Dynamic Multipliers

The SIMLIN Procedure

Dynamic Multipliers

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

$y_{t}=D{y}_{t-1}+{\Pi}_{2}x_{t}$

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

$y_{t}=D^2{y}_{t-2}+D{\Pi}_{2}x_{t-1}+{\Pi}_{2}x_{t}$

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 ith period interim multipliers are formed by Dⁱ ${\Pi}$ ₂.

The series can be expanded as

$y_{t}= D^{{\infty}}y_{t-{\infty}} +\sum_{i=0}^{{\infty}}{D^i{\Pi}_{2}x_{t-i}}$

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

$(\sum_{_{i=0}}^{{\infty}}{D^i}){\Pi}_{2}x= (I-D)^{-1}{\Pi}_{2}x$

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.

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