Identification With Imperfect Instruments
提出在工具变量与误差项相关时，通过假设相关性符号相同且工具变量相关性更弱，推导出参数的分析边界，并用差异化产品需求估计展示其有效性。
Dealing with endogenous regressors is a central challenge of applied research. The standard solution is to use instrumental variables that are assumed to be uncorrelated with unobservables. We instead allow the instrumental variable to be correlated with the error term, but we assume the correlation between the instrumental variable and the error term has the same sign as the correlation between the endogenous regressor and the error term and that the instrumental variable is less correlated with the error term than is the endogenous regressor. Using these assumptions, we derive analytic bounds for the parameters. We demonstrate that the method can generate useful (set) estimates by using it to estimate demand for differentiated products. © 2012 The President and Fellows of Harvard College and the Massachusetts Institute of Technology.