Despite reading some statistics manual and original papers I am still
not sure to have understood correctly the use and the alternative
hypotheses for several tests to be used after -xtreg-.
Command -xttest0- is referred to as "the Breusch and Pagan Lagrange-
multiplier test for random effects, a test that Var(v_i)=0". I argue
that if the test is significant, the model is better estimated
through -regress-. Am I right?
No, you have it backwards. The null for xttest0 is var(u) = 0. If
that was the case then there are no random effects. A significant
result rejects var(u)=0 in favor of var(u) > 0, in which case the
pooled OLS model that assumes that the error process has vce =
\sigma^2 I_{NT} is not the appropriate model. So a rejection
(significant test result) implies that you should NOT use OLS.
The Hausman test for FE vs RE has the null that both FE and RE are
consistent estimators because X \perp u, and since RE is more
efficient, it is to be preferred to FE. Under the null the
coefficient vectors from FE and RE should not differ significantly
for the common coefficients. Under the alternative X is correlated
with u, RE is inconsistent and you should use FE. So a significant
Hausman test statistic implies FE, as you state.