The Causal Interpretation of Two-Stage Least Squares with Multiple Instrumental Variables
研究发现，当使用多个工具变量时，标准单调性条件要求行为同质，因此提出部分单调性条件，并给出可验证的充分必要条件，使两阶段最小二乘估计量仍可解释为局部平均处理效应的正加权平均。
Empirical researchers often combine multiple instrumental variables (IVs) for a single treatment using two-stage least squares (2SLS). When treatment effects are heterogeneous, a common justification for including multiple IVs is that the 2SLS estimand can be given a causal interpretation as a positively weighted average of local average treatment effects (LATEs). This justification requires the well-known monotonicity condition. However, we show that with more than one instrument, this condition can only be satisfied if choice behavior is effectively homogeneous. Based on this finding, we consider the use of multiple IVs under a weaker, partial monotonicity condition. We characterize empirically verifiable sufficient and necessary conditions for the 2SLS estimand to be a positively weighted average of LATEs under partial monotonicity. We apply these results to an empirical analysis of the returns to college with multiple instruments. We show that the standard monotonicity condition is at odds with the data. Nevertheless, our empirical checks reveal that the 2SLS estimate retains a causal interpretation as a positively weighted average of the effects of college attendance among complier groups.