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 is to reduce the problem to a finite set of marginal moment conditions and apply 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 unwieldy and impractical if we use a growing sieve of moments. 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 based on kernel methods and neural networks, and provide appropriate conditions under which these estimators are consistent, asymptotically normal, and semiparametrically efficient in the full conditional moment model. This is in contrast to other recently proposed methods for solving conditional moment problems based on adversarial machine learning, which do not incorporate optimal weighting, do not establish asymptotic normality, and are not semiparametrically efficient.

翻译：有条件的瞬间问题是用一个强有力的提法,用可观察性来描述结构性因果参数,一个突出的例子是工具性可变回归。一个标准的方法是将问题降低到有限的一组边际时刻条件,并采用最优加权的瞬间普遍方法(OWGMMM ),但这要求我们知道一组有限的识别时间,即使确定,也可能是无效的,或者如果我们使用不断增长的瞬间隔断,仍然可能是不易操作和不切实际的。在对 OWGMM 进行变式微缩重整的激励下,我们为有条件时刻问题确定了一个非常一般的估测者类别,我们称之为时的变式方法(VMM ), 这自然可以控制无限的时段。我们提供了对多个 VMM 测算器的详细理论分析, 包括以内核方法和神经网络为基础, 并提供适当的条件, 使这些测算器在完全的瞬间模型中是正常的, 和半相对有效的。这与最近提出的其他基于对等机器学习的有条件时段问题的解决办法形成对比, 这些方法并不包含最佳的正常的加权。