# Characterizations of multinormality and corresponding tests of fit, including for Garch models

## TL;DR

This paper introduces new characterizations of multivariate normality using characteristic and moment generating functions, leading to affine invariant goodness-of-fit tests applicable to GARCH models, with theoretical and empirical validation.

## Contribution

It develops novel characterizations of multivariate normality and constructs affine invariant tests, including for GARCH model innovations, with theoretical properties and finite-sample performance analysis.

## Key findings

- New tests are affine invariant and consistent.
- Asymptotic behavior of tests is established.
- Finite-sample performance compares favorably with existing tests.

## Abstract

We provide novel characterizations of multivariate normality that incorporate both the characteristic function and the moment generating function, and we employ these results to construct a class of affine invariant, consistent and easy-to-use goodness-of-fit tests for normality. The test statistics are suitably weighted $L^2$-statistics, and we provide their asymptotic behavior both for i.i.d. observations as well as in the context of testing that the innovation distribution of a multivariate GARCH model is Gaussian. We also study the finite-sample behavior of the new tests and compare the new criteria with alternative existing tests.