st: Possibility of estimating idiosyncratic errors

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Dear members of Statalist.

I'm running a fixed-effects (within) regression on panel data consisting of 50 countries, and between 2 to 4 time periods.

One of the FE assumptions is that the idiosyncratic errors (the unobserved time-varying factors that affect the dependent variable) must be normally distributed (''conditional on the explanatory variables and the unobserved time-constant effect''), unless I have a large number of countries and few time periods (Wooldridge 2013, p. 690).

I understand that the normality assumption is not particularly important when working with large samples, but I would still like to know if it's possible to test the assumption in Stata 12. I haven't found any info on this when it comes to panel data and fixed-effects, but only for cross-sections and OLS. 

I also found a passage in Wooldridge (2013) that made me think that maybe testing this assumption after FE estimation is not possible. This because in another context, Wooldridge (2013, p. 472) writes that '’it is difficult to test whether the idiosyncratic errors u(sub-it) are serially uncorrelated after FE estimation: we can estimate the time-demeaned errors but not the u(sub-it).’’

''
Since the time-demeaning that is used when using FE estimation leaves us with time-demeaned errors (and not the idiosyncratic errors as in the ''original'' unobserved effects model), then this should imply that we cannot really estimate the idiosyncratic errors at all, and therefore that the residuals I get when writing ''predict residuals, e'' after xtreg are not really estimates of the idiosyncratic errors but only of the time-demeaned errors. Is this a correct interpretation? If so, is it impossible to test the normality assumption?

I'm sorry if this is a stupid question, but I would really appreciate any kind of short answer!

Best,
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