The UCM Procedure

CYCLE <options> ;

The CYCLE statement is used to specify a cycle component, , in the model. The stochastic equation governing a cycle component of period p and damping factor is as follows

where and are independent, zero-mean, Gaussian disturbances with variance and is the angular frequency of the cycle. Any p strictly greater than two is an admissible value for the period, and the damping factor can be any value in the interval (0, 1), including one but excluding zero. The cycles with frequency zero and , which correspond to the periods equal to infinity and two, respectively, can be specified using the AUTOREG statement. The values of less than one give rise to a stationary cycle, while gives rise to a nonstationary cycle. As a default, values of , p, and are estimated from the data. However, if necessary, you can fix the values of some or all of these parameters.

There can be multiple cycles in a model, each specified using a separate CYCLE statement. The examples that follow illustrate the use of the CYCLE statement.

The following statements request including two cycles in the model. The parameters of each of these cycles are estimated from the data.

      cycle;
      cycle;

The following statement requests inclusion of a nonstationary cycle in the model. The cycle period p and the disturbance variance are estimated from the data.

      cycle rho=1 noest=rho;

In the following statement a nonstationary cycle with a fixed period of 12 is specified. Moreover, a starting value is supplied for .

      cycle period=12 rho=1 variance=4 noest=(rho period);