Testing Gaussianity and Linearity in Multivariate Time Series and Their Applications

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Based on a characterization of orthogonality of Gaussian random variates after Hermitian polynomials transformation, we developed a Gaussianity teat for multivariate stationary time series, where a univariate Gaussianity test proposed by Kariya, Tsay Terui and Li(1998) was extended to multivariate situation. Our simulation study showed that the proposed test has reasonable power and outperforms multivariate bispectrum test available in the literature when the innovation series of the time series is symmetric, but non-Gaussian.Testing Gaussianity is closely related to testing linearity for univariate time series analysis, because nonlinearity implies non-normality in a regular time series. Extending to multivariate situation, it does not always holdbecause of their inter-relationship between marginal series. That is, non-Gaussianity of each marginal series does not always require non-linear modeling of multivariate series as a whole. We introduced a multivariate time series model, we call multivariate time series with common non-Gaussian component, which represents the above relationship and we used the proponed test to detect it. In case of that, the proposed multivariate Gaussianity test was modified so as to decompose the omnibus teats into two orthogonal tests statistics, each of which test the Gaussianity for marginal series and the inter-relational Gaussianity respectively.Further some modification of tests was proposed to accommodate the inconsistency between marginal tests and omnibus test and we showed that it gave a useful insight of inter-relationship between marginal time series for multivariate time series with common non-Gaussian component. This reduces the burden of non-Gaussian modeling, equivalently non-linear modeling in a sense, into linear Gaussian modeling.