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Johansen test

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In statistics, the Johansen test,[1] named after Søren Johansen, is a procedure for testing cointegration of several, say k, I(1) time series.[2] This test permits more than one cointegrating relationship so is more generally applicable than the Engle-Granger test which is based on the Dickey–Fuller (or the augmented) test for unit roots in the residuals from a single (estimated) cointegrating relationship.[3]

There are two types of Johansen test, either with trace or with eigenvalue, and the inferences might be a little bit different.[4] 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.

Just like a unit root test, there can be a constant term, a trend term, both, or neither in the model. For a general VAR(p) model:

There are two possible specifications for error correction: that is, two vector error correction models (VECM):

1. The longrun VECM:

where

2. The transitory VECM:

where

The two are the same. In both VECM,

Inferences are drawn on Π, and they will be the same, so is the explanatory power.[citation needed]

References

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  1. ^ Johansen, Søren (1991). "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models". Econometrica. 59 (6): 1551–1580. doi:10.2307/2938278. JSTOR 2938278.
  2. ^ For the presence of I(2) variables see Ch. 9 of Johansen, Søren (1995). Likelihood-based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press. ISBN 978-0-19-877450-1.
  3. ^ Davidson, James (2000). Econometric Theory. Wiley. ISBN 0-631-21584-0.
  4. ^ Hänninen, R. (2012). "The Law of One Price in United Kingdom Soft Sawnwood Imports – A Cointegration Approach". Modern Time Series Analysis in Forest Products Markets. Springer. p. 66. ISBN 978-94-011-4772-9.

Further reading

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