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:

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

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:


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;

turns on the checking of breaks in the level component.


fixes the value of $\sigma _{\eta }^{2}$ to the value specified in the VARIANCE= option.


requests plotting of the filtered or smoothed estimate of the level component.


requests printing of the filtered or smoothed estimate of the level component.


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.