Randomly Assigned First Differences?

TL;DR

This paper examines the potential bias in first-difference regressions caused by correlation between treatment changes and initial treatment levels, proposing a nonparametric control method and illustrating its importance with empirical data.

Contribution

It identifies a source of bias in first-difference regressions under time-varying effects and offers a nonparametric control approach to address this issue.

Findings

Strong correlation between treatment change and initial treatment level.

Controlling for this correlation can eliminate bias in estimates.

Empirical application shows the significance of the correction.

Abstract

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 $E(\Delta D|D_{1})$.