The conditional moment problem is a powerful formulation for describing structural causal parameters in terms of observables, a prominent example of which is instrumental variable regression. A standard approach is to reduce the problem to a finite set of critical moment conditions and apply the optimally weighted generalized method of moments (OWGMM). Efficient in theory, but cumbersome in practice when using sieves with increasing moment conditions. Motivated by the variational minimax reformulation of OWGMM, we define a very general class of estimators for conditional moment problems. We provide a detailed theoretical analysis of several VMM estimators, including those based on kernel methods and neural nets, that demonstrate that they are consistent, asymptotically normal, and semiparametric in full conditional moment models. Provide conditions that are efficient. Furthermore, we provide algorithms for effective statistical inference based on the same kind of variational formulation for both kernel-based and neural net-based diversity. Finally, we demonstrate the strong performance of the proposed estimation and inference algorithms in a series of detailed synthetic experiments.
