Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 221 / 2 / d / Solution

LetAssumption 2 implies ${\gamma }_0=q/d$. Although the printed $\hat{\gamma }$ need not converge to ${\gamma }_0$, its first-order effect vanishes because $\mathbb{E}[VX]=0$. Inverting the $2\times 2$ Jacobian of the remaining estimating equations gives the influence functionThe first-stage condition yieldsHence the asymptotic variance of $\sqrt{n}(\hat{\beta }-{\beta }_0)$ is the sandwich expression

There is a defect in the printed assumptions: $\mathrm{Var}(V\mid A,X)={\sigma }^2$ alone does not determine $\mathbb{E}[V^2{R}_0^2]$, because $R_0$ depends on $Z$ and $V$ can have a nonzero conditional mean given $(A,X)$ under unmeasured confounding. Under the standard intended strengthening $\mathbb{E}[V^2\mid X,Z]={\sigma }^2$, the formula simplifies toThe asymptotic variance of $\hat{\beta }$ itself is ${\mathcal{V}}_{\beta }/n$.