Introduction

In the world of financial analysis, the assumption of normal distribution plays a significant role. The Jarque-Bera test is a statistical tool used to assess whether a set of data follows a normal distribution. This chapter delves into the mechanics of the Jarque-Bera test and its application in determining the normality of returns.

Understanding the distribution of returns is pivotal for making informed investment decisions and accurately modeling financial markets. The normal distribution, also known as the Gaussian distribution, serves as a benchmark due to its well-defined properties. The Jarque-Bera test scrutinizes the skewness and kurtosis of a dataset, enabling analysts to gauge departures from the normal distribution.

Jarque-Bera Test: Principles and Procedure

The Jarque-Bera test is based on two key statistical moments: skewness and kurtosis. Skewness measures the asymmetry of the distribution, indicating whether the data is skewed to the left (negative skewness) or to the right (positive skewness). Kurtosis, on the other hand, measures the “tailedness” of the distribution.

The test statistic is calculated using skewness $(\mathrm{S})$ and excess kurtosis $(\mathrm{K})$ as follows:

$$JB=\frac{n}{6}\left(S^2+\frac{1}{4}(K-3)^2\right)$$

Where:

• $n$ is the sample size
• $S$ is the sample skewness
• $K$ is the sample excess kurtosis

Application and Interpretation

The Jarque-Bera test statistic $JB$ follows a chi-squared distribution with 2 degrees of freedom under the assumption of normality. Analysts compare the calculated $J B$ value with the critical value from the chi-squared distribution to determine whether the data is normally distributed. If the calculated JB value exceeds the critical value, it suggests non-normality.

Example: Consider a dataset of daily stock returns. After calculating the sample skewness and excess kurtosis, you find that $S=-0.15$ and $K=2.9$. With a sample size of $n=200$, the Jarque-Bera test statistic is calculated as:

$$JB=\frac{200}{6}\left((-0.15)^2+\frac{1}{4}(2.9-3)^2\right) \approx 7.42$$

Comparing this value to the chi-squared critical value at a certain significance level (e.g., 5%), you can assess whether the dataset deviates significantly from a normal distribution.