Including not all exogenous variables in the first stage IV regression

Question

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!

Answer 1

The reason to include all exogenous variable in the first-stage regression is because the endogenous variable $X$ could be correlated with other independent regressors in the $Y$ equation, let say a $\mathbf x_z$ vector. If $X$ is correlated with the order regressors $\mathbf x_z$, the other regressor would be correlated with the error term $u$, introducing bias.

Without including all exogenous variables the IV estimatior would be inconsistent. If $EX1$ is irrelevant to explain $X$ its coefficient on the first stage regression would be close to zero, so I don't see why including it would be a problem. Wooldridge (2002), Econometric Analysis of Cross Section and Panel Data, pag, 91 says: "Problem 5.11 asks you to show that omitting $x_1; . . . ; x_K$ in the first-stage regression and then explicitly doing the second-stage regression produces inconsistent estimators of the $\beta_j$"