The SSM Procedure

DEPLAG Statement

(Experimental)

DEPLAG name(response-variable) lag-term1 <lag-term2 …> ;

The DEPLAG statement defines a term, named name, that consists of a linear combination of lagged response variables. You can use name later as a right-hand-side term in the MODEL statement for the response variable, as specified in name(response-variable). For a multivariate model, a separate DEPLAG statement is needed for each MODEL statement that has a right-hand-side term that involves lagged response variables. The linear combination of lagged response variables is specified by using one or more lag-terms. Each lag-term specifies the lags that are associated with one of the response variables.

A lag-term is specified in one of the following forms:

lag-response-variable(LAGS=maximum-lag) lag-response-variable(LAGS=(integer1 integer2 …)) lag-response-variable(LAGS=maximum-lag COEFF=(number1 number2 …) | (variable1 variable2 …)) lag-response-variable(LAGS=(integer1 integer2 …) COEFF=(number1 number2 …) | (variable1 variable2 …)): The lag-response-variable in the lag term specification can be the same as the response variable that corresponds to the model equation (which is specified in name(response-variable)), or it can be a different response variable.

The first form of specification is useful when all lags up to the maximum-lag, which must be a positive integer, are present in the lag term. The second form is useful when only certain lags, which are specified as a list of positive integers in parentheses, are present. In these two cases, the lag coefficients are not specified and they are treated as unknown parameters to be estimated from the data.

The COEFF= option in the last two forms enable you to specify lag coefficients. The COEFF= option must follow the LAGS= option. You can use the COEFF=(number1 number2 …) option to specify the lag coefficients as known values. Similarly, you can use the COEFF=(variable1 variable2 …) option to specify user-defined variables as lag coefficients; the user-defined variables can be functions of parameters (which are defined by using the PARMS statement) and input variables. However, the lag coefficients cannot depend on any of the response variables. The number of coefficients specified in the COEFF= option must exactly equal the number of lags specified in the LAGS= option.

There can be at most one DEPLAG statement associated with a particular MODEL statement (you can specify all the needed lag terms in a single DEPLAG statement).

As an illustration, let lagsFORy1 and lagsFORy2 represent the following linear combinations of lagged response variables Y1, Y2, and Y3:

$\Variable{lagsFORy1} = \theta _{1 1} Y1_{t-1} + \theta _{1 2} Y1_{t-2} + \theta _{2 2} Y2_{t-2} + \theta _{2 3} Y2_{t-3} + 1.2 Y3_{t-1} - 2.1 Y3_{t-2}$

$\Variable{lagsFORy2} =\Variable{Phi1} Y1_{t-1} + \Variable{Phi2} Y1_{t-2} + \theta _{2 1} Y2_{t-1}$

where Phi1 and Phi2 denote user-defined variables and $\theta _{ij}$ denote generic parameters. You can specify lagsFORy1 (which is used in the model equation for Y1) and lagsFORy2 (which is used in the model equation for Y2) as follows:

  deplag lagsFORy1(y1) y1(lags=2) y2(lags=(2 3)) y3(lags=2 coeff=(1.2 -2.1));
  deplag lagsFORy2(y2) y1(lags=2 coeff=(phi1 phi2)) y2(lags=1);
  ... more statements ....;
  model y1 = lagsFORy1 ...;
  model y2 = lagsFORy2 ...;
  model y3 = ...;
  ... more statements ....;

assuming that the right-hand-side of the MODEL equation for Y3 does not have a term that involves lags of response variables.

The DEPLAG statement in PROC SSM has the same purpose as the DEPLAG statement in PROC UCM (see Chapter 41: The UCM Procedure). However, there are many differences in the syntax of the two statements, mainly because PROC SSM supports much more complex models. The syntax difference between the two DEPLAG statements can be illustrated by considering the differencing specification— $(1 - B)(1 - B^{12}) = (1 - B - B^{12} + B^{13})$ — in the well-known airline model (ARIMA(0, 1, 1)(0, 1, 1)12 model). You can specify the lag-term that is implied by the differencing in the airline model in PROC UCM as follows:

 deplag lags=(1)(12) phi=1 1 noest;

In PROC SSM the same specification has the following form:

 deplag airLags(y) y(lags=(1 12 13) coeff=(1 1 -1));

Both these specifications define the same lag-term: $(y_{t-1} + y_{t-12} - y_{t-13})$ .

For an example of the use of lagged response variables in a model specification, see Example 34.13. For more information about models that have dependent lags, see the section Models with Dependent Lags.

Note: Models that have lagged response variables are permitted only if the data form a time series (either univariate or multivariate). The SSM procedure adds one more restriction on the models that use lagged response variables: the variables in the list that define a component in any of the COMPONENT statements must be free of unknown parameters. This restriction is artificial and is made primarily to reduce the overall complexity of the model. In future versions of the SSM procedure, this restriction might go away.