Three different measures of sample skewness and kurtosis and their effects on the Jarque–Bera test for normality

Monte Carlo methods are used to study the size and the power of three versions of the Jarque and Bera Lagrangian multiplier test for normality, JB(g<SUB align="right"><SMALL>1</SMALL></SUB>, g<SUB align="right"><SMALL>2</SMALL></SUB>), JB(b<SUB align="right"><SMALL>1</SMALL></SUB>, b<SUB align="right"><SMALL>2</SMALL></SUB>) and, finally, JB(k<SUB align="right"><SMALL>1</SMALL></SUB>, k<SUB align="right"><SMALL>2</SMALL></SUB>). The difference between these tests comes from the different definitions (estimates) of sample skewness and kurtosis. The Jarque and Bera test has rather poor small sample properties: the slow convergence of the test statistic to its limiting distribution makes the test oversized for small nominal level and undersized for larger than 3% levels even in a reasonably large sample. However, the JB(k<SUB align="right"><SMALL>1</SMALL></SUB>, k<SUB align="right"><SMALL>2</SMALL></SUB>) for a 5% nominal level shows good properties for all samples. The power of the tests shows the same erratic form.