Instatistics, theJohansen test,^[1] named afterSøren Johansen, is a procedure for testingcointegration of several, sayk,I(1)time series.^[2] This test permits more than one cointegrating relationship so is more generally applicable than theEngle-Granger test which is based on theDickey–Fuller (or theaugmented) test forunit roots in the residuals from a single (estimated) cointegrating relationship.^[3]

There are two types of Johansen test, either withtrace or witheigenvalue, and the inferences might be a little bit different.^[4] The null hypothesis for the trace test is that the number of cointegration vectors isr = r* < k, vs. the alternative thatr = k. Testing proceeds sequentially forr* = 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 isr = r* + 1 and, again, testing proceeds sequentially forr* = 1,2,etc., with the first non-rejection used as an estimator forr.

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

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

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

1. The longrun VECM:

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

where

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

2. The transitory VECM:

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

where

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

The two are the same. In both VECM,

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

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

• 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.
• Favero, Carlo A. (2001).Applied Macroeconometrics. New York: Oxford University Press. pp. 56–71.ISBN 0-19-829685-1.
• Hatanaka, Michio (1996).Time-Series-Based Econometrics: Unit Roots and Cointegration. New York: Oxford University Press. pp. 219–246.ISBN 0-19-877353-6.
• Maddala, G. S.; Kim, In-Moo (1998).Unit Roots, Cointegration, and Structural Change. Cambridge University Press. pp. 198–248.ISBN 0-521-58782-4.

• Mathematical finance
• Time series statistical tests

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