State space representation of an AR model
Introduction
The AR process is defined by
\[\Phi\left(B\right)y_t=\epsilon_t\]where:
\[\Phi\left(B\right)=1+\varphi_1 B + \cdots + \varphi_p B^p\]is an auto-regressive polynomial.
Let $\gamma_i$ be the autocovariances of the model.
Using those notations, the state-space model can be written as follows :
State vector:
\[\alpha_t= \begin{pmatrix} y_t \\ y_{t-1} \\ \vdots \\ y_{t-p+1} \end{pmatrix}\]Dynamics
\[T_t = \begin{pmatrix}-\varphi_1 & \cdots & \cdots & -\varphi_p \\ 1 & \cdots & \cdots & 0 \\ \vdots & \ddots & \ddots & \vdots\\ 0 & 0 & 1 & 0 \end{pmatrix}\] \[S_t = \sigma_{ar} \begin{pmatrix} 1 \\ 0 \\ \vdots\\ 0 \end{pmatrix}\] \[V_t = S S'\]Measurement
\[Z_t = \begin{pmatrix} 1 & 0 & \cdots & 0\end{pmatrix}\] \[h_t = 0\]Initialization
\[\alpha_{-1} = \begin{pmatrix}0 \\ 0 \\ \vdots\\ 0 \end{pmatrix}\] \[P_{*} = \Omega\]$\Omega$ is the unconditional covariance of the state array; it can be easily derived using the MA representation. We have:
\[\Omega\left(i,0\right) = \gamma_i\] \[\Omega\left(i,j\right) = \Omega\left(i-1,j-1\right)-\psi_i \psi_j\]Implementation
AR models are implemented in the class demetra.arima.ssf.SsfArima