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 $Z$ that takes $K$ states that were originally stated in Richardson & Robins (2014), "ACE Bounds; SEMS with Equilibrium Conditions," arXiv:1410.0470.

翻译：本文为二值响应变量$Y$与二值处理变量$X$、但工具变量$Z$取$K$个状态的工具变量(IV)模型中的相关结论给出了证明。这些结论最初由Richardson & Robins (2014)在论文"ACE Bounds; SEMS with Equilibrium Conditions"（arXiv:1410.0470）中提出。