# If all hypotheses in the Johansen test are rejected, what is the order of the cointergration?

The result of the Johansen test is confusing to me. I have two time series and I'd like to test if they are cointegrated. However, in the Johansen test (performed by the function ca.jo in "urca" package in R), all hypotheses have been rejected as the test statistics are significantly bigger than the critical values.

Values of teststatistic and the critical values of the test:

           test 10pct  5pct  1pct
r <= 1 |  56.07  6.50  8.18 11.65
r = 0  | 117.13 15.66 17.95 23.52


I wonder what should be the order of cointegration for the two time series then? Since the null hypothesis r<=1 is rejected, r seems to take values greater than 1, but the order of cointegration for two time series should be 1 at most.

First, the hypotheses r = 0 and r <= 1 are about the cointegration rank rather than the order of cointegration.
Second, if you reject both r = 0 and r <= 1 that means r = 2 (r cannot be greater than the number of series in the system, which is 2). That implies there are at least two different linear combinations of the variables that are stationary (among combinations where the first weight is normalized to 1). That means the two series must be stationary to begin with, and hence there is no cointegration.