Jarque-Bera Test of Normality

4 min readNov 29, 2019

Normality

Normality is an important assumption for the regression analysis
Especially for small samples, the inference procedures depends upon the
normality assumptions of the residuals, all our Confidence intervals
Z/t-tests and F-tests would not be valid if the normality assumption was violated.

So, it is important to know/nd out if the residuals really are normally
distributed (at least for small samples).

Testing for normality:
A normality test answers the question:
Does this variable follow a normal distribution?
Is it likely that these data come from a normal distribution

We formulate the hypotheses
H0: Data is normal
H1: Data is NOT normal

Since the assumption of normality is important for many areas of
statistics, there are a large number of (univariate) normality tests, with
different ways of checking if “Data is normal”
Jarque-Bera test
Kolmogorov's test
Andersson Darling test

The normal distribution has two important properties, no matter what the
parameters µ and σ, are, we have
It is symmetrical
It has Kurtosis three

Let's take a look at these measures.

Recall the moments
First
Second
Third
Fourth

From these moments we form different measures of the distribution, such
as
Mean (first moment itself)
Variance (Second central moment itself)
Skewness = f(third moment)
Kurtosis = f(fourth moment)

Skewness measures the degree of symmetry in the distribution.

(Note that the measure of skewness given in Gujarati Appendix A page 770 is squared skewness.)

The sample estimate of skewness is

Properties of the Skewness measure:
1 Zero skewness implies asymmetric distribution (the Normal, t-distribution)
2 Positive skewness means that the distribution has a long right tail, it's skewed to the right.
3 Negative skewness means that the distribution has a long left tail, its skewed to the left.

Properties of the Kurtosis measure:
1 A distribution with kurtosis=3 is said to be mesokurtic.
2 A distribution with kurtosis>3 is said to be leptokurtic or fat-tailed.

For example, Stock returns are known to be leptokurtic, i.e more “peaked” and fat-tailed than the normal distribution.

A distribution with kurtosis <is said to be platykurtic

The Jarque-Bera test uses these two (statistical) properties of the normal
distribution, namely:
The Normal distribution is symmetric around its mean
(skewness = zero)
The Normal distribution has kurtosis three, or
Excess kurtosis = zero

The Jarque-Bera test tests the hypothesis
H0: Data is normal
H1: Data is NOT normal

using the test statistic

How to do a Jarque-Bera test in practice

1 Calculate the skewness in the sample.
2 Calculate the kurtosis in the sample.
3 Calculate the Jarque-Bera test statistic
4 Compare the Jarque-Bera test statistic with the critical values in the
chi-square table, 2 df.

Normality tests — The Jarque-Bera test — Example