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

  1. There is a path following the one-way signs from A to B

  2. There is a path following the one-way signs from B to A

  3. 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:

  1. There is no path following the one-way signs from A to B

  2. There is no path following the one-way signs from B to A

  3. 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.