Machine Learning in Empirical Asset Pricing

Introduction

Empirical asset pricing is undergoing a transformation with the advent of big data and machine learning (ML). Traditional linear factor models like the Capital Asset Pricing Model (CAPM) and Fama–French frameworks offer economic intuition but struggle to accommodate the “factor zoo” of hundreds of discovered return predictors and complex nonlinear relationships[1][2]. ML methods, by contrast, offer greater flexibility and predictive power in high-dimensional settings[1]. Recent research leverages ML to address central asset pricing tasks – from predicting stock returns to constructing better factor models – often yielding improved out-of-sample performance but raising new challenges regarding interpretability and economic insight[2]. This review synthesizes the academic literature, both foundational and cutting-edge, on applying ML in empirical asset pricing. We organize the discussion around key application areas, methodological innovations, comparisons with traditional approaches, emerging trends, data and evaluation practices, and known limitations. Throughout, we cite representative studies to illustrate how machine learning is reshaping asset pricing research.

Applications of Machine Learning in Asset Pricing

Return Prediction in the Cross-Section and Time Series

A primary focus of ML in finance has been predicting asset returns – both the cross-section of individual stock returns and the time-series of aggregate market returns. In the cross-section, researchers use ML to forecast which stocks will earn higher future returns (i.e. to measure firms’ risk premia). Pioneering work by Gu, Kelly, and Xiu (2020) compares a variety of ML algorithms on U.S. equities and finds they substantially outperform traditional regressions in predicting returns[3]. In economic terms, portfolios formed using ML forecasts can double the Sharpe ratio of strategies based on classic linear models[3][4]. For example, a long–short portfolio trading on stock-level return predictions from a neural network achieved an out-of-sample Sharpe ratio of 1.35, more than double that of a leading linear factor-based strategy (0.67) in the same sample[5]. The best-performing algorithms in this study were nonlinear models – tree ensembles and deep neural networks – which can capture predictor interactions that linear methods miss[6]. Notably, despite the flexibility of ML, there is broad agreement on which variables matter most for prediction: both linear and nonlinear methods consistently emphasize a handful of firm characteristics (e.g. past return momentum, liquidity measures, and volatility-related metrics) as the dominant return predictors[7]. This not only validates decades of anomaly research identifying these effects, but also suggests ML models are detecting many of the same risk factors or mispricing indicators identified by traditional approaches.

Beyond individual stocks, ML has also been applied to forecast aggregate market returns (the equity risk premium over time). Classic studies like Welch and Goyal (2008) showed the difficulty of predicting the market using a few economic indicators. ML offers a way to exploit many predictors and potential nonlinear dynamics. For instance, Feng, He, and Polson (2018) use deep learning on the historical U.S. macroeconomic and market data, showing that nonlinear combinations of predictors can improve out-of-sample forecasts of market returns[8]. Their deep neural network uncovers “nonlinear factors” that enhance predictability, especially at the extremes of the predictor space (e.g. during extreme market conditions), which a simple linear model would miss[8]. Gu, Kelly, and Xiu (2020) likewise apply ML to time-series market return prediction and find meaningful economic gains: timing the market with an ML-forecasted signal (e.g. a neural network using dozens of macro predictors) yields a substantially higher Sharpe ratio than a static buy-and-hold strategy[4]. These results illustrate that ML can extract signal from the noisy financial data that confound traditional time-series regressions, although the improvements are modest in absolute terms and often sensitive to how well models guard against overfitting.

Why ML helps in return prediction. Two characteristics of return prediction problems make them well-suited to ML techniques. First, the set of candidate predictor variables is enormous. Decades of research have produced hundreds of potential stock-return predictors (firm characteristics or “anomalies”) and dozens of macroeconomic variables for market timing[9]. Many of these predictors are correlated and relatively weak, making it challenging for traditional OLS or small-scale models to select the true signals. ML methods excel at high-dimensional variable selection and can handle cases where the number of predictors $P$ approaches or exceeds the number of observations $N$ by using regularization and dimension reduction[9]. Second, the relationship between predictors and returns may be complex. There is no guarantee that effects are strictly linear or additive; interactions and nonlinear functional forms can arise (e.g. an earnings yield may matter only for small firms, an interaction a linear model would miss). Machine learning algorithms (like trees and neural nets) can flexibly model such nonlinearities and interactions, searching a vast space of functional forms for patterns that improve predictions[10][11]. By addressing both the “wide data” problem (too many predictors) and the “unknown functional form” problem, ML has pushed the frontier of risk premium measurement in the cross-section and time series[12][10].

Empirically, the application of ML to return forecasting has revealed that much of the incremental predictive power comes from better handling of well-known effects and combinations thereof, rather than discovering entirely new anomalies. For example, both Gu et al. (2020) and Freyberger et al. (2020) find that past return measures (momentum or short-term reversal signals) are among the most powerful features in explaining the cross-section of stock returns[13][14]. Freyberger, Neuhierl, and Weber (2020) applied an adaptive group LASSO (with nonparametric spline terms) to 62 firm characteristics and confirmed that “past-return-based predictors stand out” as providing independent information, whereas many other touted anomalies added little incremental power[15]. In their nonparametric framework, only a small subset of characteristics – primarily those related to momentum and reversals – had robust predictive content once redundant or spurious signals were penalized[14][15]. These results reinforce the idea that markets contain a few strong cross-sectional return drivers and a long tail of weaker effects, and ML methods are useful for distinguishing the former from the latter. At the same time, studies note that the gains from complex ML models are not uniform across all stocks: the most substantial predictive improvements tend to concentrate in small and illiquid stocks[16]. For large blue-chip firms, which are closely followed by investors, even powerful ML algorithms often find little predictability beyond what simple models capture[16]. This suggests that ML is largely picking up mispricing or risk effects that are strongest where arbitrage is limited (small stocks), consistent with market efficiency imposing a lower bound on predictability for heavily traded assets.

Factor Models, the Stochastic Discount Factor, and the “Factor Zoo”

Another major application of ML in asset pricing is in factor model construction and anomaly selection – essentially, making sense of the factor zoo. Traditional empirical asset pricing revolves around explaining returns through a linear combination of factors (portfolios or characteristics) that carry risk premia. However, with hundreds of published anomalies, a key challenge is to identify which factors truly matter and to parsimoniously model the stochastic discount factor (SDF) or pricing kernel. Machine learning is being used to (a) select or reduce large sets of candidate factors, and (b) discover latent factors or functional forms that better explain asset returns.

