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