SSM Procedure

Multivariate Random Walk Trend

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

alpha alpha Subscript t plus 1 Baseline equals alpha alpha Subscript t Baseline plus eta eta Subscript t plus 1

where eta eta Subscript t is a sequence of zero mean, independent, Gaussian vectors with covariance normal upper Sigma normal upper Sigma. The specification of the associated system matrices is trivial: bold upper T is a dim-dimensional identity matrix, bold upper I Subscript d i m, bold upper Q equals normal upper Sigma normal upper Sigma, and the initial condition is fully diffuse (bold upper Q 1 equals 0 and bold upper A 1 equals bold upper I Subscript d i m).

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 less-than-or-equal-to i less-than-or-equal-to d i m) element of alpha alpha Subscript 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];
Last updated: June 19, 2025