Jarque–Bera test

In statistics, the Jarque-Bera test is a goodness-of-fit measure of departure from normality, based on the sample kurtosis and skewness. The test statistic JB is defined as

${\mathit {JB}}={\frac {n}{6}}\left(S^{2}+{\frac {(K-3)^{2}}{4}}\right),$

where n is the number of observations (or degrees of freedom in general); S is the sample skewness, K is the sample kurtosis, defined as

$S={\frac {\mu _{3}}{\sigma ^{3}}}={\frac {\mu _{3}}{\left(\sigma ^{2}\right)^{3/2}}}={\frac {{\frac {1}{n}}\sum _{i=1}^{n}\left(x-{\bar {x}}\right)^{3}}{\left({\frac {1}{n}}\sum _{i=1}^{n}\left(x-{\bar {x}}\right)^{2}\right)^{3/2}}}$

$K={\frac {\mu _{4}}{\sigma ^{4}}}={\frac {\mu _{4}}{\left(\sigma ^{2}\right)^{2}}}={\frac {{\frac {1}{n}}\sum _{i=1}^{n}\left(x-{\bar {x}}\right)^{4}}{\left({\frac {1}{n}}\sum _{i=1}^{n}\left(x-{\bar {x}}\right)^{2}\right)^{2}}}$

where μ₃ and μ₄ are the third and fourth central moments, respectively, ${\bar {x}}$ is the sample mean, and σ² is the second central moment, the variance.

The statistic JB has an asymptotic chi-square distribution with two degrees of freedom and can be used to test the null hypothesis that the data are from a normal distribution. The null hypothesis is a joint hypothesis of both the skewness and excess kurtosis being 0, since samples from a normal distribution have an expected skewness of 0 and an expected excess kurtosis of 0. As the definition of JB shows, any deviation from this increases the JB statistic.

References

• Bera, Anil K. (1980). "Efficient tests for normality, homoscedasticity and serial independence of regression residuals". Economics Letters. 6 (3): 255–259. doi:10.1016/0165-1765(80)90024-5. {{cite journal}}: Cite has empty unknown parameter: |month= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Bera, Anil K. (1981). "Efficient tests for normality, homoscedasticity and serial independence of regression residuals: Monte Carlo evidence". Economics Letters. 7 (4): 313–318. doi:10.1016/0165-1765(81)90035-5. {{cite journal}}: Cite has empty unknown parameter: |month= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Judge (1988). Introduction and the Theory and Practice of Econometrics (3rd edn. ed.). pp. 890–892. {{cite book}}: |edition= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)