We derive asymptotic expansions of the distributions of test statistics for over-identifying restrictions in a system of simultaneous equations under the null and the non-null hypotheses. We investigate the effects of the normality assumption for disturbances on the test statistics based on their asymptotic expansions. We also study the power functions of test statistics based on their asymptotic expansions. After modifying their critical regions to the same significance level, the power function of Basmann's statistic is larger than that of the likelihood ratio test when the variance of disturbances is sufficiently small. However, the difference in powers of the two test statistics disappears as the sample size grows larger.