Appendix-DoubleMachineLearningDensityEstimationfor LocalTreatmentEffectswithInstruments AIVSettingsandLTE

We will show the second. Notations We will usePϵ P(1+ϵg), whereg is a mean zero bounded random function, to denote aparametric submodel for the probability measureP. Also, we note that the causal effect 14 ψ[f(Y)] in Eq. (3) can be written asψYX[f(Y)]/ψX, where ψX and ψYX[f(Y)] are defined in Eqs.(4,7). We will first study the convergence behavior of ED[ϕm(ψh,η)]. We will show that EP[ED[ϕm(ψh,η)]] = O 1/ nhd .