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State space form of ARIMA models

Definition

The univariate ARIMA process is defined by

 ,

where

are the differencing, auto-regressive and moving average polynomials. We also write:

 Let  be the psi-weights of the Arima model,  and , the psi-weights and the autocovariances of the differenced Arma model. We also define:

State-space form

Using the usual notations and the previous definitions, we have:

State vector:

where  is the orthogonal projection of  on the subspace generated by . Thus, it is the forecast function with respect to the semi-infinite sample.

System matrices: 

The matrices of the model are

 and the initial conditions can be written: 

  is the variance/covariance of the stationary model; it can be derived by the relationships:

 is a r x r lower triangular matrix with ones on the main diagonal; other cells are defined by the recursive relationship:

 with the convention  if

  is a r x d matrix; its first d rows form an identity matrix; other cells are defined as above:

It should be noted that this representation allows models with unit roots in the moving average part and even common unit roots in both the auto-regressive and the moving average parts.