Despite reading and conducting several practical examples with IV 2SLS, I am still uncertain how, specifically and mathematically, 2SLS is able to obtain a causal coefficient, β, of an assumed endogenous variable, X, on an outcome, Y.
From what I gather, 2SLS follows this logic:
First stage: We regress the endogenous variable, X, on all exogenous variables including the instrument(-s). We then store the predicted value of X.
Second stage: In the second stage regression, the predicted value of X now replaces the endogenous variable, consequently β now represents an "isolated" causal coefficient for X on Y.
At a conceptual level, I understand that the first stage somehow removes the correlation between the X variable and the error term, ϵ. So, when we in the second stage replace X with the predicted value of X we obtain a causal coefficient, β, for the effect of X on Y. However, I am unsure about the mathematics regarding the "isolation" of the causal effect of X on Y. Thus, the main question is what is the specific mathematical operation that makes β a causal coefficient for the effect of X on Y in the second stage regression?
Another post (What is an instrumental variable?) vividly describes how 2SLS can single out the explained and unexplained variation of an endogenous variable by the two-stage procedure. However, the example is based on a first stage that regresses the endogenous variable on the instrument, thereafter you plug the predicted value of X into the second stage regression. While illustrating, I am unsure how this translates to a more conventional example where you use 2SLS with an endogenous variable, multiple explanatory variables and one instrument.