Instrumental variable estimation with first-stage heterogeneity

We propose a simple data-driven procedure that exploits heterogeneity in the first-stage correlation between an instrument and an endogenous variable to improve the asymptotic mean squared error (MSE) of instrumental variable estimators. We show that the resulting gains in asymptotic MSE can be quite large in settings where there is substantial heterogeneity in the first-stage parameters. We also show that a naive procedure used in some applied work, which consists of selecting the composition of the sample based on the value of the first-stage $t$-statistic, may cause substantial over-rejection of a null hypothesis on a second-stage parameter. We apply the methods to study (1) the return to schooling using the minimum school leaving age as the exogenous instrument and (2) the effect of local economic conditions on voter turnout using energy supply shocks as the source of identification.