One thread of research uses regularization and model selection techniques to tame the factor zoo. For example, Feng, Giglio, and Xiu (2020) propose a LASSO-based procedure to systematically test the marginal contribution of new factors in the presence of many existing ones[17]. Applying this to a universe of 99 reported anomalies, they find that only on the order of 14 factors are truly important – the rest provide redundant or negligible explanatory power once the important ones are in the model[17]. This result dramatically highlights how ML can cut through the noise of numerous candidate factors and isolate a sparse set of dominant returns drivers. In a related vein, shrinking or penalizing factor models has proven effective. Kozak, Nagel, and Santosh (2020) show that imposing heavy shrinkage toward a zero risk premium on a large set of factors results in a model that effectively uses just a few composite factors, yet prices the cross-section almost as well as using the full set of characteristics[18]. Such approaches borrow the bias–variance tradeoff perspective from ML to improve out-of-sample performance of factor models: by allowing a small bias (excluding many weak factors) they achieve lower variance and better predictive stability.

A complementary line of work uses dimension reduction and latent factor discovery via ML. These methods recognize that the plethora of firm characteristics may reflect a smaller number of underlying risk factors. Traditional tools like principal components analysis (PCA) have been used to extract latent factors, but ML offers more sophisticated variants. Kelly, Pruitt, and Su (2019) introduce Instrumented Principal Component Analysis (IPCA), an algorithm that treats firm characteristics as instruments to identify latent factors and their time-varying loadings. This technique effectively bridges characteristics and factor modeling, and was shown to outperform conventional factor models in explaining asset returns[19]. Another example is Lettau and Pelger’s (2020) “max Sharpe ratio” PCA, which is a modified PCA that weights covariance information by assets’ mean returns, thereby extracting factors that are not just statistically important but also carry high risk premia. This method, which can be seen as an ML-enhanced unsupervised learning of factors, yields improved pricing of cross-sectional returns compared to unweighted PCA factors[20]. There have also been experimental uses of autoencoders and matrix completion techniques (common in ML for uncovering low-dimensional structure) to identify factors from large panels of asset returns[21]. These approaches often find that a few nonlinear combinations of characteristics can reproduce the cross-section about as well as dozens of raw anomalies, reinforcing the idea of a low-dimensional factor structure underlying the apparent complexity[20].

Rather than building factor models in a two-step fashion (first identifying factors, then running cross-sectional regressions), some researchers directly target the stochastic discount factor using ML. The SDF $m_{t+1}$ prices assets by $E[m_{t+1}R_{i,t+1}]=1$ for all assets $i$, and it can in principle be expressed as a function of many state variables or characteristics. Machine learning can be used to estimate this function flexibly. Deep neural networks have been deployed to approximate the SDF, treating asset returns and characteristics as training data for a complex function $m_{\theta}(Z_i)$ that minimizes pricing errors. Chen, Pelger, and Zhu (2023) provide a notable example: they use a deep learning model to estimate an SDF for individual stocks, incorporating a “no-arbitrage” loss function to ensure the network’s predictions satisfy the absence of arbitrage opportunities[22]. Their innovations include using an adversarial approach to generate test assets that stress-test the model (improving robustness) and incorporating a large number of macroeconomic time series to capture the state of the economy[22]. The resulting deep SDF model significantly outperforms traditional factor models out-of-sample, achieving higher Sharpe ratios and lower pricing errors (mean absolute unexplained returns) than benchmarks like Fama–French, and it “identifies the key factors that drive asset prices.”[22]. In essence, the network discovers a few composite factors (combinations of characteristics and macro variables) that mimic the complex nonlinear SDF. This and similar efforts (e.g. autoencoder-based SDFs[21]) highlight ML’s ability to integrate information from myriad sources into a coherent pricing model. Importantly, researchers often impose economic structure in these models – for example, Chen et al. enforce the fundamental no-arbitrage condition in the training objective – to ensure the ML-driven SDF remains grounded in financial theory[22].

Handling the factor zoo. A central theme across these studies is addressing the proliferating number of factors or anomalies in a statistically rigorous way. Whereas traditional approaches might add new factors one-by-one (running the risk of p-hacking and overfitting), machine learning enables a more systematic evaluation of predictors. By penalizing complexity (LASSO, ridge, etc.), ML methods control overfit and avoid selecting spurious factors that don’t truly improve pricing. For instance, Freyberger et al. (2020) not only selected which characteristics matter, but by allowing flexible functional forms for each, they could detect non-monotonic or conditional effects (like nonlinearity in size or value effects) that fixed linear factors would miss[15]. They found that most firm characteristics contributed little once a core subset (including momentum) was in the model, and that allowing nonlinear transforms did not radically change which characteristics were chosen – it mainly refined how they were modeled[15]. Another insight from this literature is that many anomalies might be manifestations of a few broad themes (e.g. momentum, liquidity, investment, intangibles, etc.), and ML can help cluster or reduce the factors accordingly. For example, when Feng, Giglio, and Xiu (2020) winnowed 99 factors down to 14, those that survived tended to correspond to well-known categories like momentum, volatility, quality/profitability, and investment[17]. In practical terms, such results are encouraging: they suggest that despite the deluge of proposed factors, asset returns can be parsimoniously explained by a limited set of robust factors – which ML techniques can help identify. This dovetails with theoretical work arguing for a low-dimensional risk structure and provides a degree of comfort that ML isn’t just fitting noise but is often rediscovering sensible economic factors.

Theoretical and Methodological Advances: Key ML Models Employed

Methodologically, the incursion of machine learning into asset pricing has introduced a toolkit of models and algorithms beyond the traditional econometric repertoire. Here we outline the main classes of ML models used in empirical asset pricing, highlighting what they contribute:

• Penalized Regression (LASSO, Ridge, Elastic Net): Regularization methods constrain or shrink regression coefficients to prevent overfitting and perform variable selection. In asset pricing, LASSO (Least Absolute Shrinkage and Selection Operator) has been a workhorse for selecting the most important factors or characteristics from a large pool[15]. For example, studies have applied LASSO in cross-sectional return prediction to decide which firm characteristics to include, finding it improves prediction accuracy over OLS by reducing noise[23]. LASSO tends to outperform OLS especially when avoiding Type II errors (missing true predictors) is important[23]. Variants like adaptive group LASSO (used by Freyberger et al. 2020) go further by selecting whole groups of feature expansions, enabling discovery of nonlinear effects while keeping the model sparse[15]. Ridge regression (which shrinks coefficients continuously rather than zeroing them out) has been used in factor shrinkage contexts (e.g. Kozak et al. 2020) to infer a low-dimensional factor structure. Elastic net combines LASSO and ridge penalties, and is useful when there are many correlated predictors – a frequent situation with stock characteristics. Overall, penalized regressions have become a fundamental tool to address the high-dimensionality in asset pricing problems[9], effectively guarding against overfitting while selecting relevant predictors.

