Johansen 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:

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

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

1. The longrun VECM:

$\Delta X_t=\mu +\Phi D_t+\Pi X_{t-p}+{\Gamma }_{p-1}\Delta X_{t-p+1}+\cdots +{\Gamma }_1\Delta X_{t-1}+{\epsilon }_t,t=1,\dots ,T$

where

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

2. The transitory VECM:

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

where

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

The two are the same. In both VECM,

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

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