Nonlinear Johansen Cointegration Test
Also known as: nonlinear cointegration test, threshold Johansen cointegration, rank test for nonlinear cointegration, nonlinear VECM cointegration
Nonlinear Johansen cointegration extends the classical Johansen framework to detect long-run equilibrium relationships among integrated time series when the adjustment process is nonlinear. Using rank-based transformations, the approach tests for cointegration without assuming a linear error-correction mechanism, making it suitable for economic relationships characterized by asymmetric or threshold dynamics.
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When to use it
Use nonlinear Johansen cointegration when you have a multivariate system of I(1) time series and theory or diagnostic evidence suggests that the adjustment toward equilibrium is asymmetric, threshold-driven, or non-Gaussian. It is especially appropriate for commodity markets, exchange rates, interest rate spreads, and energy prices where positive and negative deviations from equilibrium revert at different speeds. Avoid it when the linear Johansen test is adequate (symmetric, Gaussian errors), when the sample is short (fewer than ~100 observations), or when unit root tests do not clearly support I(1) classification, as the rank transformation loses power in those settings.
Strengths & limitations
- Detects cointegration under nonlinear and asymmetric adjustment without requiring an explicit parametric form for the nonlinearity.
- Robust to non-Gaussian error distributions because the rank transformation removes moment-based assumptions.
- Preserves the multivariate Johansen framework, allowing estimation of multiple cointegrating vectors and cointegrating rank.
- Applicable to a wide range of economic relationships where linear adjustment is theoretically implausible.
- Rank-based statistics have known asymptotic distributions that are tractable for inference.
- Requires sufficiently long time series; rank-based tests have lower power in small samples than parametric alternatives.
- The rank transformation discards magnitude information, potentially losing efficiency when the true DGP is actually linear.
- Does not identify the specific form of nonlinearity (e.g., threshold, smooth transition); a confirmatory model must be specified separately.
- Critical values differ from the standard Johansen tables and must be obtained from the nonlinear rank-test framework.
- Software implementations are less widely available than the classical Johansen test, requiring custom code in R or Ox.
Frequently asked
How does nonlinear Johansen cointegration differ from threshold cointegration?
Threshold cointegration (Balke and Fomby, 1997) explicitly models a regime-switching error-correction term using a specified threshold variable. Nonlinear Johansen cointegration uses a nonparametric rank transformation and does not require specifying the threshold or transition function in advance, making it more flexible but less interpretable about the nature of the nonlinearity.
Can I use this test for I(2) variables?
The standard formulation targets I(1) systems. Applying it to I(2) variables requires additional differencing or a modified polynomial cointegration framework analogous to the Johansen I(2) model. Confirm integration orders carefully before proceeding.
What sample size is adequate?
As a guideline, at least 100 quarterly or 200 monthly observations are recommended for reliable asymptotic inference. Rank-based statistics converge more slowly than Gaussian parametric tests, so power is noticeably lower in shorter series.
What software can I use?
The Breitung (2001) rank test is implemented in R (via custom scripts or some time-series packages) and in Ox. The classical Johansen test with nonlinear VECM extensions is available in EViews and Stata through user-written routines. StatWise provides a guided implementation for common cases.
If the test confirms cointegration, what model should I estimate next?
Confirmation of a cointegrating relationship motivates estimation of a nonlinear VECM, a threshold VECM (TVECM), or a smooth-transition VECM depending on the suspected adjustment mechanism. Model selection should be guided by economic theory and diagnostic tests for the specific type of nonlinearity.
Sources
- Breitung, J. (2001). Rank tests for nonlinear cointegration. Journal of Business and Economic Statistics, 19(3), 331-340. DOI: 10.1198/073500101681019981 ↗
- Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica, 59(6), 1551-1580. DOI: 10.2307/2938278 ↗
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Johansen Cointegration Test. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-johansen-cointegration
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