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The Jarque–Bera test asks whether a sample’s skewness and kurtosis jointly give enough evidence to reject a normal distribution. A larger statistic means a stronger departure, but a small statistic only means that the sample has not supplied enough evidence to reject normality. For an FRM candidate, the practical task is to calculate both contributions correctly and state the conclusion at the chosen significance level.
What enters the statistic
A normal distribution has zero skewness and kurtosis of three when kurtosis is reported in its ordinary, rather than excess, form. The test squares each sample departure, so either left or right skew can raise the result, as can kurtosis above or below three. The GARP-hosted paper A Gentle Introduction to Value at Risk gives the statistic and its large-sample chi-square reference distribution with two degrees of freedom. GARP’s FRM study materials identify learning objectives as the basis for checking the current reading scope.
If a data source reports excess kurtosis instead, it has already subtracted three. In that case use the reported excess value in the squared kurtosis term; subtracting three again would be an error. The reference distribution is an asymptotic approximation, so a borderline result from a small or dependent return sample deserves care.
An illustrative decision from a sample summary
Consider an invented summary of 60 weekly returns: sample skewness is 0.6 and ordinary sample kurtosis is 4.2. Choose a 5% significance level for this example. Squared skewness contributes 0.36 inside the bracket. Excess kurtosis is 1.2, whose squared value divided by four is also 0.36. Multiplying their sum by ten gives a Jarque–Bera statistic of 7.2.
Under the stated two-degree-of-freedom chi-square approximation, a statistic of 7.2 has an upper-tail probability of about 2.7%. It falls below the illustrative 5% significance threshold, so reject the null of normality. The statistic does not say whether skewness or kurtosis matters more in every sample; here they contribute equally by construction. Nor does rejection identify a suitable replacement distribution.
Where the interpretation goes wrong
- Using ordinary kurtosis as excess kurtosis. Subtract three once when the input is ordinary kurtosis; do not subtract it from an already excess figure.
- Treating a non-rejection as proof. A result above the chosen probability threshold says the sample did not reject the null; it does not establish that future returns are normal.
- Ignoring the data behind the statistic. A few unusual observations can move skewness and kurtosis materially. The test is a diagnostic for the supplied sample, not a complete risk model.
When revising, label the kurtosis convention, calculate both squared terms and compare the result with the stated decision threshold before writing the conclusion. The FRM programme page gives the course context, and the Omni methodology sets out the study approach.
Sources
- A Gentle Introduction to Value at Risk · garp.org · Retrieved
- FRM® Study Material, Guide, Books, Practice Exams | GARP · garp.org · Retrieved
