How do I use the Jarque-Bera test to check residual normality?

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AcadiFi · Accepted Answer

The Jarque-Bera test evaluates whether a sample comes from a normal distribution by jointly examining its skewness and kurtosis. For a normal distribution, skewness is zero and excess kurtosis is zero.

The statistic is JB = (n/6) * [S^2 + (K-3)^2 / 4], where S is sample skewness, K is sample kurtosis (raw, not excess), and n is the sample size. Under the null of normality, JB follows a chi-squared distribution with 2 degrees of freedom. The 5% critical value is 5.99.

Suppose Kalani Esperanto runs a factor model on Briarclift Asset Management quarterly returns. Residuals show skewness of 0.8 and kurtosis of 5.2. With n=120, JB = (120/6) * [0.64 + (2.2)^2 / 4] = 20 * [0.64 + 1.21] = 37.0. This vastly exceeds 5.99, so normality is rejected.

Non-normality matters less in large samples because the Central Limit Theorem makes coefficient estimators asymptotically normal regardless. However, for small samples or when you need reliable confidence intervals and prediction intervals, non-normality is troubling.

Sources of non-normality in finance include outliers from crisis periods, structural breaks, nonlinear relationships, and omitted variables. Fixes include winsorizing outliers, transforming variables (log or Box-Cox), adding omitted regressors, or using bootstrap inference that avoids distributional assumptions.

Always pair JB with QQ plots and residual histograms for a complete diagnostic picture.

How do I use the Jarque-Bera test to check residual normality?

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