The SSM Procedure (Experimental)

TREND Statement

TREND name ( type ) <options> ;

The TREND statement defines a trend term in the model. Loosely speaking, a trend is a special type of component that captures the time-varying level of the data. The options in the TREND statement enable you to specify a wide variety of commonly used trend patterns. Each TREND specification in effect stands for a special pair of STATE and COMPONENT statements. You can specify more than one TREND specification. Each separate trend specification defines a component that is assumed to be independent of all other component specifications in the model.

Some of these trend specifications are applicable to all the data types—that is, they can be used for regular data types in addition to irregular data types, while the others require that the data be regular or regular-with-replication. Of course, the trend specification is only a part of the overall model specification. Therefore, the other parts of the model can imply additional constraints on the data type.

Table 27.3 lists the available trend models and their data requirements. The "type" column shows the admissible keywords that signify the particular trend type. For brevity, the "Data Type" column in Table 27.3 groups the regular and regular-with-replication data types into one category—regular. The section Predefined Trend Models provides additional details about these trend models.

Table 27.3 Summary of Trend Types
type	Data Type	Description	Parameters
RW	Regular	Random walk	Level $\text{[math]}$
LL	Regular	Local linear	Level and slope $\text{[math]}$ , $\text{[math]}$
DLL	Regular	Damped local linear	Level and slope $\text{[math]}$ , $\text{[math]}$ ,
			damping factor $\text{[math]}$
PS(order)	Irregular	Polynomial spline of order up to 3	Level $\text{[math]}$
DECAY	Irregular	A type of decay pattern	Level $\text{[math]}$ , decay rate $\text{[math]}$
DECAY(OU)	Irregular	Ornstein-Uhlenbeck decay pattern	Level $\text{[math]}$ , decay rate $\text{[math]}$
GROWTH	Irregular	A type of growth pattern	Level $\text{[math]}$ , growth rate $\text{[math]}$
GROWTH(OU)	Irregular	Ornstein-Uhlenbeck growth pattern	Level $\text{[math]}$ , growth rate $\text{[math]}$

The following example specifies polySpline as a trend of type polynomial spline of order 2:

      trend polySpline(ps(2));

Similarly, the following statement defines dampedTrend as a damped local linear trend:

      trend dampedTrend(dll) slopevar=x;

The variance parameter that governs the slope equation of this trend type is given by a variable x, which must be defined elsewhere in the program. The other parameters that define dampedTrend are left unspecified.

You can specify the following options in the TREND statement to specify the trend parameters. In addition, you can create a custom combination of given trend type by specifying the CROSS= option to create a more general trend. For an example of the use of CROSS= option, see the discussion of the second model in Example 27.3.

CROSS=(var1, var2, ...)

creates a linear combination of one or more independent trend components based on the variables in the list. If the parameters of the trend are specified by options such as the LEVELVAR= option or the PHI= option, these parameters are shared by these constituent trends. For example, suppose that the CROSS= list contains two variables $\text{[math]}$ and $\text{[math]}$ and the trend specification is of the type RW. The effect of CROSS=( $\text{[math]}$ ) is to create a component $\text{[math]}$ where $\text{[math]}$ and $\text{[math]}$ are two independent random walk trends. Moreover, if the random walk trend specification uses the LEVELVAR= option to specify the variance parameter, $\text{[math]}$ and $\text{[math]}$ share the same variance parameter; otherwise, two separate variance parameters are assigned to these random walks. The CROSS= option is useful for a variety of situations. For example, suppose $\text{[math]}$ is an indicator variable that is 1 before certain time point $\text{[math]}$ and 0 thereafter; CROSS=(X) has the effect of turning off the trend component after time $\text{[math]}$ . Similarly, suppose $\text{[math]}$ and $\text{[math]}$ are indicators for gender—for example, $\text{[math]}$ = (GENDER="M") and $\text{[math]}$ = (GENDER="F"). Then CROSS=( $\text{[math]}$ ) creates a trend that varies with the gender of the observation. The variables in the CROSS= list must be free of unknown parameters.

The CROSS= option can be computationally expensive; computationally it is equivalent to specifying as many separate trends as the number of variables in the specified list.

LEVELVAR=variable | number

specifies the disturbance variance parameter for all the trend types. For trend types LL and DLL, this option specifies $\text{[math]}$ . Any nonnegative value, including zero, is permissible. If variable contains unknown parameters, they are estimated from the data. Similarly, if the LEVELVAR= option is not specified, $\text{[math]}$ is estimated from the data.

PHI=variable | number

specifies the value of $\text{[math]}$ for trend types DLL, DECAY, DECAY(OU), GROWTH, and GROWTH(OU). For the type DLL, the specified value must be between 0.0 and 1.0. For types DECAY and DECAY(OU), $\text{[math]}$ must be strictly negative. For types GROWTH and GROWTH(OU), $\text{[math]}$ must be strictly positive. If variable contains unknown parameters, they are estimated from the data. Similarly, if the PHI= option is not specified, $\text{[math]}$ is estimated from the data.

SLOPEVAR=variable | number

specifies the second disturbance variance parameter, $\text{[math]}$ , for trend types LL and DLL. Any nonnegative value, including zero, is permissible. If variable contains unknown parameters, they are estimated from the data. Similarly, if the SLOPEVAR= option is not specified, $\text{[math]}$ is estimated from the data.

Note: This procedure is experimental.