Multivariate Time Series With Linear State Space Structure by Víctor Gómez

Author:Víctor Gómez
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

and E(v t ) = 0. The initial state vector x 1 is specified as

(4.84)

where , the matrices W, A, and are fixed and known, and is a random vector that models the unknown initial conditions. Here, the notation means that the vector v has mean m and covariance matrix .

It is assumed that the vectors x and δ are mutually orthogonal and that x 1 is orthogonal to the {u t } and {v t } sequences. As in Sect. 4.1, we will usually assume unless otherwise specified. Finally, we will further assume that if δ and β are zero in the state space model (4.82) and (4.83), then the generated data have a covariance matrix that is nonsingular. That is, if the model reduces to (4.1) and (4.2) with x 1 = x and if Y = (Y ′1, …, Y ′ n )′, then Var(Y ) is nonsingular.

Equations (4.82) and (4.83) are called the “transition equation” and the “measurement equation,” respectively.

Instead of the model (4.82) and (4.83), the following alternative single disturbance state space model can be used

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