The SSM Procedure (Experimental)

specifies that the last nd elements of the state subsection should be treated as diffuse, which becomes the dimension of the relevant subsection of the diffuse vector . The block is created by using appropriate columns of the identity matrix. nd must lie between one and dim. The absence of this option signifies that this subsection of is nondiffuse. If both COV1 and A1 options are specified, the last nd rows and columns of the matrix specified in COV1 are taken to be zero. This option cannot be used together with the RANK= option of the COV1 option.

specifies the relevant block of the disturbance covariance (for ) in the transition equation. Similar to the T option, the absence of this option signifies that this Q-block consists of only zeros. The structure of the Q-block is also similarly specified. However, the following differences exist:

• The list specified to form the covariance must result in a symmetric, positive semidefinite matrix. For an example, see Example 27.4.

• You can specify a rank constraint on the Q-block by specifying COV(RANK=integer), where the specified integer must lie between 1 and dim. A rank constraint is permissible only for the general form, and only when its elements are not specified by using a list.

• The convention of treating unset variables as structural zeros, used in specifying sparsity of the T-block, is not used in the Q-block specification. Whenever you explicitly specify the entries of the Q-block by specifying a list of variables in parentheses, all variables in the list must evaluate to nonmissing values.

The following examples illustrate different ways of specifying a Q-block. It is assumed that dim = 2.

• COV(G) specifies a general form Q-block (contributes unspecified elements to the parameter vector ) .

• COV(RANK=1) specifies a rank-one Q-block.

specifies the relevant block of the initial state covariance . The different options in this case have the same meaning as the options of the COV option. However, the following differences exist:

• If the elements of are specified by a list of variables in parentheses, then these variables must evaluate to constant values. In particular, they can depend on parameters defined by the PARMS statements; however, they cannot depend on any of the input data columns.

• If the initial condition is partially diffuse (that is, the diffuse dimension nd specified in the A1 option is nonzero), the last nd rows and columns of the matrix specified in COV1 are taken to be zero. Moreover, if the elements of are specified by a list, its number of elements must correspond to a matrix of dimension (dim – nd).

specifies the relevant block of the transition matrix . The absence of this option signifies that this block consists of only zeros. You can specify the structure of the T-block by specifying T(I) for the identity form, T(D) for the diagonal form, and T(G) for a general unstructured form. In addition, you can explicitly specify the entries of the T-block by specifying a list of numbers in parentheses, or by specifying in parentheses a list of variables that are defined by using the programming statements. The unspecified elements of the T-block are included in the list of parameters to be estimated from the data. If the elements of the T-block are supplied by a list in parentheses, the number of elements in the list depends on its structure. For the diagonal form, the list must contain exactly dim elements. In the case of the identity form—T(I)—the block is already fully specified; however, a specification T(I)=(variable) is understood to mean that the identity block is scaled by the specified variable (or a number). In the general case—T(G)—the list must consist of elements, specified in a rowwise fashion. An inappropriate number of elements in the list results in a syntax error.

The following examples illustrate different ways of specifying the transition matrix. It is assumed that dim = 2.

• T(I) specifies that the T-block is a two-dimensional identity matrix.

• T(D) specifies that the T-block is a two-dimensional diagonal matrix. The two unspecified diagonal entries become part of the parameter vector .

• T(D)=(1.1 2) fully specifies the two-dimensional diagonal T-block.

• T(D)=() specifies a two-dimensional diagonal T-block where the diagonal elements are dynamically calculated based on the values of the variables and . In this case the T-block can change with time if or change with time.

• T(G) specifies a general form T-block (with unspecified elements).

• T(G)=() specifies a general form T-block where the first row is formed by and , and the second row is formed by and .

In practice the transition matrix is often sparse—that is, many of its elements are zero. The algorithms in the SSM procedure exploit this sparsity structure for computational efficiency. Whenever you explicitly specify the entries of the T-block by specifying a list of variables in parentheses, you can leave the variables that correspond to the zero elements unset. These unset variables are treated as structural zeros by the SSM procedure. The section Sparse Transition Matrix Specification further explains how to use this sparsity convention.

specifies a state subsection that corresponds to the specified type. You can specify either a number or a variable for the RHO= and PERIOD= suboptions. The T, COV1, and A1 options are not needed. In fact they are ignored, since the transition matrix and the matrices in the initial condition ( and ) are implicitly defined by the choice of the type. However, the specification of the COV option does play a key role in the eventual form of —the covariance of the disturbance term in the transition equation. For the types LL, SEASON, and CYCLE, the dimension of the resulting state subsection is a certain multiple of dim, the dimension in the STATE statement. For example, the following specification results in a state subsection, named cycleState, of dimension 2*dim:

     state cycleState(dim) cov(g) type=cycle;

The name cycleState corresponds to the state underlying a dim-dimensional cycle component. All of these special state types require that the data be regular; the only exception is TYPE=CYCLE(CT), which defines a continuous-time cycle and is applicable to any data type. Table 27.2 summarizes some of this information for easy reference. Additional details about these state types are provided in the section Predefined Structural Models.

The TYPE=LL specification results in a state that corresponds to a multivariate local linear trend. It is governed by two covariance matrices: the COV option specifies the covariance that corresponds to the level equation, and the SLOPECOV suboption specifies the covariance used in the slope equation. The form of the SLOPECOV suboption is exactly the same as that of the COV option.

The TYPE=CYCLE option results in a state that corresponds to a (stochastic) cycle. By default, this cycle is assumed to be for the regular data type. If TYPE=CYCLE(CT), the resulting cycle is applicable to any data type. The CT option is available only for dim = 1; that is, only a univariate cycle is available for the irregular data type. The cycle specification depends on a covariance matrix and two numbers: the damping factor RHO and the cycle period PERIOD. The covariance can be specified by the COV option. The damping factor is specified by the RHO= suboption; its value must lie between 0.0 and 1.0. The cycle period can be specified by the PERIOD= suboption. If the CT option is not included, the period value must be larger than 2.0. On the other hand, if the CT option is included, its value must be strictly positive. If these parameters are not specified, they are estimated from the data.