UCM Procedure

TF Statement

TF regressor <options> ;

The TF statement specifies a regressor that has a transfer-function relationship with the dependent series. A transfer function is useful for capturing the contributions from lagged values of the regressor. Box and Jenkins (1976) popularized ARIMA models that have transfer-function inputs. In the UCM procedure, you can specify a transfer function of the following type (assuming the regression variable is x):

StartFraction left-parenthesis gamma 0 plus gamma 1 upper B Superscript l Baseline 1 Baseline plus gamma 2 upper B Superscript l Baseline 2 Baseline plus midline-horizontal-ellipsis right-parenthesis upper B Superscript d Baseline Over left-parenthesis 1 minus delta 1 upper B minus delta 2 upper B squared minus midline-horizontal-ellipsis minus delta Subscript m Baseline upper B Superscript m Baseline right-parenthesis left-parenthesis 1 minus omega 1 upper B Superscript s Baseline minus omega 2 upper B Superscript 2 s Baseline minus midline-horizontal-ellipsis minus omega Subscript n Baseline upper B Superscript n s Baseline right-parenthesis EndFraction x Subscript t

This transfer function is specified by using the ratio of polynomials in the backshift operator B. The numerator polynomial orders () are positive integers, possibly with gaps (for example, 1, 3). The numerator term signifies the delay of order d. The denominator polynomial can have two factors: a nonseasonal factor, , and a seasonal factor whose season length is s, . The orders of the terms in the denominator factors cannot have gaps; that is, if 5 is the maximum order of the nonseasonal factor, then all terms of orders 1 through 5 are present. By design, the denominator factors are restricted to be stable polynomials (their roots are strictly larger than 1 in absolute value). As an example, consider the following transfer function specification:

StartFraction left-parenthesis gamma 0 plus gamma 1 upper B Superscript 1 Baseline plus gamma 2 upper B squared right-parenthesis upper B cubed Over left-parenthesis 1 minus delta 1 upper B minus delta 2 upper B squared right-parenthesis left-parenthesis 1 minus omega 1 upper B Superscript 4 Baseline right-parenthesis EndFraction x Subscript t

You can specify this transfer function as follows:

   tf x num=(1 2) den=2 sden=1 s=4 delay=3;

Since the numerator polynomial orders do not have any gaps, the following simpler specification is also available:

   tf x num=2 den=2 sden=1 s=4 delay=3;

Because the denominator factors do not permit gaps in their orders, only the maximum orders need to be provided in their specification.

A state space representation of a transfer-function relationship is described in the section State Space Form of a Transfer Function Relationship. You can specify multiple TF statements, each one with a separate regressor. A regressor that is specified in any transfer function specification must not appear in any other regression specifications, such as in the right-hand side of the MODEL statement or in the RANDOMREG and SPLINEREG statements.

Note: The mathematical form of the transfer function considered by PROC UCM is similar to the one considered in the ARIMA procedure (Chapter 7, ARIMA Procedure). However, there are some differences:

The sign convention of the coefficients of the nonzero-order terms in the numerator polynomial in the UCM procedure is opposite to that of the ARIMA procedure.
The ARIMA procedure permits multiple polynomial factors in both the numerator and the denominator. The UCM procedure permits only one numerator factor and at most two denominator factors.
The ARIMA procedure permits full control over the terms present in each of the polynomial factors. The UCM procedure does not permit such fine control over the terms in the polynomials.
In the UCM procedure, you cannot fix the coefficients of the numerator polynomial. They are always estimated from the data.
In the UCM procedure, if both nonseasonal and seasonal factors are present in the denominator, you must specify starting values for their coefficients either for both factors or for neither.

You can specify the following options in the TF statement:

DELAY=integer

specifies the delay order, which must be a positive integer. By default, DELAY= 0.

DEN=integer

specifies the maximum order of the nonseasonal factor of the denominator polynomial. By default, DEN=0.

DENVAL=val1 val2 …

specifies the starting values of the coefficients of the nonseasonal factor of the denominator polynomial. The number of values supplied in the DENVAL= option must match the value of the DEN= option. Moreover, the resulting polynomial must be stable.

NOEST

fixes the values of the denominator polynomial coefficients to those specified in the DENVAL= and SDENVAL= specifications.

NUM=argument

specifies the positive orders of the terms in the numerator polynomial. You can specify the argument in either of the following forms:

integer: includes all orders from 1 to integer.
(lag1, lag2, …): specifies a more general list of orders.

PLOT=FILTER PLOT=SMOOTH PLOT=( <FILTER> <SMOOTH> )

requests plots of the transfer-function component. When you specify only one plot request, you can omit the parentheses around the plot request. You can use the FILTER and SMOOTH options to plot the filtered and smoothed estimates of the transfer-function component.

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

requests the printing of the filtered or smoothed estimate of the transfer-function component. When you specify only one print request, you can omit the parentheses around the print request. You can use the FILTER and SMOOTH options to print the filtered and smoothed estimates of the transfer-function component.

S=integer

specifies the season length that is used in the specification of the seasonal factor of the denominator polynomial. The season length can be any positive integer; for example, S=4 might be an appropriate value for a quarterly series. By default, S=1.

SDEN=integer

specifies the maximum order of the seasonal factor of the denominator polynomial. By default, SDEN=0.

SDENVAL=val1 val2 …

specifies the starting values of the coefficients of the seasonal factor of the denominator polynomial. The number of values supplied in this option must match the value of the SDEN= option. Moreover, the resulting polynomial must be stable.

TFSTART=value

specifies the value of the transfer function at the start of the sample (the first time ID). By default, the value of this option is a missing value that is estimated from the data. This option is often used when the past values of the transfer function can be inferred because of the structure of the problem or when it is useful to set these values (usually to 0) to achieve identifiability of the overall model. For more information, see the section State Space Form of a Transfer Function Relationship. See Example 41.10 for an example of the use of this option.

Last updated: June 19, 2025