Johansen cointegration testing: rejecting at 10% vs. 1% level

Question

I am testing for cointegration using the function ca.jo from the packeg 'urca' in R. I am struggling to understand the output.

According to my knowledge, the $H_0colon r=0$ means that there is zero cointegration relationship, and $H_acolon rleqslant1$ is that there is one cointegration relationship. Is that correct?

If this is correct, why is it then, that the $H_0$ cannot be rejected on a 1 pct confidence level but it can be rejected on a 10 pct level? I thought it should be easier to reject on a 10 pct level, than on a 1pct level. There is something I don't seem to understand here.

Any help is appreciated!

I've added some output underneath:

###################### 
# Johansen-Procedure # 
######################

Test type: trace statistic , with linear trend

Eigenvalues (lambda):
[1] 0.0087319555 0.0003334647

Values of teststatistic and critical values of test:

test 10pct  5pct  1pct
r <= 1 |  0.69  6.50  8.18 11.65
r = 0  | 18.70 15.66 17.95 23.52

Eigenvectors, normalised to first column:
(These are the cointegration relations)

EUR.3M.l1 EUR.1Y.l1
EUR.3M.l1     1.0000000      1.000000
EUR.1Y.l1    -0.8739526     -1.128037

Weights W:
(This is the loading matrix)

EUR.3M.l1 EUR.1Y.l1
EUR.3M.d  -0.003430251  0.0001479930
EUR.1Y.d  -0.001938101  0.0007150807

Dave · Answer

It is important to note that the $1%$ and $10%$ confidence levels you mention are significance levels, $alpha$, not confidence levels. Confidence level and $alpha$ are inversely related.
At $alpha=0.1$, you accept that $10%$ of the time when the null is true, you will reject. When $alpha=0.01$, you only allow that to happen $1%$ of the time. The latter requires stronger evidence to be able to reject.
Their corresponding confidence levels are $90$ and $99$, respectively. It is easier to get $90%$ confidence than $99%$ confidence, which is consistent with your results.

Richard Hardy · Answer

According to my knowledge, the H_0: r=0 means that there is zero cointegration relationship, and H_a: r<=1 is that there is one cointegration relationship. Is that correct?

Yes, this is correct.

If this is correct, why is it then, that the H_0 cannot be rejected on a 1pct confidence level but it can be rejected on a 10 pct level? I thought it should be easier to reject on a 10 pct level, than on a 1 pct level.

There is nothing wrong here (except that 1% is the significance level rather than confidence level). The test statistic is less extreme than the 1% critical value but more extreme than the 10% critical value, so you reject at 10% but not at 1%, exactly as expected.

Johansen cointegration testing: rejecting at 10% vs. 1% level

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