• Tree-Based Methods (Regression Trees, Random Forests, Gradient Boosting): Tree models recursively partition the predictor space to fit piecewise constant predictions, which can capture complex interaction effects. Regression trees by themselves are simple and interpretable (much like decision rules), though often not very predictive. However, ensemble tree methods such as Random Forests (which average many randomized trees) and Boosted Trees (which iteratively grow trees to correct errors) have shown strong performance in asset pricing tasks[6]. These methods can model nonlinear interactions between characteristics – for example, a random forest can implicitly learn that a value factor matters only among small-cap stocks, by splitting first on size then on valuation. Gu et al. (2020) report that tree-based models were among the best predictors for stock returns, rivaled only by neural networks[6]. An advantage of tree ensembles is that they provide measures of variable importance (how much each predictor improves the splits), which can be used to rank which features are most influential. This often aids interpretability: even if the model is complex, one can see that, say, momentum, size, and volatility were the top drivers – consistent with domain knowledge. Tree methods tend to be robust against outliers and can naturally accommodate different predictor types. In finance applications, gradient boosting (including XGBoost) and random forests have been used not only for return prediction but also for classification tasks like default prediction and for constructing characteristic-sorted portfolios in novel ways[14]. A limitation, however, is that ensemble trees are less interpretable than a simple linear model or a single decision tree, so their economic meaning needs careful extraction.

• Neural Networks (Deep Learning): Neural networks are flexible function approximators composed of layers of interconnected “neurons” (nonlinear transformations). They can capture extremely rich patterns, including highly nonlinear interactions, at the cost of being the least transparent model and requiring large data and tuning. In asset pricing, deep neural networks have been applied to both cross-sectional and time-series problems. For predicting stock returns, multilayer feed-forward networks (often with 2–3 hidden layers) have been used to map dozens of firm characteristics to future returns[6]. These nets often outperform other methods by fitting complex relations, but they require regularization techniques like dropout or weight decay to avoid overfitting the noise in financial data[24]. As noted, Gu et al. found dense neural networks performed on par with tree ensembles as the best return predictors[6]. For conditional factor models and SDF estimation, more specialized architectures have been explored. Chen et al. (2023) use a deep network with an input layer for stock characteristics and macro variables, multiple hidden layers, and an output that represents the SDF, trained under a no-arbitrage condition[22]. Others have tried recurrent neural networks (e.g. LSTM models) to capture time-series dynamics in return predictors[25], though this is still nascent. The great flexibility of neural networks means they can in principle approximate the true but unknown asset pricing function very well; however, their “black box” nature makes it hard to interpret what they have learned, and they are “coupled with high risks of overfitting” if not carefully constrained[24]. Ensemble approaches (e.g. averaging multiple network initializations) can improve stability but at the cost of even more complexity[24]. As a result, while deep learning has achieved top predictive performance in several studies[3][26], researchers are actively working on ways to make neural network models more transparent and aligned with financial theory (discussed more under Emerging Trends).

• Hybrid and Other Models: Some works combine elements of the above methods or embed ML into traditional frameworks. For example, Partial Least Squares (PLS) and Principal Components Regression are classical dimension reduction tools that have seen renewed use as benchmarks in the age of ML[27]. Bayesian machine learning approaches (e.g. Bayesian additive regression trees, variational autoencoders for factors) provide probabilistic predictions and have been proposed to quantify the uncertainty in return forecasts[28]. There are also two-stage learners (sometimes called “double ML”) where one stage might predict an intermediate parameter (like factor loadings or state variables) and another stage uses those for returns – these allow integrating economic structure into ML predictions[21]. Ensemble methods in general (bagging, boosting, model stacking) are popular to improve predictive robustness: for instance, averaging across many different ML models can reduce variance and deliver more stable out-of-sample performance, albeit at a loss of interpretability. In sum, the methodological advances in this literature are not about inventing brand-new algorithms, but rather adapting and selecting the right tools from machine learning to solve asset pricing problems. The choice of method often depends on the task: if the goal is interpretation and insight, a simpler model with regularization (like LASSO or a shallow tree) might be preferred; if pure prediction accuracy is the aim, ensemble trees or deep networks might be deployed and later analyzed for economic meaning.

Comparing ML Models to Traditional Approaches

A recurring question is how these ML-based models compare to the classical econometric approaches long used in asset pricing – both in terms of predictive performance and in terms of economic interpretability.

Predictive performance. Across numerous studies, ML models have demonstrated superior predictive accuracy relative to traditional linear models when evaluated out-of-sample. This is evident in metrics like out-of-sample $R^2$ (the fraction of variance of returns explained in fresh data) and investor utility gains. Gu et al. report that their machine learning methods achieved higher out-of-sample $R^2$ than any prior literature on U.S. stock returns, even after accounting for the inherent difficulty of prediction[29]. The improvements, while statistically significant, are on the order of a few percentage points in $R^2$ – reflecting the noisy nature of returns – but in economic terms these translate into substantial gains for a mean-variance investor[4]. As noted earlier, timing the market with ML forecasts or forming long–short stock portfolios based on ML yields Sharpe ratios far above those obtained by models like CAPM or Fama–French[5]. Moreover, ML can adapt to structural changes or detect regime-specific predictors more quickly than static models, potentially maintaining performance in evolving markets (though this claim is still being tested with longer samples). It’s important to emphasize that traditional factor models are typically designed for explanatory power and economic interpretation rather than pure prediction, so a degree of performance gap is expected when forecasting criteria are optimized. Even so, the magnitude of ML’s predictive edge in many studies underscores that classical models left valuable predictive information on the table – information that ML algorithms have been able to exploit by considering more variables and complex relationships.

