Multivariate Autocontours for Specification Testing in Multivariate GARCH Models

In this paper we propose a new battery of test statistics for dynamic specification and density functional form in a wide range of multivariate time series models including linear and non-linear VAR specifications with multivariate GARCH disturbances. The tests are applied to the vector of generalized errors that must be i.i.d. with a certain parametric multivariate probability density function under the null hypothesis of correct specification. The basic idea of the proposed methodology is to calculate the percentage of observations contained within the probability autocontour plots corresponding to the assumed multivariate density of the vector of independent innovations, and compare it to the population percentage. We develop t-tests based on a single autocontour and also more powerful chi-squared tests based on multiple autocontours. In the spirit of goodness-of-fit tests, we also propose an additional test that focuses on the multivariate density functional form of the vector of innovations. We explicitly consider parameter uncertainty and show that a simple bootstrap procedure overcomes this problem. We perform Monte-Carlo simulations to investigate the size and power properties of the test statistics in finite samples. We apply our tests to multivariate GARCH models fitted to excess returns on portfolios sorted according to market capitalization