## Related questions with answers

Select a stock such as Apple, Dell, or Microsoft and use the Jarque-Bera test to determine if the annual daily rates of return for a particular year follow a normal distribution.

Solution

VerifiedTo determine the normality of a given set of data using the Jarque-Bera test, we use the following:

$\begin{equation} JB=n\left[ \dfrac{\text{(skewness)}^2}{6}+\dfrac{\text{(kurtosis}-3)^2}{24} \right], \end{equation}$

where the skewness is defined by:

$\begin{equation} \text{skewness}=\dfrac{\sum_{i=1}^n(x_i-\bar x)^3}{ns^3}, \end{equation}$

and the kurtosis as:

$\begin{equation} \text{kurtosis}=\dfrac{\sum_{i=1}^n(x_i-\bar x)^4}{ns^4}. \end{equation}$

Take note that the $n$ represents the sample size, the $x$ as the observation values, $\bar x$ as the sample mean, while $s$ represents the sample standard deviation, which is given by:

$\begin{equation} s=\sqrt{\dfrac{1}{n-1}\sum^n_{i=1}(x_i-\bar x)^2}. \end{equation}$

The value of $JB$ that will be obtained will be compared to the significance points found in a statistic table of Jarque-Bera. If our test statistic $JB$ exceeds the appropriate values, we reject the null hypothesis of normality, which is the assumption that the given data exhibits a normal distribution.

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