### Example 36.2 Analysis of German Economic Variables

This example considers a three-dimensional VAR(2) model. The model contains the logarithms of a quarterly, seasonally adjusted West German fixed investment, disposable income, and consumption expenditures. The data used are in Lütkepohl (1993, Table E.1).

title 'Analysis of German Economic Variables';
data west;
date = intnx( 'qtr', '01jan60'd, _n_-1 );
format date yyq. ;
input y1 y2 y3 @@;
y1 = log(y1);
y2 = log(y2);
y3 = log(y3);
label y1 = 'logarithm of investment'
y2 = 'logarithm of income'
y3 = 'logarithm of consumption';
datalines;
180  451  415 179  465  421 185  485  434 192  493  448
211  509  459 202  520  458 207  521  479 214  540  487

... more lines ...


data use;
set west;
where  date < '01jan79'd;
keep date y1 y2 y3;
run;

proc varmax data=use;
id date interval=qtr;
model y1-y3 / p=2 dify=(1)
print=(decompose(6) impulse=(stderr) estimates diagnose)
printform=both lagmax=3;
causal group1=(y1) group2=(y2 y3);
run;


First, the differenced data is modeled as a VAR(2) with the following result:

The parameter estimates AR1_1_2, AR1_1_3, AR2_1_2, and AR2_1_3 are insignificant, and the VARX model is fitted in the next step.

The detailed output is shown in Output 36.2.1 through Output 36.2.8.

Output 36.2.1 shows the descriptive statistics.

Output 36.2.1: Descriptive Statistics

 Analysis of German Economic Variables

The VARMAX Procedure

 Number of Observations 75 0 1

Simple Summary Statistics
Variable Type N Mean Standard
Deviation
Min Max Difference Label
y1 Dependent 75 0.01811 0.04680 -0.14018 0.19358 1 logarithm of investment
y2 Dependent 75 0.02071 0.01208 -0.02888 0.05023 1 logarithm of income
y3 Dependent 75 0.01987 0.01040 -0.01300 0.04483 1 logarithm of consumption

Output 36.2.2 shows that a VAR(2) model is fit to the data.

Output 36.2.2: Parameter Estimates

 Analysis of German Economic Variables

The VARMAX Procedure

Type of Model VAR(2) Least Squares Estimation

Constant
Variable Constant
y1 -0.01672
y2 0.01577
y3 0.01293

AR
Lag Variable y1 y2 y3
1 y1 -0.31963 0.14599 0.96122
y2 0.04393 -0.15273 0.28850
y3 -0.00242 0.22481 -0.26397
2 y1 -0.16055 0.11460 0.93439
y2 0.05003 0.01917 -0.01020
y3 0.03388 0.35491 -0.02223

Output 36.2.3 shows the parameter estimates and their significance.

Output 36.2.3: Parameter Estimates Continued

Schematic Representation
Variable/Lag C AR1 AR2
y1 . -.. ...
y2 + ... ...
y3 + .+. .+.

Model Parameter Estimates
Equation Parameter Estimate Standard
Error
t Value Pr > |t| Variable
y1 CONST1 -0.01672 0.01723 -0.97 0.3352 1
AR1_1_1 -0.31963 0.12546 -2.55 0.0132 y1(t-1)
AR1_1_2 0.14599 0.54567 0.27 0.7899 y2(t-1)
AR1_1_3 0.96122 0.66431 1.45 0.1526 y3(t-1)
AR2_1_1 -0.16055 0.12491 -1.29 0.2032 y1(t-2)
AR2_1_2 0.11460 0.53457 0.21 0.8309 y2(t-2)
AR2_1_3 0.93439 0.66510 1.40 0.1647 y3(t-2)
y2 CONST2 0.01577 0.00437 3.60 0.0006 1
AR1_2_1 0.04393 0.03186 1.38 0.1726 y1(t-1)
AR1_2_2 -0.15273 0.13857 -1.10 0.2744 y2(t-1)
AR1_2_3 0.28850 0.16870 1.71 0.0919 y3(t-1)
AR2_2_1 0.05003 0.03172 1.58 0.1195 y1(t-2)
AR2_2_2 0.01917 0.13575 0.14 0.8882 y2(t-2)
AR2_2_3 -0.01020 0.16890 -0.06 0.9520 y3(t-2)
y3 CONST3 0.01293 0.00353 3.67 0.0005 1
AR1_3_1 -0.00242 0.02568 -0.09 0.9251 y1(t-1)
AR1_3_2 0.22481 0.11168 2.01 0.0482 y2(t-1)
AR1_3_3 -0.26397 0.13596 -1.94 0.0565 y3(t-1)
AR2_3_1 0.03388 0.02556 1.33 0.1896 y1(t-2)
AR2_3_2 0.35491 0.10941 3.24 0.0019 y2(t-2)
AR2_3_3 -0.02223 0.13612 -0.16 0.8708 y3(t-2)

