Instrument-residual estimator for multi-valued instruments under full monotonicity

In determining the effects of a binary treatment $D$ on an outcome $Y$, a multi-valued instrumental variable (IV) $Z=0,1,\dots ,J$ often appears. Imbens and Angrist (1994, Econometrica) showed that the IV estimator (IVE) of $Y$ on $D$ using $Z$ as an IV is consistent for a non-negatively weighted average of heterogeneous “complier” effects. However, Imbens and Angrist did not include covariates $X$. This paper generalizes their finding by explicitly allowing $X$ to appear in the linear model for the IVE, and shows that the extra condition $E\left(Z\right|X)=L\left(Z\right|X)$ is necessary for generalization, where $L\left(Z\right|X)\equiv E\left(Z{X}^{\prime }\right){\left\{E\left(X{X}^{\prime }\right)\right\}}^{-1}X$ is the linear projection. This paper therefore proposes an alternative IVE using $Z-E\left(Z\right|X)$ as an IV, which is consistent for the same estimand without the restrictive extra condition. A simulation study demonstrates that the extra condition $E\left(Z\right|X)=L\left(Z\right|X)$ is necessary for the usual IVE, but not for the alternative IVE proposed in this paper.