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Compute the Jarque-Bera test statistic and show how it lets you judge whether a set of returns is normal.

FRM® Part I · Quantitative Analysis · Measuring Returns, Volatility, and Correlation · FRM1.QA.12.d

The idea

The Jarque-Bera statistic tests whether a sample's skewness and kurtosis are consistent with a normal distribution. With n observations, sample skewness S and sample kurtosis K, not the excess kurtosis, the statistic is: JB = (n ÷ 6) × [S² + (K − 3)² ÷ 4]. The null hypothesis is that returns are normal, meaning skewness is 0 and excess kurtosis is 0. Under the null, JB follows a chi-square distribution with 2 degrees of freedom. The critical values are about 5.99 at the 5% level and 9.21 at the 1% level. If JB exceeds the critical value, reject normality. Because JB grows in proportion to n, a very large sample can reject normality for a small departure, while a small sample may fail to reject a clearly non-normal series. A frequent slip is to plug K into the formula in place of K − 3.

Formulas

Jarque-Bera statistic

JB = (n ÷ 6) × [S² + (K − 3)² ÷ 4]

n
the number of observations
S
the sample skewness
K
the sample kurtosis, not excess kurtosis

Chi-square with 2 degrees of freedom under normality: reject above about 5.99 at 5% or 9.21 at 1%. If excess kurtosis is given, it already equals K − 3.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement correctly describes the Jarque-Bera test?

  1. The null is that returns are not normal; a large statistic confirms normality.Why not: Reverses the hypotheses: normality is the null, and a large statistic rejects it.
  2. The null is zero skewness and zero excess kurtosis; a statistic above the critical value rejects normality. - correct
  3. The null is normality; a statistic below the critical value rejects it.Why not: Rejection occurs when JB is above, not below, the critical value.
  4. It tests only whether successive returns are autocorrelated.Why not: JB tests skewness and kurtosis; autocorrelation needs a different test.

JB measures how far sample skewness and excess kurtosis are from 0. Large values mean the departures are too big to be sampling noise, so normality is rejected.

A harder one

A sample of 250 daily returns has skewness of -0.30 and kurtosis of 4.2. The Jarque-Bera statistic is closest to:

  1. 63.75Why not: Omits the division by 4 on the squared excess kurtosis term.
  2. 112.50Why not: Multiplies by n instead of n / 6.
  3. 18.75 - correct
  4. 187.50Why not: Uses the kurtosis K = 4.2 instead of the excess kurtosis K - 3 = 1.2.

JB = (250 / 6) x [(-0.30)² + (4.2 - 3)² / 4] = 41.67 x (0.09 + 0.36) = 18.75. This exceeds 5.99, so normality is rejected at 5%.

The hardest tier

A risk manager finds, for a return series, skewness of -0.15 and kurtosis of 3.5 in both a 250-observation sample and a 1,000-observation sample. Critical values (2 df) are 5.99 at 5% and 9.21 at 1%. Which conclusion is correct?

  1. Rejected in both cases, with JB of about 129 and 514.Why not: Uses the kurtosis K instead of K - 3, which inflates the statistic.
  2. Not rejected in either case, because JB depends only on skewness and kurtosis, not on sample size.Why not: Ignores the n / 6 factor: JB grows in proportion to the sample size.
  3. Rejected with 250 observations but not with 1,000, because a larger sample pulls the moments toward normal values.Why not: Holds the sample moments fixed, so more data raises JB and makes rejection more likely, not less.
  4. Not rejected with 250 observations (JB about 3.5), but rejected at the 1% level with 1,000 (JB about 14.2). - correct

JB(250) = 41.67 x (0.0225 + 0.0625) = 3.54 < 5.99. JB(1000) = 166.67 x 0.085 = 14.17 > 9.21. The same sample moments are significant with more data.

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