Nonlinear Engle-Granger Cointegration
Also known as: nonlinear cointegration, threshold cointegration, KSS cointegration, ESTAR cointegration
Nonlinear Engle-Granger cointegration extends the classical two-step Engle-Granger procedure to detect long-run equilibria where adjustment toward the equilibrium is nonlinear — for example, faster above than below a threshold, or governed by a smooth transition mechanism. It is widely applied in financial economics, purchasing power parity tests, and commodity price analysis.
Key highlights
- Detects long-run relationships that linear cointegration tests miss when adjustment is nonlinear or asymmetric.
- The KSS test is straightforward to implement as a regression-based procedure extending the familiar Engle-Granger framework.
- Flexible enough to accommodate exponential smooth transition (ESTAR) and threshold (TAR, M-TAR) adjustment mechanisms.
- Grounded in well-established asymptotic theory with simulation-based critical values for reliable inference.
- Directly connects to a nonlinear error correction model for dynamic analysis once cointegration is established.
Intuition
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How it works
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When to use it
Use nonlinear Engle-Granger cointegration when theory or data inspection suggests that deviations from long-run equilibrium are corrected asymmetrically or only when they cross a threshold — common in purchasing power parity, commodity markets, interest rate spreads, and asset pricing. It is preferred over the linear Engle-Granger test when preliminary evidence (e.g., BDS test, nonlinearity tests) suggests regime switching or smooth transitions in adjustment. Do not apply it when the series are stationary I(0) rather than I(1), when the sample is very short (fewer than ~100 observations), or when you have more than two variables — use a nonlinear VECM or Johansen-based approach for multivariate systems.
Strengths & limitations
- Detects long-run relationships that linear cointegration tests miss when adjustment is nonlinear or asymmetric.
- The KSS test is straightforward to implement as a regression-based procedure extending the familiar Engle-Granger framework.
- Flexible enough to accommodate exponential smooth transition (ESTAR) and threshold (TAR, M-TAR) adjustment mechanisms.
- Grounded in well-established asymptotic theory with simulation-based critical values for reliable inference.
- Directly connects to a nonlinear error correction model for dynamic analysis once cointegration is established.
- Retains the single-equation limitation of the original Engle-Granger approach — it does not handle more than one cointegrating vector.
- Critical values are simulation-based and depend on the assumed functional form of nonlinearity; misspecification of the transition function can distort size and power.
- Requires a relatively long time series (at least 100 observations) for reliable power; common in macro and financial data but not always available.
- The first-step OLS regression ignores endogeneity of regressors; if regressors are endogenous, a DOLS or FMOLS correction in the first step is advisable.
Common pitfalls
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Applications
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Frequently asked
How does this differ from the standard Engle-Granger test?
The standard Engle-Granger test uses a linear ADF regression on the residuals, implicitly assuming that adjustment back to equilibrium is always proportional to the gap. The nonlinear version replaces the linear term with a nonlinear function (e.g., the cube of the residual in KSS), allowing the speed of adjustment to depend on the size or sign of the deviation.
What is the KSS test exactly?
The KSS test, proposed by Kapetanios, Shin and Snell (2006), tests for a unit root against a smooth-transition (ESTAR) stationary alternative. When applied to the residuals of a long-run regression, rejecting the null implies nonlinear cointegration. It uses a t-statistic on the coefficient of the cubed lagged residual, compared to tabulated simulation-based critical values.
Can I use this test with more than two variables?
The two-step Engle-Granger approach (linear or nonlinear) is designed for a single-equation setup with one cointegrating vector. For systems with multiple I(1) variables and possibly multiple cointegrating vectors, you should use a multivariate framework such as the Johansen test or a nonlinear VECM.
How large a sample do I need?
Simulation studies suggest at least 100 observations for the KSS test to have adequate power. With fewer observations the test tends to under-reject the null, making it hard to detect genuine nonlinear cointegration.
Should I demeaned or demeaned-and-detrended the residuals before the KSS regression?
Kapetanios et al. (2006) provide separate critical values for the demeaned case (constant in the long-run regression) and the demeaned-and-detrended case (constant plus trend). Choose the variant that matches the specification of your first-step OLS regression and apply the corresponding critical values.
Sources
- 1.Kapetanios, G., Shin, Y., & Snell, A. (2006). Testing for cointegration in nonlinear smooth transition error correction models. Econometric Theory, 22(2), 279-303.
- 2.Enders, W., & Granger, C. W. J. (1998). Unit-root tests and asymmetric adjustment with an example using the term structure of interest rates. Journal of Business and Economic Statistics, 16(3), 304-311.
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Cite this page
ScholarGate. (2026, June 3). Nonlinear Engle-Granger Cointegration. ScholarGate. https://scholargate.app/econometrics/nonlinear-engle-granger-cointegration