xttest3 calculates a modified Wald statistic for groupwise heteroskedasticity
in the residuals of a fixed effect regression model, following Greene
(2000, p. 598). xtreg, fe estimates this model assuming homoskedasticity.
The most likely deviation from homoskedastic errors in the context of pooled
cross-section time-series data (or panel data) is likely to be error variances
specific to the cross-sectional unit. xttest3 tests the hypothesis that
sigma^2(i)==sigma for i=1,N_g, where N_g is the number of cross-sectional
units. The resulting test statistic is distributed Chi-squared(N_g) under the
null hypothesis of homoskedasticity.