapse in deeper specifications of width schedules 1, 2, and 3.Source: Figure 4 · p. 15, §3.3 · pp. 15–16

3.4 Forecast Ranking across Long-Only Decile Portfolios

• Across long-only forecast deciles, realized returns and Sharpe ratios rise, while volatility is U-shaped and higher in the tails; residual models generally have higher Sharpe ratios in high deciles.The right-tail volatility is lower than the left-tail volatility in both comparison sets, with a more apparent decline toward the right tail for ResNet versus NN than for ResNet+ versus NN+. Raw forecasts shrink with depth for NN and NN+, but remain stable for ResNet and ResNet+; ResNet+ long-short Sharpe ratios rise from 1.92 to 1.94 to 2.07, whereas ResNet ratios move from 0.95 to 0.94 to 0.93.Source: Figure 5 · p. 16, §3.4 · pp. 16–17

3.5 Preservation and Refinement

• The decomposition compares each deeper forecast with its family- and width-specific minimum-depth anchor, separating unchanged stock-pair rankings from revisions and measuring revision accuracy and economic value.Preservation is the share of unchanged pair orderings; revision accuracy scores changed pairs against subsequent realized returns, while revision economic value weights those scores by realized return differences.Source: §3.5 · pp. 17, 19
• ResNet preserves rankings close to 1 with positive revision accuracy, whereas NN’s lower preservation and negative revision accuracy accompany negative revision economic values for both models.ResNet’s negative revision economic value is smaller, while NN does not produce statistically significant refinement; NN preservation also decreases with depth.Source: §3.5 · pp. 19, 21, Table 4 · p. 20, Figure 6 · p. 20
• ResNet+ preserves shallow rankings across depths and shows refinement, with higher revision accuracy and economic value than NN+, whose preservation falls from 0.75 to 0.64 as revisions deteriorate.These patterns align with portfolio results: ResNet+ performance improves with depth, while NN+ performance deteriorates.Source: Table 4 · p. 20, Figure 6 · p. 20, §3.5 · p. 21
• For ResNet+, most themes support preservation, while removing Low Leverage or Size reveals greater reordering alongside significantly higher revision economic value.The analysis zeros one theme in both shallow and deep models while holding fitted models and other inputs fixed; positive performance losses indicate that a theme supports the corresponding measure.Source: §3.5 · p. 21, Table 5 · p. 22
• For NN+, heavier reordering accompanies deteriorating revision economic value, and preservation also deteriorates under Low Risk, Quality, Short-Term Reversal, and Value.All NN+ revision-economic-value theme effects are negative because the shallow NN+ model uses those themes better than its deeper models.Source: §3.5 · pp. 21–22
• Theme removals show better preservation in ResNet than NN, while Low Leverage and Profit Growth significantly affect preservation across all four families but not both revision measures.ResNet has three significantly positive preservation effects versus two significantly negative effects for NN; Low Leverage generally prompts reordering, while Profit Growth generally promotes preservation.Source: §3.5 · pp. 22–23

4.1 Market Capitalization

• Excluding the largest 20% of stocks makes residual-minus-feedforward Sharpe differences significant in every depth group, while excluding the smallest 20% lowers Sharpe ratios across models and depths.Removing the largest stocks raises Sharpe ratios, partly because the remaining smaller stocks receive greater weights; every difference already significant in the full universe remains significant under either exclusion.Source: §4.1 · pp. 23–24, Table 6 · p. 24

4.2 Trading Frictions

• Transaction costs lower Sharpe ratios across model families but widen residual-minus-feedforward Sharpe differences; the residual-learning advantage persists and widens as costs rise.Residual models have turnover similar to feedforward models within depth groups, yet their Sharpe ratios decline less as costs increase; the authors relate this pattern to portfolio volatility.Source: §4.2 · pp. 25–26, Table 7 · p. 25

4.3 Market States

• ResNet+ mean-return differences are positive in all five expansions, but recession differences are mixed or statistically indistinguishable; both models lose money in the two-month COVID recession.Across subperiods, residual models outperform feedforward counterparts in most cases, but strong full-sample performance can still coincide with losses during extreme market states.Source: §4.3 · pp. 26–27, Table 8 · p. 27

