# Doing a Jarque-Bera Test In R

Skewness of $$x$$ is measured as $S = \frac{\left( E[X - \mu]^{3} \right)^{2}}{\left(E[X - \mu]^{2} \right)^{3}}$

Because the normal distribution is symmetric, the skewness (deviation from symmetry) should be zero.

Kurtosis of $$x$$ is measured as $\kappa = \frac{E[X - \mu]^{4}}{\left( E[X - \mu]^{2} \right)^{2}}$

and $$\kappa = 3$$ for a normal distribution. The Jarque-Bera statistic is $jb = T\left[ \frac{S}{6} + \frac{(\kappa - 3)^{2}}{24} \right]$

where $$T$$ is the sample size. Under the null hypothesis of normality, $$jb \sim \chi^{2}(2)$$.

The Jarque-Bera test is available in R through the package tseries. Example:

library(tseries)
set.seed(100)
x <- rnorm(5000)
jarque.bera.test(x)

The output is

    Jarque Bera Test

data:  x
X-squared = 0.046, df = 2, p-value = 0.9773

We do not reject the null hypothesis of normality for this series. That is a good thing, otherwise we would want to check if R's random number generating functions are working properly.

n <- length(x)         ## Number of observations
m1 <- sum(x)/n         ## Mean
m2 <- sum((x-m1)^2)/n  ## Used in denominator of both
m3 <- sum((x-m1)^3)/n  ## For numerator of S
m4 <- sum((x-m1)^4)/n  ## For numerator of K
b1 <- (m3/m2^(3/2))^2  ## S
b2 <- (m4/m2^2)        ## K
STATISTIC <- n*b1/6+n*(b2-3)^2/24

The test statistic (what I called $$jb$$ above) is reported as x.squared (not sure why that name was chosen), the degrees of freedom parameter is always 2, and the p-value is calculated as 1 - pchisq(STATISTIC,df = 2).

Last updated: 2015-10-07

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