Poorly Measured Confounders are More Useful on the Left than on the Right
研究发现，在回归分析中，用测量不佳的混杂变量做稳健性检验时，将其放在因变量位置比放在自变量位置能更有效地检验识别假设，并提供了估计量、检验统计量和统计功效的计算方法，对应用研究者有实际帮助。
Researchers frequently test identifying assumptions in regression-based research designs (which include instrumental variables or difference-in-differences models) by adding additional control variables on the right-hand side of the regression. If such additions do not affect the coefficient of interest (much), a study is presumed to be reliable. We caution that such invariance may result from the fact that the observed variables used in such robustness checks are often poor measures of the potential underlying confounders. In this case, a more powerful test of the identifying assumption is to put the variable on the left-hand side of the candidate regression. We provide derivations for the estimators and test statistics involved, as well as power calculations, which can help applied researchers interpret their findings. We illustrate these results in the context of estimating the returns to schooling.