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 2 is an admissible value for the period, and the damping factor can be any value in the interval (0, 1), including 1 but excluding 0. The cycles with frequency 0 and , which correspond to the periods equal to infinity and 2, respectively, can be specified using the AUTOREG statement. The values of less than 1 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 add two cycles in the model. The parameters of each of these cycles are estimated from the data.


   cycle;
   cycle;

The following statement adds 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);

You can specify the following options:

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: 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 30.9 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> ): plots the filtered or smoothed estimate of the cycle component.
PRINT=FILTER PRINT=SMOOTH PRINT=( <FILTER> <SMOOTH> ): prints 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 1 but excluding 0, 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 0, is an acceptable starting value.

Last updated: November 24, 2025