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 $i$ th response variable is defined by a component that simply picks the $i$ th ( $1 \leq i \leq dim$ ) 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 $i$ th response variable:

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