It might be a basic question but since fixed effects estimator either mean centers the data or uses first differences, is it entirely wrong to take first differences of the data and then run fixed effects regression in Stata?
Here is the explanation of the problem. So my model is the following
$\Delta LP = \alpha+\beta_1 \Delta HC+\beta_2 \Delta FDI+\beta_3 \Delta HC*\Delta FDI$
Where LP is labor productivity, HC is human capital and FDI is foreign direct investment.
Since I have panel of 21 countries for 16 years, I need to control for country specific effects and for that Hausman test suggested fixed effects model. However, when we use fixed effects model, it automatically uses first differences of the data. So my question was, is it ok to use first differenced data in fixed effects model? wouldn't it be double first differencing? thank you.