INSTRUMENTAL VARIABLES METHODS WITH HETEROGENEITY AND MISMEASURED INSTRUMENTS
研究了在有序离散选择结构系统中，用有测量误差的代理变量W替代工具变量Z时，Wald、局部工具变量和工具变量估计量的解释，发现W可识别加权平均的局部或边际处理效应，且在特定条件下与使用Z得到的结果相同。
We study the consequences of substituting an error-laden proxy W for an instrument Z on the interpretation of Wald, local instrumental variable (LIV), and instrumental variable (IV) estimands in an ordered discrete choice structural system with heterogeneity. A proxy W need only satisfy an exclusion restriction and that the treatment and outcome are mean independent from W given Z . Unlike Z , W need not satisfy monotonicity and may, under particular specifications, fail exogeneity. For example, W could code Z with error, with missing observations, or coarsely. We show that Wald, LIV, and IV estimands using W identify weighted averages of local or marginal treatment effects (LATEs or MTEs). We study a necessary and sufficient condition for nonnegative weights. Further, we study a condition under which the Wald or LIV estimand using W identifies the same LATE or MTE that would have been recovered had Z been observed. For example, this holds for binary Z and therefore the Wald estimand using W identifies the same “average causal response,” or LATE for binary treatment, that would have been recovered using Z . Also, under this condition, LIV using W can be used to identify MTE and average treatment effects for e.g., the population, treated, and untreated.