Randomly Assigned First Differences?

Abstract

We consider a first-difference regression of an outcome evolution $\hat{I}”Y$ on a treatment evolution $\hat{I}”D$. If the treatment effect changes over time, the regression residual is a function of the period-one treatment $D_1$. Then, researchers should test if $\hat{I}”D$ and $D_1$ are correlated: if they are, the regression may suffer from an omitted variable bias. To solve it, researchers may control for $E(\hat{I}”D|D_1)$. We revisit Acemoglu et al (2016), who study the effect of imports from China on US employment. $\hat{I}”D$ and $D_1$ are correlated. $\hat{I}”D$'s coefficient is less negative when controlling for $E(\hat{I}”D|D_1)$.