The UCM Procedure

LEVEL Statement

LEVEL <options> ;

The LEVEL statement is used to include a level component in the model. The level component, either by itself or together with a slope component (see the SLOPE statement), forms the trend component, $\mu _ t$ , of the model. If the slope component is absent, the resulting trend is a random walk (RW) specified by the following equations:

$\mu _{t} = \mu _{t-1} + \eta _ t , \; \; \; \; \eta _ t \; \sim \; i.i.d. \; \; N( 0, \sigma _{\eta }^{2} )$

If the slope component is present, signified by the presence of a SLOPE statement, a locally linear trend (LLT) is obtained. The equations of LLT are as follows:

$\begin{eqnarray*} \mu _{t} & = & \mu _{t-1} + \beta _{t-1} + \eta _ t , \; \; \; \; \eta _ t \; \sim \; i.i.d. \; \; N( 0, \sigma _{\eta }^{2} ) \nonumber \\ \beta _{t} & = & \beta _{t-1} + \xi _{t} , \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \xi _ t \; \sim \; i.i.d. \; \; N( 0, \sigma _{\xi }^{2} )\nonumber \end{eqnarray*}$

In either case, the options in the LEVEL statement are used to specify the value of $\sigma _{\eta }^{2}$ and to request forecasts of $\mu _ t$ . The SLOPE statement is used for similar purposes in the case of slope $\beta _ t$ . The following examples illustrate the use of the LEVEL statement. Assuming that a SLOPE statement is not added subsequently, a simple random walk trend is specified by the following statement:

      level;

The following statements specify a locally linear trend with value of $\sigma _{\eta }^{2}$ fixed at 4. It also requests printing of filtered values of $\mu _ t$ . The value of $\sigma _{\xi }^{2}$ , the disturbance variance in the slope equation, is estimated from the data.

      level variance=4 noest print=filter;
      slope;

CHECKBREAK: turns on the checking of breaks in the level component.
NOEST: fixes the value of $\sigma _{\eta }^{2}$ to the value specified in the VARIANCE= option.
PLOT=FILTER PLOT=SMOOTH PLOT=( <FILTER> <SMOOTH> ): requests plotting of the filtered or smoothed estimate of the level component.
PRINT=FILTER PRINT=SMOOTH PRINT=( <FILTER> <SMOOTH> ): requests printing of the filtered or smoothed estimate of the level component.
VARIANCE=value: specifies an initial value for $\sigma _{\eta }^{2}$ , the disturbance variance in the $\mu _ t$ equation at the start of the parameter estimation process. Any nonnegative value, including zero, is an acceptable starting value.