Dynamic Multipliers :: SAS/ETS(R) 12.1 User's Guide

Dynamic Multipliers

For models that have only first-order lags, the equation of the reduced form of the system can be rewritten

$\mb {y} _{t}=\mb {D} \mb {y} _{t-1}+{\Pi }_{2}\mb {x} _{t}$

D is a matrix formed from the columns of ${\Pi }$ $_{1}$ plus some columns of zeros, arranged in the order in which the variables meet the lags. The elements of ${\Pi }$ $_{2}$ are called impact multipliers because they show the immediate effect of changes in each exogenous variable on the values of the endogenous variables. This equation can be rewritten as

$\mb {y} _{t}=\mb {D} ^{2}\mb {y} _{t-2}+\mb {D} {\Pi }_{2}\mb {x} _{t-1}+{\Pi }_{2}\mb {x} _{t}$

The matrix formed by the product D ${\Pi }$ $_{2}$ shows the effect of the exogenous variables one lag back; the elements in this matrix are called interim multipliers and are computed and printed when the INTERIM= option is specified in the PROC SIMLIN statement. The i th period interim multipliers are formed by D $^{\mi {i}}$ ${\Pi }$ $_{2}$ .

The series can be expanded as

$\mb {y} _{t}= \mb {D} ^{{\infty }}\mb {y} _{t-{\infty }} +\sum _{i=0}^{{\infty }}{\mb {D} ^{i}{\Pi }_{2}\mb {x} _{t-i}}$

A permanent and constant setting of a value for x has the following cumulative effect:

$\left(\sum _{_{i=0}}^{{\infty }}{\mb {D} ^{i}}\right){\Pi }_{2}\mb {x} = (\mb {I} -\mb {D} )^{-1}{\Pi }_{2}\mb {x}$

The elements of (I-D) $^{-1}$ ${\Pi }$ $_{2}$ are called the total multipliers. Assuming that the sum converges and that (I-D ) is invertible, PROC SIMLIN computes the total multipliers when the TOTAL option is specified in the PROC SIMLIN statement.

The SIMLIN Procedure

Dynamic Multipliers