Johansen test

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

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:

${\displaystyle X_{t}=\mu +\Phi D_{t}+\Pi _{p}X_{t-p}+\cdots +\Pi _{1}X_{t-1}+e_{t},\quad t=1,\dots ,T}$

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

1. The longrun VECM:

${\displaystyle \Delta X_{t}=\mu +\Phi D_{t}+\Pi X_{t-p}+\Gamma _{p-1}\Delta X_{t-p+1}+\cdots +\Gamma _{1}\Delta X_{t-1}+\varepsilon _{t},\quad t=1,\dots ,T}$

where

${\displaystyle \Gamma _{i}=\Pi _{1}+\cdots +\Pi _{i}-I,\quad i=1,\dots ,p-1.\,}$

2. The transitory VECM:

${\displaystyle \Delta X_{t}=\mu +\Phi D_{t}+\Pi X_{t-1}-\sum _{j=1}^{p-1}\Gamma _{j}\Delta X_{t-j}+\varepsilon _{t},\quad t=1,\cdots ,T}$

where

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

The two are the same. In both VECM,

${\displaystyle \Pi =\Pi _{1}+\cdots +\Pi _{p}-I.\,}$

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

References

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
  .
• 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
  .
• ISBN 0-521-58782-4
  .

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