We study inference on linear functionals in the nonparametric instrumental variable (NPIV) problem with a discretely-valued instrument under a many-weak-instruments asymptotic regime, where the number of instrument values grows with the sample size. A key motivating example is estimating long-term causal effects in a new experiment with only short-term outcomes, using past experiments to instrument for the effect of short- on long-term outcomes. Here, the assignment to a past experiment serves as the instrument: we have many past experiments but only a limited number of units in each. Since the structural function is nonparametric but constrained by only finitely many moment restrictions, point identification typically fails. To address this, we consider linear functionals of the minimum-norm solution to the moment restrictions, which is always well-defined. As the number of instrument levels grows, these functionals define an approximating sequence to a target functional, replacing point identification with a weaker asymptotic notion suited to discrete instruments. Extending the Jackknife Instrumental Variable Estimator (JIVE) beyond the classical parametric setting, we propose npJIVE, a nonparametric estimator for solutions to linear inverse problems with many weak instruments. We construct automatic debiased machine learning estimators for linear functionals of both the structural function and its minimum-norm projection, and establish their efficiency in the many-weak-instruments regime. To do so, we develop a general semiparametric efficiency theory for regular estimators under weak identification and many-weak-instrument asymptotics.

翻译：本文研究在离散取值工具变量下，采用众多弱工具渐近框架的非参数工具变量问题中线性泛函的推断，其中工具变量取值的数量随样本量增长。一个关键激励案例是：在仅有短期结果的新实验中估计长期因果效应，利用历史实验作为短期结果对长期结果影响的工具变量。此处，历史实验的分配即作为工具变量：我们拥有众多历史实验，但每个实验中仅包含有限单元。由于结构函数为非参数形式但仅受有限个矩条件约束，点识别通常无法实现。为此，我们考虑矩条件最小范数解的线性泛函，该解始终良定义。随着工具变量水平数量增长，这些泛函构成目标泛函的逼近序列，从而用更适合离散工具变量的较弱渐近概念替代点识别。通过将刀切法工具变量估计量拓展至经典参数设定之外，我们提出npJIVE——一种针对具有众多弱工具变量的线性逆问题解的非参数估计量。我们构建了针对结构函数及其最小范数投影的线性泛函的自动去偏机器学习估计量，并证明其在众多弱工具变量框架下的有效性。为此，我们发展了适用于弱识别与众多弱工具变量渐近框架下常规估计量的一般半参数有效性理论。