Blog of Paper "Residual Learning in Empirical Asset Pricing"

See recent releases of OpenAI models: Luna, Terra, Sol, and Astra. The larger the model size, the better its performance. The "scaling law" applies.

Will return forecasts be more accurate with larger models? How can we enlarge empirical asset pricing models?

Year Empirical Supervised Machine Learning Model Parameters
  1. 1964CAPM2
  2. 1993Fama-French three factor model4
  3. 1997Carhart four factor model5
  4. 2013Barra USE472
  5. 2016"Factor zoo" in one model316
  6. 2020Feedforward neural network5k–6k
  7. OursNN/ResNet grid5k–25kResNet+/NN+ grid6k–66k

Let's begin with the classic CAPM.

Empirically, CAPM is a supervised machine learning model

Given in-sample explanatory variables \(X\) and the response variable \(y\), a supervised machine learning model learns a mapping from \(X\) to \(y\) by minimizing a loss function. The learned mapping can then be applied to out-of-sample \(X\) to obtain predictions \(\hat y\).

For CAPM, theoretically, we have:

E[R_i-R_f]=\beta_i E[R_m-R_f].

Let \(X_t=R_{m,t}-R_{f,t}\), and \(y_t=R_{i,t}-R_{f,t}\). Empirically, we have:

y_t=\alpha_i+\beta_i X_t+\epsilon_{i,t}.

The two trainable parameters can be estimated by minimizing MSE:

(\hat\alpha_i,\hat\beta_i)=\arg\min_{\alpha_i,\beta_i}\frac{1}{T}\sum_{t=1}^{T}(y_t-\alpha_i-\beta_i X_t)^2.

With out-of-sample data, we have:

\hat y_{T}=\hat\alpha_i+\hat\beta_i X_{T}.

So, empirically, CAPM is a supervised machine learning model. It's supervised because the model is trained with in-sample data; and the in-sample \(X\) is mapped to its corresponding in-sample \(y\). It's a machine learning model because the parameters are learned from data.

"Contemporaneous explanatory models" and "forecasting models" can both be supervised machine learning models.

Fama-French three factor model: \(y_t\) \(=\alpha_i+\beta_{i,1}X_{1,t}+\beta_{i,2}X_{2,t}+\beta_{i,3}X_{3,t}+\epsilon_{i,t}\)

Carhart four factor model: \(y_t\) \(=\alpha_i+\beta_{i,1}X_{1,t}+\beta_{i,2}X_{2,t}+\beta_{i,3}X_{3,t}+\beta_{i,4}X_{4,t}+\epsilon_{i,t}\)

Barra USE4: \(y_{i,t}\) \(=\sum_{k=1}^{73}X_{i,k,t}f_{k,t}+\epsilon_{i,t}\)

"Factor zoo" in one model: \(y_{i,t+1}\) \(=\alpha+\sum_{k=1}^{K}\beta_k X_{i,k,t}+\epsilon_{i,t+1}\)

The above machine learning models are linear models, and it's easy to see that the number of trainable parameters increases as more features are introduced. Gu et al. (2020) increase the number of trainable parameters by introducing more complex mapping functions.

Feedforward neural network: \(y_{i,t+1}\) \(=f_\theta(X_{i,t})+\epsilon_{i,t+1}\)

However, they find that the performance of neural network models peaks at relatively shallow depths, and simply adding more feedforward layers does not deliver correspondingly greater economic value. They conclude that shallow learning outperforms deeper learning.

Recent uses of large models in asset pricing typically involve LLMs. These approaches treat LLMs as given external tools that transform unstructured information into structured data. Another common approach is to fine-tune a pretrained LLM for financial (textual) question answering. The LLMs are indeed large, but they are not native asset pricing models.

(Typically) With Large Language Models

Unstructured Information↓LLMs↓Structured Data↓Asset Pricing Models↓Return Forecasts etc.

Native Large Asset Pricing Models

Structured Data↓LAPMs (Large Asset Pricing Models)↓Return Forecasts etc.

We describe native large asset pricing models (LAPMs) as models with a large number of trainable parameters that are trained directly for asset pricing tasks, such as forecasting future returns.

If we follow the second way, introducing more complex mapping functions can indeed introduce more trainable parameters exponentially. But according to Gu et al. (2020), shallow models outperform deep models. Interestingly, similar conclusions were found in many machine learning papers (Romero et al. (2015), He et al. (2015), Srivastava et al. (2015)) before He et al. (2016a).

Do shallow models really outperform deep models?

An extra layer that changes nothing

Theoretically, shallow models do not outperform deep models because of the identity transformation \(f(x)=x\).

Given a good shallow model \(g\), add one \(f(x)=x\) layer.

The new model is deeper but performs equally well.

So, "shallow is a special case of deep", or at least the deeper model has higher model capacity than that of the shallow one. The added layer could learn something other than identity mapping.

Now the question is: how can we activate the extra model capacity of the deeper model?

Residual Learning

He et al. (2016a) provide a residual block design in which shortcut paths carry the current hidden representation forward and residual branches learn residual updates to the hidden representation.