Output 36.2.4 shows the innovation covariance matrix estimates, the various information criteria results, and the tests for white noise residuals. The residuals are uncorrelated except at lag 3 for variable.

Output 36.2.4: Diagnostic Checks

Covariances of Innovations
Variable y1 y2 y3
y1 0.00213 0.00007 0.00012
y2 0.00007 0.00014 0.00006
y3 0.00012 0.00006 0.00009

Information Criteria
AICC -24.4884
HQC -24.2869
AIC -24.5494
SBC -23.8905
FPEC 2.18E-11

Cross Correlations of Residuals
Lag Variable y1 y2 y3
0 y1 1.00000 0.13242 0.28275
y2 0.13242 1.00000 0.55526
y3 0.28275 0.55526 1.00000
1 y1 0.01461 -0.00666 -0.02394
y2 -0.01125 -0.00167 -0.04515
y3 -0.00993 -0.06780 -0.09593
2 y1 0.07253 -0.00226 -0.01621
y2 -0.08096 -0.01066 -0.02047
y3 -0.02660 -0.01392 -0.02263
3 y1 0.09915 0.04484 0.05243
y2 -0.00289 0.14059 0.25984
y3 -0.03364 0.05374 0.05644

Schematic Representation
of Cross Correlations
of Residuals
Variable/Lag 0 1 2 3
y1 +.+ ... ... ...
y2 .++ ... ... ..+
y3 +++ ... ... ...

Portmanteau Test for Cross Correlations
of Residuals
Up To Lag DF Chi-Square Pr > ChiSq
3 9 9.69 0.3766

Output 36.2.5 describes how well each univariate equation fits the data. The residuals are off from the normality, but have no AR effects. The residuals for variable have the ARCH effect.

Output 36.2.5: Diagnostic Checks Continued

Univariate Model ANOVA Diagnostics
Variable R-Square Standard
Deviation
F Value Pr > F
y1 0.1286 0.04615 1.62 0.1547
y2 0.1142 0.01172 1.42 0.2210
y3 0.2513 0.00944 3.69 0.0032

Univariate Model White Noise Diagnostics
Variable Durbin
Watson
Normality ARCH
Chi-Square Pr > ChiSq F Value Pr > F
y1 1.96269 10.22 0.0060 12.39 0.0008
y2 1.98145 11.98 0.0025 0.38 0.5386
y3 2.14583 34.25 <.0001 0.10 0.7480

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 0.01 0.9029 0.19 0.8291 0.39 0.7624 1.39 0.2481
y2 0.00 0.9883 0.00 0.9961 0.46 0.7097 0.34 0.8486
y3 0.68 0.4129 0.38 0.6861 0.30 0.8245 0.21 0.9320

Output 36.2.6 is the output in a matrix format associated with the PRINT=(IMPULSE=) option for the impulse response function and standard errors. The variable in the first row is an impulse variable. The variable in the first column is a response variable. The numbers, 0.96122, 0.41555, –0.40789 at lag 1 to 3 are decreasing.

