This will be obvious to anyone with a good understanding of econometrics, but it is quite stunning really to think that all the information you see in the first set of graphs in my previous post on the EKC is thrown away by fixed effects panel estimators. That is because the graphs plot the mean value over time in each country of the dependent variable against the mean value over time in each country of the explanatory variable. Fixed effects estimation first deducts these means from the data and then estimates the regression of the two residuals using ordinary least squares. This is why fixed effects is also called the "within estimator" because the "between (country) variation" you see in these graphs is ignored. Of course, you can estimate a model that just exploits this between variation using the between estimator.*
The reason the latter estimator is rarely used is because researchers are worried about omitted variables bias. Any omitted variables are subsumed in the error term while the fixed effects estimator eliminates their country specific means and so reduces the potential bias. Hauk and Wacziarg (2009), however, found that when there is also measurement error in the explanatory variables (which can also bias the regression estimates) the between estimator performs well compared to alternatives. Fixed effects estimation tends to inflate the effect of the measurement error.
Differenced estimators sweep out any country fixed effects in the differencing operation.** So they also remove all the between variation in the data. However, they do allow us to include country characteristics that are constant over time to explain differences in growth rates across countries, which standard fixed effects does not allow.***
* The linked paper was eventually published in Ecological Economics.
** For a two period panel, fixed effects and first differences produce identical results.
*** There are variations of fixed effects that can allow this.