Studies using instrumental variables (IV) often assess the validity of their identification assumptions using falsification tests. However, these tests are often carried out in an ad-hoc manner, without theoretical foundations. In this paper, we establish a theoretical framework for negative control tests, the predominant category of falsification tests for IV designs. These tests are conditional independence tests between negative control variables and either the IV or the outcome (e.g., examining the ``effect'' on the lagged outcome). We introduce a formal definition for threats to IV exogeneity (alternative path variables) and characterize the necessary conditions that proxy variables for such unobserved threats must meet to serve as negative controls. The theory highlights prevalent errors in the implementation of negative control tests and how they could be corrected. Our theory can also be used to design new falsification tests by identifying appropriate negative control variables, including currently underutilized types, and suggesting alternative statistical tests. The theory shows that all negative control tests assess IV exogeneity. However, some commonly used tests simultaneously evaluate the 2SLS functional form assumptions. Lastly, we show that while negative controls are useful for detecting biases in IV designs, their capacity to correct or quantify such biases requires additional non-trivial assumptions.

翻译：使用工具变量（IV）的研究常通过证伪检验评估其识别假定的有效性。然而，这些检验往往以缺乏理论基础的临时性方式进行。本文为阴性对照检验建立了理论框架——这是IV设计中最主要的证伪检验类型。这类检验是对阴性对照变量与工具变量或结局变量之间的条件独立性检验（例如检验对滞后结局的"效应"）。我们为IV外生性威胁（替代路径变量）提出正式定义，并刻画未观测威胁的代理变量需满足的必要条件，使其可作为阴性对照发挥作用。该理论揭示了实施阴性对照检验时普遍存在的错误及其修正方法。我们的理论还可用于设计新的证伪检验：通过识别合适的阴性对照变量（包括当前未被充分利用的类型）并提出替代统计检验方法。理论表明，所有阴性对照检验均用于评估IV外生性，但某些常用检验会同时评估2SLS函数形式假定。最后，我们证明虽然阴性对照有助于检测IV设计中的偏倚，但其纠正或量化此类偏倚的能力需要额外的非平凡假定。