Jarque-Bera Test

The Jarque-Bera(1987) (JB) test is a relatively simple test for Normality (Gaussianity) in a univariate series. This looks at the 3rd and 4th moments of the series and compares them with the theoretical values that they would have if the series were Normal. While it's possible for a series have non-Normal residuals while matching the 3rd and 4th moments required for Normality, it is highly unlikely in practice, as the typical failure of Normality is due to fat tails, which would give large 4th moments in particular.

The J-B statistic is computed as \(jb = N\left( {\frac{{{{(Ku)}^2}}}{{24}} + \frac{{{{(Sk)}^2}}}{6}} \right)\) where \(Ku\) is the excess kurtosis (difference between kurtosis and the value of 3 that would hold under Normality) and \(Sk\) is the skewness. Under the null, this has a \(\chi _2^2\) distribution.

A few things to note:

1.There are many asymptotically equivalent formulas for the \(Ku\) and \(Sk\) statistics and different choices will give (somewhat) different results.

2.An underlying assumption is that the series is i.i.d., so in practice, this should be applied to residuals and not raw time series data.

In RATS, use the STATISTICS instruction to compute the JB test. The test statistic (and its significance level) are available in the output and as the variables %JBSTAT and %JBSIGNIF.

A multivariate version of this is available in the @MVJB procedure.