Question

Given some stock price data

"test null of symmetry at the a%"
"test null of normal tails at the b%"
"test null of normality at the c%"

What do they mean exactly?
Can we anwer this question with Jarque-Bera? Please explain how

Can we answer all 3 things with Jarque Bera?

Answer 1

Yes all the 3 can be Jaque Bera test.

When you run a Jaque Bera you get 3 values namely skewness, kurtosis and the Jaque Bera test. statistic value

First let us consider Jaque Bera test. statistic value at

JB = n [(√b1)² / 6 + (b₂ – 3)² / 24]

Where:
n is the sample size,
√b₁ is the sample skewness coefficient,
b₂ is the kurtosis coefficient.

The null hypothesis for the test is that the data is normally distributed; the alternate hypothesis is that the data does not come from a normal distribution.

If the JB value is greater than the Jaque Bera table value(statistical table value) at a% level of significance then we reject the null hypothesis which implies that the data is not normally distributed at a% level of significance .

Test null of symmetry at a% => This means to test if the distribution of the data is symmetric about the mean. The normally distributed data has skewness 0, which means the data is perfectly symmetric about the mean. So if the skewness is 0 it implies that the data is symmetric about mean

Test null of normal tails at b%=> if the kurtosis value is 3 then we accept that the tails have normal distribution. Which means that the tails are not heavy tailed (tails of normal distribution are thick) or light tailed (tails of normal distribution are thin)

extra points

• If skewness is less than -1 or greater than 1, the distribution is highly skewed.
• If skewness is between -1 and -0.5 or between 0.5 and 1, the distribution is moderately skewed.
• If skewness is between -0.5 and 0.5, the distribution is approximately symmetric.