Before revision:  In contrast to the Jarque-Bera test that is based on the third and fourth moment, our test relies on the conditional second moments.
After revision:  We derive asymptotic distributions for conditional trimmed mean and variance estimators, and use this to show asymptotic normality of the proposed test statistic. Using financial empirical data, we compare our framework to various statistical tests including the Jarque-Bera test that is based on the third and fourth moment, our test relies on the conditional second moments.