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中对于因果效应的推断至关重要。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 不存在直接影响，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 P(Y(x=1), ..., Y(x=K)) 的一个简洁刻画。随后，我们讨论潜在结局边际概率分布 P(Y(x=1)), ..., P(Y(x=K)) 的变化（不）依赖性，该性质对平均因果效应对比（如 E[Y(x=i) - Y(x=j)]）的部分识别具有直接意义。我们还提供了关于当 K 和 Q 变化时，与工具变量模型不相容的观测分布体积的模拟结果。