Part VII · Chapter 23

Impulse Response Functions

An IRF traces the dynamic ripple of a one-time shock through a VAR system. Shape, sign, and persistence reveal how shocks propagate — but identification (e.g. Cholesky ordering) matters.

Learning objectives

Read IRF shape: monotonic decay, hump, oscillation, explosive.

Understand Cholesky ordering and its caveats.

Interpret confidence bands.

Distinguish IRF from a structural causal claim.

Shock Ripple Simulator

Decay (persistence) 0.70 Initial impact 1.0 Oscillation? Show 95% band

Shape diagnosis

—

Persistence close to 1 ⇒ very slow decay. Negative coefficient ⇒ oscillating IRF. |decay|≥1 ⇒ explosive (model is non-stationary).

📐 IRF for AR(1) example

\[ \frac{\partial y_{t+h}}{\partial \varepsilon_t} = \phi^h \cdot \text{(impact)} \]

\[ \text{For VAR: } IRF_h = \Phi^h \cdot e_i \text{ where } \Phi = A_1 \text{ in VAR(1)} \]

Decay rate governed by persistence

Shape monotonic / hump / oscillating / explosive

Cholesky orders shocks via lower-triangular variance decomposition

🔍 What to look for

Positive persistence + positive coefficient ⇒ monotonic geometric decay.
Negative coefficient ⇒ alternating signs (oscillation).
|persistence| ≥ 1 ⇒ explosive — VAR is non-stationary.
Wide confidence bands ⇒ low precision; bands crossing zero ⇒ effect not significant at h.

⚠️ Pro Tip: What to Avoid

Student says

"Cholesky IRFs are assumption-free."

Why this is wrong

Cholesky imposes a recursive ordering — i.e., assumptions about which variable cannot react to another within the same period. Different orderings give different IRFs.

Correct interpretation

Always state the identification assumption (ordering). Use sign-restrictions or structural VAR for stronger identification.

📝 Mini-quiz

📋 Key Takeaways

IRF feature	Cause
Monotonic decay	0 < φ < 1
Oscillating decay	−1 < φ < 0
Hump-shaped	Higher-order dynamics
Explosive	\|φ\| ≥ 1 (unstable)
Bands crossing zero	Effect insignificant at h