Treatment effect heterogeneity with respect to covariates is common in instrumental variable (IV) analyses. An intuitive approach, which we call the interacted two-stage least squares (2sls), is to postulate a working linear model of the outcome on the treatment, covariates, and treatment-covariate interactions, and instrument it using the IV, covariates, and IV-covariate interactions. We clarify the causal interpretation of the interacted 2sls under the local average treatment effect (LATE) framework when the IV is valid conditional on the covariates. Our main findings are threefold. First, we show that the coefficients on the treatment-covariate interactions from the interacted 2sls are consistent for estimating treatment effect heterogeneity with respect to covariates among compliers for any outcome-generating process if and only if the product of the IV propensity score and covariates are linear in the covariates, referred to as the linear IV-covariate interactions condition. Second, assuming that the covariate vector has dimension K and includes a constant term, we show that the linear IV-covariate interactions condition holds only if the IV propensity score takes at most K distinct values. As a result, this condition is difficult to satisfy beyond two special cases: (a) the covariates are categorical with K levels, or (b) the IV is randomly assigned. These results underscore the difficulty of interpreting regression coefficients from specifications with treatment-covariate interactions when the covariates are not saturated and the IV is not unconditionally randomized, absent correct specification of the outcome model. Third, as an application of our theory, we show that the interacted 2sls with demeaned covariates is consistent for estimating the LATE under the linear IV-covariate interactions condition.

翻译：在工具变量（IV）分析中，处理效应相对于协变量的异质性是一种常见现象。一种直观的方法（我们称之为交互式两阶段最小二乘法，即交互式2SLS）是建立结果变量关于处理变量、协变量以及处理-协变量交互项的线性工作模型，并利用工具变量、协变量以及工具变量-协变量交互项作为工具变量进行估计。本文在工具变量条件有效的前提下，基于局部平均处理效应（LATE）框架阐明了交互式2SLS的因果解释。我们的主要发现包含三个方面。首先，我们证明当且仅当工具变量倾向得分与协变量的乘积在协变量上呈线性关系（称为线性工具变量-协变量交互条件）时，交互式2SLS中处理-协变量交互项的系数能够一致地估计依从者群体中处理效应相对于协变量的异质性，且该结论对任意结果生成过程均成立。其次，假设协变量向量维度为K且包含常数项，我们证明线性工具变量-协变量交互条件成立仅当工具变量倾向得分至多取K个不同值。因此，除以下两种特殊情况外，该条件很难满足：（a）协变量为具有K个水平的分类变量；（b）工具变量被随机分配。这些结果突显了在协变量未饱和且工具变量非无条件随机化的情形下，若未正确设定结果模型，则难以解释包含处理-协变量交互项的回归系数。第三，作为我们理论的应用，我们证明在线性工具变量-协变量交互条件成立时，采用去中心化协变量的交互式2SLS能够一致地估计局部平均处理效应。