Dear All,
I am doing cointegration and causality test among four variables. I
performed the cointegration test using ca.jo() function and the result of
trace test is:
          test 10pct  5pct  1pct
r <= 3 |  2.76  7.52  9.24 12.97
r <= 2 |  9.56 17.85 19.96 24.60
r <= 1 | 29.03 32.00 34.91 41.07
r = 0  | 62.64 49.65 53.12 60.16
which indicates that there is one cointegrating vector.
Then estimated the OLS regressions of the VECM to identify the significant
vector (variables) and the result looks like:
Response E.d :
Coefficients:
            Estimate Std. Error t value Pr(>|t|)
E.dl1       0.035197   0.243545   0.145   0.8865
PUTEP1.dl1  0.006782   0.274279   0.025   0.9805
PEPCC.dl1  -0.281715   0.273295  -1.031   0.3144
Exr.dl1     0.005339   0.146832   0.036   0.9713
E.l1       -0.709115   0.309944  -2.288   0.0326 *
PUTEP1.l1  -0.185750   0.172672  -1.076   0.2942
PEPCC.l1    0.583796   0.381307   1.531   0.1407
Exr.l1     -0.157639   0.186751  -0.844   0.4081
constant    5.922444   2.997794   1.976   0.0615 .
---
Multiple R-Squared:  0.32,      Adjusted R-squared: 0.02863
F-statistic: 1.098 on 9 and 21 DF,  p-value: 0.4051

Response PUTEP1.d :
Coefficients:
           Estimate Std. Error t value Pr(>|t|)
E.dl1       -0.2689     0.2637  -1.020    0.319
PUTEP1.dl1   0.1924     0.2970   0.648    0.524
PEPCC.dl1    0.1488     0.2959   0.503    0.620
Exr.dl1     -0.1387     0.1590  -0.872    0.393
E.l1         0.3942     0.3356   1.175    0.253
PUTEP1.l1    0.2514     0.1870   1.345    0.193
PEPCC.l1    -0.5425     0.4128  -1.314    0.203
Exr.l1       0.2921     0.2022   1.445    0.163
constant    -4.2368     3.2457  -1.305    0.206
Residual standard error: 0.1083 on 21 degrees of freedom
Multiple R-Squared: 0.4095,     Adjusted R-squared: 0.1564
F-statistic: 1.618 on 9 and 21 DF,  p-value: 0.1740

Response PEPCC.d :
Coefficients:
           Estimate Std. Error t value Pr(>|t|)
E.dl1       0.04139    0.21105   0.196   0.8464
PUTEP1.dl1  0.36818    0.23768   1.549   0.1363
PEPCC.dl1  -0.03719    0.23683  -0.157   0.8767
Exr.dl1     0.11927    0.12724   0.937   0.3592
E.l1        0.18207    0.26859   0.678   0.5053
PUTEP1.l1   0.36989    0.14963   2.472   0.0221 *
PEPCC.l1   -0.80442    0.33043  -2.435   0.0239 *
Exr.l1     -0.01621    0.16183  -0.100   0.9212
constant   -1.19117    2.59777  -0.459   0.6513
---
Residual standard error: 0.08666 on 21 degrees of freedom
Multiple R-Squared: 0.581,      Adjusted R-squared: 0.4014
F-statistic: 3.235 on 9 and 21 DF,  p-value: 0.01275

Response Exr.d :
Coefficients:
           Estimate Std. Error t value Pr(>|t|)
E.dl1       0.22674    0.25078   0.904  0.37617
PUTEP1.dl1  0.70895    0.28243   2.510  0.02032 *
PEPCC.dl1  -0.03344    0.28141  -0.119  0.90653
Exr.dl1     0.27850    0.15119   1.842  0.07964 .
E.l1       -0.70257    0.31915  -2.201  0.03903 *
PUTEP1.l1  -0.40521    0.17780  -2.279  0.03323 *
PEPCC.l1    0.74434    0.39263   1.896  0.07184 .
Exr.l1     -0.93802    0.19230  -4.878    8e-05 ***
constant    9.64131    3.08684   3.123  0.00514 **
---
Residual standard error: 0.103 on 21 degrees of freedom
Multiple R-Squared: 0.6468,     Adjusted R-squared: 0.4955
F-statistic: 4.274 on 9 and 21 DF,  p-value: 0.002914
I also performed the Granger causality test using granger.test() function
and the result looks like:
                 F-statistic      p-value
dPUTEP1 -> dE       0.2652173 0.6107489208
dPEPCC -> dE        0.3099271 0.5823104521
dExr -> dE          0.0756436 0.7853839986
dE -> dPUTEP1       0.2927093 0.5929257035
dPEPCC -> dPUTEP1   0.9240658 0.3449401105
dExr -> dPUTEP1     0.5893134 0.4493465819
dE -> dPEPCC        1.2787679 0.2680733692
dPUTEP1 -> dPEPCC  16.4966125 0.0003759451
dExr -> dPEPCC      0.0877026 0.7693843210
dE -> dExr          0.7935924 0.3808859838
dPUTEP1 -> dExr     1.8548683 0.1844741419
dPEPCC -> dExr      2.5682890 0.1206617216
The granger causality test shows that dPUTEP1 causes dPEPCC but in addition
to that the OLS of the VECM shows that PUTEP1 and E.l1 also causes dExr.
Given that there is only one cointegrating vector, which way is the right
way to identify that cointegrating vector? Do I have to use the OLS of the
VECM? If so, the OLS is showing more than one cointegrating vector. that was
my confusion. Any help is highly appreciated.
Thanks,
Bereket

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