Mardia's Tests for Multivariate Normality

Every individual variable in a dataset can pass a normality test on its own and the group as a whole can still fail to be jointly multivariate normal, because multivariate normality also requires the relationships between variables to have a specific shape — a requirement single-variable checks simply can't see. Mardia's tests extend the familiar univariate ideas of skewness and kurtosis to the multivariate case, using every pair of observations in the sample (via the squared Mahalanobis distance between them) rather than just each variable's own marginal distribution.

Mardia's skewness statistic averages a measure of asymmetry across all pairs of data points and, under multivariate normality, should be close to zero; Mardia's kurtosis statistic measures how heavy-tailed the joint distribution is relative to what multivariate normality predicts, and both are converted into test statistics with known approximate distributions under the null. A significant result on either flags a violation — most often in finance, excess multivariate kurtosis, meaning extreme joint moves across several assets happen more often together than a multivariate normal model would predict.

This matters directly for portfolio risk models: many multivariate techniques (multivariate GARCH, factor models with normal errors) assume multivariate normality, and a large Mardia kurtosis statistic on daily returns is a standard early warning that tail co-movement across assets is being understated by a normal-based model.