Sensitivity analysis and power in the presence of many weak instruments: Application to the effect of incarceration on future earnings

This article discusses a sensitivity analysis for an instrumental variable (IV) estimate in the presence of many instruments that are weakly associated with the endogenous variable. We study the effect of imprisonment on earnings using data on individuals sentenced for felony in Michigan in the years 2003–2006. Motivated by the random assignment of judges to cases, we construct a vector of instruments based on judges’ ID. Our data has two important features that cannot be handled using standard IV approaches. First, while some judges exhibit strong tendencies toward a prison or nonprison sentence, many judges do not have strong tendencies toward a particular sentence type. Second, our data includes only cases that result in sentencing, and thus the standard analyses are subject to selection bias. We develop a sensitivity analysis procedure that is robust to the presence of many weak instruments and quantifies the effect of the selection bias on the parameter of interest. A power formula for the sensitivity analysis is also provided. Analyses show that being sentenced to prison significantly reduces the offenders’ earnings. Our simulation studies highlight the value of the proposed method in terms of statistical power and also confirm the validity of our power formula.