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We generalize a recent class of tests for univariate normality that are basedon the empirical moment generating function to the multivariate setting, thusobtaining a class of affine invariant, consistent and easy-to-usegoodness-of-fit tests for multinormality. The test statistics are suitablyweighted $L^2$-statistics, and we provide their asymptotic behavior both fori.i.d. observations as well as in the context of testing that the innovationdistribution of a multivariate GARCH model is Gaussian. We study thefinite-sample behavior of the new tests, compare the criteria with alternativeexisting procedures, and apply the new procedure to a data set of monthly logreturns.

A new class of tests for multinormality with i.i.d. and Garch data based on the empirical moment generating function