Hypothesis Tests

Jarque-Bera Test

Asymptotic normality test based on skewness and kurtosis (chi-squared with 2 degrees of freedom). Reliable for large samples (n > 30).

Statusvalidated

Version1.0.0

Minitab equivalentStat > Basic Statistics > Normality Test (not directly available — Jarque-Bera is common in econometrics, e.g. EViews, Stata)

Description

The Jarque-Bera test checks whether the skewness and excess kurtosis of a sample are consistent with a normal distribution. The test statistic JB = (n/6)(S² + K²/4) is asymptotically chi-squared distributed with 2 degrees of freedom under H₀, where S is the sample skewness and K is the excess kurtosis. The test is especially suited for large samples; for small samples (n < 30), Shapiro-Wilk and Anderson-Darling have higher power.

Formulas

J B = \frac{n}{6} (S^{2} + \frac{K ^{2}}{4})

Test statistic: Combines skewness S and excess kurtosis K

S = \frac{m _{3}}{m _{2}^{3/2}} = \frac{\frac{1}{n} \sum _{i = 1}^{n} ( x _{i} - x ˉ ) ^{3}}{( \frac{1}{n} \sum _{i = 1}^{n} ( x _{i} - x ˉ ) ^{2} ) ^{3/2}}

Sample skewness (biased, using population moments m₂ and m₃)

K = \frac{m _{4}}{m _{2}^{2}} - 3

Excess kurtosis (0 for normal distribution)

H₀

H_{0} : S = 0 and K = 0 \Rightarrow J B \sim χ^{2} (2)

Null hypothesis: Skewness and excess kurtosis are consistent with a normal distribution

Assumptions

Data are independent and identically distributed (i.i.d.)
Sample size: n ≥ 3 (recommended: n ≥ 30 for reliable asymptotic approximation)
Continuous data

Limitations

Asymptotic test — low power for small samples (n < 30)
Sensitive to outliers since skewness and kurtosis are based on higher-order moments
Uses population moments (dividing by n), not sample moments (dividing by n-1)

References

Jarque, C. M. & Bera, A. K. (1987), A test for normality of observations and regression residuals, International Statistical Review, 55(2), 163–172
Jarque, C. M. & Bera, A. K. (1980), Efficient tests for normality, homoscedasticity and serial independence of regression residuals, Economics Letters, 6(3), 255–259
D'Agostino, R. B. & Stephens, M. A. (1986), Goodness-of-Fit Techniques, Marcel Dekker, New York