Interpretation of inverse mills ratio and generalized residuals

[Alexander Huber]

Dear experts,

currently I am working on a project that evaluates the role of a binary endogenous variable towards the effect on firm growth. The regressions include both the usage of the inverse mills ratio (to account for selection bias) as well as the generalized residuals (after Gourieroux et al 1987) within a control function approach. I derive both control factors using a probit model in which I regress variables which I think have an effect on the chance to "receive" the endogeneous binary variable.
-> probit Y2, X1, X2, X3, Xn.From that regression I calculate the inverse mills ratio to account for a possible selection bias using the following formula:

• predict Dummy, xb
  gen Invmills = normalden(Dummy)/normal(Dummy)

In addition, from the estimates of the probit regression, I calculate the generalized residuals using the following formula (source: http://www.stata.com/statalist/archi...sg00650.html):

• predict xb, xb
• gen Lambda = cond(Endog. binary regressor == 1,normalden(xb)/normal(xb), -normalden(xb)/(1-normal(xb)))

The inverse mills ratio is for example used in a regular OLS including the binary edogenous regressor. Knwoing that the estimates of this regression are biased, I nevertheless report them for comparison. Further, I include the IMR in an instrumental variable approach too. Further, I add the IMR together with the generalized residual control fcator in a control function approach to compare the results with the outcomes of the IV regression. Now I am wondering if both control factors really control for different things. Can I use both control factors in one regression (such as the control function approach) to account for a) selection bias and b) how the error terms of the probit regression correlate with the error terms of my growth equation (to see how firm growth is affected)

Your advice is much appreciated.

Thank you and best regards,
Alex