In a causal diagram, there is an easy graphical condition (d-connectedness) to see whether (and how) X and Y are related given Z:
We need to have a path from X to Y that satisfies certain properties:
That path can start out moving upstream (i.e. against the causal arrows); it may switch from moving upstream to downstream at any time (including at the start); it must switch direction whenever it hits a node in Z; and it may only switch from moving downstream to upstream when it hits a node in Z.
I think this is wrong, because it would imply that X and Y are d-connected in [X ← Z → Y]. It should say:
That path can start out moving upstream (i.e. against the causal arrows); it may switch from moving upstream to downstream at any time (including at the start); it can only connect to a node in Z if it’s currently moving downstream; and it must (and can only) switch from moving downstream to upstream when it hits a node in Z.
I think this is wrong, because it would imply that X and Y are d-connected in [X ← Z → Y]. It should say:
That path can start out moving upstream (i.e. against the causal arrows); it may switch from moving upstream to downstream at any time (including at the start); it can only connect to a node in Z if it’s currently moving downstream; and it must (and can only) switch from moving downstream to upstream when it hits a node in Z.