Seats

Canonical decomposition of ARIMA models

We consider in the description of SEATS a simplified model with two components (the signal and the noise):

$y_t = s_t + n_t$

$\phi(B) \Delta(B) y_t = \theta(B)\epsilon_t$

$\phi_s(B) \Delta_s(B) s_t = \theta_s(B)\epsilon_{st}$

$\phi_n(B) \Delta_n(B) n_t = \theta_n(B)\epsilon_{nt}$

The variances of the innovations are respectively $\sigma^2$ , $\sigma^2_s$ and $\sigma^2_n$

The aggregation constraint yields:

$\phi(B) = \phi_s(B)\phi_n(B)$

$\Delta(B) = \Delta_s(B)\Delta_n(B)$

$\sigma^2\theta(B)\theta(F)=\sigma^2_s\phi_n(B)\Delta_n(B)\theta_s(B)\phi_n(F)\Delta_n(F)\theta_s(F)$ $+\sigma^2_n\phi_s(B)\Delta_s(B)\theta_n(B)\phi_s(F)\Delta_s(F)\theta_n(F)$

The estimator of the signal is obtained as

$\hat{s}_t = k_s \frac{\Psi_s(B)\Psi_s(F)}{\Psi(B)\Psi(F)}y_t$