Interpreting Johansen Trace and Maximum-Eigenvalue Tests

Summary

This exchange explains how to read Johansen test statistics against their critical values when testing whether two stock price series share a cointegrating relationship. The example shows separate null hypotheses for rank zero and rank at most one, and compares each reported statistic with critical values at selected significance levels. A statistic below its corresponding critical value means the test does not reject that null at that level; the example’s reported values are below the listed thresholds.

The answer also frames the test as estimating the cointegration rank and notes that two series permit testing for no cointegration and for a higher rank. It does not explain how to reconcile conflicting trace and maximum-eigenvalue conclusions, despite that being part of the question. The response is brief and its wording about accepting a hypothesis is stronger than standard hypothesis-testing language: failing to reject is not proof that the null is true. It also omits model specification, diagnostics, and practical guidance for using the result in a trading strategy.

Key ideas

• Johansen tests assess the rank of the long-run relation among time series.
• Each reported rank hypothesis has its own test statistic and critical values.
• A statistic below its critical value means the null is not rejected at that significance level.
• The example’s statistics fall below their corresponding reported thresholds.
• The answer does not resolve how to interpret disagreement between the trace and maximum-eigenvalue tests.

Tags

# Interpretation of Johansen cointegration test in R


# Interpretation of Johansen cointegration test in R












I am using urca package of R for Johansen Cointegration test in 2 stocks datas( A and B.

My question is very elementar, but have cause some problems for me. How I interpret the critical values, for exemple, `H1 <- ca.jo(ll,type = "eigen", ecdet= "const", K = 4,spec = "longrun")` produce:

```
Values of teststatistic and critical values of test:

      test 10pct  5pct  1pct
r <= 1 |  6.39  7.52  9.24 12.97
r = 0  | 11.62 13.75 15.67 20.20
```

1 - In this case, for 10pct I have 11.62 < 13.75, then I can accept the hipotesis that A and B is not co-intregrate ? (whatever if for r <= 1 test is smaller that critical value ?).

2 - If for result in Trace statistic I find that A and B is cointegrate and for eigenvalue test A and B is not ? what does that mean it ? I reject the result of Trace Statistics in this case and admit A and Bnot cointegrate ?

## Answer by Nod Ulus (score 2)

https://quant.stackexchange.com/a/25361

Johansen test estimates the rank (r) of given matrix of time series with confidence level. In your example you have 2 time series, therefore Johansen tests null hypothesis of r=0 < (no cointegration at all), r<1 (till n-1, where n=2 in your example). If r<=1 test value (6.39) was greater than a confidence level's value (say 10%: 7.52), we would assume there is a cointegration of r time series (in this case r<=1). But as you see, none of your test values are greater than than critical values at r<0 and r<=1, therefore there is no cointegration.

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