Following the residual learning convention in He et al. (2016a, 2016b), we add residual blocks with shortcut paths to feedforward asset pricing models. A residual block is a combination of a shortcut path and a residual branch, in which the shortcut path keeps an earlier representation intact through an identity mapping and the residual branch learns a residual update.

For return forecasting, the shortcut path carries the current pricing signal forward, while the residual branch searches for residual updates to that pricing signal.

h_0=\mathcal I(X),\qquad h_d=h_{d-1}+g_d(h_{d-1})

In a feedforward block, the learned transformation becomes the next representation. A residual block supplies a second route, in which the shortcut path carries the input and the residual branch contributes a residual update.

For any shallower specification in this idealized recursion with depth \(d_0<D\), the deeper function class can reproduce it by setting \(g_d(\cdot)\equiv 0\) for all \(d>d_0\), so the shallow specification is a subset of the deeper specifications, and is nested inside the idealized deeper function class with residual learning.

We build an empirical sequence that begins with a shallow model, then add layers under residual and feedforward rules, and examine whether these models retain, deteriorate, or improve upon the out-of-sample performance of the shallow model.

Experiments design

Two comparison sets are developed to evaluate the residual-learning framework across varying depths. The first is the raw benchmark comparison set, which compares a feedforward neural network model (NN) with the same model directly equipped with residual learning (ResNet). The second is the main comparison set, which compares the asset-pricing-tailored feedforward neural network model (NN+) with the same model directly equipped with residual learning (ResNet+). In each comparison set, we ensure that the models are evaluated under the same conditions such as data, preprocessing, training protocol, and portfolio evaluation methods to isolate the effect of residual learning.

However, our ResNet+ allows the hidden width to change while still being able to maintain the residual learning structure by using projection paths.

To isolate the sole effect of residual learning from the combined effect of the tailored architecture and residual learning, we remove the residual-learning components from ResNet+ and replace them with tailored feedforward blocks to form its counterpart NN+, which is identical in overall structure to ResNet+ except that it replaces residual blocks with feedforward blocks.

Empirical Design

With expanding training windows, the model parameters are optimized using an initial training sample containing predictor observations from January 1963 through December 1986 and their one-month-ahead returns. The last response observation in the initial training sample is the realized return in January 1987. After the January 1987 returns are observed, model training is completed, and the model uses forecast origin observations from January through December 1987 to forecast out-of-sample future returns from February 1987 through January 1988.

Subsequent annual refits follow this expanding approach, yielding 443 forecast origins from January 1987 through November 2023 and realized returns from February 1987 through December 2023. The expanding training window is designed to avoid forward-looking bias while preserving the mapping \(X_t \rightarrow R_{t+1}\) throughout model training and out-of-sample forecasting.

On the last day of each month, eligible stocks are sorted into deciles by their out-of-sample forecasts. The analysis mainly uses value-weighted decile portfolios, with stock weights proportional to market capitalization.

Does additional depth help?

We then calculate the Sharpe ratio for each width and depth specification and average these ratios within each model family and depth group to produce the reported Sharpe ratios.

Long-Short Portfolio Performance by Depth Group and Model Family

The out-of-sample performance of ResNet remains nearly the same across all three groups, while NN shows a monotonic decline in these performance metrics, indicating that directly adding the residual learning provides protection against deterioration with depth.

The mean monthly long-short return and volatility of ResNet+ also change little across depths, but as depth increases, the value-weighted Sharpe ratio increases monotonically and gradually from 1.92 in the shallow group to 1.94 in the medium group, reaching 2.07 in the deep group. NN+ begins with a value-weighted Sharpe ratio of 1.84 and then falls to 1.47 and 0.89 as the specifications move from shallow through medium to deep. With the tailored architecture, the residual learning provides an improvement in out-of-sample performance as network depth increases, while the performance of NN+ deteriorates as depth increases.

Long-only value-weighted portfolios across all ten forecast deciles show how realized performance varies over the full ranking.

The return profiles in each subfigure are close in the middle deciles and separate toward the tails, showing that the residual learning is more effective at distinguishing the most promising and the most undesirable stocks.

The Sharpe ratios for all models also show a common rising trend across deciles, indicating that every model is economically useful in ranking stocks. For stocks forecast to perform poorly in the low deciles, the Sharpe ratios of the residual models are generally lower than those of the feedforward models; for stocks forecast to perform promisingly in the high deciles, the Sharpe ratios of the residual models are generally higher than those of the feedforward models.

Preserve and refine

A deeper model may reproduce the ranking of the shallow model, discard it, or preserve most of the ranking and revise the rest.

Let \(n_t\) be the total number of pairs and \(m_t\) be the number whose ordering changes. For each changed pair \(k\), let \(c_{k,t+1}\) denote its pairwise revision score based on next month's realized returns.

\begin{aligned} P_t&=1-\frac{m_t}{n_t}, & C_{t+1}&=\frac{1}{m_t}\sum_{k=1}^{m_t}c_{k,t+1},\\ && V_{t+1}&=\frac{1}{n_t}\sum_{k=1}^{m_t}c_{k,t+1}\left|R_{i_k,t+1}-R_{j_k,t+1}\right|,\end{aligned}

where the preservation score \(P_t\) measures the fraction of stock pairs whose forecast ordering is unchanged from the corresponding minimum-depth shallow anchor forecast, ranging from \(0\) to \(1\), with \(0\) indicating no preservation and \(1\) indicating complete preservation.

