Fixed effect error term assumptions Does the error term in the fixed model follow the assumptions of the OLS (normality, autocorrelation, homoscedasticity)? 
Say there are two countries 1 and 2 and I get the residuals for each country. 
What are the assumptions on the two error terms? Should the error term for each country follow the OLS assumption or the stacked ones (stacking one residual on another) follow the OLS assumptions? Could you please explain this? 
 A: See Wooldridge Econometric Analysis of Cross Section and Panel Data
The fixed-effects model 
$y_{it} = X_{it}\beta + c_i + u_{it}$
assumes (necessary for consistency)


*

*E$\big(u_{it}\,|\,x_{it},c_i\big)=0$.

*rank$\Big[\text{E}\big(\ddot{X}^T\ddot{X}\big)\Big]=$ K
where $\ddot{x}_{it}=x_{it}-\bar{x}_i$, and for efficiency


*E$\big(u_iu_i^T\,|\,x_{it},c_i\big)=\sigma^2_u I_T$


If you look at the time-demeaned equation
$\ddot{y}_{it}=\ddot{x}_{it}\beta+\ddot{u}_{it}$
it can be easily shown that
$E\big(\ddot{u}_{it},\ddot{u}_{is}\big)=-\frac{1}{T}\sigma^2_u\stackrel{!}{=}0$
i.e. there issome autocorrelation which tends to go 
to zero for $T\rightarrow\infty$.
The residual variance $\hat{\sigma}^2_{u^\text{FE}}$ can be estimated via
$\hat{\sigma}^2_{u^\text{FE}}=\frac{\sum\sum \ddot{u}_{it}}{N(T-1)-K}$
which differs from the residual variance of the pooled OLS estimator 
(because of the time-demeaning) which is 
$\hat{\sigma}^2_{u^\text{POLS}}=\frac{\sum\sum u_{it}}{NT-K}$
But you can easily correct for the standard error if you assume 
assumption 3. If not, you can calculate robust standard error. 
A: If the original error terms from your model are IID, then fixed effects is appropriate. If they instead follow a random walk, first differences will be efficient rather than fixed effects. 
But first differences is consistent under the slightly more general assumption of sequential exogeneity, whereas fixed effects require the error terms to be strictly exogenous. This rules out so-called feedback effects, where a regressor in one period is correlated with an error term in the future or past. This is the only sense in which the required assumptions are stricter for fixed effects than for pooled OLS. 
