Start-of period controls in first difference estimation

[Lea Birm]

Dear Stata-Users,
I am currently trying to understand the econometrics of the Paper of Dippel et al. (2015), where the authors estimate the causal effect of imports on voting behavior in Germany. Therefore, they estimate the following first-difference specification, where Yit refers to electoral outcomes, NetExposure refers to import - export exposure, τtr are time-varying fixed effects and Xit is a of control variables:



What I do not really get is why the authors include "undifferenced" controls (Xit) in the first difference model. Differencing the variables allows to get rid of unobserved time constant effects, so why would one not difference the control variables as well?

Can anyone explain the initiation behind it ?

Thank you in advance for any answers!

[Andrew Musau]

First-differencing can be used to eliminate the time-invariant individual heterogeneity in much the same way as fixed effects does. But if someone puts in some differenced variables in a model with other variables in levels, it does not mean that they are running a first-differences regression. \(\Delta y\) is literally "the change in y" and \(\Delta \; NetExposure\) is "the change in net exposure". So they are most likely interested in how a one unit increase in the change of net exposure affects the change in y after controlling for a number of things.

τtr are time-varying fixed effects

Fixed effects at what level? If this was a panel dataset and the fixed effects were at the panel-identifier level, assuming that this is indexed by \(i\), then you would expect the term to include the subscript \(i\). It may be poor notation on the part of the authors, but I suspect that this is not a panel dataset, but a repeated time-series. Ultimately, instead of asking a bunch of strangers, my advice is that you need to ask the authors (get it straight from the horse's mouth!).