I have an instrumental variable (IV) estimation where I use Z as an instrument for treatment D, to estimate a treatment effect of D on Y. After certain discussion, I find that there might be another variable, X, which is correlated both with Y and with Z.
I then decide to include X into my IV model, to control for what seems to be a violation of exclusion assumption (i.e. Z doesn't affect Y solely through X).
After re-doing the first stage of 2SLS, I find that Z coefficient is not statistically significant, as standard errors are high. I deduced that this might be due to high collinearity of X with Z, which is is 0.75. Still, however, X is significant in the first stage regression.
However, consequently using Z to instrument D, and including X as a control variable yields absolutely no statistical significance.
I therefore concluded that Z is a weak instrument if we do not control for X, and after we control for X, Z is still weak because there is no statistical significance in first stage. I am relatively new to IV estimation, and I wanted to get an opinion on whether my logic is flawed and what the next steps could entail:
- Is omitting X equivalent an example of OVB and does its inclusion remove it?
- What is the significance of high collinearity between instrument Z and control variable X?
- Does the insignificant first stage (with inclusion of X) mean that we cannot use Z as instrument?