5 Conclusions

• 2.07 is the out-of-sample Sharpe ratio for deep residual value-weighted long-short portfolios, versus 1.92 for corresponding shallow models and 0.89 for deep feedforward models.The authors report that the residual models’ advantage sharpens with depth, while feedforward counterparts deteriorate as depth increases.Source: §5 · pp. 27–28
• The paper tailors feedforward architecture to financial panel data and its training protocol, then compares it with residual counterparts across depths and width schedules.The comparisons hold data, preprocessing, training protocol, and portfolio evaluation methods fixed within each set.Source: §5 · pp. 27–28
• Residual models preserve shallow forecasts and refine them as depth increases; the reported results remain robust to market-capitalization exclusions and transaction costs, and persist across most NBER business cycles.The authors describe residual learning as a way to train deeper networks with more parameters directly on structured panel data to forecast returns, forming native “large asset pricing models”.Source: §5 · p. 28

A.1 Normalization and Training Protocol

• Firm characteristics are median-imputed by month and rank-normalized; entirely missing monthly cross-sections receive zero, while macroeconomic predictors are not rank-normalized across stocks.The passage describes rank-normalizing firm characteristic x_i,t after filling missing values with monthly cross-sectional medians.Source: §A.1 · p. 29
• NN and ResNet use batch normalization, whereas NN+ and ResNet+ use layer normalization in corresponding blocks and before the final forecast map.Batch normalization uses observations in a mini-batch for each feature; layer normalization uses features within one hidden vector.Source: §A.1 · p. 29
• Annual January refits start from the preceding year’s fitted parameters, train with mean squared error for at most 100 epochs, and average forecasts across ten random seeds.Training stops after five consecutive epochs without improvement on the most recent five years of training data; NN and ResNet use Adam with ℓ1 regularization, while NN+ and ResNet+ use AdamW with ℓ2 regularization.Source: §A.1 · p. 29

A.2 ResNet+ Specification Details

• ResNet+ combines an identity shortcut with residual branches; at width changes, projection paths can add depth while preserving the residual structure.Figure 7 distinguishes unchanged-width Type I blocks from Type II width changes; the circled plus denotes elementwise addition.Source: §A.2 · p. 29, Figure 7 · p. 30

A.3 Predictor List

• Forecasts use 153 firm characteristics and eight macroeconomic predictors; firm characteristics are median-imputed monthly and rank-normalized.The firm characteristics are the JKP variables; the macroeconomic predictors are used by Gu et al. (2020) and drawn from Welch and Goyal (2008).Source: §A.3 · p. 36

B.1 Forecast and Portfolio Measures

• Stocks with forecasts outside the monthly 2.5th–97.5th percentile range are excluded before portfolio construction, which sorts retained stocks into ten forecast deciles.The appendix defines long-short portfolio returns and annualized Sharpe ratios, along with turnover and factor-alpha measures.Source: §B.1 · pp. 37–38

B.2 Comparison Sets, Depth Summaries, and Heatmaps

• Depth summaries compare residual and feedforward specifications over defined depth groups, exclude constant forecasts, and equally weight completed regular specifications within each model family.The shallow, medium, and deep groups cover hidden depths 1–5, 6–10, and 11–20; the appendix also defines Sharpe-ratio and monthly-return differences.Source: §B.2 · p. 38

B.3 Sharpe Ratio Difference Inference

• Sharpe-ratio contrasts use a studentized circular block bootstrap with 1,000 draws, 12-month blocks, and Newey-West(12) standard errors.Each bootstrap draw samples the same months for all return series.Source: §B.3 · p. 39

B.4 Robustness Measures

• Robustness tests use the full stock universe and universes excluding the smallest or largest 20% of stocks by formation-month market capitalization.The passage also introduces transaction-cost rate κ in basis points per dollar traded.Source: §B.4 · p. 39

C.1 Definitions and Relation to Revision Economic Value

• Shallow and deep ranking economic values measure forecast rankings, not portfolio returns.The section defines these values for a common monthly stock sample using forecast-ordering signs.Source: §C.1 · p. 40
• Revision economic value is the difference between deep and shallow ranking economic value, obtained by subtracting their separate changes.The derivation focuses on pairs whose shallow and deep ordering signs differ; unchanged pairs have equal signs.Source: §C.1 · p. 40

