State space representation of an ARIMA model

We consider that the auto-regressive polynomial contains some unit roots. It can be factorized in a stationary polynomial (defined by the roots outside the unit circle) and in a non-stationary polynomial (defined by the roots on the unit circle), which is notated:

$\Delta\left(B\right) = 1 + \delta_1 B + \cdots + \delta_d B^d$

The state space form of an ARIMA model is similar to the state space form of an ARMA model except for its initialization.

Initialization

The initial conditions can be written as follows:

$\alpha_{-1} = \begin{pmatrix}1 \\ 0 \\ \vdots\\ 0 \end{pmatrix}$

$P_{*} = \Sigma \Omega \Sigma'$

$B = \Lambda$

$P_{\infty}= \Lambda \Lambda'$

$\Omega$ is the unconditional covariance of the state array of the stationary model.

$\Sigma = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ \lambda_1 & 1 & \cdots & 0 \\ \lambda_2 & \lambda_1 & \cdots & \vdots\\ \vdots & \vdots & \vdots & \vdots \\ \lambda_{r-1} & \lambda_{r-2} & \cdots & \lambda_{1} \end{pmatrix}$

where $\lambda_{i}$ are generated by $\frac{1}{\Delta\left(B\right)}$

$\Lambda$ is a r x d matrix; its first d rows form an identity matrix; other cells are defined by the recursive relationship:

$\Lambda \left(i,j\right) = -\sum_{k=1}^d {\delta_k \Lambda \left(i-k,j\right)}$

Implementation

ARIMA models are implemented in the class demetra.arima.ssf.SsfArima