Local linear trend
Description
The local linear trend block describes the following trend component:
\[l_{t+1} = l_t + n_t + \epsilon_t\] \[n_{t+1} = n_t + \mu_t\] \[\epsilon_t \sim N(0, \sigma^2_l)\] \[\mu_t \sim N(0, \sigma^2_n)\]State block
The state block is
\[\alpha_t=\begin{pmatrix} l_t \\ n_t \end{pmatrix}\]Diffuse initialization
\[a_0 = 0\] \[P_*= 0\] \[B= \begin{pmatrix} 1 && 0 \\ 0 && 1 \end{pmatrix}\] \[P_{\infty}= \begin{pmatrix} 1 && 0 \\ 0 && 1 \end{pmatrix}\]Dynamics
\[T_t = \begin{pmatrix} 1 && 1 \\ 0 && 1 \end{pmatrix}\] \[V_t = \begin{pmatrix} \sigma^2_l && 0 \\ 0 && \sigma^2_n \end{pmatrix}\] \[S_t = \begin{pmatrix} \sigma_l && 0 \\ 0 && \sigma_n \end{pmatrix}\]Default measurement
\[Z_t= \begin{pmatrix} 1 && 0 \end{pmatrix}\]Parameters
\[\sigma^2_l \ge 0,\quad \sigma^2_n \ge 0\]The block is represented by
\[\text{llt}(\sigma^2_l,\sigma^2_n)\]Remarks
On occasion, the local linear trend can be split into two components:
- A smooth trend $=\text{llt}(0, \sigma^2_n)$
- A short-term component $=\text{ll}_0(\sigma^2_l)$