 Solution

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.