\(c_{k,t+1}\) is assigned \(+1\) if the changed deeper model's ordering agrees with realized returns ordering, which means the deeper model corrects the wrong ordering of the shallow anchor, and \(c_{k,t+1}\) is assigned \(-1\) if the shallow anchor's ordering is correct but the deeper model's ordering is wrong.

Averaging \(c_{k,t+1}\) gives the revision accuracy \(C_{t+1}\), with \(100\%\) indicating every revision is correct and \(-100\%\) indicating every revision is wrong.

Because a correct revision may be more valuable when the realized returns of the two stocks are far apart, we weight each \(c_{k,t+1}\) by the realized return difference to get the revision economic value \(V_{t+1}\). Revision economic value \(V_{t+1}\) measures the economic value of these revisions over all \(n_t\) stock pairs, where \(i_k\) and \(j_k\) are the stocks in changed pair \(k\) and unchanged pairs contribute zero.

A positive revision economic value indicates refinement on average, and a negative value indicates deterioration. We report \(P_t\) on its original scale, \(C_{t+1}\) in percent, and \(V_{t+1}\) in basis points.

Revision economic value is the deep ranking economic value minus the shallow ranking economic value.

Forecast Preservation and Refinement by Depth Group

Preservation scores \(P_t\) for ResNet are close to 1 and nearly the same throughout the depth range, and its revision accuracy \(C_{t+1}\) is positive for both the medium and the deep group; for NN, the preservation scores are also close to 1 but lower than those for ResNet, while its revision accuracy is negative. However, the revision economic values \(V_{t+1}\) for ResNet and NN are both negative, with ResNet showing a smaller negative value.

For ResNet+, the preservation scores are nearly the same throughout the depth range, while its revision accuracy and revision economic values show refinement and are higher than those of NN+. For NN+, by contrast, the preservation scores fall from 0.75 to 0.64, and its revision accuracy shows deterioration as depth increases.

To further explore the economic intuition underlying the preservation and refinement, we examine which firm characteristics the deeper models rely on to retain the shallow forecast ranking and make accurate and economically valuable revisions. We group the firm characteristics into 13 economic themes according to Jensen et al. (2023) and set the characteristics in each theme to zero, one theme at a time. Holding the fitted models and other inputs fixed, we set the same theme to zero in both shallow and deep models, recompute their forecasts, and measure the resulting changes.

Forecast Preservation and Revision Losses by Theme

For the “Low Leverage” and “Size” themes, the residual-learning framework in deep ResNet+ reorders the ranking made by the shallow anchor more extensively and achieves a significantly higher revision economic value, showing economic refinement. By contrast, in the NN+ model, the significantly heavier reordering is accompanied by deterioration in revision economic value, and preservation is accompanied by deterioration under the “Low Risk”, “Quality”, “Short-Term Reversal” and “Value” themes.

Do the results hold up?

Robustness tests change the stock universe, trading cost, or market state one at a time, with other settings kept the same throughout each exercise.

The ResNet+ versus NN+ gap stays large and distinct after excluding extremely large- or small-cap stocks, survives transaction costs, and persists across most of the NBER business cycles.

Market Capitalization

We remove either the smallest or the largest 20 percent of stocks from the stock universe each month based on current market capitalization, then give the models the trimmed universe to generate raw forecasts, and finally form value-weighted long-short portfolios based on the forecast deciles.

These results show that residual models make the result more robust when a particular market capitalization segment is excluded.

Trading Frictions

To examine whether the advantage of residual learning persists after accounting for trading frictions, we consider two linear transaction cost rates as trading frictions: 10 and 50 basis points (bps) per dollar traded.

The difference between ResNet+ and NN+ in the shallow group is not statistically significant without transaction costs but becomes significant at 50 bps. Every significant \(\Delta\) SR without transaction costs, remains significant after costs are introduced. These results show that the advantage of residual learning persists under transaction costs, and the advantage widens as the cost rate rises.

Market States

The market states analysis examines how the model mean return varies across expansions and recessions.

Four of the five differences are significant at the 5 percent level. The two longest expansions account for more than half of all months, and both have significantly positive differences. During recessions, however, only three of the four mean return differences are positive, and these differences are either statistically insignificant or based on too few months for inference. The negative difference occurs in March and April 2020, the two months of the COVID recession; both ResNet+ and NN+ have negative mean returns in this subperiod, showing that the return patterns learned from earlier data do not translate into profitable portfolios during this short period.

Takeaway

"Shallow is a special case of deep".

Residual learning enables asset pricing models to benefit from greater depth.

Greater depths preserve and refine stock rankings.

Our code is open source, models are modular, and the components can be reused to improve other neural-network-based asset pricing models.

Native large asset pricing models are feasible.

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