Johansen test

Short description: Time series statistical test

In statistics, the Johansen test, named after Søren Johansen, is a procedure for testing cointegration of several, say k, I(1) time series. 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.

Types

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

$X_t=\mu +\mathrm{\Phi }{D}_t+{\mathrm{\Pi }}_p{X}_{t-p}+\cdots +{\mathrm{\Pi }}_1{X}_{t-1}+e_t,t=1,\dots ,T$

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

1. The longrun VECM:

$\mathrm{\Delta }{X}_t=\mu +\mathrm{\Phi }{D}_t+\mathrm{\Pi }{X}_{t-p}+{\mathrm{\Gamma }}_{p-1}\mathrm{\Delta }{X}_{t-p+1}+\cdots +{\mathrm{\Gamma }}_1\mathrm{\Delta }{X}_{t-1}+{\epsilon }_t,t=1,\dots ,T$

where

${\mathrm{\Gamma }}_i={\mathrm{\Pi }}_1+\cdots +{\mathrm{\Pi }}_i-I,i=1,\dots ,p-1.$

2. The transitory VECM:

$\mathrm{\Delta }{X}_t=\mu +\mathrm{\Phi }{D}_t+\mathrm{\Pi }{X}_{t-1}-\sum _{j=1}^{p-1}{\mathrm{\Gamma }}_j\mathrm{\Delta }{X}_{t-j}+{\epsilon }_t,t=1,\cdots ,T$

where

${\mathrm{\Gamma }}_i=({\mathrm{\Pi }}_{i+1}+\cdots +{\mathrm{\Pi }}_p),i=1,\dots ,p-1.$

The two are the same. In both VECM,

$\mathrm{\Pi }={\mathrm{\Pi }}_1+\cdots +{\mathrm{\Pi }}_p-I.$

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

Further reading

• Banerjee, Anindya et al. (1993). Co-Integration, Error Correction, and the Econometric Analysis of Non-Stationary Data. New York: Oxford University Press. pp. 266–268. ISBN 0-19-828810-7. https://archive.org/details/cointegrationerr00bane_133. 
• Favero, Carlo A. (2001). Applied Macroeconometrics. New York: Oxford University Press. pp. 56–71. ISBN 0-19-829685-1. https://archive.org/details/appliedmacroecon00cafa. 
• Hatanaka, Michio (1996). Time-Series-Based Econometrics: Unit Roots and Cointegration. New York: Oxford University Press. pp. 219–246. ISBN 0-19-877353-6. https://books.google.com/books?id=ZQsaRNl5J60C&pg=PA219. 
• Maddala, G. S.; Kim, In-Moo (1998). Unit Roots, Cointegration, and Structural Change. Cambridge University Press. pp. 198–248. ISBN 0-521-58782-4. https://books.google.com/books?id=llXBvougICMC&pg=PA198.