Output 36.2.6: Impulse Response Function

Simple Impulse Response by Variable
Variable
Response\Impulse
Lag y1 y2 y3
y1 1 -0.31963 0.14599 0.96122
STD 0.12546 0.54567 0.66431
2 -0.05430 0.26174 0.41555
STD 0.12919 0.54728 0.66311
3 0.11904 0.35283 -0.40789
STD 0.08362 0.38489 0.47867
y2 1 0.04393 -0.15273 0.28850
STD 0.03186 0.13857 0.16870
2 0.02858 0.11377 -0.08820
STD 0.03184 0.13425 0.16250
3 -0.00884 0.07147 0.11977
STD 0.01583 0.07914 0.09462
y3 1 -0.00242 0.22481 -0.26397
STD 0.02568 0.11168 0.13596
2 0.04517 0.26088 0.10998
STD 0.02563 0.10820 0.13101
3 -0.00055 -0.09818 0.09096
STD 0.01646 0.07823 0.10280

The proportions of decomposition of the prediction error covariances of three variables are given in Output 36.2.7. If you see the variable in the first column, then the output explains that about 64.713% of the one-step-ahead prediction error covariances of the variable is accounted for by its own innovations, about 7.995% is accounted for by innovations, and about 27.292% is accounted for by innovations.

Output 36.2.7: Proportions of Prediction Error Covariance Decomposition

Proportions of Prediction Error Covariances by Variable
y1 1 1.00000 0.00000 0.00000
2 0.95996 0.01751 0.02253
3 0.94565 0.02802 0.02633
4 0.94079 0.02936 0.02985
5 0.93846 0.03018 0.03136
6 0.93831 0.03025 0.03145
y2 1 0.01754 0.98246 0.00000
2 0.06025 0.90747 0.03228
3 0.06959 0.89576 0.03465
4 0.06831 0.89232 0.03937
5 0.06850 0.89212 0.03938
6 0.06924 0.89141 0.03935
y3 1 0.07995 0.27292 0.64713
2 0.07725 0.27385 0.64890
3 0.12973 0.33364 0.53663
4 0.12870 0.33499 0.53631
5 0.12859 0.33924 0.53217
6 0.12852 0.33963 0.53185

The table in Output 36.2.8 gives forecasts and their prediction error covariances.

Output 36.2.8: Forecasts

Forecasts
Variable Obs Time Forecast Standard
Error
95% Confidence Limits
y1 77 1979:1 6.54027 0.04615 6.44982 6.63072
78 1979:2 6.55105 0.05825 6.43688 6.66522
79 1979:3 6.57217 0.06883 6.43725 6.70708
80 1979:4 6.58452 0.08021 6.42732 6.74173
81 1980:1 6.60193 0.09117 6.42324 6.78063
y2 77 1979:1 7.68473 0.01172 7.66176 7.70770
78 1979:2 7.70508 0.01691 7.67193 7.73822
79 1979:3 7.72206 0.02156 7.67980 7.76431
80 1979:4 7.74266 0.02615 7.69140 7.79392
81 1980:1 7.76240 0.03005 7.70350 7.82130
y3 77 1979:1 7.54024 0.00944 7.52172 7.55875
78 1979:2 7.55489 0.01282 7.52977 7.58001
79 1979:3 7.57472 0.01808 7.53928 7.61015
80 1979:4 7.59344 0.02205 7.55022 7.63666
81 1980:1 7.61232 0.02578 7.56179 7.66286

Output 36.2.9 shows that you cannot reject Granger noncausality from to using the 0.05 significance level.

Output 36.2.9: Granger Causality Tests

Granger-Causality Wald Test
Test DF Chi-Square Pr > ChiSq
1 4 6.37 0.1734

Test 1: Group 1 Variables: y1 y2 y3

The following SAS statements show that the variable is the exogenous variable and fit the VARX(2,1) model to the data.

proc varmax data=use;
id date interval=qtr;
model y2 y3 = y1 / p=2 dify=(1) difx=(1) xlag=1 lagmax=3
print=(estimates diagnose);
run;


The fitted VARX(2,1) model is written as

The detailed output is shown in Output 36.2.10 through Output 36.2.13.

Output 36.2.10 shows the parameter estimates in terms of the constant, the current and the lag one coefficients of the exogenous variable, and the lag two coefficients of the dependent variables.

