Part VI · Chapter 21
Error Correction Models
When variables are cointegrated, deviations from long-run equilibrium are corrected over time. The speed-of-adjustment coefficient λ must be NEGATIVE for stability.
\( \Delta y_t = \gamma \Delta x_t + \lambda\, e_{t-1} + u_t,\;\; -1<\lambda<0 \)
Learning objectives
Estimate ECM by 2-step Engle-Granger.
Interpret λ as fraction of disequilibrium corrected per period.
Compute half-life of disequilibrium.
Avoid sign-of-λ trap.
Equilibrium Correction Simulator
Seed:
Adjustment
λ: —
Fraction corrected/period: — %
Half-life of disequilibrium: — periods
📐 ECM
\[ \Delta y_t = \gamma \Delta x_t + \lambda\,(y_{t-1}-\alpha-\beta x_{t-1}) + u_t \]
λ < 0 convergence to equilibrium
λ = 0 no correction (cointegration fails)
λ > 0 divergence — unstable
Half-life ln(0.5) / ln(1+λ)
🔍 What to look for
- After a shock, the ECT (y − βx) returns to zero — the rubber band pulls back.
- Larger |λ| ⇒ faster correction.
- λ > 0 produces visible divergence (the chart EXPANDS away).
⚠️ Pro Tip: What to Avoid
Student says
"Positive λ means strong correction."
Why this is wrong
For stability, λ must be NEGATIVE: when residual is above equilibrium (positive), λ·e_{t-1} pushes Δy DOWN to bring it back.
Correct interpretation
Need −1 < λ < 0. Magnitude |λ| measures correction speed; sign must be negative.
📝 Mini-quiz
📋 Key Takeaways
| λ value | Interpretation |
|---|---|
| −1 | Full correction in one period |
| −0.3 | 30% of gap corrected per period |
| 0 | No error correction — likely no cointegration |
| +0.2 | UNSTABLE — moves away from equilibrium |