Instrumental variable models are central to the inference of causal effects in many settings. We consider the instrumental variable model with discrete variables where the instrument (Z), exposure (X) and outcome (Y) take Q, K, and M levels respectively. We assume that the instrument is randomized and that there is no direct effect of Z on Y so that Y(x,z) = Y(x). We first provide a simple characterization of the set of joint distributions of the potential outcomes P(Y(x=1), ..., Y(x=K)) compatible with a given observed distribution P(X, Y | Z). We then discuss the variation (in)dependence property of the marginal probability distribution of the potential outcomes P(Y(x=1)), ..., P(Y(x=K)) which has direct implications for partial identification of average causal effect contrasts such as E[Y(x=i) - Y(x=j)]. We also include simulation results on the volume of the observed distributions not compatible with the IV model as K and Q change.

翻译：工具变量模型在许多场景下是推断因果效应的核心方法。本文研究离散变量下的工具变量模型，其中工具变量（Z）、处理变量（X）与结果变量（Y）分别取Q、K和M个水平。我们假设工具变量是随机化的，且Z对Y没有直接效应，即满足Y(x,z) = Y(x)。我们首先给出了与给定观测分布P(X, Y | Z)相容的潜在结果联合分布P(Y(x=1), ..., Y(x=K))集合的简明刻画。随后讨论了潜在结果边际概率分布P(Y(x=1)), ..., P(Y(x=K))的变差（不）依赖性，该性质对平均因果效应对比（如E[Y(x=i) - Y(x=j)]）的部分可识别性具有直接意义。我们还通过模拟展示了当K和Q变化时，与工具变量模型不相容的观测分布所占的体积。