Ordinary smoother (univariate)
Notations
$a_t, P_t, M_t \left(=P_t Z_t’ \right), f_t, e_t$ are quantities obtained in the filtering process. $r_t, r_{0t}, r_{1t}$ are auxiliary row-matrices, while $N_t,N_{0t},N_{1t}, N_{2t}$ are auxiliary square matrices. Except $r_{0t},N_{0t}$ which are initialised with the last values of $r_t,N_t$, those objects are set to 0 at the beginning of the process.
The following notations are used:
\[\tilde a_t=E\left(\alpha_{t} \vert y_0 \cdots y_{n}\right)\] \[\tilde P_t=var\left(\alpha_{t} \vert y_0 \cdots y_{n}\right)\] \[\tilde e_t=E\left(\epsilon_{t} \vert y_0 \cdots y_{n}\right)\]Normal recursions
Observed t, with $f_t \neq 0$
\[K_t = T_t M_t / f_t\] \[\tilde e_t = e_t / f_t - r_t K_t\] \[\left[\text{var }\tilde e_t = 1/f_t + K_t' N_t K_t \right]\] \[r_{t-1} = \tilde e_t Z_t + r_t T_t\] \[\left[L_t = T_t - K_t Z_t \right]\] \[\left[N_{t-1} = Z_t' Z_t / f_t + L_t' N_t L_t \right]\]Missing t or observed t, with $f_t = 0$
\[r_{t-1} = r_t T_t\] \[\left[N_{t-1} = T_t' N_t T_t \right]\]Smoothed states
\[\tilde a_t' = a_t' + r_t P_t\] \[\left[\tilde P_t = P_t + P_t N_t P_t \right]\]Smoothed disturbances
\[\tilde u_t = r_t S_t\] \[\left[var\left(\tilde u_t \right) = V_t-S_t' N_t S_t \right]\]Implementation details
The ordinary smoother for univariate models is implemented in the class demetra.ssf.univariate.OrdinarySmoother of the library demetra-ssf
The current implementation computes successively the following quantities (observed case; operations marked with an asterisk don’t imply actual matrix computations; they use functional forms):
\[x = r_t T_t \quad *\] \[\tilde e_t =\left( e_t-x M_t\right)/f_t\] \[r_{t-1} = x + \tilde e_t Z_t \quad *\] \[A = xl(xl(N_t)')\] \[N_{t-1}= ZZ'/f_t+A \quad *\]xl operator
The operator $xl(y) = y(T_t - K_t Z_t)$ is computed as follows
\[q = y T_t \quad *\] \[w = q M_t\] \[xl(y) = q - w/f_t Z_t \quad *\]It can be applied on each row of a matrix
Multivariate model
$a_t, P_t, \tilde M_t \left(=P_t Z_t’ R’^{-1}_t \right), R_t, u_t\left( = e_t R’^{-1}_t \right)$ are quantities obtained in the filtering process. $R_t$ is the Cholesky factor of $F_t$ ($R_t R’_t = F_t$),
The previous algorithm becomes:
\[\tilde K_t = T_t \tilde M_t\] \[\tilde e_t = (u_t - r_t \tilde K_t)R_t^{-1}\] \[\left[\text{var }\tilde e_t = R'^{-1}_t(I+ K_t' N_t K_t)R^{-1}_t \right]\] \[r_{t-1} = \tilde e_t Z_t + r_t T_t\] \[\left[L_t = T_t - K_t Z_t \right]\] \[\left[N_{t-1} = Z_t' F^{-1}_t Z_t + L_t' N_t L_t \right]\]Missing t or observed t, with $f_t = 0$
\[r_{t-1} = r_t T_t\] \[\left[N_{t-1} = T_t' N_t T_t \right]\]Implementation details
We compute the following steps:
\[x = r_t T_t\] \[\tilde e_t =\left( u_t-x \tilde M_t\right)R_t^{-1}\] \[r_{t-1} = x + \tilde e_t Z_t\] \[A = xl(xl(N_t)')\] \[N_{t-1}= Z_tR_t'^{-1}R_t^{-1}Z_t'+A\]xl operator
The operator $xl(y) = y(T_t - K_t Z_t)$ is computed as follows
\[q = y T_t\] \[w = q \tilde M_t\] \[xl(y) = q - (w R_t^{-1}) Z_t\]It can be applied on each row of a matrix
The ordinary smoother is implemented in the classes demetra.ssf.univariate.OrdinarySmoother and demetra.ssf.multivariate.MultivariateOrdinarySmoother.