Kurtosis

Kurtosis

Kurtosis is a statistical measure that describes data distribution in a dataset. Kurtosis indicates whether the data are more or less extreme than a normal distribution.

There are typically three common types of kurtosis:

1. Mesokurtic (Normal Kurtosis):If the kurtosis is near zero, the data distribution has tails similar to a normal distribution. In this case, the distribution is neither too peaked nor too flat.
2. Leptokurtic (Positive Kurtosis):When the kurtosis is greater than zero,This means that the distribution has heavier tails and a more pronounced peak compared to a normal distribution
3. Platykurtic (Negative Kurtosis):If the kurtosis is less than zero, This suggests that the distribution has lighter tails and is flatter compared to a normal distribution. In this case, the data have fewer extreme values or outliers.

Kurtosis is calculated using the fourth central moment of the data distribution and is typically defined as:

Where:

• $n$ is the number of data points.
• xi​ represents each data point.
• $\mu$ is the mean (average) of the data.
• $\sigma$ is the standard deviation of the data.

Heteroscedasticity

Heteroscedasticity is a term used in regression analysis and statistics to describe a situation where the variability of the errors (residuals) in a regression model is not constant across all levels of the independent variables.

The Breusch-Pagan test is a statistical test used in regression analysis to check for the presence of heteroscedasticity in a regression model.
Specifically, you estimate a regression model where the dependent variable is the squared residuals from your original regression, and the independent variables are the variables you suspect might be related to heteroscedasticity. If the coefficients of these independent variables are statistically significant, it suggests the presence of heteroscedasticity. Then we perform a hypothesis test on the coefficients of the independent variables in the regression model. The null hypothesis typically states that there is no heteroscedasticity (i.e., the coefficients are equal to zero), and the alternative hypothesis is that there is heteroscedasticity.

 

Calculate the p-value: The test produces a p-value associated with the null hypothesis. If the p-value is smaller than a predetermined significance level (e.g., 0.05), you would reject the null hypothesis and conclude that heteroscedasticity is present.

If the Breusch-Pagan test indicates the presence of heteroscedasticity, you may need to address it by considering model adjustments, transformations, or robust regression techniques, as mentioned in the previous response.