Instrumental variables (IV) are a common means to identify treatment effects. But standard IV methods do not allow us to unpack the complex treatment effects that arise when a treatment and its outcome together cause a second outcome of interest. For example, IV methods have been used to show...

This paper considers the problem of making inferences about the effects of a program on multiple outcomes when the assignment of treatment status is imperfectly randomized. By imperfect randomization we mean that treatment status is reassigned after an initial randomization on the basis of...