Economic interpretability. On the flip side of the performance coin is the question of interpretability. Traditional asset pricing models have a clear economic narrative: each factor has a story (market risk, size, value, etc.), and linear exposures (betas) measure an asset’s sensitivity to those factors. This transparency allows researchers and practitioners to debate the economic mechanisms (risk-based or behavioral) behind asset returns. Machine learning models, especially high-dimensional or nonlinear ones, often sacrifice this clarity. A random forest or a neural network with dozens of inputs is essentially a black box – it may yield accurate predictions, but understanding why a particular stock is predicted to have high returns can be difficult. As Bagnara (2024) notes in a critical review, “probably the biggest downside of ML methods is their lack of interpretability.”[30] The complexity that gives ML its edge also means the models do not readily produce human-interpretable parameters analogous to factor premia or betas. This lack of interpretability is not just an academic concern; it matters for practical implementation (e.g. portfolio managers need to trust and explain their models) and for connecting empirical results back to theory.

That said, researchers are finding ways to extract economic insight from ML models. One straightforward approach is to examine which predictors the model relies on most. As discussed, even complex ML models tend to highlight familiar predictors like momentum, liquidity, and volatility as important[13], providing reassurance that ML isn’t finding arbitrary patterns that contradict known finance principles. Some studies report that ML-predicted returns still conform to known risk factor structures. For example, portfolios sorted on ML forecasts have been shown to have large exposures to factors like momentum and quality, implying the ML is in part repackaging these known strategies[13][14]. Another strategy to improve interpretability is to impose economic structure on the model ex ante. Chuan Shi (2025) advocates a unified framework based on the stochastic discount factor that integrates machine learning while preserving economic interpretability[31]. By forcing the ML model to take the form of an SDF (essentially a weighted combination of asset payoffs or characteristics that prices all assets), one can ensure the outputs have the interpretation of state prices or factor risk prices. This approach can bridge the gap between traditional theory and ML: the ML model can be very flexible in how it forms the SDF from inputs, but the end object is still an SDF that one can analyze in terms of priced risk dimensions.

There is also a growing use of explainable AI (XAI) techniques to interpret ML models in finance. Methods like partial dependence plots, Shapley value decomposition, and LIME (Local Interpretable Model-agnostic Explanations) can be applied to complex models to gauge the effect of each input on the prediction. For instance, one can compute the Shapley values for each characteristic on a particular stock’s predicted return to see which traits contributed most to a high or low forecast. These tools can sometimes reveal economically sensible patterns (e.g. the model gives a boost to stocks with high momentum or low valuations, consistent with anomaly literature) or detect nonlinear thresholds (e.g. maybe the model only values an earnings yield once it’s above a certain level, indicating a nonlinear value effect). Although these techniques provide insight, they are essentially post hoc and do not replace having a simple model. Thus, a tension remains: ML models excel in predictive power, but traditional models excel in transparency. The consensus in recent literature is that ML and classical approaches should be seen as complements: ML can uncover patterns and push the frontier of predictability, while economic theory and simple models are vital for interpreting those patterns and ensuring they make sense. As one survey put it, the goal is to harness ML’s power “with rigor, robustness, and power to make new asset pricing discoveries” in a way that advances our understanding of markets[32].

Emerging Trends and Future Directions

The intersection of machine learning and asset pricing is evolving rapidly. We highlight several emerging trends and research frontiers that are shaping this field:

• Interpretability and Explainability: Addressing the black-box nature of ML models has become a priority. Recent studies explicitly call for enhancing the interpretability of deep learning models in finance to increase their utility for both practitioners and academics[33]. This has led to a surge in work on explainable AI tailored to asset pricing. Researchers are exploring restriction of model complexity (e.g. using monotonic neural networks that enforce economically plausible monotonic relations between characteristics and returns) and developing visualization tools to interpret ML predictions. For example, partial dependence analysis can illustrate how predicted returns change as one characteristic varies (holding others fixed), shedding light on whether the ML model’s inferred relationship is linear, threshold-based, or something more complex. Another trend is proximate modeling, where a simpler model is trained to approximate the predictions of a complex model – essentially distilling the ML model’s knowledge into an interpretable form. The literature is also embracing the idea of “opening the black box” by examining learned representations: for instance, in a deep network that produces an SDF, one might try to interpret the intermediate layers as latent factors or economic signals. While full interpretability remains challenging, the intent is clear: ML models in asset pricing will not gain full acceptance until we can understand and trust the basis of their forecasts. We expect future work to incorporate more economic constraints or priors (e.g. no-arbitrage, positivity of certain effects, diminishing returns, etc.) directly into model architectures to ensure outputs are not only accurate but also make economic sense[33].

• Integration with Economic Theory: A related trend is combining machine learning models with traditional economic theory to create hybrid approaches. Rather than viewing ML and theory-driven models as separate, researchers are finding ways to enforce theory within ML or use ML to test theory. One example is the no-arbitrage regularization used by Chen et al. (2023), where the network is penalized based on pricing errors to adhere to arbitrage-free pricing[22]. Another example is constraining ML models to respect known equilibrium relations – for instance, ensuring that if a characteristic is theoretically linked to higher expected returns (say via a risk story), the ML model is structured to capture a monotonically increasing relationship. There is growing interest in “structural ML” where machine learning methods are used to estimate models that come from solving economic agents’ problems (such as consumption-based asset pricing models with investor heterogeneity), blending deep learning with deep theory. While still in early stages, such approaches could allow estimation of complex general equilibrium models that were previously intractable, using techniques like reinforcement learning to solve for equilibrium policies or pricing kernels. More immediately, researchers like Chuan Shi (2025) propose integrating ML into the stochastic discount factor framework, which is fundamentally grounded in asset pricing theory, thereby “preserving economic interpretability” and rigor even as we unleash flexible algorithms[31]. This kind of integration will likely intensify, as it offers a path to ensure ML advances are not just statistical tricks but contribute to the core understanding of what drives asset prices.

