
A1(nd)

specifies that the last nd elements of the state subsection be treated as diffuse. This becomes the dimension of the relevant subsection of the diffuse
vector . The block is created by using appropriate columns of the identity matrix. The value of nd must lie between 1 and the nominal dimension, dim. The absence of this option signifies that this subsection of is nondiffuse. If both the COV1 and A1 options are specified, the last nd rows and columns of the matrix specified in the COV1 option are taken to be 0. This option cannot be used together with the
RANK= option of the COV1 option.

COV( D ) <= (var1 var2 ...)  (number1 number2 ...)>
COV( G ) <= (var1 var2 ...)  (number1 number2 ...)>
COV( I ) <= (variable)  (number)>
COV( RANK=integer )

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 Qblock consists of only zeros. The structure of the Qblock is also
similarly specified. However, the following differences exist:

The list that is specified to form the covariance must result in a symmetric, positive semidefinite matrix. For an example,
see Example 27.5.

You can specify a rank constraint on the Qblock 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, which is used in specifying sparsity of the Tblock, is not
used in the Qblock specification. Whenever you explicitly specify the entries of the Qblock 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 Qblock. It is assumed that dim = 2.

COV(G) specifies a generalform Qblock, which contributes unspecified elements to the parameter vector .

COV(RANK=1) specifies a rankone Qblock.

COV1( D ) <= (var1 var2 ...)  (number1 number2 ...)>
COV1( G ) <= (var1 var2 ...)  (number1 number2 ...)>
COV1( I ) <= (variable)  (number)>
COV1( RANK=integer )

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 that are specified 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).

PRINT=AR  COV  COV1  MA  T
PRINT=(<AR> <COV> <COV1> <MA> <T> )

requests printing of the respective system matrices. You can specify PRINT=AR or PRINT=MA only if you specify the TYPE=VARMA
option. If any of these matrices are timevarying, the matrix that corresponds to the first time instance is printed.

SINPUT = (var1 var2 ...)  (number1 number2 ...)

specifies the relevant dimdimensional block of the state input vector . The absence of this option signifies that this block of the vector consists of only zeros. If the elements of are specified by a list of variables in parentheses, then these variables must be independent of unknown parameters. In particular,
they cannot be functions of parameters that are defined by the PARMS statements.

T( D ) <= (var1 var2 ...)  (number1 number2 ...)>
T( G ) <= (var1 var2 ...)  (number1 number2 ...)>
T( I ) <= (variable)  (number)>

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 Tblock
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 Tblock 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 Tblock
are included in the list of parameters to be estimated from the data. If the elements of the Tblock 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.
In practice the transition matrix is often sparse—that is, many of its elements are 0. The algorithms in the SSM procedure
exploit this sparsity structure for computational efficiency. Whenever you explicitly specify the entries of the Tblock 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.

TYPE=WN
TYPE=RW
TYPE=LL (SLOPECOV(I  D  G) <= (var1, var2, ...)  (number1, number2, ...)> )
TYPE=LL (SLOPECOV( RANK=integer) )
TYPE=SEASON(LENGTH=integer)
TYPE=CYCLE <( <CT> <RHO=variable  number> <PERIOD=variable  number> )>
TYPE=VARMA ( <p <(I  D)> =integer> <q<(D)>=integer> )

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. When TYPE=VARMA, the autoregressive and moving average orders can be at most 1 ( and ). Moreover, by using the D and I flags with the order specification, you can impose additional structure on the autoregressive
and moving average coefficient matrices—for example, specifying TYPE=VARMA(P=1) implies a VAR(1) model with general autoregressive
coefficient matrix, whereas specifying TYPE=VARMA(P(D)=1) implies a VAR(1) model with diagonal autoregressive coefficient
matrix. If you specify the TYPE= option, the T, COV1, SINPUT, 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, CYCLE, SEASON, and VARMA, the dimension
of the resulting state subsection is a certain multiple of dim, the nominal 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 dimdimensional cycle component. All of these special state types require that the data be regular (replication is permissible);
the only exception is TYPE=CYCLE(CT), which defines a continuoustime cycle and is applicable to any data type. Table 27.2 summarizes some of this information for easy reference. For more information about these state types, see 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.
Table 27.2: Summary of Predefined State Types
Type

Description

Parameters

State Dimension

WN

dimvariate white noise

COV

dim

RW

dimvariate random walk

COV

dim

LL

dimvariate local linear

COV, SLOPECOV

2*dim

SEASON(LENGTH=length)

dimvariate season

COV

(length–1)*dim

CYCLE

dimvariate cycle

COV, RHO, PERIOD

2*dim

VARMA(P=p Q=q)

dimvariate VARMA(p, q)

COV, AR, MA

dim*max( p, q+1)
