Part IV · Chapter 11
Autoregressive (AR) Models
The current value depends on its own past plus a shock. The parameter φ controls how long memory of past values persists.
\( y_t = c + \phi\, y_{t-1} + \varepsilon_t \)
Learning objectives
Simulate an AR(1) and interpret φ.
Compute the half-life of a shock.
Recognize AR(p) by PACF cutoff at lag p.
Forecast: AR(1) decays geometrically toward the mean.
AR(1) Persistence Slider
Seed:
Live interpretation
φ: —
Half-life of a shock: — periods
Unconditional variance: —
📐 AR(1) properties
\[ y_t=c+\phi y_{t-1}+\varepsilon_t,\quad |\phi|<1 \text{ for stationarity} \]
\[ \rho(h)=\phi^h,\quad \text{Var}(y_t)=\frac{\sigma^2}{1-\phi^2},\quad \text{half-life}=\frac{\ln 0.5}{\ln \phi} \]
φ near 0 weak memory
φ near 1 strong persistence — near unit root
φ negative oscillating series
φ = 1 UNIT ROOT — not stationary
🔍 What to look for
- Forecast curve decays geometrically toward the mean — fast if φ small, slow if near 1.
- Forecast intervals widen and saturate at the unconditional variance.
- PACF should have a single significant spike at lag 1 (AR(1) signature).
⚠️ Pro Tip: What to Avoid
Student says
"φ = 0.99 is stationary, so it behaves like any other AR(1)."
Why this is wrong
Near-unit-root persistence creates very long-lived shocks, near-non-stationary inference, and unreliable standard asymptotic properties.
Correct interpretation
Treat |φ| close to 1 as "near-integrated" — use robust inference, expect long half-lives, and consider unit-root testing.
📝 Mini-quiz
📋 Key Takeaways
| Quantity | Formula |
|---|---|
| Mean | c / (1 − φ) |
| Variance | σ² / (1 − φ²) |
| ACF | ρ(h) = φ^h |
| Half-life | ln(0.5) / ln(φ) |
| PACF | cuts off at lag 1 |