How to implement Hansen J-test for selection model "by hand"

[Doro Kuebler]

Dear all,

I am estimating a 2 stage model of the form
stage1: P(Event==1) = f(Z,X)
stage2: Y = f(X, inverse mills ratio from the first stage)

Z includes up to three instruments, there is one variable to be instrumented (selection).

I do this "by hand", i.e. not with the heckman command, but rather a linear probability model in the first stage with several dummy varibles, calculating the IMR and including it in the linear second stage. I am wondering how to test overidentifying restrictions.

I read a lot of different approaches on this, ranging from simple a pairwise correlation of the 2nd stage residuals (having used all instruments to calculate them) and the instruments themselves (but I guess this is wrong), to, say calculating the model with Z1,Z2 only and then regressing the second stage residuals on Z3 with OLS and checking significance of Z3's slope parameter.

could you please help me with his, the older threads I find concerning related issues typically use estat overid / ivergress.... which does not apply to my specific case.

Thank you!
AF

[Doro Kuebler]

hello! so now I understand that I should calculate 2nd stage residuals and that the test statistic, N*R^2, is put to a chi2 test.

The one thing that I do not yet fully understand, here is where some sources are unclear/contradictory: do I calculate the 2nd stage residuals with all the instruments, or do I always have to leave at least one out for it to be overidentifying? as an example, imagine I have 2 selection variables Z1 and Z2. will I use

a) both of them to get 2nd stage residuals via the inverse mills ratio

or

b) do I calculate two inverse mills ratios, one with Z1 only (and one with Z2 only), and then regress the 2nd stage residuals on Z2 only (Z1 only)

I know this is more of a general statistics question, rather than a Stata question, but I hope you can tell me nevertheless.