Nonlinear Vector Error Correction Model (Nonlinear VECM)
Nonlinear Vector Error Correction Model · Also known as: nonlinear VECM, NVECM, threshold VECM, asymmetric VECM
The Nonlinear VECM extends the standard linear VECM by allowing the speed of adjustment toward long-run equilibrium to differ depending on the sign, magnitude, or regime of deviations from that equilibrium. It captures asymmetric or threshold-driven dynamics in cointegrated time-series systems that a standard VECM would miss.
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When to use it
Use the Nonlinear VECM when theory or preliminary diagnostics suggest that adjustment toward long-run equilibrium is asymmetric — for example, upward and downward price corrections that differ in speed, or adjustment that activates only beyond a transaction-cost threshold. It is appropriate for multivariate I(1) series that are cointegrated but whose residuals from a linear VECM show evidence of nonlinearity (RESET test, BDS test, or sign-bias test). Do not apply it when cointegration itself is not established, when the series are stationary (use a nonlinear VAR instead), when sample size is small (fewer than about 100 observations per regime), or when a simpler linear VECM passes all misspecification tests — adding nonlinearity without evidence inflates parameter uncertainty and invites overfitting.
Strengths & limitations
- Captures asymmetric or threshold-driven adjustment dynamics that a linear VECM systematically masks.
- Grounded in economic theory: transaction costs, menu costs, and downward nominal rigidities naturally generate threshold or asymmetric error correction.
- Preserves the long-run cointegrating relationship while modelling flexible short-run adjustment.
- Allows testing whether asymmetry is statistically significant rather than imposing it a priori.
- Encompasses the linear VECM as a special case, enabling formal model comparison.
- Requires a larger sample than the linear VECM — reliable threshold estimation typically needs at least 100 observations and a minimum proportion of observations in each regime.
- Threshold parameter estimation is computationally intensive and introduces additional model-selection uncertainty.
- Specification of the nonlinearity form (threshold, smooth-transition, Markov-switching) must be guided by theory or pre-tests; wrong form leads to misspecification.
- Inference can be nonstandard: critical values for the Phi statistics differ from those of the linear VECM and must be tabulated or bootstrapped.
- Impulse-response functions and forecasts are regime-dependent and more complex to construct and interpret than in the linear case.
Frequently asked
How does a Nonlinear VECM differ from a standard VECM?
A standard VECM assumes the same constant adjustment speed regardless of whether the system is above or below equilibrium. A Nonlinear VECM allows these speeds — and sometimes the short-run dynamics — to differ by regime, capturing asymmetric or threshold-triggered adjustment that the linear model cannot detect.
Which test should I use to choose between TAR and M-TAR specifications?
Enders and Granger (1998) recommend estimating both, selecting the one with the lower Akaike Information Criterion (AIC) or Schwarz Criterion (BIC). The M-TAR is preferable when momentum (the direction of recent change in the ECT) drives adjustment; TAR is preferable when the level of the ECT is the relevant trigger.
Can I use the standard Johansen critical values for cointegration in a Nonlinear VECM?
No. If you first test for threshold cointegration using the Enders-Granger framework, you must use their simulated critical values for the Phi statistic. The Johansen trace and max-eigenvalue critical values apply only to the linear VECM setup and are not valid under the TAR or M-TAR structure.
How many observations do I need?
As a rough guideline, you need at least 100 time-series observations and should ensure that at least 10–15 percent of observations fall in each regime after threshold estimation. Small samples produce unstable threshold estimates and size-distorted inference; bootstrap critical values are strongly recommended when n < 150.
What should I do if the nonlinearity tests are insignificant?
If formal tests for asymmetric adjustment (F-test on α⁺ = α⁻, BDS test on VECM residuals, RESET test) are all insignificant, the additional complexity of the Nonlinear VECM is not supported by the data. Retain the simpler linear VECM to avoid overfitting and report the test results as evidence of adequacy of the linear specification.
Sources
- 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 & Economic Statistics, 16(3), 304–311. DOI: 10.1080/07350015.1998.10524769 ↗
- Granger, C. W. J., & Lee, T. H. (1989). Investigation of production, sales and inventory relationships using multicointegration and non-symmetric error correction models. Journal of Applied Econometrics, 4(S1), S145–S159. DOI: 10.1002/jae.3950040508 ↗
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Vector Error Correction Model. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-vecm
Which method?
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