Instatistics, theJarque–Bera test is agoodness-of-fit test of whether sample data have theskewness andkurtosis matching anormal distribution. The test is named afterCarlos Jarque andAnil K. Bera.The test statistic is always nonnegative. If it is far from zero, it signals the data does not have a normal distribution.

Thetest statisticJB is defined as

$${\displaystyle \mathit{J}\mathit{B}=\frac{n}{6}\left(S^2+\frac{1}{4}(K-3{)}^2\right)}$$

wheren is the number of observations (or degrees of freedom in general);S is the sampleskewness,K is the samplekurtosis :

$${\displaystyle S=\frac{{\hat{\mu }}_3}{{\hat{\sigma }}^3}=\frac{\frac{1}{n}\sum _{i=1}^n{\left(x_i-\overline{x}\right)}^3}{{\left(\frac{1}{n}\sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2\right)}^{3/2}},}$$$${\displaystyle K=\frac{{\hat{\mu }}_4}{{\hat{\sigma }}^4}=\frac{\frac{1}{n}\sum _{i=1}^n{\left(x_i-\overline{x}\right)}^4}{{\left(\frac{1}{n}\sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2\right)}^2},}$$where${\displaystyle {\hat{\mu }}_3}$ and${\displaystyle {\hat{\mu }}_4}$ are the estimates of third and fourthcentral moments, respectively,${\displaystyle \overline{x}}$ is the samplemean, and${\displaystyle {\hat{\sigma }}^2}$ is the estimate of the second central moment, thevariance.

If the data comes from a normal distribution, theJB statisticasymptotically has achi-squared distribution with twodegrees of freedom, so the statistic can be used totest the hypothesis that the data are from anormal distribution. Thenull hypothesis is a joint hypothesis of the skewness being zero and theexcess kurtosis being zero. 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 ofJB shows, any deviation from this increases the JB statistic.

For small samples the chi-squared approximation is overly sensitive, often rejecting the null hypothesis when it is true. Furthermore, the distribution ofp-values departs from auniform distribution and becomes aright-skewedunimodal distribution, especially for smallp-values. This leads to a largeType I error rate. The table below shows somep-values approximated by a chi-squared distribution that differ from their true alpha levels for small samples.

(These values have been approximated usingMonte Carlo simulation inMatlab)

InMATLAB's implementation, the chi-squared approximation for the JB statistic's distribution is only used for large sample sizes (> 2000). For smaller samples, it uses a table derived fromMonte Carlo simulations in order to interpolatep-values.^[1]

The statistic was derived by Carlos M. Jarque and Anil K. Bera while working on their Ph.D. Thesis at the Australian National University.

According to Robert Hall, David Lilien, et al. (1995) when using this test along with multiple regression analysis the right estimate is:

$${\displaystyle \mathit{J}\mathit{B}=\frac{n-k}{6}\left(S^2+\frac{1}{4}(K-3{)}^2\right)}$$

wheren is the number of observations andk is the number of regressors when examining residuals to an equation.

• ALGLIB includes an implementation of the Jarque–Bera test in C++, C#, Delphi, Visual Basic, etc.
• gretl includes an implementation of the Jarque–Bera test
• Julia includes an implementation of the Jarque-Bera testJarqueBeraTest in theHypothesisTests package.^[2]
• MATLAB includes an implementation of the Jarque–Bera test, the function "jbtest".
• Python statsmodels includes an implementation of the Jarque–Bera test, "statsmodels.stats.stattools.py".
• R includes implementations of the Jarque–Bera test:jarque.bera.test in the packagetseries,^[3] for example, andjarque.test in the packagemoments.^[4]
• Wolfram includes a built in function called, JarqueBeraALMTest^[5] and is not limited to testing against a Gaussian distribution.

• D'Agostino's K-squared test, another test based on kurtosis and skewness.

• Jarque, Carlos M.; 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.
• Jarque, Carlos M.; 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.
• Jarque, Carlos M.; Bera, Anil K. (1987). "A test for normality of observations and regression residuals".International Statistical Review.55 (2):163–172.doi:10.2307/1403192.JSTOR 1403192.
• Judge; et al. (1988).Introduction and the theory and practice of econometrics (3rd ed.). pp. 890–892.
• Hall, Robert E.; Lilien, David M.; et al. (1995).EViews User Guide. p. 141.

• Normality tests

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