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：工具变量模型是许多因果推断场景中的核心方法。本文考虑离散变量下的工具变量模型，其中工具变量（Z）、暴露变量（X）和结局变量（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无直接效应，即Y(x,z)=Y(x)。首先，我们给出与给定观测分布P(X,Y|Z)相容的潜在结局联合分布族P(Y(x=1),...,Y(x=K))的简洁刻画。然后讨论潜在结局边缘概率分布P(Y(x=1)),...,P(Y(x=K))的变分（非）依赖性性质，该性质直接蕴含平均因果效应对比（如E[Y(x=i)-Y(x=j)]）的部分识别。最后，我们通过模拟研究展示当K和Q变化时，与IV模型不相容的观测分布体积变化。