We address the issue of the testability of instrumental variables derived from observational data. Most existing testable implications are centered on scenarios where the treatment is a discrete variable, e.g., instrumental inequality (Pearl, 1995), or where the effect is assumed to be constant, e.g., instrumental variables condition based on the principle of independent mechanisms (Burauel, 2023). However, treatments can often be continuous variables, such as drug dosages or nutritional content levels, and non-constant effects may occur in many real-world scenarios. In this paper, we consider an additive nonlinear, non-constant effects model with unmeasured confounders, in which treatments can be either discrete or continuous, and propose an Auxiliary-based Independence Test (AIT) condition to test whether a variable is a valid instrument. We first show that, under the completeness condition, if the candidate instrument is valid, then the AIT condition holds. Moreover, we illustrate the implications of the AIT condition and demonstrate that, under certain additional conditions, the AIT condition is necessary and sufficient to detect all invalid IVs. We also extend the AIT condition to include covariates and introduce a practical testing algorithm. Experimental results on both synthetic and three different real-world datasets show the effectiveness of our proposed condition.

翻译：本文探讨了从观测数据中推导出的工具变量的可检验性问题。现有的大多数可检验性含义集中于处理变量为离散变量的情形（例如工具变量不等式，Pearl, 1995），或假设效应为常数的情形（例如基于独立机制原理的工具变量条件，Burauel, 2023）。然而，处理变量常常是连续变量，例如药物剂量或营养成分水平，且非恒定效应可能出现在许多现实场景中。在本文中，我们考虑了一个存在未测量混杂因素的可加性非线性非恒定效应模型，其中处理变量可以是离散的或连续的，并提出了一种基于辅助变量的独立性检验（AIT）条件，用以检验一个变量是否为有效的工具变量。我们首先证明，在完备性条件下，若候选工具变量有效，则AIT条件成立。此外，我们阐述了AIT条件的含义，并证明了在某些附加条件下，AIT条件是检测所有无效工具变量的充分必要条件。我们还将AIT条件扩展到包含协变量的情形，并引入了一种实用的检验算法。在合成数据以及三个不同的真实世界数据集上的实验结果表明了我们所提条件的有效性。