I am implementing a standard control function regression. I start with the equations:
$y_1 = \alpha \; y_2 + X \beta + \varepsilon$
$y_2 = \gamma z + X \beta + \nu$
$y_2$ is endogenous in the first equation, $X$ contains exogenous predictors, and $z$ is my exogenous instrument. Technically, the first equation is an OLS, whereas the second is a logit, though I'm not certain if that is super relevant here. I use simple replacement bootstrapping to calculate the errors on this two-stage model.
I get my residuals from the regression of $y_2$ on predictors then run the following version of the regression for $y_1$:
$y_1 = \alpha \; y_2 + \delta \hat{\nu} + X \beta + \varepsilon$.
I know that I can use a significance test on $\hat{\delta}$ in the second stage equation to test whether $y_2$ is indeed endogenous. (i.e. see Wooldridge, Panel Data, 2010, p.126ff).
In my particular situation, I am frequently observing that $\hat{\alpha}$ and $\hat{\delta}$ are very similar - namely, they are quite similar in magnitude and statistical significance, but just opposite in their sign. This effect persists across a variety of formulations in my models (e.g. changing composition of $X$, varying sample composition, etc.).
Is this a very normal occurrence? Is there any reason that this should give me pause, either regarding the validity of my instrument or somehow my implementation of the control function approach?
