Deep Learning with Non-Linear Factor Models: Adaptability and Avoidance of Curse of Dimensionality

In this paper, we connect the deep learning literature with non-linear factor models and show that deep learning estimation leads to a substantial improvement in the non-linear factor model literature. We provide bounds on the expected risk and prove that these upper bounds are uniform over a set of multiple response variables. We extend our results to an additive model setting and show its connection to non-linear factor models for financial applications. Compared to traditional factor models which assume rigid linear relations between the factors and the observed variables, our deep neural network factor model (DNN-FM) offers major improvements in modeling flexibility. We develop a novel data-dependent estimator of the error covariance matrix in deep neural networks and prove that the estimator is consistent in spectral norm. Moreover, we show the consistency and provide the rates of convergence of the covariance matrix and precision matrix estimators for asset returns. The rates of convergence do not depend on the number of factors. Various Monte Carlo simulations confirm our large sample findings and reveal superior accuracies of the DNN-FM in estimating the true underlying functional form, as well as the covariance and precision matrix compared to competing approaches. Moreover, in an out-of-sample portfolio forecasting application it outperforms in most of the cases alternative portfolio strategies in terms of out-of-sample portfolio standard deviation and Sharpe ratio.

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