What to explore
Change parameters and watch the model adjust.
- Persistence φ across the unit root, and the innovation SD σ
- Sample length T and the long-run mean μ
Intermediate econometrics
The AR(1) Process: Persistence, Mean Reversion, and the Unit Root
A first-order autoregressive process carries a fraction φ of last period's deviation into this period. One parameter separates three worlds: mean reversion, the random walk, and explosive growth.
Focus
The simulated path, the ACF, and the phase change at φ = 1
Drag the persistence parameter through one and watch a stationary, mean-reverting series become a random walk and then explosive — while the sample ACF tracks the theoretical geometric decay φᵏ.Interactive diagram
AR(1) Time Series — explore it instantly
How to read this
The top panel simulates yₜ = μ + φ(yₜ₋₁ − μ) + σεₜ — this period's deviation from the mean is a fraction φ of last period's, plus a fresh shock. With |φ| < 1 the series is stationary: shocks fade geometrically and the path keeps returning to μ, mostly staying inside the ±2·SD band.
Drag φ toward 1 and watch mean reversion weaken: the long-run SD σ/√(1−φ²) grows, the shock half-life stretches, and the sample ACF bars in the bottom panel decay more slowly (their theoretical heights are exactly φᵏ). At φ = 1 the process has a unit root — a random walk whose shocks never fade — and beyond 1 it is explosive, leaving the chart entirely. That qualitative break at one number is why unit-root tests exist and why macro series are usually differenced before regression.
The innovations are a fixed draw, so the sliders deform the same realisation: φ, σ and μ reshape history rather than rerolling it, and T extends or truncates it. Negative φ makes the path oscillate — and the ACF alternate in sign.
Core ideas
Interpret the mechanics before you chase the graphs.
- This period's deviation from the mean is φ times last period's, plus a fresh shock — so shocks fade geometrically when |φ| < 1.
- The stationary distribution has SD σ/√(1−φ²), which grows without bound as φ approaches one.
- At φ = 1 shocks never fade (a random walk); beyond one the path is explosive — behaviour changes qualitatively, not gradually.
Learning goals
What this model should help students internalize.
- Interpret φ as persistence: how fast shocks fade and how far the series wanders from its mean.
- Read an autocorrelation function and connect the sample bars to the theoretical decay φᵏ.
- Explain why the unit root at φ = 1 is a qualitative break — and why macro series are differenced before regression.
Prerequisites
Concepts to review before diving in.
- Simple regression and the idea of a data-generating process
- Basic comfort with sequences and geometric decay
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