Randomly Assigned First Differences?

I consider treatment-effect estimation with a two-periods panel, using a first-difference regression of the outcome evolution $\Delta Y_g$ on the treatment evolution $\Delta D_g$. To justify this regression, one may assume that $\Delta D_g$ is as good as randomly assigned, namely uncorrelated to the residual of the first-differenced model and to the treatment's effect. This paper shows that if one posits a causal model in levels between the treatment and the outcome, then the residual of the first-differenced model is a function of the period-one treatment $D_{g,1}$, so the first-difference regression may suffer from an omitted variable bias whenever $\Delta D_g$ is correlated to $D_{g,1}$. To solve this problem, one can control for $E(\Delta D_g|D_{g,1})$ in the regression. I apply these results to a first-difference regression of the 1999 to 2007 employment evolutions of US industries on the evolution of their Chinese imports penetration ratio, estimated on the data of Acemoglu et al (2016). The first-difference regression indicates a large negative effect of Chinese imports on US employment, but $\Delta D_g$ and $D_{g,1}$ are very strongly correlated, so this regression may suffer from an omitted variable bias. If one controls for $E(\Delta D_g|D_{g,1})$, the effect of Chinese imports penetration on US employment is 33\% to 41\% smaller than in the first-difference regression, and less significant.