SSM Procedure

Multivariate Local Linear Trend

The STATE statement option TYPE=LL specifies a (2*dim)-dimensional alpha alpha Subscript t, needed for defining a dim-dimensional local linear trend. The first dim elements of alpha alpha Subscript t correspond to the needed multivariate trend, and the subsequent dim elements are needed to capture the slope vector of this trend. alpha alpha Subscript t can be defined as

alpha alpha Subscript t plus 1 Baseline equals bold upper T 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 D normal i normal a normal g left-parenthesis normal upper Sigma normal upper Sigma comma normal upper Sigma normal upper Sigma Subscript normal s normal l normal o normal p normal e Baseline right-parenthesis and bold upper T is a 2*dim-dimensional block matrix bold upper T equals left-parenthesis bold upper I Subscript d i m Baseline bold upper I Subscript d i m Baseline comma bold 0 bold upper I Subscript d i m Baseline right-parenthesis. The initial condition is fully diffuse (bold upper Q 1 equals 0 and bold upper A 1 equals bold upper I Subscript 2 asterisk d i m). This is a multivariate generalization of the univariate local linear trend.

The multivariate local linear trend 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 element (1 less-than-or-equal-to i less-than-or-equal-to d i m) of alpha alpha Subscript t. For example, the component ll_i defined as follows can be used as a trend term in the MODEL statement of the ith response variable: 

  state localLin(dim) type=ll(slopecov..) ...;
  component ll_3 = localLin[3];
Last updated: June 19, 2025