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.
翻译:在本文中,我们将深层学习文献与非线性要素模型联系起来,并表明深层学习估计使非线性添加要素模型文献大有改进。我们提供了预期风险的界限,并通过扩展Schmidt-Hieber(202020年)理论显示,这些上界在一系列多重响应变量上是统一的。我们显示,我们的风险约束并不取决于因素的数量。为了构建一个资产回报的共变矩阵估计符,我们开发了一个新的数据依赖的变量估计,以显示深层神经网络的错误共变差矩阵。估计数字指的是灵活的调整功能阈值临界值技术,对创新的外端者来说是强有力的。我们证明,根据光谱标准变差矩阵和精确矩阵估算资产回报的一致程度并不取决于因素的数量。我们获得的因素模型文献显示,在更深层神经变差异变差中,最精确的变差率是比更精确的变差率和预测结果。除了精确值外,整个货币变差的变差率分析结果显示,所有货币变差的变差率速度并不取决于各种因素的变差率,因此获得的变差的变差和变差的变差性数据是新的结果,在更难性变差的变差的变差率、最深的变差率、最深的变差差差的计算结果。