Testing Constancy of the Error Covariance Matrix in Vector Models against Parametric Alternatives using a Spectral Decomposition

I consider multivariate (vector) time series models in which the error covariance matrix may be time-varying. I derive a test of constancy of the error covariance matrix against the alternative that the covariance matrix changes over time. I design a new family of Lagrange-multiplier tests against the alternative hypothesis that the innovations are time-varying according to several parametric specifications. I investigate the size and power properties of these tests and find that the test with smooth transition specification has satisfactory size properties. The tests are informative and may suggest to consider multivariate volatility modelling.