Part VI · Chapter 18
Dynamic Regression & ARDL Models
Short-run impact ≠ long-run multiplier. Lagged dependent and lagged independent terms allow the response to a shock to unfold over time.
\( y_t = \alpha + \phi y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} + u_t \)
Learning objectives
Compute short-run vs long-run multipliers.
Trace the dynamic response of y to a unit shock in x.
Recognize stability condition |φ| < 1.
Distinguish ARDL from static OLS.
Long-Run Multiplier Lab
Multipliers
Short-run (β₀): —
Cumulative (3-period): —
Long-run multiplier: —
LR = (β₀+β₁+β₂) / (1−φ). Requires |φ|<1.
📐 ARDL(1,2)
\[ y_t = \alpha + \phi y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} + \beta_2 x_{t-2} + u_t \]
\[ \text{Long-run multiplier} = \frac{\beta_0+\beta_1+\beta_2}{1-\phi} \]
β₀ instant impact
β_j impact j periods later
φ persistence of y
Stability |φ| < 1
🔍 What to look for
- Pulse shock at t=10 propagates through both immediate β₀ and lagged β₁, β₂.
- The y response decays through the persistence channel (φ y_{t-1}).
- Sum of lagged β / (1−φ) gives the EVENTUAL response to a permanent change in x.
⚠️ Pro Tip: What to Avoid
Student says
"β₀ is the effect of x on y."
Why this is wrong
β₀ is ONLY the immediate effect. The full long-run effect accumulates through lagged x AND lagged y persistence.
Correct interpretation
Report short-run (β₀), cumulative-h-period, and long-run multiplier separately.
📝 Mini-quiz
📋 Key Takeaways
| Effect | Formula |
|---|---|
| Short-run | β₀ |
| 3-period cumulative | β₀ + β₁ + β₂ |
| Long-run | (Σβ) / (1 − φ) |
| Stability | |φ| < 1 |