Using predicted probabilities as regressors I am working on a project where I investigate growth in wages due to migration. I correct for the endogeneity in the decision to migrate (only those that are most likely to gain from migration will migrate) by first using a probit model to predict the probabilities of migration based on various characteristics. I then use the predicted probabilities in a second step as a proxy for migration (this in effect is an instrumental variables regression).
My problem is that I get unreasonably high estimates - wages are predicted to increase up to 200%. My concern is that since my predicted probabilities are very low (on average 3%, 25% at the 99th percentile), which is reasonable as in the sample only about 5% migrate, the results that I get come from the marginal increase of probability to migrate from 0 to 1. As far as the predicted probabilities go in my sample, an increase from 0 to 1 is very extreme. Could this be causing the huge estimates? Am I interpreting this correctly? Or should I rather look at the strength of my instruments, etc.? 
 A: If you are interested in an approximation of the average partial effect you could just use a linear probability model in the first stage, i.e. do your instrumental variables estimation via 2SLS, for instance, in the usual way. However, due to the non-linearities involved this is not the efficient approach but it can give a good initial idea of the effect under study. For a more in-depth treatment of this argument see Wooldridge (2010) "Econometric Analysis of Cross-Section and Panel Data" in section 15.7.3 from page 594 onward. On page 265-268 he explains the forbidden regression and its problems.
Another procedure that you might be interested in was used by Adams et al. (2009). They use a three-step procedure where they have a probit "first stage" and an OLS second stage without falling for the forbidden regression problem. Their general approach is:


*

*use probit to regress the endogenous variable on the instrument(s) and exogenous variables

*use the predicted values from the previous step in an OLS first stage together with the exogenous (but without the instrumental) variables

*do the second stage as usual


This procedure will yield unbiased estimates and generally is more efficient than doing 2SLS with a linear probability model in the first stage.
