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 Subscript t, of the model. If the slope component is absent, the resulting trend is a random walk (RW) specified by the following equations:

mu Subscript t Baseline equals mu Subscript t minus 1 Baseline plus eta Subscript t Baseline comma eta Subscript t Baseline tilde normal i normal i normal d upper N left-parenthesis 0 comma sigma Subscript eta Superscript 2 Baseline right-parenthesis

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:

StartLayout 1st Row 1st Column mu Subscript t 2nd Column equals 3rd Column mu Subscript t minus 1 Baseline plus beta Subscript t minus 1 Baseline plus eta Subscript t Baseline comma eta Subscript t Baseline tilde normal i normal i normal d upper N left-parenthesis 0 comma sigma Subscript eta Superscript 2 Baseline right-parenthesis 2nd Row 1st Column beta Subscript t 2nd Column equals 3rd Column beta Subscript t minus 1 Baseline plus xi Subscript t Baseline comma xi Subscript t Baseline tilde normal i normal i normal d upper N left-parenthesis 0 comma sigma Subscript xi Superscript 2 Baseline right-parenthesis EndLayout

In either case, the options in the LEVEL statement are used to specify the value of sigma Subscript eta Superscript 2 and to request forecasts of mu Subscript t. The SLOPE statement is used for similar purposes in the case of slope beta Subscript 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 Subscript eta Superscript 2 fixed at 4. It also requests printing of filtered values of mu Subscript t. The value of sigma Subscript xi Superscript 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 Subscript eta Superscript 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 Subscript eta Superscript 2, the disturbance variance in the mu Subscript t equation at the start of the parameter estimation process. Any nonnegative value, including zero, is an acceptable starting value.

Last updated: June 19, 2025