• Combining ML with Traditional Models (Ensemble or Two-Stage Approaches): A practical emerging trend is to use ML in conjunction with simpler models rather than outright replacing them. For example, one might use machine learning to screen or transform inputs to a traditional regression – essentially using ML for feature engineering – and then use a linear model for the final pricing equation. Alternatively, ML could generate candidate factors which are then tested in a classic Fama–MacBeth two-pass setting for statistical significance. This hybrid approach is appealing because it leverages ML’s ability to sift through data while retaining the familiarity of linear factors for the end result. An illustration is the work of Bryzgalova, Pelger, and Zhu (2021), who combine ML-based portfolio optimization with the Fama–MacBeth procedure to construct factors that maximize Sharpe ratio under certain constraints[19][20]. By design, those factors are interpretable (they are portfolios of stocks) even if the way they were found involved an ML optimization. Another area of interest is Bayesian machine learning in asset pricing, which offers a probabilistic ensemble of models. Bayesian approaches can incorporate prior beliefs about, say, the sign of coefficients or the number of significant factors, blending expert knowledge with data-driven learning. They also provide credible intervals for predictions and estimated risk premia, addressing the need to quantify uncertainty in ML forecasts (a topic of recent research)[28]. The ability to produce not just point predictions but also confidence intervals or distributions (e.g. via Bayesian neural nets or quantile regression forests) is likely to become more important, since understanding the uncertainty is crucial for risk management applications of ML forecasts.

• New Data Sources and Alternative Data: While most academic studies to date focus on structured data (prices, accounting ratios, etc.), a burgeoning trend is to incorporate alternative data into asset pricing models using ML. Textual data from news, earnings calls, or social media, for instance, can be converted into sentiment or topic scores via natural language processing (NLP) models and used as predictors for returns or risk factors. Machine learning is particularly well-suited to handling unstructured data like text and images, and we are seeing early research that brings such data into pricing models (e.g. using news sentiment to predict factor returns, or satellite images to gauge economic activity impacting asset prices). While this area is more prevalent in industry, academic work is catching up, often using ML to bridge the gap between large alternative datasets and asset pricing theory. For example, researchers might use NLP-derived signals as additional characteristics in a return prediction ML model to see if they capture information beyond traditional fundamentals. The keyword “alternative data” has started to appear in asset pricing and ML contexts[34], indicating that the scope of empirical asset pricing is expanding beyond standard databases. One must be cautious, however, as alternative data can introduce new pitfalls (survivorship bias, lookahead bias, etc.), and its economic interpretation is not always clear. But as these data become more available, ML will be the primary tool to extract their value for asset pricing insights.

• Open Platforms and Reproducibility: Given the complexity of ML models, ensuring results are robust and reproducible is an emerging concern. There is a movement toward open-source asset pricing research and public testing platforms. For instance, Chen et al. (2025) introduce the FactorWiki project as a public platform providing an expansive dataset of asset returns and anomalies for researchers to test new models[35]. The goal is to enable apples-to-apples comparisons of different ML models on a common dataset and to reduce the “secret sauce” aspect of data processing that might give false impressions of one method’s superiority. By analyzing 782 articles, Chen et al. also emphasize the need for standardized evaluation practices and data engineering, noting that time-series factors are highly noisy and careful feature engineering (such as standardizing and de-noising inputs) is crucial[36]. The FactorWiki and similar efforts reflect a trend towards collaborative validation of ML asset pricing models. Researchers are increasingly sharing code and data, which helps in identifying what truly works versus what might have been an artefact of a particular sample or set of choices. As this culture grows, we can expect a tighter feedback loop where promising ML methods are verified by multiple teams, solidifying the findings.

In summary, the future of ML in empirical asset pricing is likely to be characterized by more interpretability, more theory-driven innovations, the inclusion of novel data, and greater collaboration. The initial excitement of “let’s throw a complex model at financial data and see if R\^2 improves” is giving way to a more mature approach: integrating domain knowledge, assessing economic significance, and ensuring that the ML models not only predict well but also enhance our understanding of asset pricing mechanisms.

Data Sets and Evaluation Metrics in ML-Based Asset Pricing

Empirical studies in this domain draw on a range of data sets and have developed specific metrics to evaluate model performance. Here we outline the most notable data sources and the common evaluation criteria:

Data Sets and Universes:

• Equity Cross-Section Data: The prototypical data for many studies is the CRSP–Compustat merged database of U.S. stocks, often supplemented by additional data on firm characteristics (accounting ratios, price-based signals, etc.). Many papers examine decades of monthly U.S. stock returns (e.g. 1963–present) along with a large set of firm characteristics compiled from prior anomaly studies. For example, Gu et al. (2020) use an extensive list of 94 characteristics spanning fundamental, technical, and market variables[9]. Similarly, Freyberger et al. (2020) focus on 62 characteristics drawn from prominent anomaly papers[15]. Some researchers rely on already synthesized anomaly portfolios – e.g. the 100+ “characteristic-sorted” test portfolios often used in factor model tests (such as size-value sorted portfolios, momentum-sorted deciles, etc.). These provide a lower-dimensional target (portfolio returns instead of individual stocks) which can reduce estimation noise. However, with ML’s ability to handle high dimensions, many studies work directly at the stock level despite the noise, aiming to fully exploit the cross-sectional richness. In recent years, international data have also been used to assess generalizability – for instance, there are “global” versions of the empirical asset pricing via ML exercise that include stocks from many countries, as well as studies on specific markets like China to verify that patterns are not U.S.-centric. By and large, the data requirement for ML models is large: one needs a long history or a broad cross-section (ideally both) to train complex models without overfitting. This is a challenge in asset pricing, where time-series are limited (we rarely have more than ~100 years of stock data at a monthly frequency) and cross-sectional expansion (to thousands of stocks) comes with the cost of heterogeneity and missing data. Nonetheless, the assembly of anomaly databases by academia (e.g. the datasets from Hou, Xue, and Zhang or from Green et al., which curate hundreds of anomaly variables in a consistent way) has been a catalyst for ML research – providing a fertile ground for algorithms to test themselves against a broad menu of potential predictors.

