Part VI · Chapter 20
Cointegration
Two I(1) variables can drift forever, yet their linear combination stays stationary — they share a long-run equilibrium. Think of a rubber band: variables move apart but are pulled back.
\( e_t = y_t - \alpha - \beta x_t \;\;\text{is}\;\; I(0) \)
Learning objectives
Distinguish cointegrated from spuriously co-moving I(1) variables.
Run the Engle-Granger two-step procedure.
Read the cointegrating residual / spread.
Recognize the use of correct critical values.
Rubber Band Simulator
Seed:
Engle-Granger diagnostics
Estimated β: —
R² of levels OLS: —
Residual lag-1 ACF: —
Verdict: —
Stationary residual ⇒ cointegration. Non-stationary residual ⇒ spurious.
📐 Engle-Granger 2-step
\[ 1.\;\hat e_t = y_t - \hat\alpha - \hat\beta x_t \qquad 2.\;\text{ADF on } \hat e_t \]
H₀ (step 2) residual has unit root (no cointegration)
5% critical (k=1) ≈ −3.37 (shifted ADF)
5% critical (k=2) ≈ −3.74
Johansen multivariate, multiple cointegrating vectors
🔍 What to look for
- Cointegrated: y and x wander together; their spread oscillates around zero (stationary).
- Independent: spread wanders just like the levels — no equilibrium force.
- High R² in levels means little if residuals are non-stationary.
⚠️ Pro Tip: What to Avoid
Student says
"R² = 0.98 ⇒ cointegrated."
Why this is wrong
High R² in levels regression is necessary but NOT sufficient. The test is on residual stationarity, not on fit.
Correct interpretation
Run Engle-Granger ADF on residuals (use shifted critical values). Stationary residuals ⇒ cointegration. Then estimate ECM.
📝 Mini-quiz
📋 Key Takeaways
| Case | Action |
|---|---|
| Both I(0) | OLS in levels OK |
| Both I(1), residual I(0) | Cointegrated → ECM |
| Both I(1), residual I(1) | Spurious → difference |
| Mixed I(0)/I(1) | ARDL bounds test |