Properties of State Space Models
This chapter briefly discusses three properties of dynamic systems in state space form that are important in our model-building procedure. They are stability, observability, and reachability (controllability). The notion of stability is well known. A dynamic system is asymptotically stable if the effects of initial conditions vanish asymptotically over time. This property holds if and only if all the eigenvalues of the transition matrix A of a state space model are less than one in magnitude. The other two properties are less familiar to statisticians and econometricians. They may be motivated by requiring that state space model representations be parsimonious—i.e., by the “minimality” of state vector dimension. We briefly mentioned observability in connection with the observability canonical form in Chapter 4. The pair of matrices A and C in a state space model determines this property. More on these properties, as well as economic examples of state space models, can be found in Aoki . The pair of matrices A and B jointly determines the reachability property. When the model possesses both of these properties, then the minimality of the state vector dimension is assured (see Kalman , or Lindquist and Picci ).
KeywordsManifold Covariance Assure Estima
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