JarqueBera

The Jarque-Bera test is a statistical test used to assess whether a sample of data has the skewness and kurtosis of a normal distribution. It was introduced by Carlos M. Jarque and Anil K. Bera in the 1980s and has become a widely used diagnostic in econometrics and statistics for evaluating normality of data or regression residuals.

The test uses two sample moments: skewness and kurtosis. Let S be the sample skewness and K the sample kurtosis (often expressed as excess kurtosis, K − 3). The Jarque-Bera statistic is JB = n/6 [ S^2 + ( (K − 3)^2 / 4 ) ], where n is the sample size. Some formulations report JB = n/6 [ S^2 + (G2^2 / 4) ] using the excess kurtosis G2. Under the null hypothesis that the data come from a normal distribution, JB is asymptotically distributed as a chi-square random variable with 2 degrees of freedom.

Interpretation of the result follows a standard significance test. A large JB statistic yields rejection of normality at the chosen significance level, indicating departures in either skewness, kurtosis, or both. The test is omnibus for these two aspects but may have limited power against certain alternatives and can be sensitive to outliers. Its accuracy improves with larger samples; for small samples, bootstrap or simulation-based p-values may be preferred.

Jarque-Bera is commonly used to assess normality of residuals in regression analyses and financial return series. It is implemented in many statistical packages, including R, Python, and MATLAB, often under names like jarque.bera.test or similar.