The SSM Procedure

Multivariate Season

The STATE statement option TYPE=SEASON(LENGTH=s) specifies a ((s–1)*dim)-dimensional $\pmb {\alpha }_{t}$, needed for defining a dim-dimensional trigonometric season component with season length s. A (multivariate) trigonometric season component, $\pmb {\zeta }$, is a sum of (multivariate) cycles of different frequencies,

\[  \pmb {\zeta } = \sum _{j = 1}^{[s/2]} \pmb {\zeta }_{j}  \]

where the constituent cycles $\pmb {\zeta }_{j}$, called harmonics, have frequencies $\lambda _ j = 2 \pi j/$s. All the harmonics are assumed to be statistically independent, have the same damping factor $\rho = 1$, and are governed by the disturbances with the same covariance matrix $\pmb {\Sigma }$. The number of harmonics, $[\Argument{s}/2]$, equals $\Argument{s}/2$ if s is even and $(\Argument{s}-1)/2$ if it is odd. This means that specifying TYPE=SEASON(LENGTH=s) is equivalent to specifying $[\Argument{s}/2]$ cycle specifications with correct frequencies, damping factor $\rho = 1$, and the COV option restricted to the same covariance $\pmb {\Sigma }$. The resulting $\pmb {\alpha }_{t}$ is necessarily ((s–1)*dim)-dimensional. When the season length $\Argument{s}$ is even, the last harmonic cycle, $ \pmb {\zeta }_{\Argument{s}/2}$, has frequency $\pi $ and requires special attention. It is of dimension dim rather than 2*dim because its underlying state equation simplifies to a dim-variate autoregression with autoregression coefficient $-\mb{I}_{dim}$. As a result of this discussion, it is clear that the system matrices $ \mb{T}$ and $ \mb{Q}$ associated with the ((s–1)*dim)-dimensional $\pmb {\alpha }_{t}$ are block-diagonal with the blocks corresponding to the harmonics. The initial condition is fully diffuse.

For all the models discussed so far, the first dim elements of $\pmb {\alpha }_{t}$ provided the needed (multivariate) component. This is not the case for the (multivariate) season component. Extracting the ith seasonal component from $\pmb {\alpha }_{t}$ requires accumulating the contributions from the $[\Argument{s}/2]$ harmonics that are associated with this ith seasonal, which are not organized contiguously in $\pmb {\alpha }_{t}$. For example, suppose that dim is 2 and the season length s is 4. In this case $[\Argument{s}/2] $ is 2, and the bivariate seasonal component is a sum of two independent bivariate cycles, $ \pmb {\zeta }_{1}$ and $ \pmb {\zeta }_{2}$. The cycle $\pmb {\zeta }_{1}$ has frequency $\pi /2$ and its underlying state, say $\pmb {\alpha }_{t}^{a}$, has dimension $2 * dim = 4$. The last harmonic, $\pmb {\zeta }_{2}$, has frequency $\pi $, and therefore its underlying state, say $\pmb {\alpha }_{t}^{b}$, has dimension 2. The combined state $\pmb {\alpha }_{t} = ( \pmb {\alpha }_{t}^{a}, \pmb {\alpha }_{t}^{b} )$ has dimension $6 = 4 + 2$. In order to extract the first bivariate seasonal component, you must extract the first components of bivariate cycles $\pmb {\zeta }_{1}$ and $\pmb {\zeta }_{2}$, which in turn implies the first elements of $ \pmb {\alpha }_{t}^{a}$ and $ \pmb {\alpha }_{t}^{b}$, respectively. Thus, obtaining the first bivariate seasonal component requires extracting the first and the fifth elements of the combined state $ \pmb {\alpha }_{t}$. Similarly, obtaining the second bivariate seasonal component requires extracting the second and the sixth elements of the combined state $ \pmb {\alpha }_{t}$. All this can be summarized by the dot product expressions

\begin{eqnarray*}  s_{ 1 t} &  = &  ( 1 \;  0 \;  0 \;  0 \;  1 \;  0 ) \; \pmb {\alpha }_{t} \nonumber \\ s_{2 t} &  = &  ( 0 \;  1 \;  0 \;  0 \;  0 \;  1 )\;  \pmb {\alpha }_{t} \nonumber \end{eqnarray*}

where $ s_{ 1 t}$ and $ s_{ 2 t}$ denote the first and second components, respectively, of the bivariate seasonal component. Note that $ s_{ 1 t}$ and $ s_{ 2 t}$ are univariate seasonal components, each of season length 4, in their own right. They are correlated components; their correlation structure depends on $\pmb {\Sigma }$.

Obtaining the desired components of the multivariate seasonal component is made easy by a special syntax convention of the COMPONENT statement. Continuing with the previous example, the following examples illustrate two equivalent ways of obtaining $ s_{ 1 t}$ and $ s_{ 2 t}$. The first set of statements explicitly specify the linear combinations needed for defining $ s_{ 1 t}$ and $ s_{ 2 t}$:

     state seasonState(2) type=season(length=4)  ...;
     component s_1 =( 1  0  0  0  1  0 ) * seasonState;
     component s_2 =( 0  1  0  0  0  1 ) * seasonState;

The following simpler specification achieves the same result:

     state seasonState(2) type=season(length=4)  ...;
     component s_1 = seasonState[1];
     component s_2 = seasonState[2];

In the latter specification, the meaning of the element operator [] changes if the state in question is defined by using the TYPE= option.