I am testing for cointegration using the Johansen test. I have seen questions like how to interpret the test results, but when I am interpreting mine I have some doubts. In my results r = 3 since 4.10 < 10.49, so I cannot form a stationary series. It is the same for r = 2 and r = 1. But for r = 0, 86.12 > 59.14, so there is a stationary combination.
But r = 0 implies that there are zero cointegrating vectors. Does that mean that my data are not cointegrated and therefore I cannot build a VECM?
Please find my results below.
> cointegration <- ca.jo(Canada, type="trace",ecdet="trend",spec="transitory")
> summary(cointegration)
######################
# Johansen-Procedure #
######################
Test type: trace statistic , with linear trend in cointegration
Eigenvalues (lambda):
[1] 4.483918e-01 2.323995e-01 1.313250e-01 4.877895e-02 -1.859499e-17
Values of teststatistic and critical values of test:
test 10pct 5pct 1pct
r <= 3 | 4.10 10.49 12.25 16.26
r <= 2 | 15.65 22.76 25.32 30.45
r <= 1 | 37.33 39.06 42.44 48.45
r = 0 | 86.12 59.14 62.99 70.05
Eigenvectors, normalised to first column:
(These are the cointegration relations)
e.l1 prod.l1 rw.l1 U.l1 trend.l1
e.l1 1.0000000 1.0000000 1.00000000 1.00000000 1.00000000
prod.l1 0.3685667 -0.1582521 2.01545971 0.06122231 -0.09644538
rw.l1 -0.1369713 -0.5035147 -0.08233586 -0.15589592 -0.47523051
U.l1 3.2569951 2.4162383 2.98414327 1.57795960 1.54780259
trend.l1 -0.1539863 0.1477376 -0.53596432 -0.20898570 0.16907450
Weights W:
(This is the loading matrix)
e.l1 prod.l1 rw.l1 U.l1 trend.l1
e.d 0.01520061 0.10989739 0.04306410 -0.01664954 -6.999563e-13
prod.d 0.06282619 0.17899905 -0.05415524 -0.10283813 -5.525444e-12
rw.d -0.22958927 0.17308184 -0.03869293 0.06509098 -6.034107e-12
U.d -0.05230297 -0.08731406 -0.01833898 -0.03719022 1.367902e-12
Best Answer
In Johansen cointegration test, the alternative hypothesis for the eigenvalue test is that there are $r+1$ cointegration relations.
The test is therefore sequential: you test first for $r=0$, then $r=1$, etc.
The test concludes on the value of $r$ when the test fails to reject $H_0$ for the first time. In your case, the test fails to reject the null hypothesis for the first time when $r=1$.
Therefore, you have one cointegration relationship.