The Mardia's kurtosis of a multivariate GARCH model

The Mardia's kurtosis of a random vector with nonsingular covariance matrix and finite fourth-order moments is the fourth moment of the Mahalanobis distance of the random vector from its mean. In particular, the Mardia's kurtosis of a nondegenerate random variable with finite fourth moment coincides with its fourth standardized moment. The Mardia's kurtosis is the best known measure of multivariate kurtosis and appears in normality testing, robustness studies and outlier detection. Under mild assumptions, we show that an observation generated by a multivariate GARCH model has a Mardia's kurtosis which is greater than the Mardia's kurtosis of the innovation in the same model. The result generalizes to the multivariate case a well-known feature of univariate GARCH models. The practical relevance of the result is assessed with real data.