### Impulse Response Function

#### Simple Impulse Response Function (IMPULSE=SIMPLE Option)

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 .

#### Accumulated Impulse Response Function (IMPULSE=ACCUM Option)

The accumulated impulse response function is the cumulative sum of the impulse response function, .

#### Orthogonalized Impulse Response Function (IMPULSE=ORTH Option)

The MA representation of a VARMA(,) model with a standardized white noise innovation process offers another way to interpret a VARMA(,) model. Since is positive-definite, 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 .

#### Impulse Response of Transfer Function (IMPULSX=SIMPLE Option)

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.

#### Impulse Response of Transfer Function (IMPULSX=ACCUM Option)

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 y1-y3 = 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 long-run 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)

The VARMAX Procedure

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 36.28 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 .

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 one-unit change in after 3 periods. As the lag gets higher, you can see the long-run responses of to an impulse in itself.

Figure 36.30: Impulse Response Function (IMPULSE= Option)

The VARMAX Procedure

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 36.32 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 36.34 is the output in a univariate format associated with the PRINT=(IMPULSE=) option for the orthogonalized impulse response function. The two right-hand side columns, and , represent the and variables. These are the impulses variables. The left-hand 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 long-run 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.