UCM Procedure

SLOPE Statement

  • SLOPE <options>;

The SLOPE statement is used to include a slope component in the model. The slope component cannot be used without the level component (see the LEVEL statement). The level and slope specifications jointly define the trend component of the model. A SLOPE statement without the accompanying LEVEL statement is ignored. The equations of the trend, defined jointly by the level mu Subscript t and slope beta Subscript t, 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

The SLOPE statement is used to specify the value of the disturbance variance, sigma Subscript xi Superscript 2, in the slope equation, and to request forecasts of beta Subscript t. The following examples illustrate this statement: 

   level;
   slope;

The preceding statements fit a model with a locally linear trend. The disturbance variances sigma Subscript eta Superscript 2 and sigma Subscript xi Superscript 2 are estimated from the data. You can request a locally linear trend with fixed slope by using the following statements: 

   level;
   slope variance=0 noest;
NOEST

fixes the value of the disturbance variance, sigma Subscript xi 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 slope component.

PRINT=FILTER
PRINT=SMOOTH
PRINT=( <FILTER> <SMOOTH> )

requests printing of the filtered or smoothed estimate of the slope component beta Subscript t.

VARIANCE=value

specifies an initial value for the disturbance variance, sigma Subscript xi Superscript 2, in the beta 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