ABSTRACT
Our object here is to describe new results in the analysis of multivariate (nonGaussian) time series and in particular bilinear models. In Sec. 2, we obtain expressions for cumulants of random matrices which are extensions of the results well known in univariate case, Leonov and Shiryaev (1959). Using the results obtained in Sec. 2, we derive expressions for higher-order cumulant spectra of vector time series satisfying a linear process representation. This is analogous to the well-known relation given by Bartlett, Brillinger and Rosenblatt in the univariate case. This relation can be used for constructing a test for linearity of vector time series similar to the Subba Rao and Gabr ( 1980) test. In Sec. 4, symmetries of cumulants of random vectors are discussed. Multivariate bilinear models are introduced in Sec. 5, and the higher order properties of bilinear processes are studied in Sec. 6 and we show that they satisfy Yule-Walker type difference equations. A test for Gaussianity is briefly discussed in Sec. 7. The estimation of the parameters of the bilinear models is considered in Sec. 8 and the methods are illustrated with real data in Sec. 9.
