Using an IV and control function approach to account for endogeneity

[Alexander Huber]

Dear experts,

I am working on a project that analyses the effect of a binary endogenous regressor in combination with other independent variables and their effect on a non-binary outcome variable (Y1). In order to control for endogeneity, I employ two different methods. In a first step, I use the instrumental variables approach to account for the endogenous nature of my dummy regressor. In a second step, I account for endogeneity by using a control function (CF) approach and an inverse mills type of control factor (generalized residuals (Gourieroux et.al. (1987)). However, I was unable to find a real "instruction" on the application of the CF within stata. Hence, I would like to ask whether my current setting correctly accounts for I) endogeneity and II) to test the robustness of my IV estimates. Here is what I do:

1) I calculate 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

2) 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)

2) In addition, from the estimates of the probit regression (Step 1), I calculate the generalized residuals (Gourieroux et.al. (1987)) using the following formula:

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

3) I use the inverse mills ratio, the generalized residuals (Lambda) and my endogenous binary variable in my regressions. For illustration purposes, I attached a table of my models. First (column 2), I use regular OLS without controls for endogeneity for consistency reasons. If Iam not mistaken, the results should be baised due to the use of the endogneous regressor ("Endogenous Variable") in my model. I only include the inverse mills ratio to account for potential selection bias. Second (column 3), I use the IV approach where the predicted probabilities "Endogenous Variable (Predicted)" are regressed against my dependent variable. Again, I also include the inverse mills ratio to account for a potential selection bias. Third (column 4), I use the control function approach to test the robustnes of the IV approach. Thus, I use regular OLS (similar to column 2) but add the binary endogenous variable ("Endogenous Variable") as well as the generalized residuals ("Lambda (VC)") to the regression.

I am wondering if my approach justifies my assumtpion to account for endogeneity correctly and that my results are not (as) biased as they would be when using my OLS model with just the endogenous binary regressor and no controls for endogeneity (as I do in column 2). Second, I wonder if my results are somehow biased since I include both, the inverse mills ratio as well as the generalized residuals which are both based on the same probit regression.

Would you agree with my approach or do you find improvements for my argumentation?

Your advice is much appreciated.

Thank you and best regards,
Alex

[Ibai Ostolozaga Falcon]

Hello Alexander,

Did you find an answer for your question? I have used the same methodology as you but I am not sure about the results.