content_image/317206.jpg content_image/317208.jpg content_image/317207.jpg
## Appendix 

## Proof of Lemma 1 

I provide a proof only for sharp bounds on P$_{1}$ ( y, 0 | z ). Sharp bounds on P$_{0}$ ( y, 1 | z ) are obtained similarly. 

<img src='content_image/317206.jpg'>

The model (1) under M. 1 − M. 4 is uninformative about the counterfactual distribution term P [ Y$_{1}$ ≤ y | p < U ] . Therefore by plugging 0 and 1 into the term, bounds on P [ Y$_{1}$ ≤ y, 0 | z ] can be obtained as follows: 

where 

<img src='content_image/317208.jpg'>

<img src='content_image/317207.jpg'>

