What is an impulse response function in a VAR model?

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AcadiFi · Accepted Answer

An impulse response function traces the dynamic response of each variable in a VAR to a one-time shock to one of the error terms, holding other shocks at zero. It answers questions like: what happens to inflation over the next 12 quarters if the policy rate is shocked upward by 25 basis points today?

The raw VAR residuals are typically correlated across equations, so a shock to one equation is accompanied by contemporaneous changes in others. To isolate individual structural shocks, researchers orthogonalize the residuals, most commonly via a Cholesky decomposition, which imposes a recursive ordering: the first variable responds only to its own shock contemporaneously, the second responds to shocks in the first and itself, and so on.

Example: central bank researcher Yusuf Balogun-Olatunji at Western African Monetary Institute estimates a VAR with output, inflation, and interest rate in that Cholesky order. The IRF of inflation to a rate shock shows a gradual decline peaking at minus 0.4 percentage points after six quarters, then dissipating by quarter 16. Confidence bands (usually 68 or 95 percent bootstrap) are plotted alongside the point estimate.

```mermaid
flowchart LR
    S[Shock t0] --> R1[Response t1]
    R1 --> R2[Response t2]
    R2 --> R3[Response t3]
    R3 --> D[Decay to 0]
```

Different Cholesky orderings yield different IRFs, so the ordering must be justified theoretically (e.g. slow-moving variables first). Structural VARs (SVARs) use theory-based restrictions beyond simple recursive ordering.

What is an impulse response function in a VAR model?

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