Rules for Parsing Unsigned Arrow Diagrams
For OLS to be unbiased, you need Cov(X,epsilon) = 0 (exclusion restriction for OLS)
For IV to be unbiased, you need
Cov(Z,epsilon) = 0 (exclusion restriction for IV)
AND
Cov(Z,X) is not zero (relevance of the instrument)
How do you tell if a covariance is zero or not zero?
It works the same way for all 3 cases. Let me call the two things you want to know if the covariance between is zero or not A and B. (A and B come from the set {X,epsilon,Z).
The covariance is NOT zero if EITHER
There is a path following the one-way signs from A to B
There is a path following the one-way signs from B to A
There is a path following the one-way signs from something else to A, and a path following the one-way signs from that same thing to B.
To show a covariance is zero, you have to check a lot of things. You need:
There is no path following the one-way signs from A to B
There is no path following the one-way signs from B to A
There is no other thing from which there is a path following the one-way signs to A, and from which there is a path following the one-way signs from that same thing to B.