alternative invariant endogenous variable in conditional logit

[Lei Zhang]

Dear all,

I have a case-control sample in which one case has several alternatives and only one will be chosen. My main analysis is a conditional logit model, in which Zi is an endogenous variable invariant across alternatives. All variables X1ij, X2ij, and Zi are continuous variables. Xii and X2ij are NOT dependent on Zi since they only come into play after Zi emerges. My conditional logit model looks like this.
πij=exp{yij}/∑exp{yik}
yij=beta0+beta1*X1ij+beta2*X2ij+beta3*Zi+beta4*Zi* X1ij+ uit
beta3 will not be estimated since Zi does not vary across alternatives, but beta4 will be.
I tried to deal with endogeneity issues in this model using control function (two-stage residual inclusion). In the first stage, I run a following model where Wi is the instrumental variable for Zi.
Zi=gama0+gama1*X1ij+gama2*X2ij+gama3*Wi+vij
Here are my questions:
1) Should I include X1ij and X2ij in the first stage model? In my case, Xii and X2ij are NOT dependent on Zi since they only come into play AFTER Zi emerges.
2) If I do not include X1ij and X2ij in first stage model, then the residual generated from first stage model will be invariant across alternatives. It will be dropped out of model when I include it in second stage model (conditional logit model). In this case, how should I deal with the issue of endogeneity?
3) If I do include X1ij and X2ij in first stage model, then I got "nonconcave" results. What might be a possible reason for that?

Thank you very much!

[Phil Bromiley]

I'm not sure what you mean by "x2ij are not dependent in Z since they only come into play after Z" - I would have thought this would mean they are dependent on Z.

While control function approaches seem extremely general, they must have some limitations. Has anyone dealt with this kind of model to show control functions give you consistent estimates?

You might look at doing this with SEM instead where you can model all of these directly.

[Lei Zhang]

I'm not sure what you mean by "x2ij are not dependent in Z since they only come into play after Z" - I would have thought this would mean they are dependent on Z.

While control function approaches seem extremely general, they must have some limitations. Has anyone dealt with this kind of model to show control functions give you consistent estimates?

You might look at doing this with SEM instead where you can model all of these directly.

Thank you!
X2ij is a variable comes after Zij happens in time, therefore it is not dependent on Zij.
I am not sure how I may do SEM considering I am using conditional logit for the matched sample.