Statistical
Jarque-Bera Normality Test: Skewness & Excess Kurtosis Chi-Square P-Value
Test whether regression residuals or asset returns follow a normal distribution using the Jarque-Bera statistic JB = (n / 6) * (S^2 + (K_excess^2 / 4)) and exact df = 2 exponential p-value EXP(-JB / 2).
Jointly test fat tails and asymmetry against the normal bell curve!
A normal distribution has theoretical skewness S = 0 (symmetric tails) and kurtosis = 3, corresponding to excess kurtosis K_excess = 0! Financial return series and skewed operational cycle times often fail OLS regression and parametric VaR assumptions because of fat tails (leptokurtosis, K_excess > 0) or skewness. Carlos Jarque and Anil Bera derived the omnibus Lagrange-multiplier test statistic combining sample size n, sample skewness S, and sample excess kurtosis K_excess: JB = (n / 6) * (S^2 + (K_excess^2 / 4))!
Under the null hypothesis of normality, JB converges to a Chi-Square distribution with exactly df = 2 degrees of freedom (1 for skewness + 1 for excess kurtosis). Because a Chi-Square(df = 2) distribution is mathematically identical to an Exponential(1/2) distribution, its exact right-tail p-value simplifies in closed form to p = EXP(-JB / 2) and its critical threshold at significance alpha is -2 * LN(alpha) (= 5.991 at alpha = 0.05)! For n = 120 returns with S = 0.65 and K_excess = 1.20, JB = 20 * (0.4225 + 0.36) = 15.65 (p = 0.0004 < 0.05 -> Reject Normality!).
| Sample Profile (n = 120, alpha = 0.05) | Skewness S & Excess Kurtosis Ke | Jarque-Bera Statistic JB | Exact P-Value & Decision |
|---|---|---|---|
| Fat-Tailed Equity Returns | S = 0.65, Ke = 1.20 | JB = 15.65 (> 5.991) | p = 0.0004 -> Reject Normality |
| Well-Behaved Sensor Noise | S = 0.15, Ke = -0.20 | JB = 0.65 (<= 5.991) | p = 0.7225 -> Fail to Reject (Normal) |
| Critical Cutoff (df = 2) | -2 * LN(0.05) | JB_crit = 5.991 | alpha = 5% Significance Boundary |
Jarque-Bera Test Statistic JB = (n / 6) * (S^2 + K_excess^2 / 4)
Validates sample size n >= 8 before combining squared skewness B3 and squared excess kurtosis B4 / 4.
=IF(B2 < 8, "Sample Too Small (n >= 8)", ROUND((B2 / 6) * ((B3 ^ 2) + ((B4 ^ 2) / 4)), 3))Exact Asymptotic Chi-Square(df = 2) Right-Tail P-Value = EXP(-JB / 2)
Uses the exact Chi-Square(2) = Exponential(1/2) survival identity EXP(-JB / 2) to return the p-value.
=IF(B2 < 8, "Sample Too Small (n >= 8)", ROUND(EXP(-((B2 / 6) * ((B3 ^ 2) + ((B4 ^ 2) / 4))) / 2), 4))Try the example
Use the original values or change them to test your understanding.