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); output lead=5; 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 35.2.1 through Output 35.2.8.
Output 35.2.1 shows the descriptive statistics.
Output 35.2.1: Descriptive Statistics
Analysis of German Economic Variables |
Number of Observations | 75 |
---|---|
Number of Pairwise Missing | 0 |
Observation(s) eliminated by differencing | 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 35.2.2 shows that a VAR(2) model is fit to the data.
Output 35.2.2: Parameter Estimates
Analysis of German Economic Variables |
Type of Model | VAR(2) |
---|---|
Estimation Method | 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 35.2.3 shows the parameter estimates and their significance.
Output 35.2.3: Parameter Estimates Continued
Schematic Representation | |||
---|---|---|---|
Variable/Lag | C | AR1 | AR2 |
y1 | . | -.. | ... |
y2 | + | ... | ... |
y3 | + | .+. | .+. |
+ is > 2*std error, - is < -2*std error, . is between, * is N/A |
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 35.2.4 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 35.2.4: 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 | +++ | ... | ... | ... |
+ is > 2*std error, - is < -2*std error, . is between |
Portmanteau Test for Cross Correlations of Residuals |
|||
---|---|---|---|
Up To Lag | DF | Chi-Square | Pr > ChiSq |
3 | 9 | 9.69 | 0.3766 |
Output 35.2.5 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 35.2.5: 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 35.2.6 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 35.2.6: 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 35.2.7. 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 35.2.7: Proportions of Prediction Error Covariance Decomposition
Proportions of Prediction Error Covariances by Variable | ||||
---|---|---|---|---|
Variable | Lead | y1 | y2 | y3 |
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 35.2.8 gives forecasts and their prediction error covariances.
Output 35.2.8: 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 35.2.9 shows that you cannot reject Granger noncausality from to using the 0.05 significance level.
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 35.2.10 through Output 35.2.13.
Output 35.2.10 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 35.2.10: Parameter Estimates
Analysis of German Economic Variables |
Type of Model | VARX(2,1) |
---|---|
Estimation Method | 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 35.2.11 shows the parameter estimates and their significance.
Output 35.2.11: 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 35.2.12 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 35.2.12: 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 | ++ | .. | .. | .. |
+ is > 2*std error, - is < -2*std error, . is between |
Portmanteau Test for Cross Correlations of Residuals |
|||
---|---|---|---|
Up To Lag | DF | Chi-Square | Pr > ChiSq |
3 | 4 | 8.38 | 0.0787 |
Output 35.2.13 describes how well each univariate equation fits the data. The residuals are off from the normality, but have no ARCH and AR effects.
Output 35.2.13: 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 |