Generic state space form
We write
\[y^C_t=\sum_{k=t-t \textsf{ mod }c}^t{y_k}\]It represents the cumulator variable, from the beginning of each benchmarking period to the current period (included).
So, $y^C_t=\mathbf{y}_{t/c}$ 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 $\alpha_t=\begin{pmatrix}y_t^{\underline C} \ \tilde\alpha_t \end{pmatrix}$
Initialization
\[a_{-1} = \begin{pmatrix} 0 & \tilde a_{-1}\end{pmatrix}\]Dynamics
\[T_t =\begin{cases} \begin{pmatrix} 0 & 0 \\ 0 &\tilde T_t\end{pmatrix} & if\; {t+1} \textsf{ mod } c = 0 \\\begin{pmatrix} 0 & \tilde Z_t \\ 0 &\tilde T_t\end{pmatrix} & if\; t \textsf{ mod } c = 0 \\\begin{pmatrix} 1 & \tilde Z_t \\ 0 &\tilde T_t\end{pmatrix} & otherwise \end{cases}\]Measurement
\[Z_t = \begin{cases} \begin{pmatrix} 0 & \tilde Z_t \end{pmatrix} & if \; t \textsf{ mod } c = 0 \\ \begin{pmatrix} 1 & \tilde Z_t \end{pmatrix} & if \; t \textsf{ mod } c \neq 0\end{cases}\]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.