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 ith response variable is defined by a component that simply picks the ith 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 ith response variable: 

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