The VARMAX(,,) model has a convergent representation

where and .
The elements of the matrices from the operator , called the impulse response, can be interpreted as the impact that a shock in one variable has on another variable. Let be the element of at lag , where is the index for the impulse variable, and is the index for the response variable (impulse response). For instance, is an impulse response to , and is an impulse response to .
The accumulated impulse response function is the cumulative sum of the impulse response function, .
The MA representation of a VARMA(,) model with a standardized white noise innovation process offers another way to interpret a VARMA(,) model. Since is positivedefinite, there is a lower triangular matrix such that . The alternate MA representation of a VARMA(,) model is written as

where , , and .
The elements of the matrices , called the orthogonal impulse response, can be interpreted as the effects of the components of the standardized shock process on the process at lag .
The coefficient matrix from the transfer function operator can be interpreted as the effects that changes in the exogenous variables have on the output variable at lag ; it is called an impulse response matrix in the transfer function.
The accumulated impulse response in the transfer function is the cumulative sum of the impulse response in the transfer function, .
The asymptotic distributions of the impulse functions can be seen in the section VAR and VARX Modeling.
The following statements provide the impulse response and the accumulated impulse response in the transfer function for a VARX(1,0) model.
proc varmax data=grunfeld plot=impulse; model y1y3 = x1 x2 / p=1 lagmax=5 printform=univariate print=(impulsx=(all) estimates); run;
In Figure 36.26, the variables and are impulses and the variables , , and are responses. You can read the table matching the pairs of such as , , , , , and . In the pair of , you can see the longrun responses of to an impulse in (the values are 1.69281, 0.35399, 0.09090, and so on for lag 0, lag 1, lag 2, and so on, respectively).
Figure 36.26: Impulse Response in Transfer Function (IMPULSX= Option)
Simple Impulse Response of Transfer Function by Variable 


Variable Response\Impulse 
Lag  x1  x2 
y1  0  1.69281  0.00859 
1  0.35399  0.01727  
2  0.09090  0.00714  
3  0.05136  0.00214  
4  0.04717  0.00072  
5  0.04620  0.00040  
y2  0  6.09850  2.57980 
1  5.15484  0.45445  
2  3.04168  0.04391  
3  2.23797  0.01376  
4  1.98183  0.01647  
5  1.87415  0.01453  
y3  0  0.02317  0.01274 
1  1.57476  0.01435  
2  1.80231  0.00398  
3  1.77024  0.01062  
4  1.70435  0.01197  
5  1.63913  0.01187 
Figure 36.27 shows the responses of , , and to a forecast error impulse in .
Figure 36.27: Plot of Impulse Response in Transfer Function
Figure 36.28 shows the accumulated impulse response in transfer function.
Figure 36.28: Accumulated Impulse Response in Transfer Function (IMPULSX= Option)
Accumulated Impulse Response of Transfer Function by Variable 


Variable Response\Impulse 
Lag  x1  x2 
y1  0  1.69281  0.00859 
1  2.04680  0.00868  
2  2.13770  0.01582  
3  2.18906  0.01796  
4  2.23623  0.01867  
5  2.28243  0.01907  
y2  0  6.09850  2.57980 
1  11.25334  3.03425  
2  14.29502  3.07816  
3  16.53299  3.06440  
4  18.51482  3.04793  
5  20.38897  3.03340  
y3  0  0.02317  0.01274 
1  1.55159  0.02709  
2  3.35390  0.02311  
3  5.12414  0.01249  
4  6.82848  0.00052  
5  8.46762  0.01135 
Figure 36.29 shows the accumulated responses of , , and to a forecast error impulse in .
Figure 36.29: Plot of Accumulated Impulse Response in Transfer Function
The following statements provide the impulse response function, the accumulated impulse response function, and the orthogonalized impulse response function with their standard errors for a VAR(1) model. Parts of the VARMAX procedure output are shown in Figure 36.30, Figure 36.32, and Figure 36.34.
proc varmax data=simul1 plot=impulse; model y1 y2 / p=1 noint lagmax=5 print=(impulse=(all)) printform=univariate; run;
Figure 36.30 is the output in a univariate format associated with the PRINT=(IMPULSE=) option for the impulse response function. The keyword STD stands for the standard errors of the elements. The matrix in terms of the lag 0 does not print since it is the identity. In Figure 36.30, the variables and of the first row are impulses, and the variables and of the first column are responses. You can read the table matching the pairs, such as , , , and . For example, in the pair of at lag 3, the response is 0.8055. This represents the impact on y1 of oneunit change in after 3 periods. As the lag gets higher, you can see the longrun responses of to an impulse in itself.
Figure 36.30: Impulse Response Function (IMPULSE= Option)
Simple Impulse Response by Variable  

