Question on Implementation of First-Difference Regression and Interpretation

[Leon Schmidt]

Dear All,

I am trying to estimate the effect of tariffs changes on employment within firms. Thus, I run a regression like

xtreg ln_employment ln_tariffs i.year, fe robust

where ln is the natural logarithm. The interpretation of that would be that a one percentage point increase in tariffs is associated with a b1 percent change in employment.

What I see quite often in the literature is that they estimate some regression like DELTA(ln_employment) = DELTA(ln_tariffs) + error to account for the variables being non-stationary.

Would the following code implement this correctly in Stata

xtreg d.ln_employment d.ln_tariffs i.year, fe, robust

? Further, I am unsure how to interpret it since the variables are differenced and in natural logarithms. Intuitively, I´d say that "an increase of one percentage point in the growth of tariffs is associated with a b1 percent increase in the growth of employment". Would that be correct?

Thank you very much for your help!

All the best
Leon

[Jesse Wursten]

The Stata syntax is correct. The interpretation of variables remains the same. That is, a doubling of the tariff (-> d.ln_tariffs = 1) leads to a b1 percent change in ln_employment (d.ln_emp = +b1). I personally find the wording in growth misleading, as that implies a permanent effect (to me, but YMMV). That's only the case if emp is in differences but tariff is in levels. Whereas if they are both in differences, then the effect of the change in tariffs dies out immediately in the next period. If you do not find this plausible, you could include lags and leads of the tariff variable, or switch to the local projection method (Jorda, 2005).

[Leon Schmidt]

Thank you very much Jesse for the very helpful response!

Is it generally the case that the interpretation of a regression in first difference doesn´t change? I´m not exactly sure why it would stay the same. So also if the tariffs are not in ln (as in this regression

xtreg d.ln_employment d.tariffs i.year, fe robust

) I could say a one percentage point increase in tariffs is associated with a 100*b1 percent change in tariffs?

Thank you again very much!

[Jesse Wursten]

An increase of one unit in d.tariffs will lead to an increase of b1 units in d.ln_employment. If d.tariffs increases by one, that means tariffs increases by one. If d.ln_employment increases by one, ln_employment increases by one. As a result, you can interpret the first difference model coefficient the same as their levels equivalent.

[Leon Schmidt]

Thanks a lot!

[Eric de Souza]

Jeffrey Wooldridge has repeated several times here that we must carefully distinguish between the model and estimation methoed.
If the variables in your model is in levels, and the transformation into first differences is just for estimation purposes (to put numbers to the coefficients, so to say), the coefficients should be interpreted using the levels model. If your model is in first differences -- for instance, you take the log and then the first difference to obtain growih rates -- then the coefficients should be interpreted in terms of growth rates.

[Leon Schmidt]

Thank you very much for this explanation!