Output 36.2.10: Parameter Estimates

 Analysis of German Economic Variables

The VARMAX Procedure

Type of Model VARX(2,1) Least Squares Estimation

Constant
Variable Constant
y2 0.01542
y3 0.01319

XLag
Lag Variable y1
0 y2 0.02520
y3 0.05130
1 y2 0.03870
y3 0.00363

AR
Lag Variable y2 y3
1 y2 -0.12258 0.25811
y3 0.24367 -0.31809
2 y2 0.01651 0.03498
y3 0.34921 -0.01664

Output 36.2.11 shows the parameter estimates and their significance.

Output 36.2.11: Parameter Estimates Continued

Model Parameter Estimates
Equation Parameter Estimate Standard
Error
t Value Pr > |t| Variable
y2 CONST1 0.01542 0.00443 3.48 0.0009 1
XL0_1_1 0.02520 0.03130 0.81 0.4237 y1(t)
XL1_1_1 0.03870 0.03252 1.19 0.2383 y1(t-1)
AR1_1_1 -0.12258 0.13903 -0.88 0.3811 y2(t-1)
AR1_1_2 0.25811 0.17370 1.49 0.1421 y3(t-1)
AR2_1_1 0.01651 0.13766 0.12 0.9049 y2(t-2)
AR2_1_2 0.03498 0.16783 0.21 0.8356 y3(t-2)
y3 CONST2 0.01319 0.00346 3.81 0.0003 1
XL0_2_1 0.05130 0.02441 2.10 0.0394 y1(t)
XL1_2_1 0.00363 0.02536 0.14 0.8868 y1(t-1)
AR1_2_1 0.24367 0.10842 2.25 0.0280 y2(t-1)
AR1_2_2 -0.31809 0.13546 -2.35 0.0219 y3(t-1)
AR2_2_1 0.34921 0.10736 3.25 0.0018 y2(t-2)
AR2_2_2 -0.01664 0.13088 -0.13 0.8992 y3(t-2)

Output 36.2.12 shows the innovation covariance matrix estimates, the various information criteria results, and the tests for white noise residuals. The residuals is uncorrelated except at lag 3 for variable.

Output 36.2.12: Diagnostic Checks

Covariances of Innovations
Variable y2 y3
y2 0.00014 0.00006
y3 0.00006 0.00009

Information Criteria
AICC -18.3902
HQC -18.2558
AIC -18.4309
SBC -17.9916
FPEC 9.91E-9

Cross Correlations of Residuals
Lag Variable y2 y3
0 y2 1.00000 0.56462
y3 0.56462 1.00000
1 y2 -0.02312 -0.05927
y3 -0.07056 -0.09145
2 y2 -0.02849 -0.05262
y3 -0.05804 -0.08567
3 y2 0.16071 0.29588
y3 0.10882 0.13002

Schematic Representation
of Cross Correlations
of Residuals
Variable/Lag 0 1 2 3
y2 ++ .. .. .+
y3 ++ .. .. ..

Portmanteau Test for Cross Correlations
of Residuals
Up To Lag DF Chi-Square Pr > ChiSq
3 4 8.38 0.0787

Output 36.2.13 describes how well each univariate equation fits the data. The residuals are off from the normality, but have no ARCH and AR effects.

Output 36.2.13: Diagnostic Checks Continued

Univariate Model ANOVA Diagnostics
Variable R-Square Standard
Deviation
F Value Pr > F
y2 0.0897 0.01188 1.08 0.3809
y3 0.2796 0.00926 4.27 0.0011

Univariate Model White Noise Diagnostics
Variable Durbin
Watson
Normality ARCH
Chi-Square Pr > ChiSq F Value Pr > F
y2 2.02413 14.54 0.0007 0.49 0.4842
y3 2.13414 32.27 <.0001 0.08 0.7782

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
y2 0.04 0.8448 0.04 0.9570 0.62 0.6029 0.42 0.7914
y3 0.62 0.4343 0.62 0.5383 0.72 0.5452 0.36 0.8379