Deep Learning with Non-Linear Factor Models: Adaptability and Avoidance of Curse of Dimensionality
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In this paper, we connect deep learning literature with non-linear factor models and show that deep learning estimation makes a substantial improvement in the non-linear additive factor model literature. We provide bounds on the expected risk and show that these upper bounds are uniform over a set of multiple response variables by extending Schmidt-Hieber (2020) theorems. We show that our risk bound does not depend on the number of factors. In order to construct a covariance matrix estimator for asset returns, we develop a novel data-dependent estimator of the error covariance matrix in deep neural networks. The estimator refers to a flexible adaptive thresholding technique which is robust to outliers in the innovations. We prove that the estimator is consistent in spectral norm. Then using that result, we show consistency and rate of convergence of covariance matrix and precision matrix estimator for asset returns. The rate of convergence in both results do not depend on the number of factors, hence ours is a new result in the factor model literature due to the fact that number of factors are impediment to better estimation and prediction. Except from the precision matrix result, all our results are obtained even with number of assets are larger than the time span, and both quantities are growing. Various Monte Carlo simulations confirm our large sample findings and reveal superior accuracies of the DNN-FM in estimating the true underlying functional form which connects the factors and observable variables, 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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