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,…,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(Z|X)=L(Z|X) is necessary for generalization, where L(Z|X)≡E(ZX^′){E(XX^′)}⁻¹X is the linear projection. This paper therefore proposes an alternative IVE using Z−E(Z|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(Z|X)=L(Z|X) is necessary for the usual IVE, but not for the alternative IVE proposed in this paper.