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# H16 - H16 The discussion of bivariate processes is readily...

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H16 The discussion of bivariate processes is readily extended to the general multivariate case. If we have m processes, , each having zero mean, we define ,1 ,2 , { },{ }, . ..,{ } tt t XX X m m the covariance matrix at lag h by ( ) [ ( )], 1,. .., , 1,. .., ij hh i m j γ Γ= = = , where . For ,, () ij t h i t j hE X X + ⎡⎤ = ⎣⎦ ,( ii ij h ) = denotes the ACVF of , , ti X while for ij ) denotes the cross-covariance function between and . , X , tj X We assume that the m processes are jointly stationary , i.e. that for all ij ij h is a function of h only and does not depend on t . Each of the three main types of univariate models has its corresponding multivariate extension, which is obtained, essentially, by replacing the scalar parameters in the univariate model by matrix parameters. Let and be m -dimensional random vectors. , [ ,..., ]' t m = X , [ ,..., t m ZZ = Z A zero-mean, m -variate model is defined by ARMA( , ) pq (*) 11 ... ... p t p t t q −− = Φ ++ Φ + + Θ Θ X Z Z Z t q q where AR and MA coefficients, { and { } ( 1,. .., ) i ip Φ= } ( 1,. .., ) i i Θ = , are real matrices, and mm × is the multivariate white noise, i.e., with zero mean vector and uncorrelated values at different times

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H16 - H16 The discussion of bivariate processes is readily...

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