State-Space Model for Time Series

The complete state-space model can be written as a set of two coupled equations 

It can be noted that, in Eq. (5.127), an additional term, called the input, has been included in the model. This term allows the model to consider deterministic or other stochastic signals that have an impact on the overall process. When dealing with state-space models, it is traditional to consider the full form from the start rather 

Finally, a state-space model is said to be causal (or stable) if the absolute value of the eigenvalues of the state matrix ( -matrix) lie inside the unit circle. 

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