Vector Error Correction Model (VECM)
Vector Error Correction Model · Also known as: VECM, error correction VAR, cointegrated VAR, vector equilibrium correction model
The Vector Error Correction Model extends the Vector Autoregression (VAR) framework to a system of variables that share one or more long-run equilibrium relationships. It jointly models short-run dynamics and the speed at which each variable corrects back toward equilibrium after a shock, making it the standard tool for analysing cointegrated multivariate time series.
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When to use it
Use VECM when you have two or more time-series variables that are each individually non-stationary (I(1)) but are cointegrated — that is, they share a stable long-run relationship. VECM is the correct model when Johansen tests confirm at least one cointegrating vector; estimating a plain VAR in levels would be inefficient, and a VAR in first differences would discard the valuable long-run information. Do NOT use VECM if the variables are stationary (use VAR in levels), if there is no cointegration (use VAR in first differences), or if you have fewer than about 50 observations per variable (the rank tests lack power and estimation is unreliable).
Strengths & limitations
- Simultaneously models short-run dynamics and long-run equilibrium adjustment, capturing richer economic relationships than a VAR in differences.
- The adjustment coefficients α provide a direct, interpretable measure of how fast each variable corrects toward equilibrium.
- Johansen's procedure determines the number of cointegrating relationships from the data without imposing a single-equation structure.
- Avoids spurious regression by explicitly incorporating the cointegrating relationship rather than differencing away long-run information.
- Widely accepted in applied macroeconomics and finance, with rich theoretical justification via Granger's representation theorem.
- Requires that all variables be integrated of the same order (I(1)); mixing I(0) and I(2) variables invalidates the framework.
- Lag-length and rank selection strongly affect results; wrong choices lead to misspecified dynamics or missed cointegrating vectors.
- Demands moderately large samples — Johansen trace tests have low power in small samples (n < 50 per variable), making rank determination unreliable.
- The cointegrating vector β is only identified up to normalisation; economic interpretation requires imposing identifying restrictions.
- Sensitive to structural breaks: a break in the cointegrating relationship can make cointegration appear absent or create a spurious relationship.
Frequently asked
What is the difference between VECM and VAR?
A VAR models stationary variables or first differences, ignoring long-run equilibrium. VECM is a restricted VAR that adds error-correction terms to capture how variables adjust toward a shared long-run equilibrium. When cointegration exists, VECM is more efficient; when it does not, VAR in differences is correct.
How do I determine the number of cointegrating vectors?
Use Johansen's trace test and maximum-eigenvalue test sequentially. The trace test evaluates the null that there are at most r cointegrating vectors; reject until you cannot, and the last r you rejected is your rank. Cross-check with the max-eigenvalue test; disagreement calls for sensitivity analysis.
Can VECM be used with more than two variables?
Yes, and this is one of its advantages over the Engle-Granger two-step method. VECM handles n variables with up to n−1 cointegrating vectors using Johansen's full-system approach, making it standard for multivariate macro models.
What if a structural break is present in the cointegrating relationship?
A structural break can cause standard cointegration tests to fail to reject the null of no cointegration even when a long-run relationship exists. Use the Gregory-Hansen test, which allows for a single break in the cointegrating vector, or a Fourier-VECM approach for smooth regime changes.
How do I interpret a negative adjustment coefficient?
A negative α for variable i means it adjusts back toward equilibrium when it overshoots — the expected sign for a variable that 'corrects.' A positive or insignificant α means that variable does not respond to the disequilibrium, implying the other variables in the system carry the adjustment burden.
Sources
- Engle, R. F., & Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation, and testing. Econometrica, 55(2), 251–276. DOI: 10.2307/1913236 ↗
- 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). Vector Error Correction Model. ScholarGate. https://scholargate.app/en/econometrics/vector-error-correction-model
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- ARIMA modelEconometrics↔ compare
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- Vector AutoregressionEconometrics↔ compare