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.