自由度为2

问题如下：

If an asset has zero skewness, what is the maximum kurtosis it can have to not reject normality with a sample size of 100 using a 5% test? What if the sample size is 2,500?

选项：

解释：

The Jarque-Bera has $\chi_2^2$ distribution, and the critical value for a test with a size of 5% is 5.99. The Jarque-Bera statistic is

$JB = (T - 1)(\widehat S^2/6+(\widehat\kappa-3)^2/24)$

so that when the skewness $\widehat S = 0$, the test statistic is $(T - 1)(\widehat\kappa-3)^2/24$.

In order to not reject the null, we need $(T - 1)(\widehat\kappa-3)^2/24$ ≤ 5.99

and so $(\widehat\kappa-3)^2 ≤ 24*5.99/(T-1)$

and $\widehat\kappa ≤ 3+\sqrt {24*5.99/(T-1)}$.

When T = 100, this value is 4.20. When T = 2,500 this value is 3.24. This shows that the JB test statistic is sensitive to even mild excess kurtosis.