The SSM Procedure

Multivariate Random Walk Trend

The STATE statement option TYPE=RW specifies a dim-dimensional random walk

\[  \pmb {\alpha }_{t+1} = \pmb {\alpha }_{t} + \pmb {\eta }_{t+1}  \]

where $ \pmb {\eta }_{t}$ is a sequence of zero mean, independent, Gaussian vectors with covariance $\pmb {\Sigma }$. The specification of the associated system matrices is trivial: $\mb{T}$ is a dim-dimensional identity matrix, $ \mb{I}_{dim}$, $\mb{Q} = \pmb {\Sigma }$, and the initial condition is fully diffuse ($\mb{Q}_{1} = 0$ and $\mb{A}_{1} = \mb{I}_{dim}$).

The multivariate random walk is a useful trend model for multivariate time series data. The trend term for the ith response variable is defined by a component that simply picks the ith ($1 \leq i \leq dim$) element of $\pmb {\alpha }_{t}$. For example, the component rw_i defined as follows can be used as a trend term in the MODEL statement of the ith response variable:

     state randomWalk(3) type=rw ...;
     component rw_2 = randomWalk[2];