The Sargan-Hansen test is a test of overidentifying restrictions. The joint
null hypothesis is that the instruments are valid instruments, i.e.,
uncorrelated with the error term,
The underidentification test is an LM test of whether the
equation is identified, i.e., that the excluded instruments are "relevant",
meaning correlated with the endogenous regressors. The test is essentially the
test of the rank of a matrix: under the null hypothesis that the equation is
underidentified
When errors are assumed to be i.i.d., ivreg2 automatically reports an LM version of the Anderson (1951) canonical correlations test. ivreg2 also reports the closely-related Cragg-Donald (1993) Wald test statistic. a result of rejection of the null should be treated with caution,
because weak instrument problems may still be present.
When the i.i.d. assumption is dropped,ivreg2 reports the LM and Wald versions of the Kleibergen-Paap (2006) rk statistic,