Abstract Suppose we are interested in the effect of variable X on variable Y . If X and Y both influence, or are associated with variables that influence, a common outcome, called a collider , then conditioning on the collider (or on a variable influenced by the collider – its “child”)...

Abstract Manski (Monotone treatment response. Econometrica 1997;65:1311–34) and Manski and Pepper (Monotone instrumental variables: with an application to the returns to schooling. Econometrica 2000;68:997–1010) gave sharp bounds on causal effects under the assumptions of monotone treatment response...