2112 ✦ Chapter 32: The VARMAX Procedure Impulse Response of Transfer Function (IMPULSX=SIMPLE Option) The coefficient matrix ‰ j from the transfer function operator ‰ .B/ can be interpreted as the effects that changes in the exogenous variables x t have on the output variable y t at lag j ; 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, ‰ a l D P l j D0 ‰ j . The asymptotic distributions of the impulse functions can be seen in the section “VAR and VARX Modeling” on page 2133. 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; Impulse Response Function ✦ 2113 In Figure 32.26, the variables x1 and x2 are impulses and the variables y1 , y2 , and y3 are responses. You can read the table matching the pairs of impulse ! response such as x1 ! y1 , x1 ! y2 , x1 ! y3 , x2 ! y1 , x2 ! y2 , and x2 ! y3 . In the pair of x1 ! y1 , you can see the long-run responses of y1 to an impulse in x1 (the values are 1.69281, 0.35399, 0.09090, and so on for lag 0, lag 1, lag 2, and so on, respectively). Figure 32.26 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 2114 ✦ Chapter 32: The VARMAX Procedure Figure 32.27 shows the responses of y1, y2, and y3 to a forecast error impulse in x1. Figure 32.27 Plot of Impulse Response in Transfer Function Impulse Response Function ✦ 2115 Figure 32.28 shows the accumulated impulse response in transfer function. Figure 32.28 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 2116 ✦ Chapter 32: The VARMAX Procedure Figure 32.29 shows the accumulated responses of y1, y2, and y3 to a forecast error impulse in x1. Figure 32.29 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 32.30, Figure 32.32, and Figure 32.34. proc varmax data=simul1 plot=impulse; model y1 y2 / p=1 noint lagmax=5 print=(impulse=(all)) printform=univariate; run; Impulse Response Function ✦ 2117 Figure 32.30 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 32.30, the variables y1 and y2 of the first row are impulses, and the variables y1 and y2 of the first column are responses. You can read the table matching the impulse ! response pairs, such as y1 ! y1 , y1 ! y2 , y2 ! y1 , and y2 ! y2 . For example, in the pair of y1 ! y1 at lag 3, the response is 0.8055. This represents the impact on y1 of one-unit change in y1 after 3 periods. As the lag gets higher, you can see the long-run responses of y1 to an impulse in itself. Figure 32.30 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 2118 ✦ Chapter 32: The VARMAX Procedure Figure 32.31 shows the responses of y1 and y2 to a forecast error impulse in y1 with two standard errors. Figure 32.31 Plot of Impulse Response Impulse Response Function ✦ 2119 Figure 32.32 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 32.32 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 2120 ✦ Chapter 32: The VARMAX Procedure Figure 32.33 shows the accumulated responses of y1 and y2 to a forecast error impulse in y1 with two standard errors. Figure 32.33 Plot of Accumulated Impulse Response Impulse Response Function ✦ 2121 Figure 32.34 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, y1 and y2 , represent the y1_i nnovation and y2_innovat ion variables. These are the impulses variables. The left-hand side column contains responses variables, y1 and y2 . You can read the table by matching the impulse ! response pairs such as y1_i nnovation ! y1 , y1_i nnovation ! y2 , y2_innovat ion ! y1, and y2_i nnovat ion ! y2. Figure 32.34 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 32.4, there is a positive correlation between " 1t and " 2t . Therefore, shock in y1 can be accompanied by a shock in y2 in the same period. For example, in the pair of y1_i nnovation ! y2 , you can see the long-run responses of y2 to an impulse in y1_i nnovat ion. . 2.1 597 7 -0.51058 STD 0.05508 0.05 898 2 3 .225 89 -1. 299 29 STD 0.21684 0 .227 76 3 4.03144 -2.14728 STD 0. 5221 7 0.536 49 4 4.50241 -2.88504 STD 0 .96 922 0 .97 088 5 4.64556 -3.4 095 3 STD 1.51137 1.47 122 y2. 1. 692 81 -0.008 59 1 0.35 399 0.01727 2 0. 090 90 0.00714 3 0.05136 0.00214 4 0.04717 0.00072 5 0.04620 0.00040 y2 0 -6. 098 50 2.5 798 0 1 -5.15484 0.45445 2 -3.04168 0.04 391 3 -2.23 797 -0.01376 4 -1 .98 183. x2 y1 0 1. 692 81 -0.008 59 1 2.04680 0.00868 2 2.13770 0.01582 3 2.1 890 6 0.01 796 4 2.23623 0.01867 5 2.28243 0.0 190 7 y2 0 -6. 098 50 2.5 798 0 1 -11.25334 3.03425 2 -14. 295 02 3.07816 3 -16.53 299 3.06440 4