Fixed effects in first-difference model with percentage changes? I'm estimating the following panel data model (years $t$, regions $i$) in Stata:
$$\%Δy_{ti} = \beta_0 + \beta_1 \%Δx_{ti} +  \beta_2 \%Δcontrols_{ti} \dots + e_{ti}$$
As far as I know, this is a first-difference model with the only distinction being that I use % change ($\%Δ$) instead of absolute change ($Δ$). There are about 125 regions $i$ and 3 three-year periods $t$.
My questions:


*

*I understood that region-fixed effects shouldn't be
included as this part is already taken care of by this differencing
method. Is this right? I'm a bit worried about this, since adding those fixed effects leads to a loss of significance of all coefficients (thus the choice is of huge impact). 

*I deliberately chose for the $\%Δ$ of the variables instead of the $Δ$, as this would yield elasticities in my results, and it felt like this would put the changes in perspective. Is this indeed better than looking at just $Δ$?

 A: First, assuming you do xtreg y x controls, fe, let's answer with a simple specification. Say you have $\forall \{t,i\}\in[1,T]\times[1,I]$ the following model specification
$y_{ti} = \beta_0 + X_{ti}\beta + e_{ti}= \beta_0 + X_{ti}\beta + \alpha_i + \varepsilon_{ti}$
where arbitrarily $\alpha_1$ is not specified (which is different from being $0$), and $\varepsilon_{ti}$ is the error component varying both over regions and over time, said to be idiosyncratic, .

The non-relative (and traditional) first-difference would have been

$y_{ti}-y_{(t-1)i} = (\beta_0 -\beta_0)+(X_{ti}-X_{(t-1)i})\beta + (\alpha_i-\alpha_i) + (\varepsilon_{ti}-\varepsilon_{(t-1)i})$
Instead of this, you choose to compute the following relative first-difference.
$-1 + \frac{y_{ti}}{y_{(t-1)i}} = \left(-1 + \frac{\beta_0}{\beta_0}\right)+\left(-1 + \frac{X_{ti}}{X_{(t-1)i}}\right)\beta + \left(-1 + \frac{\alpha_i}{\alpha_i}\right) + \left(-1 + \frac{\varepsilon_{ti}}{\varepsilon_{(t-1)i}}\right)$
or in continuous terms
$\ln\left(\frac{y_{ti}}{y_{(t-1)i}}\right) = \ln\left(\frac{\beta_0}{\beta_0}\right) + \ln\left(\frac{X_{ti}}{X_{(t-1)i}}\right)\beta + \ln\left(\frac{\alpha_i}{\alpha_i}\right) + \ln\left(\frac{\varepsilon_{ti}}{\varepsilon_{(t-1)i}}\right)$


*

*All these regions fixed-effects are removed by the first difference. So, this is right, region-fixed effects are already taken care of by the differencing method.



Thus, your first-difference model is

$\%\Delta y_{ti} = \%\Delta X_{ti}\beta + \%\Delta \varepsilon_{ti}$
And Stata's FE actually does 
$\%\Delta y_{ti} - \overline{\%\Delta y_{i} } + \overline{\overline{\%\Delta y}} = \left(\%\Delta X_{ti} - \overline{\%\Delta X_{i} } + \overline{\overline{\%\Delta X}} \right)\beta + \left(\%\Delta \varepsilon_{ti} - \overline{\%\Delta \varepsilon_{i} } + \overline{\overline{\%\Delta \varepsilon}} \right)$
Where $\overline{Z_i} \forall i$ stand for within region over-time averages, and $\overline{\overline{Z}}$ stand for a grand average over time and regions.


*If your question of research is about estimating short-run elasticities (between two years) controling for effects that are entailed in all regions over time (actually all what the FE model is about since it puts out of $\beta$ any region and region-time common denominator), your choice of using relative variations as input to an FE estimation procedure is fair. 



Note that you should check beforehand with a Durbin–Wu–Hausman test how such using an FE (vs RE) won't return consitent but inefficient estimates.
