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I'm struggling with Reinsel's book "Elements of Multivariate Time Series Analysis," because I thought that it would be a good idea to switch from Vector ARMA to state-space representations; particularly after reading What are disadvantages of state-space models and Kalman Filter for time-series modelling?, and after struggling with parameter estimation of VARMA processes.

The problem is that I can't see how correlation between two time series can be modeled in state-space representations. Is it that the correlation appears in the error covariance matrices? It sounds weird to me, particularly coming from VARMA. (Switching to state-space models is being harder than I thought!)

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Correlation can be introduced via the covariance matrices, but also through non zero off-diagonal terms in the transition or observation matrix.

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  • $\begingroup$ About your kind answer: (1) if I introduce the correlation in the transition matrix (off-diagonal elements), isn't it that I'm forcing a deterministic correlation? I mean that it seems that I'm forcing say x1 and x2 to be always varying by the same amount. (If the off-diagonal matrix entry is 2, I am saying that x2 will "always" be double the value of x1.) (2) Do you know of any good resource where i can find how to estimate the state-space of two cross-correlated time series? $\endgroup$ – Sonntag Apr 16 '14 at 8:54
  • $\begingroup$ (1) no, because the total correlation will include the error covariance. (2) see my answer here $\endgroup$ – Aksakal Apr 16 '14 at 13:35
  • $\begingroup$ @Sonntag, as pointed in the previous comment, if in one of your state equations you have x1(t+1) = ax1(t) + bx2(t) + eps(t), the connection between x1(t) and x2(t) is not deterministic, for it is affected by the state equation noise eps(t). $\endgroup$ – F. Tusell Apr 16 '14 at 13:40

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