The conditional moment problem is a powerful formulation for describing structural causal parameters in terms of observables, a prominent example being instrumental variable regression. A standard approach reduces the problem to a finite set of marginal moment conditions and applies the optimally weighted generalized method of moments (OWGMM), but this requires we know a finite set of identifying moments, can still be inefficient even if identifying, or can be theoretically efficient but practically unwieldy if we use a growing sieve of moment conditions. Motivated by a variational minimax reformulation of OWGMM, we define a very general class of estimators for the conditional moment problem, which we term the variational method of moments (VMM) and which naturally enables controlling infinitely-many moments. We provide a detailed theoretical analysis of multiple VMM estimators, including ones based on kernel methods and neural nets, and provide conditions under which these are consistent, asymptotically normal, and semiparametrically efficient in the full conditional moment model. We additionally provide algorithms for valid statistical inference based on the same kind of variational reformulations, both for kernel- and neural-net-based varieties. Finally, we demonstrate the strong performance of our proposed estimation and inference algorithms in a detailed series of synthetic experiments.

翻译：条件矩问题是用可观测量描述结构性因果参数的有力形式，一个典型例子是工具变量回归。标准方法将该问题简化为有限个边际矩条件，并应用最优权重广义矩方法（OWGMM），但这需要我们事先知道一组具有识别性的矩条件，即使具备识别性仍可能效率低下，或者若使用不断增广的矩条件筛子，则理论上有效但实际操作困难。受OWGMM变分极小化极大重构的启发，我们为条件矩问题定义了一类极为广泛的估计量，称之为变分矩方法（VMM），该方法自然能够控制无穷多个矩条件。我们对多种VMM估计量（包括基于核方法和神经网络的估计量）进行了详细理论分析，并给出了在完整条件矩模型下这些估计量具有一致性、渐近正态性和半参数有效性的条件。此外，我们基于相同的变分重构思想，为基于核方法和神经网络的估计量提供了有效统计推断算法。最后，通过一系列详细的合成实验，我们证明了所提估计与推断算法的优异性能。