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Jarque-Bera Test

Overview

The Jarque-Bera test is a statistical test used to assess whether a dataset follows a normal distribution by evaluating two key features of the data: skewness and kurtosis. It is widely used in econometrics and financial applications, where normality is a crucial assumption for many statistical models and methods.

Hypotheses

  • Null Hypothesis (\(H_0\)): The data is normally distributed.
  • Alternative Hypothesis (\(H_1\)): The data is not normally distributed.

Computation of the Jarque-Bera Test Statistic

We compute the test statistic \(JB\) using the following formula:

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

where

  • \(n\) is the sample size,
  • \(S\) is the sample skewness,
  • \(K\) is the sample kurtosis,
  • The factor 6 in the denominator is a scaling constant to normalize the contribution of skewness and kurtosis to the statistic.

Steps to Compute the Test Statistic

  1. Compute the Sample Mean:

    \[ \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \]

  2. Compute the Sample Skewness (\(S\)):

    \[ S = \frac{1}{n} \sum_{i=1}^{n} \left( \frac{X_i - \bar{X}}{\sigma} \right)^3 \]

    where \(\sigma^2\) is the sample variance:

    \[ \sigma^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2 \]

  3. Compute the Sample Kurtosis (\(K\)):

    \[ K = \frac{1}{n} \sum_{i=1}^{n} \left( \frac{X_i - \bar{X}}{\sigma} \right)^4 \]

  4. Calculate the Jarque-Bera Statistic (\(JB\)):

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

    This statistic combines the squared skewness and the squared deviation of the kurtosis from 3 (the kurtosis of a normal distribution), weighted by sample size \(n\).

Deriving the p-Value

The Jarque-Bera test statistic \(JB\) follows a chi-square (\(\chi^2\)) distribution with 2 degrees of freedom under the null hypothesis (since we compute it based on two components: skewness and kurtosis). To obtain the \(p\)-value:

  1. Compare the computed \(JB\) statistic to the critical values of the chi-square distribution with 2 degrees of freedom.

  2. The \(p\)-value is the probability that the test statistic \(JB\) would be as extreme as or more extreme than the observed value, under the assumption that the null hypothesis (\(H_0\)) is true.

    • A small \(p\)-value (typically less than a significance level \(\alpha = 0.05\)) suggests that the data deviates significantly from normality, and we reject the null hypothesis.
    • A large \(p\)-value suggests insufficient evidence to reject the null hypothesis, meaning the data could reasonably come from a normal distribution.

Decision Rule

  • If the \(p\)-value is less than the significance level \(\alpha\) (e.g., 0.05), reject \(H_0\) and conclude that the data is not normally distributed.
  • If the \(p\)-value is greater than or equal to \(\alpha\), fail to reject \(H_0\) and conclude that the data may be normally distributed.

Interpretation

  • A high \(JB\) statistic implies that the data has skewness or kurtosis (or both) that deviates significantly from a normal distribution.
  • A low \(JB\) statistic indicates that the sample data's skewness and kurtosis are consistent with a normal distribution.

Python Implementation

import numpy as np
from scipy import stats

np.random.seed(0)

n = 1000

# Generate a sample dataset
data = np.random.normal(0, 1, n)
# data = np.random.exponential(1, n)

skewness_value = stats.skew(data)
kurtosis_value = stats.kurtosis(data)
JB = n / 6 * (skewness_value**2 + kurtosis_value**2 / 4)
print(f"{JB = }")

# Perform Jarque-Bera test
stat, p_value = stats.jarque_bera(data)
print(f"Jarque-Bera Test: Statistic={stat}, p-value={p_value}")

# Interpretation
alpha = 0.05
if p_value <= alpha:
    print("Reject H_0: The data is not normally distributed.")
else:
    print("Fail to reject H_0: The data is normally distributed.")

Jarque-Bera Test vs D'Agostino's K-Squared Test

The Jarque-Bera test is not an approximation of D'Agostino's K-squared test. While both tests assess normality by examining skewness and kurtosis, they are fundamentally different in how they compute the test statistics.

Key Differences

Jarque-Bera Test:

  • Directly uses the sample's skewness and kurtosis to compute its test statistic.

  • The test statistic is:

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

  • Combines skewness and kurtosis into a single test statistic, assuming the sample follows a chi-square distribution with 2 degrees of freedom under the null hypothesis.

D'Agostino's K-Squared Test:

  • Transforms both skewness and kurtosis into independent Z-scores:
    • \(Z_{\text{skewness}}\): A transformation that normalizes the sample's skewness.
    • \(Z_{\text{kurtosis}}\): A transformation that normalizes the sample's kurtosis.

  • The test statistic is:

    \[ K^2 = Z_{\text{skewness}}^2 + Z_{\text{kurtosis}}^2 \]

  • Like the Jarque-Bera test, \(K^2\) follows a chi-square distribution with 2 degrees of freedom, but D'Agostino's test uses separate transformations for skewness and kurtosis, which makes it more robust and sensitive to deviations from normality.

Why They Are Distinct

  • Different Approaches: The Jarque-Bera test applies a simple, direct formula based on the raw skewness and kurtosis values, whereas D'Agostino's K-squared test applies transformations that adjust for sample size and normalize the distribution of skewness and kurtosis.

  • Test Statistic Construction: Jarque-Bera straightforwardly combines skewness and kurtosis into a single statistic, while D'Agostino's test separates them, transforming them into individual test statistics, which are then squared and summed.

  • Sensitivity: D'Agostino's K-squared test is more sensitive to deviations from normality than the Jarque-Bera test, especially in larger sample sizes. The Z-transformations make it more accurate when the sample size increases, whereas Jarque-Bera's performance may degrade in small samples or be less sensitive to tail deviations.

Conclusion

The Jarque-Bera test does not approximate D'Agostino's K-squared test; they are distinct methods for testing normality with different underlying statistical foundations. Both tests use skewness and kurtosis, but D'Agostino's test is considered more robust due to its transformation-based approach, while Jarque-Bera is more straightforward and commonly used in econometrics.