Does the direction of correlation matter for Instrumental Variable?

Question

For a valid instrumental variable:

(1) the instrument must be correlated with the endogenous explanatory variables, and,

(2) the instrument cannot be correlated with the error term in the explanatory equation.

My question is about point 1. Does the direction of the correlation matter? Suppose I choose Z as an instrument for the endogenous variable X. Z is highly correlated with X (and seemingly unrelated to the outcome of interest), but in the opposite direction that I expected. My hunch is that it undermines my assumptions about point 2, but I wasn't quite sure.

Answer 1

No, direction of correlation does not matter for the purpose at hand. In the simplest IV-estimation case, the estimator emerges from the assumption of no correlation,

$$E(Z'u) = 0 \implies E[Z'(y-Xb)] = 0 \implies E(Z'y) = E(Z'X)b$$

Using sample averages instead of expected values as we do with method of moments estimation, and ignoring the scaling $1/n$ we arrive at

$$\hat b_{IV} = (Z'X)^{-1}Z'y = (Z'X)^{-1}Z'(Xb+u) = b + (Z'X)^{-1}Z'u$$

For this estimator to be feasible we want $E(Z'X) \neq 0 $ so that we can invert the matrix. The sign of the relation does not matter.