Using the Johansen Test to Estimate Multivariate Cointegration

Summary

The article introduces the Johansen procedure for testing cointegration among multiple time series and estimating stationary linear combinations. It describes expressing a vector autoregressive model as a vector error correction model, then using the rank of its long-run coefficient matrix to assess the number of cointegrating relationships. Trace and maximum-eigenvalue tests examine rank hypotheses sequentially; estimated eigenvectors provide candidate portfolio weights. Unlike a two-series regression approach, this procedure estimates the relationships within the multivariate test.

Examples apply the trace test to simulated series with a shared random-walk component and to baskets of ETFs. The simulated case illustrates how a stationary combination can be checked with an additional unit-root test. The financial examples are presented as exploratory evidence, not a strategy backtest. The article cautions that rank conclusions can be borderline and that its short historical sample may not represent persistent relationships. A detected cointegrating vector alone does not establish trading profitability or stability out of sample.

Key ideas

• The Johansen test can assess cointegration across more than two time series and estimate multiple relationships.
• A vector error correction model links short-run changes to lagged levels through a long-run coefficient matrix.
• The matrix rank represents the number of cointegrating relationships, and trace tests evaluate rank hypotheses in sequence.
• Eigenvectors associated with the estimated relationships provide weights for candidate stationary combinations.
• Borderline test results and short samples call for caution, and statistical cointegration does not establish a profitable strategy.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.