Debiased Machine Learning: Identification, Estimation, and Shape Constraints

We develop a general framework of identification and estimation for automatic debiased machine learning (DML) where the parameter of interest $\theta_0$ is identified by a moment condition involving a nuisance $\gamma_0$ that may be high dimensional. DML leverages machine learning to estimate $\gamma_0$ while correcting for regularization and overfitting biases that may otherwise transmit to biased estimation of $\theta_0$. We establish conditions under which the Riesz representer $\alpha_0$, which is at the core of DML, is identified, and show that the identification occurs precisely when $\alpha_0$ uniquely optimizes a quadratic functional. This characterization enables us to develop a general estimation procedure for $\alpha_0$ that allows for generic $\gamma_0$ including those defined by models with endogeneity and encompasses both classical sieves and modern architectures such as deep neural networks. To improve estimation precision and mitigate the curse of dimensionality, we incorporate shape constraints on $\gamma_0$ by embedding them into a possibly nonlinear parameter space. We illustrate our estimation procedure through simulations and empirical applications.