C.2 Ranking Economic Values and Theme Losses

• Table 10 reports shallow and deep ranking economic values using the same specification samples and aggregation weights as Table 4.The table includes forecast preservation and ranking economic values by depth group.Source: §C.2 · p. 40, Table 10 · p. 41
• Table 10’s ranking economic values are monthly basis points, and their difference is the revision economic value reported in Table 4.Target depths are 6–10 for Medium and 11–20 for Deep; shallow anchors use the minimum depth rather than an average across shallow depths.Source: §C.2 · p. 41
• Table 11 separates shallow and deep ranking economic value losses underlying the theme-loss measures reported in Table 5.The difference between the separate shallow and deep losses is the table’s ∆V.Source: §C.2 · pp. 41–42, Table 11 · p. 42
• Table 11 compares full-input and theme-zeroed measures with the same theme zeroed in both models while fitted models and other inputs remain fixed.Theme effects are not additive, and the ranking economic values are not portfolio returns.Source: §C.2 · p. 42

D Theoretical Performance Bounds

• Unlike nonresidual networks whose composed Jacobians can shrink exponentially with depth, residual updates can preserve gradient magnitude when the residual-branch derivative is small.The passage describes the backpropagated gradients as approximately equal across adjacent layers in that case.Source: D Theoretical Performance Bounds · p. 43
• Residual blocks add a residual update to the preceding hidden representation while retaining that representation through an identity path.The analysis studies this residual structure to establish stability of ResNet+.Source: D Theoretical Performance Bounds · p. 43
• LayerNorm’s Jacobian has a uniformly bounded operator norm, making LayerNorm Lipschitz with a depth-independent constant 2√ε∥γ∥∞.The stated bound assumes bounded gain and bias parameters.Source: D Theoretical Performance Bounds · p. 44
• The weight assumption bounds the second moment of each residual block’s affine weight matrix.The text describes this assumption as standard in random-matrix-based analyses and notes it holds automatically for certain initializations.Source: D Theoretical Performance Bounds · p. 45
• With positive probability, gradients through ResNet+ remain bounded above and below, with no exploding or vanishing updates under Assumption 1.The theorem gives 0 < r∥∇hᴰL∥ ≤ ∥∇h⁰L∥ < s∥∇hᴰL∥ < ∞ for positive constants r and s.Source: D Theoretical Performance Bounds · p. 46

E.1 Forecast Ranking Diagnostic

• Table 12 examines forecast collapse for width schedule 1 at depth 7; forecast dispersion indicates whether fitted models can distinguish stocks for portfolio sorting.The comparison holds width schedule and training protocol fixed while varying architecture and depth.Source: §E.1 · pp. 47–48, Table 12 · p. 48

E.2 Equal-Weighted Portfolios

• Table 13 repeats paired portfolio tests using equal-weighted decile returns and full-universe long-short portfolios.The table reports annualized Sharpe ratios for residual and feedforward models and their differences.Source: Table 13 · p. 48, §E.2 · p. 48
• The reported differences include Sharpe ratio, FF5 alpha, turnover, and out-of-sample R² differences between residual and feedforward models.FF5 alpha and turnover differences are monthly percentages; out-of-sample R² differences are in percentage points.Source: §E.2 · p. 48
• Depth groups are shallow at 1–5 hidden layers, medium at 6–10, and deep at 11–20; jointly completed width and depth specifications receive equal weight.Sharpe-ratio differences are tested with a two-sided Ledoit–Wolf studentized circular block bootstrap.Source: §E.2 · p. 48

E.3 Width and Depth Profiles

• Tables 14 and 15 report results by width and depth for completed specifications in grouped summaries and heatmaps.The tables include available full-universe specifications after constant forecasts are excluded.Source: §E.3 · pp. 49, 51, Table 14 · p. 49, Table 15 · p. 51
• Out-of-sample R2 is calibrated after monthly cross-sectional demeaning, 2.5% forecast trimming from each tail, and nonnegative scalar rescaling.The fitted nonnegative coefficient is reported as Scale.Source: §E.3 · pp. 51, 53
• VW SR denotes the annualized Sharpe ratio of the value-weighted long-short portfolio; Fama-French five-factor α, turnover, decile returns, and H−L are monthly percentages.D1 and D10 are the lowest and highest forecast deciles, and H−L is D10 minus D1; t(α) uses Newey-West standard errors with 12 lags.Source: §E.3 · p. 53

F Model Scale Reference

• Figure 8 compares fitted parameter ranges for the paper’s four models with familiar asset-pricing specifications and published neural networks.Its horizontal axis is logarithmic; rows use counts of regression parameters, factors, predictors, or neural-network parameters, depending on the model.Source: F Model Scale Reference · p. 54, Figure 8 · p. 54