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The homogeneous treatment effect and consistency implyInstrument validity gives $Z\perp R\mid X$.

If Assumption 1 holds, put $V=R-{\alpha }_{0}X$. Then $\mathbb{E}[V\mid X]=0$, so conditional instrument independence givesConsequently both population estimating equations vanish at $({\beta }_0,{\alpha }_0)$ for any probability limit of $\hat{\gamma }$.

If Assumption 2 holds, choose the linear-projection coefficientThen $\mathbb{E}[(R-{\alpha }_{\ast }X)X]=0$, whileThe first term is zero by conditional instrument independence and the second by the definition of ${\alpha }_{\ast }$. Thus $({\beta }_0,{\alpha }_{\ast })$ solves the population equations. Under the nonsingularity condition from part b, the root is unique, so standard estimating equation consistency proveswhenever either Assumption 1 or Assumption 2 holds. This is double robustness.

A sufficient causal condition is conditional exchangeabilitytogether with consistency of potential outcomes and positivity in causal inference. For the ordinary-least-squares coefficient itself to equal one common causal effect, also require the correctly specified additive conditional-mean modelThen $\beta$ is the homogeneous treatment effect, and the adjusted coefficient consistently estimates both conditional effects and the average treatment effect in the analyzed population.