UCM Procedure

CYCLE Statement

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

StartBinomialOrMatrix psi Subscript t Baseline Choose psi Subscript t Superscript asterisk Baseline EndBinomialOrMatrix equals rho Start 2 By 2 Matrix 1st Row 1st Column cosine lamda 2nd Column sine lamda 2nd Row 1st Column minus sine lamda 2nd Column cosine lamda EndMatrix StartBinomialOrMatrix psi Subscript t minus 1 Baseline Choose psi Subscript t minus 1 Superscript asterisk Baseline EndBinomialOrMatrix plus StartBinomialOrMatrix nu Subscript t Baseline Choose nu Subscript t Superscript asterisk EndBinomialOrMatrix

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);

NOEST=PERIOD NOEST=RHO NOEST=VARIANCE NOEST=( <RHO> <PERIOD> <VARIANCE> ): fixes the values of the component parameters to those specified in the RHO=, PERIOD=, and VARIANCE= options. This option enables you to fix any combination of parameter values.
ORDER=integer ^Experimental: enables you to specify a higher-order cycle. A higher-order cycle (a cycle whose order is greater than 1) is a generalization of the stochastic cycle described at the beginning of this section, which can be thought of as a first-order cycle. Higher-order cycles are well explained in Trimbur (2005) and Pelagatti (2015, sect. 3.3.3). A cycle whose order is greater than 2 is rarely needed, and specifying cycles of large orders (for example, an order greater than 4) can lead to computational instability. See Example 41.9 for an example of the use of higher-order cycles.
PERIOD=value: specifies an initial value for the cycle period during the parameter estimation process. Period value must be strictly greater than 2.
PLOT=FILTER PLOT=SMOOTH PLOT=( <FILTER> <SMOOTH> ): requests plotting of the filtered or smoothed estimate of the cycle component.
PRINT=FILTER PRINT=SMOOTH PRINT=( <FILTER> <SMOOTH> ): requests the printing of a filtered or smoothed estimate of the cycle component .
RHO=value: specifies an initial value for the damping factor in this component during the parameter estimation process. Any value in the interval (0, 1), including one but excluding zero, is an acceptable initial value for the damping factor.
VARIANCE=value: specifies an initial value for the disturbance variance parameter, , to be used during the parameter estimation process. Any nonnegative value, including zero, is an acceptable starting value.

Last updated: June 19, 2025