Ordinary smoother

Ordinary smoother (univariate)

Notations

$a_t, P_t, C_t \left(=P_t Z_t’ \right), f_t, e_t$ 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

$K_t = T_t C_t / f_t$ $\tilde e_t = v_t / f_t - r_t K_t$ $\left[var \tilde e_t = 1/f_t + K_t' N_t K_t \right]$ $r_{t-1} = \tilde e_t Z_t + r_t T_t$ $\left[L_t = T_t - K_t Z_t \right]$ $\left[N_{t-1} = Z_t' Z_t / f_t + L_t' N_t L_t \right]$

Missing t

$r_{t-1} = r_t T_t$ $\left[N_{t-1} = T_t' N_t T_t \right]$

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):

$x = r_t T_t \quad *$ $\tilde e_t =\left( v_t-x C_t\right)/f_t$ $r_{t-1} = x + \tilde e_t Z_t \quad *$ $A = xl(xl(N_t)')$

xl operator

The operator $xl(y) = y(T_t - K_t Z_t)$ is computed as follows

$q = y T_t \quad *$ $w = q C_t$ $xl(y) = q - w/f_t Z_t \quad *$

It can be applied on each row of a matrix

Implementation

The ordinary smoother is implemented in the classes demetra.ssf.univariate.OrdinarySmoother and demetra.ssf.multivariate.MultivariateOrdinarySmoother.