Randomly Assigned First Differences?

We consider treatment-effect estimation using a first-difference regression of an outcome evolution $\Delta Y$ on a treatment evolution $\Delta D$. Under a causal model in levels with a time-varying effect, the regression residual is a function of the period-one treatment $D_{1}$. Then, researchers should test if $\Delta D$ and $D_{1}$ are correlated: if they are, the regression may suffer from an omitted variable bias. To solve it, researchers may control nonparametrically for $E(\Delta D|D_{1})$. We use our results to revisit first-difference regressions estimated on the data of \cite{acemoglu2016import}, who study the effect of imports from China on US employment. $\Delta D$ and $D_{1}$ are strongly correlated, thus implying that first-difference regressions may be biased if the effect of Chinese imports changes over time. The coefficient on $\Delta D$ is no longer significant when controlling for $E(\Delta D|D_{1})$.