• Macro and Market Time-Series: For predicting aggregate market returns or factors, researchers often use databases of economic indicators. A well-known example is the Welch–Goyal (2008) dataset, which contains dozens of macroeconomic and market variables (dividend yield, yield curve measures, credit spreads, etc.) often used to forecast the U.S. equity premium. ML studies (like Feng et al. 2018; Gu et al. 2020) use expanded versions of such datasets, sometimes adding more recent predictors (e.g. volatility indices, sentiment indices, international macro variables) to cast a wide net. Macroeconomic panel data (e.g. many quarterly series on consumption, output, unemployment, etc.) have been fed into ML models that attempt to learn the “state of the economy” affecting expected returns[37]. One novel approach (Chen et al. 2023) is extracting “hidden states of the economy” via unsupervised learning on a large panel of macro series, and then using those as inputs to asset pricing models[37]. The idea is that ML can distill a few latent factors from dozens of macro variables more effectively than economists manually can. For bonds and other asset classes, analogous datasets (like yield curve data, inflation expectations, etc.) have been used in ML forecasting contexts, though equity has been the primary focus.

• Alternative and Unstructured Data: As noted, some emerging studies incorporate alternative data. Examples include textual data (news articles, earnings call transcripts), which require NLP processing to turn into numerical features, or even market microstructure data (order books, high-frequency trading data) for short-term prediction. While not mainstream in published asset pricing papers yet, these data are being explored using ML in working papers and could become more common. Options data is another domain: some work looks at option-implied information (like implied volatility skew, etc.) as additional features for predicting stock or factor returns[38]. The broad availability of data and the ease of merging different sources via machine learning pipelines mean that the definition of “asset pricing data” is expanding. To keep research grounded, though, many scholars stick to well-vetted datasets (like CRSP/Compustat, macro data from the Federal Reserve, etc.) for the core analysis, to avoid the pitfalls of obscure or lightly-scrutinized data sources that might introduce biases.

• Public Benchmarks: An important development is the creation of public benchmark datasets specifically for testing asset pricing models. The FactorWiki project mentioned earlier is one such effort, aiming to provide a large, cleaned, and regularly updated dataset of asset returns and candidate factors for researchers[35]. It includes not only U.S. equities but potentially other asset classes and is meant to serve as a common ground for evaluating new ML models. Similarly, some papers release their processed datasets (for instance, Gu et al. provided an Internet Appendix with details of their data and code). This improves comparability: if two ML papers use the same underlying data and period, differences in performance can more credibly be attributed to the method rather than data quirks. We anticipate that as the field matures, the community will converge on certain benchmark tasks (e.g. predicting monthly U.S. stock returns using a given list of characteristics from 1965–2015) that every new ML method is tested on, much like standardized datasets in machine learning literature (e.g. ImageNet in computer vision). This will facilitate an apples-to-apples comparison of models.

Evaluation Metrics:

To gauge the success of ML models in asset pricing, researchers employ a variety of evaluation metrics, often distinguishing between statistical performance and economic significance:

• Out-of-Sample $R^2$: This measures the fraction of variance of the outcome (returns) explained by the model’s predictions in a holdout sample, defined as $R^2{\text{OOS}} = 1 - \frac{\text{MSE}{\text{model}}}{\text{MSE}_{\text{benchmark}}}$. A common benchmark is either a simple historical mean (for time-series) or a cross-sectional mean or CAPM baseline (for stocks). While $R^2$ values in predicting stock returns are typically low (because most return variation is noise), ML papers have reported OOS $R^2$ on the order of 1%–5% in challenging settings, which often dwarfs the essentially zero or negative OOS $R^2$* of naive models or non-ML approaches[39]. For example, Gu et al. found an OOS $R^2$ around 5% for their best stock-level model, compared to essentially 0% for a simple linear model – a notable improvement in this context[29]. In factor models, a related metric is the explained variation in average returns; some studies report how much cross-sectional variance in asset mean returns is captured by the ML factors versus Fama–French (e.g. “pricing $R^2$” from cross-sectional regressions of mean returns on beta estimates). Chen et al. (2023) note that their deep learning SDF explains a greater portion of cross-sectional return variance and yields smaller pricing errors than traditional models[26].

• Sharpe Ratios and Investor Utility: Because a model’s value ultimately lies in improving investment decisions, many papers translate predictions into portfolio returns and compute Sharpe ratios, information ratios, or utility gains. A typical exercise is to form a long-short portfolio that goes long the top decile of stocks (with highest predicted returns) and short the bottom decile, then evaluate its out-of-sample performance. The Sharpe ratio of this strategy is a convenient single statistic summarizing performance. As mentioned, ML-based strategies have achieved Sharpe ratios that are 1.3–1.5 in some studies, compared to about 0.5–0.7 for strategies based on simpler models[5]. Another approach is to compute the certainty-equivalent return for an investor using the model’s forecasts to optimize a portfolio (subject to risk aversion). This directly measures how much a risk-averse investor would pay for the model’s information. Such utility-based metrics often reveal that even modest improvements in prediction $R^2$ can translate to sizeable utility gains, because the model consistently tilts the portfolio in the right direction (avoiding big losses, capturing more gains)[4]. Some research also looks at turnover and trading costs in these strategies, ensuring that the theoretical performance net of reasonable costs still shows an improvement – an important reality check for economic significance.

• Classification Metrics: In some contexts (like predicting the direction of return, or default/no-default), classification metrics such as accuracy, precision/recall, AUC (Area Under ROC Curve) are relevant. For equity returns, which are continuous, these are less commonly reported, but one might consider the frequency with which the model correctly signs the return or identifies the top-quartile performers, etc. By and large, regression metrics (MSE, $R^2$) and portfolio performance metrics are favored, since they align with how an investor would use the model.

• Pricing Error Metrics: For factor models and SDF estimates, evaluation focuses on pricing errors. A popular metric is the mean absolute error (MAE) or root mean squared error (RMSE) of pricing, i.e. the difference between actual average returns and those implied by the model (given the estimated factor risk premia). The Hansen–Jagannathan distance is another measure used – it essentially quantifies the distance between the model’s SDF and any true SDF that could price all assets, with smaller distances indicating a better fit. Studies that propose new factor constructions often report these metrics relative to benchmarks. For instance, an ML factor model might achieve an pricing RMSE of only 1% per month on test assets compared to 2% for Fama–French, indicating a tighter fit to the cross-section of returns. Additionally, GRS tests (Gibbons, Ross, Shanken test) can be used to statistically test if residual alphas (unexplained returns) are jointly zero; an ML model that significantly reduces these alphas would likely pass a GRS test where a simpler model fails. Recent work also considers multiple testing adjustments when evaluating many alphas (since using hundreds of test assets can lead to false rejections by chance) – linking back to Harvey, Liu, and Zhu (2016) concerns about the factor zoo and statistical significance[40].

