The VARMAX Procedure

## Example 30.1 Analysis of U.S. Economic Variables

Consider the following four-dimensional system of U.S. economic variables. Quarterly data for the years 1954 to 1987 are used (Lütkepohl 1993, Table E.3.).

```   title 'Analysis of U.S. Economic Variables';
data us_money;
date=intnx( 'qtr', '01jan54'd, _n_-1 );
format date yyq. ;
input y1 y2 y3 y4 @@;
y1=log(y1);
y2=log(y2);
label y1='log(real money stock M1)'
y2='log(GNP in bil. of 1982 dollars)'
y3='Discount rate on 91-day T-bills'
y4='Yield on 20-year Treasury bonds';
datalines;
450.9 1406.8  0.010800000 0.026133333
453.0 1401.2 0.0081333333 0.025233333

... more lines ...

```

The following statements plot the series and proceed with the VARMAX procedure.

```   proc timeseries data=us_money vectorplot=series;
id date interval=qtr;
var y1 y2;
run;
```

Output 30.1.1 shows the plot of the variables and .

Output 30.1.1 Plot of Data

The following statements plot the variables and .

```   proc timeseries data=us_money vectorplot=series;
id date interval=qtr;
var y3 y4;
run;
```

Output 30.1.2 shows the plot of the variables and .

Output 30.1.2 Plot of Data

```   proc varmax data=us_money;
id date interval=qtr;
model y1-y4 / p=2 lagmax=6 dftest
print=(iarr(3) estimates diagnose)
cointtest=(johansen=(iorder=2))
ecm=(rank=1 normalize=y1);
cointeg rank=1 normalize=y1 exogeneity;
run;
```

This example performs the Dickey-Fuller test for stationarity, the Johansen cointegrated test integrated order 2, and the exogeneity test. The VECM(2) is fit to the data. From the outputs shown in Output 30.1.5, you can see that the series has unit roots and is cointegrated in rank 1 with integrated order 1. The fitted VECM(2) is given as

The prefixed to a variable name implies differencing.

Output 30.1.3 through Output 30.1.14 show the details. Output 30.1.3 shows the descriptive statistics.

Output 30.1.3 Descriptive Statistics
 Analysis of U.S. Economic Variables

The VARMAX Procedure

 Number of Observations 136 0

Simple Summary Statistics
Variable Type N Mean Standard
Deviation
Min Max Label
y1 Dependent 136 6.21295 0.07924 6.10278 6.45331 log(real money stock M1)
y2 Dependent 136 7.77890 0.30110 7.24508 8.27461 log(GNP in bil. of 1982 dollars)
y3 Dependent 136 0.05608 0.03109 0.00813 0.15087 Discount rate on 91-day T-bills
y4 Dependent 136 0.06458 0.02927 0.02490 0.13600 Yield on 20-year Treasury bonds

Output 30.1.4 shows the output for Dickey-Fuller tests for the nonstationarity of each series. The null hypotheses is to test a unit root. All series have a unit root.

Output 30.1.4 Unit Root Tests
Unit Root Test
Variable Type Rho Pr < Rho Tau Pr < Tau
y1 Zero Mean 0.05 0.6934 1.14 0.9343
Single Mean -2.97 0.6572 -0.76 0.8260
Trend -5.91 0.7454 -1.34 0.8725
y2 Zero Mean 0.13 0.7124 5.14 0.9999
Single Mean -0.43 0.9309 -0.79 0.8176
Trend -9.21 0.4787 -2.16 0.5063
y3 Zero Mean -1.28 0.4255 -0.69 0.4182
Single Mean -8.86 0.1700 -2.27 0.1842
Trend -18.97 0.0742 -2.86 0.1803
y4 Zero Mean 0.40 0.7803 0.45 0.8100
Single Mean -2.79 0.6790 -1.29 0.6328
Trend -12.12 0.2923 -2.33 0.4170

