Part VII · Chapter 22

VAR Models

In a vector autoregression each variable depends on its own lags AND lags of all other variables — every variable is endogenous. Individual coefficients are hard to interpret; use IRFs (Ch 23).

Learning objectives

Write a 2-variable VAR(1) system.

Simulate joint feedback between inflation and interest rate.

Recognize VAR stability via eigenvalues.

Understand why IRFs are needed for interpretation.

Two-Variable Feedback

β₁₁ (π → π) 0.6 β₁₂ (r → π) -0.2 β₂₁ (π → r) 0.3 β₂₂ (r → r) 0.7 Shock to inflation at t=20 2 Shock to rate at t=40 0

Seed:

Stability check

Max |eigenvalue|: —

VAR is stable if all eigenvalues of the companion matrix have modulus < 1.

📐 VAR(1) 2-variable

\[ \pi_t = \beta_{11}\pi_{t-1} + \beta_{12} r_{t-1} + u_{1t} \]

\[ r_t = \beta_{21}\pi_{t-1} + \beta_{22} r_{t-1} + u_{2t} \]

β_{ij} effect of lagged j on i

Diagonal own-persistence

Off-diagonal cross-feedback

IRFs see Ch 23 for proper interpretation

🔍 What to look for

Shocks propagate through both variables via cross-lags.
Strong own-persistence (β₁₁, β₂₂ near 1) creates long-lived responses.
Negative cross terms can produce oscillating responses.

⚠️ Pro Tip: What to Avoid

Student says

"β₁₂ = −0.2 means a 1pp rise in r reduces π by 0.2pp."

Why this is wrong

Individual VAR coefficients ignore the feedback. A "shock to r" propagates through π's reaction, then r reacts to that, etc. Use IRFs.

Correct interpretation

For policy interpretation, report orthogonalized IRFs and FEVD over horizon h, with ordering assumption stated.

📝 Mini-quiz

📋 Key Takeaways

Concept	Note
VAR(p)	k variables, p lags, each variable on all lags
Stability	All companion eigenvalues \|λ\| < 1
Interpretation	IRFs + FEVD, not individual coefficients
Lag selection	AIC, BIC, HQ + diagnostics