Dear Stack Exchange users,
Let's say that my independent variable of interest is expected to be endogenous and that I want to investigate its effect on 3 separate outcomes. I run separate 2SLS's (one per outcome) and in each of them I use the same set of instrumental variables (and same set of control variables).
In two 2SLS's, ancillary tests tell me that my instruments are neither weak nor uncorrelated with the endogenous variable (under and weak identification tests), they are not correlated with the residuals in the second stage either (over-identification test). Finally, the Hausman test confirms that the instrumented variable is indeed endogenous.
In the third 2SLS's (the only difference is the outcome), ancillary tests tell me that my instruments are correlated with the residuals in the second stage, and that the instrumented variable is not endogenous. (under and weak identification tests give reassuring results, as for the other two outcomes).
Based on this background, I have two related questions:
What could be the cause of the difference between the first two 2SLS' and the third one?
Is it possible that the independent variable of interest is not endogenous in the third 2SLS, that the instruments do not predict well this exogenous variable, and, thus, that trying to predict an exogenous variable causes the instruments to be correlated with the residuals in the second stage?