The SSM Procedure

Continuous-Time Cycle

The STATE statement option TYPE=CYCLE(CT) specifies a two-dimensional $\pmb {\alpha }_{t}$ , needed for defining a univariate continuous time cycle. In this case the nominal dimension, dim, must be 1. In particular, $\pmb {\Sigma }$ becomes one-dimensional, which is denoted by $\sigma ^{2}$ . This cycle can be used for any data type. As before, the parameters of the cycle are a damping factor $\rho$ , $0 < \rho \leq 1$ , and period $> 0$ . Unlike in the discrete-time cycle described in the section Multivariate Cycle, the period is not required to be larger than 2. Let $\lambda = 2\pi /\Argument{period}$ , and let $h_ t = (\tau _{t+1} - \tau _{t})$ denote the difference between successive time points. In this case, the system matrices $\mb {T}$ and $\mb {Q}$ that govern $\pmb {\alpha }_{t}$ depend on $h_ t$ . They are:

$\begin{eqnarray*} \mb {T} & = & \rho ^{h} \left( \cos (\lambda h) \; \sin (\lambda h), \; -\sin (\lambda h) \; \cos (\lambda h) \right) \nonumber \\ \mb {Q} & = & \frac{\sigma ^{2}(1 - \rho ^{2h})}{-2\ln (\rho )}*\mb {I}_{2} \qquad \text {if}\; \rho < 1 \nonumber \\ \mb {Q} & = & \sigma ^{2}h\mb {I}_{2} \qquad \text {if}\; \rho = 1 \nonumber \end{eqnarray*}$

If $\rho < 1$ , the initial condition is nondiffuse: $\mb {Q}_{1} = \frac{\sigma ^{2}}{-2\ln (\rho )}\mb {I}_{2}$ . For $\rho = 1$ , the initial condition is fully diffuse.

The first element of $\pmb {\alpha }_{t}$ corresponds to the needed cycle, and the second element is an auxiliary quantity. You can define a cycle term based on this state as follows:

     state cycleState(1) type=cycle(CT)  ...;
     component cycle = cycleState[1];

The CT option must be included in the use of TYPE=CYCLE.