Aggregation of state space models
Introduction
We consider the aggregation of $n$ state space blocks such that the innovations of the transition equations are independent.
State block
The state vector is defined as the stack of the state vectors of the different models
\[\alpha_t = \begin{pmatrix} \alpha_{1t} \\ \vdots \\ \alpha_{nt} \end{pmatrix}\]Dynamics and initialization
\[T_t = \begin{pmatrix} T_{1t} & 0 & \cdots & 0 \\ 0 & T_{2t} & 0 & \vdots \\ \vdots & \ddots & \ddots & \vdots\\ 0 & \cdots & 0 & T_{nt} \end{pmatrix}\]The other matrices of the transition equation and of the initialization are defined in a similar way.
Measurement
\[Z_t = \begin{pmatrix} Z_{1t} & \cdots & Z_{nt} \end{pmatrix}\] \[H_t = H_{1t}\]Implementation
Aggregate models are implemented in the classes demetra.ssf.implementations.CompositeSsf and demetra.ssf.implementations.MultivariateCompositeSsf