Local level (= random walk)

Description

The local level block describes a random walk component. When the innovation variance is set to 0, it also describes a constant term (known when the initialization is specified, estimated otherwise)

\[l_{t+1} = l_t + \epsilon_t\] \[\epsilon_t \sim N(0, \sigma^2_l)\]

State block

The state block is $\alpha_t=\begin{pmatrix} l_t \end{pmatrix}$

Diffuse initialization

\[a_0 = 0\] \[P_*= 0\] \[B= 1\] \[P_\infty= 1\]

Dynamics

\[T_t = 1\] \[V_t = \sigma^2_l\] \[S_t = \sigma_l\]

Default measurement

\[Z_t = 1\]

Parameters

\[\sigma^2_l \ge 0\]

The block is represented by $\text{ll}(\sigma^2_l)$ or $\text{ll}_0(\sigma^2_l)$ in case of 0-initialization