Johansen test

In statistics, the Johansen test,{{cite journal |last=Johansen |first=Søren |title=Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models |journal=Econometrica |volume=59 |issue=6 |year=1991 |pages=1551–1580 |doi=10.2307/2938278 |jstor=2938278 }} named after Søren Johansen, is a procedure for testing cointegration of several, say k, I(1) time series.For the presence of I(2) variables see Ch. 9 of {{Cite book|url=https://books.google.com/books?id=BH7nCwAAQBAJ |title=Likelihood-based Inference in Cointegrated Vector Autoregressive Models|last=Johansen|first=Søren|publisher=Oxford University Press|year=1995|isbn=978-0-19-877450-1 }} 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.{{cite book |last=Davidson |first=James |title=Econometric Theory|publisher=Wiley |year=2000 |isbn=0-631-21584-0 |url={{Google books|shWtvsFbxlkC|Econometric Methods|page=|plainurl=yes}} }}

Types

There are two types of Johansen test, either with trace or with eigenvalue, and the inferences might be a little bit different.{{cite book |first=R. |last=Hänninen |chapter=The Law of One Price in United Kingdom Soft Sawnwood Imports – A Cointegration Approach |title=Modern Time Series Analysis in Forest Products Markets |publisher=Springer |year=2012 |page=66 |chapter-url=https://books.google.com/books?id=DuL6CAAAQBAJ&pg=PA66 |isbn=978-94-011-4772-9 }} 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:

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

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

1. The longrun VECM:

:: $\Delta X_t =\mu+\Phi D_{t}+\Pi X_{t-p}+\Gamma_{p-1}\Delta X_{t-p+1}+\cdots+\Gamma_{1}\Delta X_{t-1}+\varepsilon_t,\quad t=1,\dots,T$

:where

:: $\Gamma_i = \Pi_1 + \cdots + \Pi_i - I,\quad i=1,\dots,p-1. \,$

2. The transitory VECM:

:: $\Delta X_{t}=\mu+\Phi D_{t}+\Pi X_{t-1}-\sum_{j=1}^{p-1}\Gamma_{j}\Delta X_{t-j}+\varepsilon_{t},\quad t=1,\cdots,T$

:where

:: $\Gamma_i = \left(\Pi_{i+1}+\cdots+\Pi_p\right),\quad i=1,\dots,p-1. \,$

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.{{cn|date=February 2019}}

Johansen test

Types

References

Further reading