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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:

  1. Do I now chose its corresponding Eigenvalue Statistic which would be the second one, 3.88167814?
  2. 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?

Thank you

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    $\begingroup$ I am not sure you can conclude that there is cointegration from the second step if you were not able to do it in the first one, if you reject it in the first step you should stop the test and accept your first null hypothesis: no cointegration between the time series. The logic of my assertion is the following: If you cannot reject the null hypothesis in the first step (your matrix has a rank equal to zero), you cannot go to the second step, because the second step has a null hypothesis of rank equal to one and an alternative one which states that the rank is bigger than one. $\endgroup$
    – Mauro
    Commented Jun 28, 2022 at 15:17
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    $\begingroup$ This article helped me understand better how to interpret the Johansen results: letianzj.github.io/cointegration-pairs-trading.html. Maybe it helps you too. $\endgroup$
    – Tamás Pap
    Commented Jul 20, 2022 at 7:26

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