Inspired by the probit 2SLS estimation (see e.g. Wooldridge p.623, procedure 18.1 or check here Probit two-stage least squares (2SLS)), I am wondering if instead of running a Probit in the very first step, I run a propensity score matching regression in the very first step and get predicted values of the endogeneous variable from the regression, and then use the predicted values (as instrument) together with the exogenous variables to run a conventional 2SLS in the second and third steps, whether it will give me consistent estimates? Any possible references about this?
What you describe in your question is exactly the three-step procedure outlined in the linked answer regarding the other question about probit two stage least squares. The first of these steps can be thought of estimating a propensity score. One way of estimating the propensity score is via a probit regression so what you call "propensity score matching regression" boils down to that.