Johansen test
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:
There are two possible specifications for error correction: that is, two vector error correction models (VECM):
1. The longrun VECM:
- where
2. The transitory VECM:
- where
The two are the same. In both VECM,
Inferences are drawn on Π, and they will be the same, so is the explanatory power.[citation needed]
References
[edit]- ^ Johansen, Søren (1991). "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models". Econometrica. 59 (6): 1551–1580. doi:10.2307/2938278. JSTOR 2938278.
- ^ For the presence of I(2) variables see Ch. 9 of Johansen, Søren (1995). Likelihood-based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press. ISBN 978-0-19-877450-1.
- ^ Davidson, James (2000). Econometric Theory. Wiley. ISBN 0-631-21584-0.
- ^ Hänninen, R. (2012). "The Law of One Price in United Kingdom Soft Sawnwood Imports – A Cointegration Approach". Modern Time Series Analysis in Forest Products Markets. Springer. p. 66. ISBN 978-94-011-4772-9.
Further reading
[edit]- 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.