alternative invariant endogenous variable in conditional logit

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!