Definition
The Johansen test is a statistical procedure used in econometrics to assess the presence and number of cointegrating relationships among multiple non‑stationary time series. It implements a maximum‑likelihood estimation framework for vector autoregressive (VAR) models to test for long‑run equilibrium linkages.
Overview
Developed by Søren Johansen in the late 1980s, the test extends the concept of cointegration, originally introduced by Engle and Granger for bivariate series, to a multivariate setting. It is widely applied in macroeconomics, finance, and other fields where researchers examine whether several integrated variables share a common stochastic trend.
The procedure involves estimating a VAR model in levels, determining the appropriate lag order, and then rewriting the system in its error‑correction form. Two test statistics are derived:
- Trace statistic – tests the null hypothesis that the number of cointegrating vectors is less than or equal to a given value.
- Maximum eigenvalue statistic – tests the null that the number of cointegrating vectors is exactly a given value against the alternative of one additional vector.
Critical values for both statistics are obtained from asymptotic distributions that depend on the deterministic components (e.g., intercepts, trends) included in the model.
Etymology/Origin
The test is named after Søren Johansen, a Danish econometrician who introduced the methodology in his 1988 and 1991 papers and later formalized it in the monograph Likelihood‑Based Inference in Cointegrated Vector Autoregressive Models. The term “Johansen test” thus directly references its creator.
Characteristics
| Feature | Description |
|---|---|
| Model framework | Vector autoregression (VAR) with integrated (I(1)) variables. |
| Assumptions | – Variables are integrated of order one (I(1)). – Linear relationships among variables. – Error terms are Gaussian and serially uncorrelated (or appropriately modeled). |
| Deterministic components | Allows inclusion of intercepts and linear trends in the cointegration space and/or the VAR drift. |
| Estimation method | Maximum‑likelihood estimation under the reduced‑rank restriction on the long‑run impact matrix. |
| Output | Number of cointegrating vectors, their normalized loading matrices, and speed‑of‑adjustment coefficients. |
| Software implementation | Available in major econometric packages such as EViews, Stata (johansen command), R (urca package), MATLAB, and Python (statsmodels.tsa.vector_ar.vecm). |
| Limitations | – Sensitive to lag‑length selection and deterministic term specification. – Large sample sizes are needed for reliable asymptotic critical values. – Not robust to structural breaks unless extended versions are used. |
Related Topics
- Cointegration – The broader concept of long‑run equilibrium relationships among integrated series.
- Engle‑Granger two‑step method – An earlier, bivariate approach to testing for cointegration.
- Vector Error Correction Model (VECM) – The dynamic representation derived from a cointegrated VAR, incorporating short‑run adjustments to the long‑run equilibrium.
- Unit root tests (e.g., Augmented Dickey‑Fuller, Phillips‑Perron) – Preliminary procedures to determine the integration order of individual series.
- Structural break tests – Extensions of the Johansen framework that account for regime shifts (e.g., Gregory‑Hansen test).
The Johansen test remains a cornerstone technique for empirical analysis of long‑run relationships in multivariate time‑series data.