The SSM Procedure

Multivariate Local Linear Trend

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

$\pmb {\alpha }_{t+1} = \mb {T} \pmb {\alpha }_{t} + \pmb {\eta }_{t+1}$

where $\pmb {\eta }_{t}$ is a sequence of zero mean, independent, Gaussian vectors with covariance $\mr {Diag}(\pmb {\Sigma }, \; \pmb {\Sigma }_{slope})$ and $\mb {T}$ is a 2*dim-dimensional block matrix $\mb {T} = (\mb {I}_{dim} \; \mb {I}_{dim}, \; \mb {0} \; \mb {I}_{dim} )$ . The initial condition is fully diffuse ( $\mb {Q}_{1} = 0$ and $\mb {A}_{1} =\mb {I}_{2*dim}$ ). 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 $i$ th response variable is defined by a component that simply picks the $i$ th element ( $1 \leq i \leq dim$ ) of $\pmb {\alpha }_{t}$ . For example, the component ll_i defined as follows can be used as a trend term in the MODEL statement of the $i$ th response variable:

     state localLin(dim) type=ll(slopecov..) ...;
     component ll_3 = localLin[3];