Jarque-Bera test

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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

$S = \frac{ \mu_3 }{ \sigma^3 } = \frac{ \mu_3 }{ \left( \sigma^2 \right)^{3/2} } = \frac{ \frac{1}{n} \sum_{i=1}^n \left( x_i - \bar{x} \right)^3}{ \left( \frac{1}{n} \sum_{i=1}^n \left( x_i - \bar{x} \right)^2 \right)^{3/2}}$

$K = \frac{ \mu_4 }{ \sigma^4 } = \frac{ \mu_4 }{ \left( \sigma^2 \right)^{2} } = \frac{\frac{1}{n} \sum_{i=1}^n \left( x_i - \bar{x} \right)^4}{\left( \frac{1}{n} \sum_{i=1}^n \left( x_i - \bar{x} \right)^2 \right)^2}$

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 the skewness being zero and the excess kurtosis being 0, since samples from a normal distribution have an expected skewness of 0 and an expected excess kurtosis of 0 (which is the same as a kurtosis of 3). As the definition of JB shows, any deviation from this increases the JB statistic.

[edit] References

Bera, Anil K.; Carlos M. Jarque (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.
Bera, Anil K.; Carlos M. Jarque (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.
Judge; et al. (1988). Introduction and the Theory and Practice of Econometrics, 3rd edn., 890–892.