A VECM is used when non-stationary series share one or more long-run equilibrium relations. The short-run dynamics are modeled in differences, while deviations from the long-run relation enter through an error-correction term.
Rank Testing
The lab uses Johansen trace and max-eigenvalue tests with a selected lag order and deterministic specification. The output reports eigenvalues, rank-test rows, and a rank reading.
Maximum-Likelihood Estimate
When the chosen rank is positive, the lab reports:
- beta, the cointegration matrix
- alpha, the adjustment matrix
- Pi, the error-correction matrix
- residual covariance
- cointegration series
- mean-reversion half-life by dimension
- concentrated log likelihood
Forecasting
Rank-restricted forecasts use the fitted VECM dynamics. If rank is zero, there is no cointegration relation to enforce. If the sample is short or the system is close to singular, the estimate should be read as fragile.
Interpretation
Cointegration is an economic claim as well as a statistical finding. A high rank statistic does not by itself identify a meaningful equilibrium relation. Variable choice, sample period, deterministic terms, and breaks matter.
References
- Engle and Granger, 1987.
- Johansen, 1988.
- Johansen, 1991.
- Juselius, 2006.