Variable Response\Impulse 
Lag  y1  y2 
y1  1  1.15977  0.51058 
STD  0.05508  0.05898  
2  1.06612  0.78872  
STD  0.10450  0.10702  
3  0.80555  0.84798  
STD  0.14522  0.14121  
4  0.47097  0.73776  
STD  0.17191  0.15864  
5  0.14315  0.52450  
STD  0.18214  0.16115  
y2  1  0.54634  0.38499 
STD  0.05779  0.06188  
2  0.84396  0.13073  
STD  0.08481  0.08556  
3  0.90738  0.48124  
STD  0.10307  0.09865  
4  0.78943  0.64856  
STD  0.12318  0.11661  
5  0.56123  0.65275  
STD  0.14236  0.13482 
Figure 36.31 shows the responses of and to a forecast error impulse in with two standard errors.
Figure 36.31: Plot of Impulse Response
Figure 36.32 is the output in a univariate format associated with the PRINT=(IMPULSE=) option for the accumulated impulse response function. The matrix in terms of the lag 0 does not print since it is the identity.
Figure 36.32: Accumulated Impulse Response Function (IMPULSE= Option)
Accumulated Impulse Response by Variable  

Variable Response\Impulse 
Lag  y1  y2 
y1  1  2.15977  0.51058 
STD  0.05508  0.05898  
2  3.22589  1.29929  
STD  0.21684  0.22776  
3  4.03144  2.14728  
STD  0.52217  0.53649  
4  4.50241  2.88504  
STD  0.96922  0.97088  
5  4.64556  3.40953  
STD  1.51137  1.47122  
y2  1  0.54634  1.38499 
STD  0.05779  0.06188  
2  1.39030  1.25426  
STD  0.17614  0.18392  
3  2.29768  0.77302  
STD  0.36166  0.36874  
4  3.08711  0.12447  
STD  0.65129  0.65333  
5  3.64834  0.52829  
STD  1.07510  1.06309 
Figure 36.33 shows the accumulated responses of and to a forecast error impulse in with two standard errors.
Figure 36.33: Plot of Accumulated Impulse Response
Figure 36.34 is the output in a univariate format associated with the PRINT=(IMPULSE=) option for the orthogonalized impulse response function. The two righthand side columns, and , represent the and variables. These are the impulses variables. The lefthand side column contains responses variables, and . You can read the table by matching the pairs such as , , , and .
Figure 36.34: Orthogonalized Impulse Response Function (IMPULSE= Option)
Orthogonalized Impulse Response by Variable  

Variable Response\Impulse 
Lag  y1  y2 
y1  0  1.13523  0.00000 
STD  0.08068  0.00000  
1  1.13783  0.58120  
STD  0.10666  0.14110  
2  0.93412  0.89782  
STD  0.13113  0.16776  
3  0.61756  0.96528  
STD  0.15348  0.18595  
4  0.27633  0.83981  
STD  0.16940  0.19230  
5  0.02115  0.59705  
STD  0.17432  0.18830  
y2  0  0.35016  1.13832 
STD  0.11676  0.08855  
1  0.75503  0.43824  
STD  0.06949  0.10937  
2  0.91231  0.14881  
STD  0.10553  0.13565  
3  0.86158  0.54780  
STD  0.12266  0.14825  
4  0.66909  0.73827  
STD  0.13305  0.15846  
5  0.40856  0.74304  
STD  0.14189  0.16765 
In Figure 36.4, there is a positive correlation between and . Therefore, shock in can be accompanied by a shock in in the same period. For example, in the pair of , you can see the longrun responses of to an impulse in .
Figure 36.35 shows the orthogonalized responses of and to a forecast error impulse in with two standard errors.
Figure 36.35: Plot of Orthogonalized Impulse Response