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 = μ + Φ D t + Π p X t − p + ⋯ + Π 1 X t − 1 + e t , t = 1 , … , T {\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:
Δ X t = μ + Φ D t + Π X t − p + Γ p − 1 Δ X t − p + 1 + ⋯ + Γ 1 Δ X t − 1 + ε t , t = 1 , … , T {\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
Γ i = Π 1 + ⋯ + Π i − I , i = 1 , … , p − 1. {\displaystyle \Gamma _{i}=\Pi _{1}+\cdots +\Pi _{i}-I,\quad i=1,\dots ,p-1.\,}
2. The transitory VECM:
Δ X t = μ + Φ D t + Π X t − 1 − ∑ j = 1 p − 1 Γ j Δ X t − j + ε t , t = 1 , ⋯ , T {\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
Γ i = ( Π i + 1 + ⋯ + Π p ) , i = 1 , … , p − 1. {\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,
Π = Π 1 + ⋯ + Π p − I . {\displaystyle \Pi =\Pi _{1}+\cdots +\Pi _{p}-I.\,}
Inferences are drawn on Π, and they will be the same, so is the explanatory power.
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. Maddala, G. S.; Kim, In-Moo (1998). Unit Roots, Cointegration, and Structural Change. Cambridge University Press. pp. 198–248. ISBN 0-521-58782-4.
