This has been asked a few times before, but no answer was in my opinion satisfactory. My test also contains more details than in other question.
After using the Johansen test for two time-series in Python ( statsmodels.tsa.vector_ar.vecm.JohansenTestResult, link here), I get the following results:
Trace Statistic Crit 90% Crit 95% Crit 99% 10.55896424 13.4294 15.4943 19.9349 3.88167814 2.7055 3.8415 6.6349 Eigenvalue Statistic Crit 90% Crit 95% Crit 99% Eigenvectors 6.67728609 12.2971 14.2639 18.52 9.1332911 -0.15266422 3.88167814 2.7055 3.8415 6.6349 -11.72787276 13.49364426
From my understanding, we would first look at each Trace Statistic and compare it to the chosen critical level, for example lets say Crit 95%. So the first Trace Statistic is smaller than the Crit 95% level so we accept the null hypothesis that there is no cointegration, but the second Trace Statistics is bigger than its Crit 95% level so we reject the null hypothesis and accept that there exists cointegration.
Now to find the coefficients in order construct a stationary time-series from the two time-series I have, I would need to find the eigenvectors $A$ and $B$ so that $U_t=AS_1 + BS_2$ where $S_1$ and $S_2$ are given time series.
Having chosen the second Trace Statistic, my questions are:
- Do I now chose its corresponding Eigenvalue Statistic which would be the second one, 3.88167814?
- Having chosen the Eigenvalue Statistic, do I again need to compare that statistic to its Crit 95% level if I would like to take its corresponding Eigenvectors? What does this comparison tell us?