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:

:$X_t=\mu+\Phi D_t+\Pi_p X_+\cdots+\Pi_1 X_+e_t,\quad t=1,\dots,T$

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

1. The longrun VECM:
::$\Delta X_t =\mu+\Phi D_+\Pi X_+\Gamma_\Delta X_+\cdots+\Gamma_\Delta X_+\varepsilon_t,\quad t=1,\dots,T$
:where
::$\Gamma_i = \Pi_1 + \cdots + \Pi_i - I,\quad i=1,\dots,p-1. \,$

2. The transitory VECM:
::$\Delta X_=\mu+\Phi D_+\Pi X_-\sum_^\Gamma_\Delta X_+\varepsilon_,\quad t=1,\cdots,T$
:where
::$\Gamma_i = \left(\Pi_+\cdots+\Pi_p\right),\quad i=1,\dots,p-1. \,$

The two are the same. In both VECM,

: $\Pi=\Pi_+\cdots+\Pi_-I. \,$

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

References

Further reading

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{{Statistics , analysis

Mathematical finance
Time series statistical tests