IV as a Solution to Omitted Variables

We think of instrumental variables as a solution to Y causing X, but it also can help when there is an omitted variable. In that case, X ends up being correlated with the error term, because the omitted variable X2 is correlated with both Y and X. So what we can do is find a Z which is correlated with X but not with X2 or Y. We can do a first-stage regression of X on Z, and then use the fitted value Xhat in our main regression, Y on Xhat.