The Johansen cointegration rank test shows whether the series is integrated order either 1 or 2 as shown in Output 30.1.5. The last two columns in Output 30.1.5 explain the cointegration rank test with integrated order 1. The results indicate that there is the cointegrated relationship with the cointegration rank 1 with respect to the 0.05 significance level because the test statistic of 20.6542 is smaller than the critical value of 29.38. Now, look at the row associated with . Compare the test statistic value and critical value pairs such as (219.62395, 29.38), (89.21508, 15.34), and (27.32609, 3.84). There is no evidence that the series are integrated order 2 at the 0.05 significance level.

Output 30.1.5 Cointegration Rank Test
Cointegration Rank Test for I(2)
r\k-r-s 4 3 2 1 Trace
of I(1)
5% CV of I(1)
0 384.60903 214.37904 107.93782 37.02523 55.9633 47.21
1   219.62395 89.21508 27.32609 20.6542 29.38
2     73.61779 22.13279 2.6477 15.34
3       38.29435 0.0149 3.84
5% CV I(2) 47.21000 29.38000 15.34000 3.84000

Output 30.1.6 shows the estimates of the long-run parameter, , and the adjustment coefficient, .

Output 30.1.6 Cointegration Rank Test Continued
Beta
Variable 1 2 3 4
y1 1.00000 1.00000 1.00000 1.00000
y2 -0.46458 -0.63174 -0.69996 -0.16140
y3 14.51619 -1.29864 1.37007 -0.61806
y4 -9.35520 7.53672 2.47901 1.43731

Alpha
Variable 1 2 3 4
y1 -0.01396 0.01396 -0.01119 0.00008
y2 -0.02811 -0.02739 -0.00032 0.00076
y3 -0.00215 -0.04967 -0.00183 -0.00072
y4 0.00510 -0.02514 -0.00220 0.00016

Output 30.1.7 shows the estimates and .

Output 30.1.7 Cointegration Rank Test Continued
Eta
Variable 1 2 3 4
y1 52.74907 41.74502 -20.80403 55.77415
y2 -49.10609 -9.40081 98.87199 22.56416
y3 68.29674 -144.83173 -27.35953 15.51142
y4 121.25932 271.80496 85.85156 -130.11599

Xi
Variable 1 2 3 4
y1 -0.00842 -0.00052 -0.00208 -0.00250
y2 0.00141 0.00213 -0.00736 -0.00058
y3 -0.00445 0.00541 -0.00150 0.00310
y4 -0.00211 -0.00064 -0.00130 0.00197

Output 30.1.8 shows that the VECM(2) is fit to the data. The ECM=(RANK=1) option produces the estimates of the long-run parameter, , and the adjustment coefficient, .

Output 30.1.8 Parameter Estimates
 Analysis of U.S. Economic Variables

The VARMAX Procedure

Type of Model VECM(2) Maximum Likelihood Estimation 1

Beta
Variable 1
y1 1.00000
y2 -0.46458
y3 14.51619
y4 -9.35520

Alpha
Variable 1
y1 -0.01396
y2 -0.02811
y3 -0.00215
y4 0.00510

Output 30.1.9 shows the parameter estimates in terms of the constant, the lag one coefficients () contained in the estimates, and the coefficients associated with the lag one first differences ().

Output 30.1.9 Parameter Estimates Continued
Constant
Variable Constant
y1 0.04076
y2 0.08595
y3 0.00518
y4 -0.01438

Parameter Alpha * Beta' Estimates
Variable y1 y2 y3 y4
y1 -0.01396 0.00648 -0.20263 0.13059
y2 -0.02811 0.01306 -0.40799 0.26294
y3 -0.00215 0.00100 -0.03121 0.02011
y4 0.00510 -0.00237 0.07407 -0.04774

AR Coefficients of Differenced Lag
DIF Lag Variable y1 y2 y3 y4
1 y1 0.34603 0.09131 -0.35351 -0.96895
y2 0.09936 0.03791 0.23900 0.28661
y3 0.18118 0.07859 0.02234 0.40508
y4 0.03222 0.04961 -0.03292 0.18568

Output 30.1.10 shows the parameter estimates and their significance.