• Stability and Robustness Checks: While not formal metrics, a critical part of evaluation is testing robustness – e.g. how sensitive the results are to the training sample period, to different hyperparameter choices, or to subsetting the universe of stocks. Many papers do subsample analyses (e.g. train the model in earlier decades and test in later decades) to ensure the performance isn’t driven by one period. They may also evaluate stability of variable importance: do the same predictors keep emerging as important in different folds of the data? Consistency in what the model finds predictive lends credence to its economic meaningfulness. Another diagnostic is to compare model-implied factor premia or betas to known values – for example, if an ML-SDF assigns a high price of risk to a factor that we know is likely unpriced, that could indicate overfitting or a problem. Hence, a mix of quantitative metrics and qualitative economic checks is used to evaluate ML models in asset pricing. The literature has been increasingly diligent about such checks, especially after early enthusiasm was tempered by the realization that complex models can overfit if not rigorously validated.

In summary, the evaluation toolkit for ML asset pricing models spans both the statistical domain (forecast errors, $R^2$) and the economic domain (portfolio returns, Sharpe ratios, pricing errors). A model is considered truly successful if it improves predictive accuracy and those improvements lead to economically meaningful gains without generating unreasonable or nonrobust implications. Because finance ultimately cares about risk and return trade-offs, metrics like Sharpe ratio and pricing error are often given more weight than pure statistical metrics. This ensures that ML models are judged not just by how well they fit the data, but by whether they would actually benefit investors or help explain asset prices in a way consistent with financial theory.

Limitations and Critiques of Machine Learning in Asset Pricing

While the marriage of machine learning and asset pricing has been fruitful, it is not without its critics and clear limitations. We discuss several key concerns that emerge from the literature:

• Lack of Interpretability: As mentioned, opacity is a fundamental drawback of complex ML models. Unlike a linear factor model where one can interpret coefficients as risk premia, ML models often defy simple explanation. This opacity leads to skepticism: If we cannot understand the why behind a prediction, can we trust it in high-stakes settings? Moreover, lack of interpretability makes it harder to cumulatively build knowledge. In traditional asset pricing, if many studies find a particular factor is important, it spurs theoretical explanations and further tests. If an ML model finds a pattern but cannot articulate it, it’s difficult to translate that into an advance in theory. This has led to calls for caution – some argue that ML might produce “accurate predictions of returns as point estimates, but without advancing our understanding of the underlying risks or mispricings”. The literature is addressing this by, as noted, bringing interpretability methods and theory constraints into models, but the issue remains that a lot of ML results are initially a black box. Bagnara (2024) summarizes that ML techniques “offer great flexibility and prediction accuracy but require special care as they strongly depart from traditional econometrics”[41]. The implicit warning is that one should not blindly trust an ML model’s output without interrogating it and ensuring it aligns with reasonable economic rationale.

• Overfitting and False Discoveries: Finance data is notoriously noisy – much of the variation in returns is random noise or one-off events that do not repeat. ML models, especially highly flexible ones like deep networks, run the risk of overfitting this noise and mistaking it for signal. Even with regularization, the sheer complexity of some models means they could latch onto patterns that are artifacts of the particular sample. This is exacerbated by the factor zoo problem: if one is not careful, using ML to search through hundreds of predictors could easily result in some combination that fits past data well but has no true predictive power (a manifestation of data mining). Harvey, Liu, and Zhu (2016) famously cautioned that many published anomalies may be false positives, and with ML one could data-mine even more intensely if proper validation is not used[40]. Researchers have learned to mitigate this via strict out-of-sample testing, cross-validation, and even multiple-hypothesis testing corrections. For example, strict separation of training and test sets and reporting of out-of-sample performance (not just in-sample fit) is now standard – a cultural shift partly imported from machine learning’s best practices. Some studies use white-noise simulations or bootstrap to sanity-check that their methods aren’t finding spurious structure (e.g. applying the ML model on randomized returns to see if it still “finds” something – it shouldn’t). Nonetheless, the risk remains that a complex model might be over-tuned to past market regimes and may fail if conditions change. Indeed, stability over time is a concern: some papers have found that certain ML models’ performance decays in more recent years or needs frequent re-estimation, suggesting they might be capturing ephemeral patterns.

• Non-Stationarity and Regime Changes: Financial markets evolve. Relationships that held in one period (say, the value premium in the 1980s-1990s) might attenuate or even reverse in another (2000s-2010s) due to changing investor behavior, regulations, or technology. ML models, unless specifically designed for adaptability, can struggle with such non-stationarity. A static model trained on 1960–2000 data might not perform well post-2000 if new regimes emerge. While rolling re-estimation can help, it also introduces the possibility of overfitting on shorter windows. A critique here is that ML, with its emphasis on historical data, might be inherently backward-looking and not robust to structural breaks. This is not a new issue (econometric models face it too), but the complexity of ML can make diagnosing failure modes harder when a regime shift happens. For instance, if an ML model’s performance deteriorates, is it because of a regime change or because it overfit the past? Researchers are beginning to address this by using techniques from online learning or by incorporating regime indicators, but it remains a challenge. In practice, it underscores the need for continuous monitoring of ML model performance and a willingness to retrain or rethink models as markets change.

• Economic Meaning and Theoretical Consistency: A more philosophical critique is that ML models might lack a grounding in economic theory, raising the danger that they pick up patterns that lack any economic meaning. A classic asset pricing tension is between risk-based explanations (where return predictors proxy for exposures to some systematic risk) and mispricing explanations (where predictors exploit behavioral or structural market inefficiencies). An ML model could exploit mispricings that are profitable in-sample but get arbitraged away once discovered (leading to lack of stability), or it could mix together so many effects that it’s unclear what the investor is being compensated for. Without theory, we might end up with models that work until they don’t, and we won’t know why. Some critics from the traditional camp argue that “predictive power alone is not enough; we need to know the why.” This has spurred efforts to ensure ML models are at least consistent with no-arbitrage and other fundamental constraints. For example, a model that implied a free lunch (arbitrage opportunity) would be deemed suspect even if it fit the data well. By incorporating theory (as discussed in Emerging Trends), researchers aim to rule out grossly illogical results. But a lingering critique is that ML models might be capturing mostly mispricing rather than risk. If that’s the case, one might worry about whether those mispricings will persist. On the other hand, if ML is capturing risk in a complicated way, we’d like to identify what risk it is (e.g. is it learning some nonlinear exposure to recession risk? or an interaction of leverage and volatility that indicates financial distress risk?). The literature is only starting to unravel these questions.

