Generic SSF

Generic state space form

We write

$y_t^{\underline{C}}=\sum_{k=t-t\mod{c}}^{t-1}{y_k}$

It represents the cumulator variable, from the beginning of each benchmarking period (included) to the current period (excluded). and

$y_t^{C}=\sum_{k=t-t\mod{c}}^t{y_k} = y_t^{\underline{C}}+y_t$

So that $y_t^C=\mathbf{y}_{t/c}$ when $t+1$ is a multiple of $c$ and is unobserved otherwise.

We consider that the unobserved disaggregated series has a state space form (SSF) identified by the state $\tilde\alpha_t$ and the system matrices $\left[ \tilde Z_t,\tilde H_t,\tilde T_t,\tilde V_t, \tilde S_t, \tilde a_{-1}, \tilde P_*,\tilde B,\tilde P_\infty \right]$ (see Ssf model ).

The benchmarking SSF is the original SSF extended by the cumulator variable

State vector

The state is $\alpha_t=\begin{pmatrix}y_t^C & \tilde\alpha_t \end{pmatrix}$

Initialization

$a_{-1} = \begin{pmatrix} 0 & \tilde a_{-1}\end{pmatrix}$

$B = \begin{pmatrix} 0 \\ \tilde B\end{pmatrix}$

$P_*= \begin{pmatrix} 0 & 0 \\ 0 &\tilde P_*\end{pmatrix}$

$P_\infty= \begin{pmatrix} 0 & 0 \\ 0 &\tilde P_\infty\end{pmatrix}$

Dynamics

$T_t =\begin{cases} \begin{pmatrix} 0 & 0 \\ 0 &\tilde T_t\end{pmatrix} if\; {t+1} \mod c = 0 \\\begin{pmatrix} 0 & \tilde Z_t \\ 0 &\tilde T_t\end{pmatrix} if\; t \mod c = 0 \\\begin{pmatrix} 1 & \tilde Z_t \\ 0 &\tilde T_t\end{pmatrix} otherwise \end{cases}$

$V_t= \begin{pmatrix} 0 & 0 \\ 0 &\tilde V_t\end{pmatrix}$

$S_t = \begin{pmatrix} 0 \\ \tilde S_t\end{pmatrix}$

Measurement

$Z_t = \begin{cases} \begin{pmatrix} 0 & \tilde Z_t \end{pmatrix} if \; t \mod c = 0 \\ \begin{pmatrix} 1 & \tilde Z_t \end{pmatrix} if \; t \mod c \neq 0\end{cases}$

Regression model

The regression model is now built on the cumulated series $y_t^C = X_t^C \beta+\mu_t^C$.

The problem is then a simple problem of missing observations, which can be easily computed by means of the Kalman smoother.