Interpretable machine learning for asset pricing

Deep neural networks are used to estimate time-varying equity risk premia. Two main innovations are introduced. First, a nonlinear generalisation of Fama-MacBeth regressions is developed in which partial derivatives of the conditional mean with respect to firm characteristics play the role of local risk prices. Second, Jacobian regularisation is introduced in the objective function used to estimate the neural network model. The derivative estimates provide observation-level, first-order summaries of how predicted returns respond to characteristics, while derivative penalisation encourages sparse and stable sensitivities. The approach is motivated by asset pricing but applies more generally to nonlinear economic models in which marginal effects of inputs are substantive objects of interest. Theoretical results on the estimation and inference of the neural network model are provided. Simulation and extensive empirical results provide support for the methodology.