# Deep Partial Least Squares for Empirical Asset Pricing

arXiv: https://arxiv.org/abs/2206.10014 · HTML: https://mlsanity.com/paper/2206.10014/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 61 references that are in the map.
- First posted on arXiv 20 Jun 2022.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Deep Learning and the Information Bottleneck Principle](https://mlsanity.com/paper/1503.02406/) (2015)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning](https://mlsanity.com/paper/1810.01075/) (2018)
- [Posterior Concentration for Sparse Deep Learning](https://mlsanity.com/paper/1803.09138/) (2018)
- [Deep Learning in Asset Pricing](https://mlsanity.com/paper/1904.00745/) (2019)
- [Classification-based Financial Markets Prediction using Deep Neural Networks](https://mlsanity.com/paper/1603.08604/) (2016)
- [Significance Tests for Neural Networks](https://mlsanity.com/paper/1902.06021/) (2019)
- [Horseshoe Regularization for Machine Learning in Complex and Deep Models](https://mlsanity.com/paper/1904.10939/) (2019)
- [Deep Learning for Predicting Asset Returns](https://mlsanity.com/paper/1804.09314/) (2018)
- [Deep Learning Partial Least Squares](https://mlsanity.com/paper/2106.14085/) (2021)
- [Industrial Forecasting with Exponentially Smoothed Recurrent Neural Networks](https://mlsanity.com/paper/2004.04717/) (2020)

## Built on by (most cited first, 1 of 1)

- [Portfolio Optimization Proxies under Label Scarcity and Regime Shifts via Bayesian and Deterministic Students under Semi-Supervised Sandwich Training](https://mlsanity.com/paper/2604.14206/) (2026)

## Abstract (arXiv)

We use deep partial least squares (DPLS) to estimate an asset pricing model for individual stock returns that exploits conditioning information in a flexible and dynamic way while attributing excess returns to a small set of statistical risk factors. The novel contribution is to resolve the non-linear factor structure, thus advancing the current paradigm of deep learning in empirical asset pricing which uses linear stochastic discount factors under an assumption of Gaussian asset returns and factors. This non-linear factor structure is extracted by using projected least squares to jointly project firm characteristics and asset returns on to a subspace of latent factors and using deep learning to learn the non-linear map from the factor loadings to the asset returns. The result of capturing this non-linear risk factor structure is to characterize anomalies in asset returns by both linear risk factor exposure and interaction effects. Thus the well known ability of deep learning to capture outliers, shed lights on the role of convexity and higher order terms in the latent factor structure on the factor risk premia. On the empirical side, we implement our DPLS factor models and exhibit superior performance to LASSO and plain vanilla deep learning models. Furthermore, our network training times are significantly reduced due to the more parsimonious architecture of DPLS. Specifically, using 3290 assets in the Russell 1000 index over a period of December 1989 to January 2018, we assess our DPLS factor model and generate information ratios that are approximately 1.2x greater than deep learning. DPLS explains variation and pricing errors and identifies the most prominent latent factors and firm characteristics.