I am using this forum for the first time and hope to get comments on my question that I have been struggling to answer for quite some time now.
My dependent variable is a bounded variable Y [0 to 1] and I am using a GLM with a probit link. This variable is measured at time t.
The main independent variable is a binary variable X [0 and 1] and is believed to be endogenous. This variable is measured at t-1. Note that this does not mean periods. t-1 means few days before t in some, or few months before t in other cases. There are other independent exogenous variables. I would like to control for endogeneity using a two stage IV approach.
1st stage: I probit regress X against IVs that affect X but uncorrelated with Y directly. Most of the posts in this forum strongly suggest to also include all exogenous variables in the first stage because of econometric reasons. My problem is that some of these exogenous variables (EX1) are measured after t-1 but before t. Others (EX2) are measured before t-1.
Basically, it does not make sense to include these EX1 in the first stage regression as they cannot explain something (X) that happened before. I will try to illustrate this:
In this situation, would be OK to include EX2 only in the first stage regression and to include all exogenous variables in the second regression as shown below?
1 stage (Probit): X = aIVs + bEX2 + e
2 stage (GLM with Probit): Y = Xhat + aEX1 + bEX2 + e
Thank you very much in advance! Your help is much appreciated!