In this note we give proofs for results relating to the Instrumental Variable (IV) model with binary response $Y$ and binary treatment $X$, but with an instrument $Z$ with $K$ states. These results were originally stated in Richardson & Robins (2014), "ACE Bounds; SEMS with Equilibrium Conditions," arXiv:1410.0470.

翻译：本注记给出了针对二元响应变量$Y$、二元处理变量$X$以及具有$K$个状态的工具变量$Z$的IV模型相关结果证明。这些结果最初由Richardson & Robins (2014)在论文《ACE边界；带均衡条件的结构方程模型》（arXiv:1410.0470）中提出。