Testing for Multivariate Distributions

Abstract

In this paper, we consider testing distributional assumptions based on residual em-pirical distribution functions. The method is stated for general distributions, but at-tention is centered on multivariate normal and multivariate t-distributions, as they are widely used, especially in financial time series models such as GARCH. Using the fact that joint distribution carries the same amount of information as the marginal together with conditional distributions, we first transform the multivariate data into univariate independent data based on the marginal and conditional cumulative distribution func-tions. We then apply the Khmaladze's martingale transformation (K-transformation) to the empirical process in the presence of estimated parameters. The K-transformation purges the effect of parameter estimation, allowing a distribution free test statistic to be constructed. We show that the K-transformation takes a very simple form for test-ing multivariate normal and multivariate t distributions. For example, when testing normality, we show that K-transformation for multivariate data coincides with that of univariate data. For multivariate t, the transformation depends on the dimension of the data but in a very simple way. We also extend the test to serially correlated observations, including multivariate GARCH models. Finally, we present a practical application of our test procedure on a real multivariate financial time series data set.