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:
Xt=\mu+\PhiDt+\PipXt-p+ … +\Pi1Xt-1+et, t=1,...,T
There are two possible specifications for error correction: that is, two vector error correction models (VECM):
1. The longrun VECM:
\DeltaXt=\mu+\PhiDt+\PiXt-p+\Gammap-1\DeltaXt-p+1+ … +\Gamma1\DeltaXt-1+\varepsilont, t=1,...,T
where
\Gammai=\Pi1+ … +\Pii-I, i=1,...,p-1.
2. The transitory VECM:
\DeltaXt=\mu+\PhiDt+\PiXt-1
| p-1 | |
| -\sum | |
| j=1 |
\Gammaj\DeltaXt-j+\varepsilont, t=1, … ,T
where
\Gammai=\left(\Pii+1+ … +\Pip\right), i=1,...,p-1.
The two are the same. In both VECM,
\Pi=\Pi1+ … +\Pip-I.
Inferences are drawn on Π, and they will be the same, so is the explanatory power.