ARMA Type Models: Multivariate

The above discussion of the AR( p) and MA(q) models can be extended to the case where the observable process is a vector {Zt, t Є T}, Zt : (m x 1). This stochastic vector process is said to be second-order stationary if:

E(Z t) = p, cov(Z t, Z t-T) = E((Z t – p)(Z t_T – p)T) = I (t).

Note that I (t) is not symmetric since aij (t) = а;ї(-т); see Hamilton (1994).

The ARMA representations for the vector process {Zt, t Є T} take the form:

VAR( p): Z t = a 0 + A1Z t-1 + A2Z t-2 + … + ApZ t-p +

VMA(q):. Z t = p + фlЄt-l + ф2^ t-2 + … + фч£^ч + єt,

VARMA(p, q): Z t = Y0 + A1Zt-1 + … + ApZt-p + ®1£t-1 + … + £t-q + e t,

where the vector error process is of the form: et ~ NI(0, Q).

In direct analogy to the univariate case, the probabilistic assumptions are:

Подпись: 1e zero mean: О II CO Ш 2e constant variance: E(et eT) = Q, 3e no autocorrelation: E(eteP) = 0, T Ф 0, 4e normality: et ~ N(v), Подпись: > t Є T.(28.27)

Looking at the above representations from the PR perspective we need to trans­late 1e-4e into assumptions in terms of the observable vector {Zt, t Є T}.

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