SSM Procedure

Multivariate Season

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

zeta zeta equals sigma-summation Underscript j equals 1 Overscript left-bracket s slash 2 right-bracket Endscripts zeta zeta Subscript j

where the constituent cycles zeta zeta Subscript j, called harmonics, have frequencies lamda Subscript j Baseline equals 2 pi j slashs. All the harmonics are assumed to be statistically independent, have the same damping factor rho equals 1, and are governed by the disturbances with the same covariance matrix normal upper Sigma normal upper Sigma. The number of harmonics, left-bracket sans-serif-italic s slash 2 right-bracket, equals sans-serif-italic s slash 2 if s is even and left-parenthesis sans-serif-italic s negative 1 right-parenthesis slash 2 if it is odd. This means that specifying TYPE=SEASON(LENGTH=s) is equivalent to specifying left-bracket sans-serif-italic s slash 2 right-bracket cycle specifications with correct frequencies, damping factor rho equals 1, and the COV option restricted to the same covariance normal upper Sigma normal upper Sigma. The resulting alpha alpha Subscript t is necessarily ((s–1)*dim)-dimensional. When the season length sans-serif-italic s is even, the last harmonic cycle, zeta zeta Subscript sans-serif-italic s slash 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 minus bold upper I Subscript d i m. As a result of this discussion, it is clear that the system matrices bold upper T and bold upper Q associated with the ((s–1)*dim)-dimensional alpha alpha Subscript 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 alpha alpha Subscript t provided the needed (multivariate) component. This is not the case for the (multivariate) season component. Extracting the ith seasonal component from alpha alpha Subscript t requires accumulating the contributions from the left-bracket sans-serif-italic s slash 2 right-bracket harmonics that are associated with this ith seasonal, which are not organized contiguously in alpha alpha Subscript t. For example, suppose that dim is 2 and the season length s is 4. In this case left-bracket sans-serif-italic s slash 2 right-bracket is 2, and the bivariate seasonal component is a sum of two independent bivariate cycles, zeta zeta 1 and zeta zeta 2. The cycle zeta zeta 1 has frequency pi slash 2 and its underlying state, say alpha alpha Subscript t Superscript a, has dimension 2 asterisk d i m equals 4. The last harmonic, zeta zeta 2, has frequency pi, and therefore its underlying state, say alpha alpha Subscript t Superscript b, has dimension 2. The combined state alpha alpha Subscript t Baseline equals left-parenthesis alpha alpha Subscript t Superscript a Baseline comma alpha alpha Subscript t Superscript b Baseline right-parenthesis has dimension 6 equals 4 plus 2. In order to extract the first bivariate seasonal component, you must extract the first components of bivariate cycles zeta zeta 1 and zeta zeta 2, which in turn implies the first elements of alpha alpha Subscript t Superscript a and alpha alpha Subscript t Superscript b, respectively. Thus, obtaining the first bivariate seasonal component requires extracting the first and the fifth elements of the combined state alpha alpha Subscript t. Similarly, obtaining the second bivariate seasonal component requires extracting the second and the sixth elements of the combined state alpha alpha Subscript t. All this can be summarized by the dot product expressions

StartLayout 1st Row 1st Column s Subscript 1 t 2nd Column equals 3rd Column left-parenthesis 1 0 0 0 1 0 right-parenthesis alpha alpha Subscript t 2nd Row 1st Column s Subscript 2 t 2nd Column equals 3rd Column left-parenthesis 0 1 0 0 0 1 right-parenthesis alpha alpha Subscript t EndLayout

where s Subscript 1 t and s Subscript 2 t denote the first and second components, respectively, of the bivariate seasonal component. Note that s Subscript 1 t and s Subscript 2 t are univariate seasonal components, each of season length 4, in their own right. They are correlated components; their correlation structure depends on normal upper Sigma normal upper 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 Subscript 1 t and s Subscript 2 t. The first set of statements explicitly specify the linear combinations needed for defining s Subscript 1 t and s Subscript 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.

Last updated: June 19, 2025