# Johansen’s Test: Simple Definition

Johansen’s test is a way to determine if three or more time series are cointegrated. More specifically, it assesses the validity of a cointegrating relationship, using a maximum likelihood estimates (MLE) approach. It is also used to find the number of relationships and as a tool to estimating those relationships (Wee & Tan, 1997).

## Types of Johansen’s Test

There are two types of Johansen’s test: one uses trace (from linear algebra), the other a maximum eigenvalue approach (an eigenvalue is a special scalar; When you multiply a matrix by a vector and get the same vector as an answer, along with a new scalar, the scalar is called an eigenvalue).

Both forms of the test will determine if cointegration is present. The null hypothesis for both forms of test is that there are no cointegrating equations. The difference is in the alternate hypothesis: the trace test alternate hypothesis is simply that the number of cointegrating relationships is at least one (shown by the number of linear combinations). The maximum eigenvalue test has an alternate hypothesis of K0 + 1 (instead of K > K0). Rejecting the null hypothesis in this situation is basically stating there is only one combination of the non-stationary variables that gives a stationary process.

Many authors agree that Johansen’s Test is an improvement over the Engle-Granger test and Stock & Watson’s test (in Introduction to Econometrics). It avoids the issue of choosing a dependent variable as well as issues created when errors are carried from one step to the next. As such, the test can detect multiple cointegrating vectors and is more appropriate than Engle-Granger for multivariate analysis. Another desirable property is that Johansen’s test treats every test variable as endogenous variables (Wassell & Saunders, 2008).

However, the test is far from perfect. Researchers Gonzalo & Lee (1997) reported that, for most situations, Engle-Granger was more robust than Johansen’s likelihood ratio test. The authors recommend using both Engle-Granger and Johansen tests to discover (or avoid) any pitfalls.

## References

Gonzalo, J. & Lee, T. “Pitfalls in testing for long run relationships.” Journal of Econometrics. 86 (1998) 129—154.
Engle, R. F. and Granger, C. W. J. (1991) Long-run Economic Relationships: Readings in Cointegration, Oxford University Press.Johansen, S. “Statistical Analysis of Cointegrating Vectors.” Journal of Economic Dynamics and
Control, 12 (1988), 231-54.
Wassell, C. & Saunders, P. (2008). Time series evidence on social security and private saving: The issue Revisited. Retrieved March 31, 2020 from: http://www.cwu.edu/business/sites/cts.cwu.edu.business/files/Soc%20Sec%20Final%20Draft.pdf
Wee, P. & Tan, R. (1997). Performance of Johansen’s Cointegration Test. In East Asian Economic Issues (Applied Economics Research Series) (v. 3).