• Practical Constraints and Implementability: From a practitioner’s viewpoint, a strategy that looks great on paper may falter in reality due to costs, turnover, and other frictions. ML strategies often involve frequent trading (since they continuously update forecasts and may chase short-term signals) and can be heavy in small stocks (where many anomalies reside). This can lead to high transaction costs that erode theoretical returns. Indeed, some studies note that the impressive Sharpe ratios of ML long–short portfolios are partly driven by microcap stocks with limited capacity[16]. Once one accounts for realistic trading costs, the net advantage of ML can shrink. Moreover, ML models can be computationally intensive and complex to maintain. In an industry setting, the most complex models might be deemed too unstable or hard to justify to risk managers. Simpler but slightly less predictive models might be preferred for operational robustness. These considerations mean that not every academic ML innovation will translate to an immediate industry application, at least without modification. However, we do see quantitative finance firms adopting ML techniques, often in a piecemeal way (e.g. using ML signals as inputs to a human-driven strategy, or using ML to optimize execution of trades). The critique here is simply that academic studies sometimes focus on the maximum theoretical improvement and less on the nitty-gritty of implementation – a gap that future work is starting to fill by considering costs, market impact, etc., more explicitly.

• Biases in Data and ML Models: Finally, one should be aware of biases that can creep in. Look-ahead bias or survival bias in the data can fool an ML model into overstating its performance. If a researcher is not careful in aligning accounting data with returns (ensuring the data was available at the time of the return prediction), an ML model might be “cheating” by using information that investors wouldn’t have had. Traditional asset pricing has long warned about these biases, and they equally apply in ML research. ML models also have their own quirks: they can be sensitive to how data is normalized or encoded. A characteristic like market capitalization, which ranges over several orders of magnitude, needs scaling or else a neural network might struggle to learn from it. Fortunately, most studies handle this by normalizing characteristics (often rank-transforming to uniform [0,1] distributions)[15]. Another issue is model uncertainty – there are so many algorithms and hyperparameter choices that results could inadvertently be cherry-picked (consciously or not). The community is addressing this by emphasizing that one should either fix a procedure before seeing results or use techniques like cross-validation to choose parameters objectively. In any case, transparency in reporting (e.g. reporting if many models were tried and only the best presented) is crucial to maintain credibility.

In light of these critiques, the overall sentiment in recent literature is one of guarded optimism. Machine learning is recognized as a powerful tool that can yield genuine improvements in asset pricing empirics, but its use must be accompanied by rigor and skepticism. As one review put it, ML is “not a substitute for theory” but a tool to explore the data; blindly applied, it could lead us astray, but tailored and interpreted wisely, it can help “fortify the relevance and robustness of financial models.”[33]. The responsibility lies with researchers to ensure that ML models are robust, interpretable, and economically sensible – a standard that, if met, could truly augment the field of asset pricing.

Conclusion

Machine learning has undeniably opened a new frontier in empirical asset pricing. It has enabled researchers to grapple with the challenges of high-dimensional data (the myriad firm characteristics and macro signals) and complex relationships, yielding improved predictive performance in measuring risk premia and pricing assets. Key areas of impact include more accurate return prediction models, refined identification of which factors or anomalies are truly important, and novel approaches to estimating factor structures and the stochastic discount factor. Equally important, ML methods have prompted a re-examination of long-held models – for instance, by highlighting that a few well-chosen factors (often found via ML) can explain returns nearly as well as an encyclopedia of anomalies, they encourage a more disciplined understanding of asset pricing signals.

The literature to date demonstrates that machine learning can complement and enhance traditional asset pricing methods, rather than wholesale replace them. ML models often rediscover established factors like momentum, size, and value, but also show how interactions or nonlinear effects add nuance to these well-known patterns[7][14]. In comparing approaches, we see a trade-off: ML delivers greater predictive accuracy and the ability to model complex phenomena, while classical models provide clear economic interpretations. The current trajectory of research is to get the best of both worlds – integrating economic theory into ML and using ML insights to inform theory. Emerging trends such as improving model interpretability, enforcing theoretical structure (no-arbitrage, equilibrium constraints), and utilizing new data sources all point toward a synthesis of machine learning techniques with financial economic reasoning[36][31].

Nonetheless, caution is warranted. The enthusiasm for machine learning is tempered by reminders of past episodes in finance where overly complex models or overfit strategies failed in real time. The community is actively addressing these issues by emphasizing out-of-sample testing, reproducibility (through public datasets like FactorWiki[35]), and the development of statistical inference for ML estimates (to know when an improvement is significant or just noise). The intersection of ML and asset pricing is still relatively young, and many open questions remain – for example, how to best quantify the uncertainty in ML predictions[28], how to handle regime shifts gracefully, or how to design algorithms that yield economically interpretable factors by construction. As researchers continue to explore these questions, the dialogue between data science and financial economics will deepen.

In conclusion, machine learning has proven to be a powerful catalyst for empirical asset pricing research. It has led to better-performing models and sharpened our tools for dealing with the factor zoo and prediction problems. Perhaps most encouraging, it has sparked fresh perspectives on age-old questions: What drives asset returns? ML encourages us to let the data speak with minimal assumptions, but the ultimate goal is to translate what it says into the language of economic insight. By focusing on interpretability, theoretical consistency, and robustness – not just raw predictive power – researchers aim to ensure that the ML revolution in asset pricing yields lasting knowledge and practical advances. The journey is ongoing, but the potential for machine learning to help unravel asset pricing puzzles, in partnership with theory, marks an exciting era for the field[33][31].

Sources: The review draws upon a range of recent studies and surveys in this area, including Gu et al. (2020)[3][4], Bagnara (2024)[2][30], Chuan Shi (2025)[42][31], Chen et al. (2025)[43][36], Freyberger et al. (2020)[15], Feng et al. (2020)[17], Chen, Pelger & Zhu (2023)[22], among others. These and numerous other works collectively illustrate the promises and challenges of applying machine learning to empirical asset pricing, as synthesized above.

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https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3350138

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https://arxiv.org/html/2503.00549v1

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[38] Asset Pricing: Cross-Section Predictability

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