Jarque Bera test

The Jarque-Bera test is a statistical test that uses the skewness and kurtosis in the data to check whether the distribution is normal . It is therefore a special adaptation test . The test was suggested by Carlos M. Jarque and Anil K. Bera .

definition

The test statistic JB of the Jarque Bera test is defined as

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

Where is the number of observations ; with is the skewness and with the kurtosis . ${\displaystyle n}$ ${\displaystyle x_{1},\ldots ,x_{n}}$ ${\displaystyle S}$ ${\displaystyle K}$

The skewness in the data is defined as follows: ${\displaystyle S}$

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

With symmetrical distributions like the normal distribution, the theoretical value of the skewness is zero.

The kurtosis , a measure of the curvature of a distribution, has a value of three for a normal distribution. Values ​​greater than this indicate that the distribution has bold distribution ends (see distribution with heavy borders ); This means that the density of a distribution at the edges, for example outside the usual ± 2σ-bounds, is greater and therefore lower in the middle areas than in the normal distribution. This applies, for example, to the t-distribution . The kurtosis is defined as follows: ${\displaystyle K}$

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

where and represent the third and fourth central moment , is the mean of the sample and symbolizes the second moment, i.e. the variance . ${\displaystyle \mu _{3}}$ ${\displaystyle \mu _{4}}$ ${\displaystyle {\bar {x}}}$ ${\displaystyle \sigma ^{2}}$

It applies , d. that is, the test statistic is asymptotically chi-square distributed with two degrees of freedom . ${\displaystyle {\mathit {JB}}\sim \chi _{2}^{2}}$ ${\displaystyle {\mathit {JB}}}$

The pair of hypotheses is:

${\displaystyle H_{0}\colon }$ The sample is normally distributed.

${\displaystyle H_{1}\colon }$ The sample is not normally distributed.

At a significance level, the following applies: For values ​​of the test statistic above 4.6, the hypothesis of normal distribution is rejected; for the significance levels , and the bounds 6, 7.8 and 9.2 result. ${\displaystyle \alpha =0{,}10}$ ${\displaystyle \alpha =0{,}05}$ ${\displaystyle \alpha =0{,}02}$ ${\displaystyle \alpha =0{,}01}$

literature

• Anil K. Bera, Carlos M. Jarque : Efficient tests for normality, homoscedasticity and serial independence of regression residuals . In: Economics Letters . 6, No. 3, 1980, pp. 255-259. doi : 10.1016 / 0165-1765 (80) 90024-5 .
• Anil K. Bera, Carlos M. Jarque : Efficient tests for normality, homoscedasticity and serial independence of regression residuals: Monte Carlo evidence . In: Economics Letters . 7, No. 4, 1981, pp. 313-318. doi : 10.1016 / 0165-1765 (81) 90035-5 .
• George Judge, et al .: Introduction and the Theory and Practice of Econometrics , 3rd edn. 1988 edition, pp. 890-892.

See also

• Shapiro-Wilk test
• Kolmogorov-Smirnov test
• statistical test