The standard scheme of instrumental variable in terms of causality (
Z -> X -> Y
Where Z is an instrument, X an endogenous variable, and Y a response.
Is it possible, that following relations:
Z <- X ->Y
Z <-> X ->Y
are also valid?
While correlation between instrument and variable is satisfied, how may I think of exclusion restriction in such cases?
NOTE: The notation
<-> is not explicit and might lead to different understandings of the problem. Still, the answers highlight this issue and use it to show important aspects of the problem. When reading, please proceed with caution about this part of the question.