With discussion with Prof W and his student M., I tried permutation to see how Rt ~ tweet sentiment co-integrate. To my surprise, after permutation, Johansen's test becomes more significant. So, HQ thought that cointegration of a signal with a random signal may likely to co-integrate. So, this means that significant co-integration of Rt versus some factors that we thought are significant maybe be good news.
We need to demonstrate that both factors are not stationary first, and only their combination are stationary.
According to https://en.wikipedia.org/wiki/Johansen_test , Johansen test seems to only consider of I(1) order of 1 co-integration.
For the number of $k$ time series, "The null hypothesis for the trace test is that the number of cointegration vectors is r = r* < k, vs. the alternative that r = k. Testing proceeds sequentially for r* = 1,2, etc. and the first non-rejection of the null is taken as an estimate of r. The null hypothesis for the "maximum eigenvalue" test is as for the trace test but the alternative is r = r* + 1 and, again, testing proceeds sequentially for r* = 1,2,etc., with the first non-rejection used as an estimator for r."
A time series is integrated of order d if
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