Part VII · Chapter 24
Vector Error Correction Models
VECM = VAR + cointegration. When I(1) variables share a long-run equilibrium, modeling in differences alone throws away that information. VECM keeps both short-run dynamics AND long-run pull-back.
\( \Delta Y_t = \Pi\, Y_{t-1} + \Gamma_1 \Delta Y_{t-1} + u_t,\;\; \Pi = \alpha \beta' \)
Learning objectives
Recognize when VAR / VAR-in-differences / VECM is correct.
Read the rank r of Π as the number of cointegrating vectors.
Interpret α (adjustment speeds) and β (long-run coefficients).
Avoid using VECM when cointegration fails.
VAR vs VECM Decision
Seed:
Decision output
—
If I(0): plain VAR. If I(1) & not cointegrated: VAR in differences. If I(1) & cointegrated: VECM with rank r.
📐 VECM
\[ \Delta Y_t = \alpha\, \beta' Y_{t-1} + \sum_{i=1}^{p-1}\Gamma_i \Delta Y_{t-i} + u_t \]
β long-run cointegrating vectors (r × k)
α adjustment speeds (k × r)
Π = αβ' long-run impact matrix; rank r
Johansen test determines r (trace / max eigenvalue)
🔍 What to look for
- Cointegrated series drift together; their ECT (β'Y_{t-1}) mean-reverts.
- Negative α brings y back to equilibrium after shocks.
- Rank 0 → use VAR in differences. Rank k → series already stationary (VAR in levels).
⚠️ Pro Tip: What to Avoid
Student says
"All variables are I(1) so I'll fit VAR in differences."
Why this is wrong
Differencing throws away long-run cointegrating information. Always TEST for cointegration first (Johansen). If r ≥ 1, fit VECM.
Correct interpretation
Workflow: (1) ADF/KPSS on each variable. (2) Johansen trace + max-eigenvalue tests. (3) Choose VAR / VAR-diff / VECM based on rank.
📝 Mini-quiz
📋 Key Takeaways
| Case | Model |
|---|---|
| All I(0) | VAR in levels |
| All I(1), r = 0 | VAR in first differences |
| All I(1), 1 ≤ r < k | VECM with rank r |
| Mixed I(0)/I(1) | ARDL bounds approach |