First differences model: How to control for unobserved variables that vary over time but remain constant across entities?

[Cla Tuor]

Hello everyone,

I use a stacked first differences model to estimate the impact of globalization on the relative unemployment rate of unskilled labor in Western Europe.
I have a panel dataset with 16 countries observed at four different times. The estimated model is of the following form:

∆yit = β1∆xit + Ct-1 + vt + eit

∆yit = change in the relative unemployment rate (unskilled vs. skilled) in country i from time t-1 to t
∆xit = change in Chinese import exposure per worker in country i from time t-1 to t
C: a vector of control variables at the start of each period
vt: time dummy for each period

My understanding is that using such a stacked first difference model eliminates unobserved time-invariant variables that vary across countries. Hence, I don't have to include country-specific fixed effects.
How do I control for unobserved variables that are constant across countries but vary over time in such a model? Can one include time fixed effects in a first differences model?

Thank you for your help!

[Sebastian Kripfganz]

[SIZE=14px]How do I control for unobserved variables that are constant across countries but vary over time in such a model? Can one include time fixed effects in a first differences model?

You can simply include time dummies.

[Cla Tuor]

Thank you Sebastian!

Hence, I estimate the following model for t = 2, 3, 4 (change from t-1 to t is stored in the time t record):

∆yit = α0 +α3D3t + α4D4t + β1∆xit + γCit-1 + ∆εit

• I control for time-invariant variables that vary across countries by using the first differences model.
• I control for time-variant variables that are constant across Western European countries (e.g. technological change) by using time dummies (D3t, D4t).
• I added an intercept (α0)since Woolridge recommends adding an intercept in the first-differenced equation if the time intercepts are not of direct interest.

Have I specified the model correctly?
Are the first two bullet points correct?

Thank you!

[Sebastian Kripfganz]

That sounds alright.

In principle, you could still add country-fixed effects to the first-differenced model. They would capture different time trends across countries because a linear time trend in levels becomes a constant in first differences. If you think that time trends are similar across countries, then first differencing is sufficient.

[Cla Tuor]

Thank you!

Adding country-fixed effects to my model yields following model: ∆yit = β1∆xit + γCt-1 + αi + vt + ∆εit

However, adding country-fixed effects (αi) cancels them out in a first-difference model, don't they?
What am I missing?

Best,
Cla

[Fei Wang]

Cla Tuor , yes, the αi in your original model is canceled out in the FD, but as Sebastian suggests, you may have a country-specific linear time trend in your original model (αi*t) -- After FD, this interaction becomes αi, as in #5.

[Cla Tuor]

Fei Wang Thank you!

Do I have to note αi or αi*t when specifying the first difference model in my paper?

[Fei Wang]

Fei Wang Thank you!

Do I have to note αi or αi*t when specifying the first difference model in my paper?

If your original model (before FD) contains αi*t, then the FD model needs to contain αi. If the original model doesn't contain αi*t, then no αi is needed in your FD model. You may not contain these terms if you assume in the original model that all countries have common trend.

[Cla Tuor]

Thanks for your help!