Output 30.1.10 Parameter Estimates Continued
Model Parameter Estimates
Equation Parameter Estimate Standard
Error
t Value Pr > |t| Variable
D_y1 CONST1 0.04076 0.01418 2.87 0.0048 1
AR1_1_1 -0.01396 0.00495     y1(t-1)
AR1_1_2 0.00648 0.00230     y2(t-1)
AR1_1_3 -0.20263 0.07191     y3(t-1)
AR1_1_4 0.13059 0.04634     y4(t-1)
AR2_1_1 0.34603 0.06414 5.39 0.0001 D_y1(t-1)
AR2_1_2 0.09131 0.07334 1.25 0.2154 D_y2(t-1)
AR2_1_3 -0.35351 0.11024 -3.21 0.0017 D_y3(t-1)
AR2_1_4 -0.96895 0.20737 -4.67 0.0001 D_y4(t-1)
D_y2 CONST2 0.08595 0.01679 5.12 0.0001 1
AR1_2_1 -0.02811 0.00586     y1(t-1)
AR1_2_2 0.01306 0.00272     y2(t-1)
AR1_2_3 -0.40799 0.08514     y3(t-1)
AR1_2_4 0.26294 0.05487     y4(t-1)
AR2_2_1 0.09936 0.07594 1.31 0.1932 D_y1(t-1)
AR2_2_2 0.03791 0.08683 0.44 0.6632 D_y2(t-1)
AR2_2_3 0.23900 0.13052 1.83 0.0695 D_y3(t-1)
AR2_2_4 0.28661 0.24552 1.17 0.2453 D_y4(t-1)
D_y3 CONST3 0.00518 0.01608 0.32 0.7476 1
AR1_3_1 -0.00215 0.00562     y1(t-1)
AR1_3_2 0.00100 0.00261     y2(t-1)
AR1_3_3 -0.03121 0.08151     y3(t-1)
AR1_3_4 0.02011 0.05253     y4(t-1)
AR2_3_1 0.18118 0.07271 2.49 0.0140 D_y1(t-1)
AR2_3_2 0.07859 0.08313 0.95 0.3463 D_y2(t-1)
AR2_3_3 0.02234 0.12496 0.18 0.8584 D_y3(t-1)
AR2_3_4 0.40508 0.23506 1.72 0.0873 D_y4(t-1)
D_y4 CONST4 -0.01438 0.00803 -1.79 0.0758 1
AR1_4_1 0.00510 0.00281     y1(t-1)
AR1_4_2 -0.00237 0.00130     y2(t-1)
AR1_4_3 0.07407 0.04072     y3(t-1)
AR1_4_4 -0.04774 0.02624     y4(t-1)
AR2_4_1 0.03222 0.03632 0.89 0.3768 D_y1(t-1)
AR2_4_2 0.04961 0.04153 1.19 0.2345 D_y2(t-1)
AR2_4_3 -0.03292 0.06243 -0.53 0.5990 D_y3(t-1)
AR2_4_4 0.18568 0.11744 1.58 0.1164 D_y4(t-1)

Output 30.1.11 shows the innovation covariance matrix estimates, the various information criteria results, and the tests for white noise residuals. The residuals have significant correlations at lag 2 and 3. The Portmanteau test results into significant. These results show that a VECM(3) model might be better fit than the VECM(2) model is.

Output 30.1.11 Diagnostic Checks
Covariances of Innovations
Variable y1 y2 y3 y4
y1 0.00005 0.00001 -0.00001 -0.00000
y2 0.00001 0.00007 0.00002 0.00001
y3 -0.00001 0.00002 0.00007 0.00002
y4 -0.00000 0.00001 0.00002 0.00002

Information Criteria
AICC -40.6284
HQC -40.4343
AIC -40.6452
SBC -40.1262
FPEC 2.23E-18

Schematic Representation of Cross Correlations
of Residuals
Variable/Lag 0 1 2 3 4 5 6
y1 ++.. .... ++.. .... +... ..-- ....
y2 ++++ .... .... .... .... .... ....
y3 .+++ .... +.-. ..++ -... .... ....
y4 .+++ .... .... ..+. .... .... ....

Portmanteau Test for Cross Correlations
of Residuals
Up To Lag DF Chi-Square Pr > ChiSq
3 16 53.90 <.0001
4 32 74.03 <.0001
5 48 103.08 <.0001
6 64 116.94 <.0001

Output 30.1.12 describes how well each univariate equation fits the data. The residuals for and are off from the normality. Except the residuals for , there are no AR effects on other residuals. Except the residuals for , there are no ARCH effects on other residuals.

Output 30.1.12 Diagnostic Checks Continued
Univariate Model ANOVA Diagnostics
Variable R-Square Standard
Deviation
F Value Pr > F
y1 0.6754 0.00712 32.51 <.0001
y2 0.3070 0.00843 6.92 <.0001
y3 0.1328 0.00807 2.39 0.0196
y4 0.0831 0.00403 1.42 0.1963

Univariate Model White Noise Diagnostics
Variable Durbin
Watson
Normality ARCH
Chi-Square Pr > ChiSq F Value Pr > F
y1 2.13418 7.19 0.0275 1.62 0.2053
y2 2.04003 1.20 0.5483 1.23 0.2697
y3 1.86892 253.76 <.0001 1.78 0.1847
y4 1.98440 105.21 <.0001 21.01 <.0001

Univariate Model AR Diagnostics
Variable AR1 AR2 AR3 AR4
F Value Pr > F F Value Pr > F F Value Pr > F F Value Pr > F
y1 0.68 0.4126 2.98 0.0542 2.01 0.1154 2.48 0.0473
y2 0.05 0.8185 0.12 0.8842 0.41 0.7453 0.30 0.8762
y3 0.56 0.4547 2.86 0.0610 4.83 0.0032 3.71 0.0069
y4 0.01 0.9340 0.16 0.8559 1.21 0.3103 0.95 0.4358

The PRINT=(IARR) option provides the VAR(2) representation in Output 30.1.13.

Output 30.1.13 Infinite Order AR Representation
Infinite Order AR Representation
Lag Variable y1 y2 y3 y4
1 y1 1.33208 0.09780 -0.55614 -0.83836
y2 0.07125 1.05096 -0.16899 0.54955
y3 0.17903 0.07959 0.99113 0.42520
y4 0.03732 0.04724 0.04116 1.13795
2 y1 -0.34603 -0.09131 0.35351 0.96895
y2 -0.09936 -0.03791 -0.23900 -0.28661
y3 -0.18118 -0.07859 -0.02234 -0.40508
y4 -0.03222 -0.04961 0.03292 -0.18568
3 y1 0.00000 0.00000 0.00000 0.00000
y2 0.00000 0.00000 0.00000 0.00000
y3 0.00000 0.00000 0.00000 0.00000
y4 0.00000 0.00000 0.00000 0.00000

Output 30.1.14 shows whether each variable is the weak exogeneity of other variables. The variable is not the weak exogeneity of other variables, , , and ; the variable is not the weak exogeneity of other variables, , , and ; the variable and are the weak exogeneity of other variables.

Output 30.1.14 Weak Exogeneity Test
Testing Weak Exogeneity of
Each Variables
Variable DF Chi-Square Pr > ChiSq
y1 1 6.55 0.0105
y2 1 12.54 0.0004
y3 1 0.09 0.7695
y4 1 1.81 0.1786

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